DayDreamin’ Comics https://ddcomics.org/ Have you ever seen a dream walking? Well i did. Sun, 21 Jun 2026 21:38:53 +0000 en-US hourly 1 https://wordpress.org/?v=7.0 https://i0.wp.com/ddcomics.org/wp-content/uploads/2023/07/cropped-DD-icon.jpg?fit=32%2C32&ssl=1 DayDreamin’ Comics https://ddcomics.org/ 32 32 230705254 Write up on Fritz Leiber (revision 2) https://ddcomics.org/2026/06/21/write-up-on-fritz-leiber-revision-2/ https://ddcomics.org/2026/06/21/write-up-on-fritz-leiber-revision-2/#respond Sun, 21 Jun 2026 21:17:35 +0000 https://ddcomics.org/?p=6903 Background of Study Introduction Fritz Leiber was an incredibly prolific author, who won numerous awards during his life for works that nowadays seem more obscure than they should be. In addition to writing, he was a chess master, fencer, and Shakespearean actor. While most popular for the sword and sorcery duo Fafhrd and the Gray […]

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Background of Study

Introduction

Fritz Leiber was an incredibly prolific author, who won numerous awards during his life for works that nowadays seem more obscure than they should be. In addition to writing, he was a chess master, fencer, and Shakespearean actor. While most popular for the sword and sorcery duo Fafhrd and the Gray Mouser, he wrote horror, urban fantasy, and science fiction just as easily.

 Born Fritz Reuter Leiber, Jr. in Chicago, Illinois, on December 24, 1910, to Fritz Leiber, Sr. and Virginia Bronson Leiber, both Shakespearean actors. Toured with father’s repertory company in 1928 before entering the University of Chicago, from which he graduated in 1932; went on to study at General Theological Seminary in New York, and was briefly a candidate for ordination in the Episcopal Church. Toured intermittently with father’s company and appeared with him in films Camille (1936) and The Great Garrick (1937). Married Jonquil Stephens in 1936 and moved to Hollywood; they soon had a son. Corresponded with horror writer H. P. Lovecraft, who encouraged and influenced his literary development; wrote a supernatural novella,

Pelan also mentions in Horrible Imaginings (2004): “‘The Automatic Pistol” was so overshadowed by “Smoke Ghost” that many have forgotten what an excellent early story this was.”

 The Dealings of Daniel Kesserich (1936; published posthumously in 1997), and showed Lovecraft early stories. Returning to Chicago, took job as staff writer for Consolidated Book Publishing (1937–41), contributing to the Standard American Encyclopedia.

 His first publication as a professional writer, “Two Sought Adventure” (in John W. Campbell, Jr.’s Unknown in 1939), introduced popular characters Fafhrd and the Gray Mouser, developed with his friend Harry Fischer and modeled on their relationship; the story inaugurated a series he would continue for more than fifty years, helping to define the subgenre he labeled “Sword and Sorcery.” (Fafhrd and the Gray Mouser stories were later collected in Two Sought Adventure, 1957; Swords in the Mist, 1968; Swords Against Wizardry, 1968; The Swords of Lankhmar, 1968; Swords and Deviltry, 1970; Swords and Ice Magic, 1977; The Knight and Knave of Swords, 1988; and other volumes.)

Worked as a drama and speech instructor at Occidental College in 1941, and during the war as an inspector at Douglas Aircraft. His first novel, Conjure Wife—about secret witchcraft on a college campus—appeared in Unknown in 1943 (but not as a book until 1952; it was filmed three times). His first science fiction novel, Gather, Darkness!, was also serialized in 194 (book version, 1950). From 1945 to 1956, he worked as an editor at Science Digest in Chicago. Published science fiction novels Destiny Times Three (in Astounding, 1945; as book, 1957); The Green Millennium (1953); and The Big Time (in Galaxy, 1958; as book, 1961), the last winning a Hugo Award and inaugurating his popular “Change War” series. Moved back to Los Angeles in 1958, and turned to writing full-time; published science fiction novels The Silver Eggheads (1961), The Wanderer (1964), and A Specter Is Haunting Texas (1969). Lived in San Francisco after the death of his wife in 1969; the city forms the setting of his fantasy novel Our Lady of Darkness (1977). In 1976, he received a World Fantasy Award for Life Achievement, and in 1981 a Grand Master Award from Science Fiction Writers of America. Married Margo Skinner in May 1992; died on September 5, 1992, in San Francisco, of an apparent stroke. In 2001 he was inducted posthumously into the Science Fiction Hall of Fame.

Fritz Leiber’s appearances in Weird Tales are both surprising and disappointing. As an outer member of the Lovecraft Circle, it was only natural that Leiber wanted to found in “The Unique Magazine”

After initial rejections, Fritz finally appeared in Weird Tales with “The Automatic Pistol” (Weird Tales, May 1940). Shortly before this he sold a classic to John W. Campbell, “The Jewels in the Forest” (Unknown, August 1939), the first Fafhrd & Grey Mouser tale, making “The Automatic Pistol” his second major publication. (Campbell paid a penny a word or better and on acceptance. Weird Tales on publication.) “The Automatic Pistol” is a story of a murdered man’s gun pursuing his killer, which seems like pretty usual fare for Weird Tales. But as John Pelan points out in his introduction to The Black Gondalier and Other Stories (2000):

Significance of the Study

From the very start his stories took on a modern attitude quite unlike that of his contemporaries in Weird Tales, who were busily scrambling to pen stories of improbably-named cosmic monstrosities and babbling aliens in a misguided homage to H.P. Lovecraft…

Many science fiction writers have written fantasy or horror fiction as well, although few have ex celled in more than one genre. Fritz Leiber is probably the only writer to have an enviable reputation in all three branches of fantastic fiction. His novel Conjure Wife (1953) and the short stories in Night’s Black Agents (1947) and elsewhere established him as an important horror writer, his Fafhrd and the Gray Mouser sword and sorcery series rivals even Leiber, Fritz 225 Conan in popularity among fantasy readers, and his science fiction includes several award winning stories as well as the excellent Change War time travel series. Leiber’s first story sale was in 1939, but he wrote no significant science fiction until 1943, when Gather Darkness! first appeared in serial form. The setup is a future world dictatorship with the rulers cloaking themselves in the costume of an organized religion in order to frighten the mass of the population into obedience.

 The inevitable resistance movement springs up, and appropriately they adopt the guise of demons and devils in a dramatic, if not entirely credible, symbolic gesture. Despite its occasional lack of plausibility, the novel is a rousing adventure story with some clever plot twists; and the policy of the government to awe the populace by mimicking supernatural intervention is a not particularly veiled swipe at human gullibility. His next novel, Destiny Times Three (1945), was a lackluster effort about a man who discovers that he exists in three different although interlocking realities, but The Sinful Ones (1950, also published as You’re All Alone) was much better.

The protagonist in this case discovers one day that he is one of the few remaining human beings in a world in which robots are masquerading as people. The Green Millennium (1953), like many of Leiber’s short stories during the early 1950s, was satirical, following the adventures of a man who is con cerned that a robot might make him obsolete. Among Leiber’s targets was contemporary sexual mores, which he lampooned in a fashion somewhat daring for its time. The corrupt American govern ment is secretly in league with organized crime in an association reminiscent of that in The Syndic (1953) by Cyril M. KORNBLUTH. The Change War series appeared in the late 1950s, and despite the small number of titles in the series, it ranks with Poul ANDERSON’s Time Patrol as the best of its type.

The BIG TIME (1958) won a Hugo Award, and the shorter “Try and Change the Past” is also excellent. The premise is that two organizations, known familiarly as the Snakes and the Spiders, are battling back and forth through time in an effort to maintain or change the existing course of history. The quality of Leiber’s short fiction in general improved dramatically, and the themes were wide-ranging. Leiber appeared equally adept at satire and adventure, serious themes and humor. Stories like “A Deskful of Girls,” “The Big Trek,” and “Night of the Long Knives” made him a fre quent and welcome contributor to the magazines. His next novel was The Silver Eggheads (1962), a satire on the writing community. Robots have been programmed to act like people, and authors use machines to produce their fiction, rather than doing it themselves, feeding in basic ideas but leav ing the prose and plot construction to their me chanical servants. The brains of prominent citizens—including a handful of actual writers— are preserved in smooth metal receptacles where they remain conscious.

 When a crisis threatens to disrupt the flow of new novels and stories, radical methods are used to save the situation. The Silver Eggheads is Leiber’s most underrated novel. Leiber became an even more productive short story writer during the 1960s, producing such minor classics as “Kreativity for Kats,” “The Man Who Made Friends with Electricity,” “The Secret Songs,” and “Far Reach to Cygnus.” His major col lections from this period are A Pail of Air (1964), Night of the Wolf (1966), and The Night Monsters (1969). He also produced his most praised novel, The Wanderer (1964), in which the world is rav aged by the near passage of another astronomical body. The story follows the separate stories of vari ous survivors, concentrating on realistic, common experiences rather than on the usual heroic efforts to reestablish civilization. His characters are deliberately flawed and occasionally fail, and the result is a much more convincing blend of tragedy and hope than is common in that form. Although The Wanderer is certainly one of the outstanding disaster novels, it is somewhat surprising that it was more popular than Leiber’s more original work. A Specter Is Haunting Texas (1969) was an other superb satire.

A visitor from the Moon— where the lower gravity has resulted in very tall, thin body types—visits a future independent Texas that dominates North America, and where genetic engineering has made Texans into virtual giants who tower over their Mexican slave population. The visitor becomes the inadvertent inspiration for a revolution in what is clearly a parody of a long-standing and often used science fiction 226 Leinster, Murray plot. Leiber’s knife-edged wit was at its best, and he handles the occasionally uneasy mix of sarcasm and light humor deftly. His short fiction continued to appear with regularity and was rarely less than excellent during the 1970s; the best of his work of that decade is probably the Hugo Award–winning “CATCH THAT ZEPPELIN.”

 Several major collec tions appeared during that period including The Book of Fritz Leiber (1974), The Second Book of Fritz Leiber (1975), and The Worlds of Fritz Leiber (1976). Although Leiber continued to write short fiction throughout the 1980s, his output dropped dramatically at that point. The Change War series has been assembled in its entirety as Changewar (1983). Other late collec tions include The Ghost Light (1984), The Leiber Chronicles (1990), and Kreativity for Kats and Other Feline Fantasies (1990). Leiber also wrote the autho rized novelization of the film Tarzan and the Valley of Gold (1966), only marginally science fiction, but probably the best-known and most successful addition to the chronicles of Edgar Rice BURROUGHS’s most famous character. Leiber con tinued to contribute to all three genres throughout his career, and his last Fafhrd and the Gray Mouser story appeared in 1988. The Dealings of Daniel Kesserich (1997) was a previously unpublished novel written in the 1930s and is only of historical inter est. Fritz Leiber will probably be best remembered for his short fiction, but several of his novels deserve an equal place of honor

Sample of his Writing

Big Time : Excerpt

Big Time : Excerpt of  Literature Sample as he incorporates Illustration


CHAPTER 4

De Bailhache, Fresca, Mrs. Cammel, whirledBeyond the circuit of the shuddering BearIn fractured atoms.

—Eliot

SOS FROM NOWHERE

I REALIZED the piano had deserted Erich and I cranked my head up and saw Beau, Maud and Sid streaking for the control divan. The Major Maintainer was blinking emergency-green and fast, but the code was plain enough for even me to recognize the Spider distress call and for a second I felt just sick. Then Erich blew out his reserve breath in the middle of “Door” and I gave myself another of those helpful mental boots at the base of the spine and we hurried after them toward the center of the Place along with Mark.

The blinks faded as we got there and Sid told us not to move because we were making shadows. He glued an eye to the telltale and we held still as statues as he caressed the dials like he was making love.

One sensitive hand flicked out past the Introversion switch over to the Minor Maintainer and right away the Place was dark as your soul and there was nothing for me but Erich’s arm and the knowledge that Sid was nursing a green light I couldn’t even see, although my eyes had plenty time to accommodate.

Then the green light finally came back very slowly and I could see the dear reliable old face—the green-gold beard making him look like a merman—and then the telltale flared bright and Sid flicked on the Place lights and I leaned back.

“That nails them, lads, whoever and whenever they may be. Get ready for a pick-up.”

Beau, who was closest of course, looked at him sharply. Sid shrugged uneasily. “Meseemed at first it was from our own globe a thousand years before our Lord, but that indication flickered and faded like witchfire. As it is, the call comes from something smaller than the Place and certes adrift from the cosmos. Meseemed too at one point I knew the fist of the caller—an antipodean atomicist named Benson-Carter—but that likewise changed.”

Beau said, “We’re not in the right phase of the cosmos-Places rhythm for a pick-up, are we, sir?”

Sid answered, “Ordinarily not, boy.”

Beau continued, “I didn’t think we had any pick-ups scheduled. Or stand-by orders.”

Sid said, “We haven’t.”

Mark’s eyes glowed. He tapped Erich on the shoulder. “An octavian denarius against ten Reichsmarks it is a Snake trap.”

Erich’s grin showed his teeth. “Make it first through the Door next operation and I’m on.”


IT didn’t take that to tell me things were serious, or the thought that there’s always a first time for bumping into something from really outside the cosmos. The Snakes have broken our code more than once. Maud was quietly serving out weapons and Doc was helping her. Only Bruce and Lili stood off. But they were watching.

The telltale brightened. Sid reached toward the Maintainer, saying, “All right, my hearties. Remember, through this Doorway pass the fishiest finaglers in and out of the cosmos.”

The Door appeared to the left and above where it should be and darkened much too fast. There was a gust of stale salt seawind, if that makes sense, but no stepped-up Change Winds I could tell—and I had been bracing myself against them. The Door got inky and there was a flicker of gray fur whips and a flash of copper flesh and gilt and something dark and a clump of hoofs and Erich was sighting a stun gun across his left forearm, and then the Door had vanished like that and a tentacled silvery Lunan and a Venusian satyr were coming straight toward us.

The Lunan was hugging a pile of clothes and weapons. The satyr was helping a wasp-waisted woman carry a heavy-looking bronze chest. The woman was wearing a short skirt and high-collared bolero jacket of leather so dark brown it was almost black. She had a two-horned petsofa hairdress and she was boldly gilded here and there and wore sandals and copper anklets and wristlets—one of them a copper-plated Caller—and from her wide copper belt hung a short-handled double-headed ax. She was dark-complexioned and her forehead and chin receded, but the effect was anything but weak; she had a face like a beautiful arrowhead—and a familiar one, by golly!

But before I could say, “Kabysia Labrys,” Maud shrilly beat me to it with, “It’s Kaby with two friends. Break out a couple of Ghostgirls.”

And then I saw it really was old-home week because I recognized my Lunan boy friend Ilhilihis, and in the midst of all the confusion I got a nice kick out of knowing I was getting so I could tell the personality of one silver-furred muzzle from another.

They reached the control divan and Illy dumped his load and the others let down the chest, and Kaby staggered but shook off the two ETs when they started to support her, and she looked daggers at Sid when he tried to do the same, although she’s his “sweet Keftian friend” he’d mentioned to Bruce.


SHE leaned straight-armed on the divan and took two gasping breaths so deep that the ridges of her spine showed through her brown-skinned waist, and then she threw up her head and commanded, “Wine!”

While Beau was rushing it, Sid tried to take her hand again, saying, “Sweetling, I’d never heard you call before and knew not this pretty little fist,” but she ripped out, “Save your comfort for the Lunan,” and I looked and saw—Hey, Zeus!—that one of Ilhilihis’ six tentacles was lopped off halfway.

That was for me, and, going to him, I fast briefed myself: “Remember, he only weighs fifty pounds for all he’s seven feet high; he doesn’t like low sounds or to be grabbed; the two legs aren’t tentacles and don’t act the same; uses them for long walks, tentacles for leaps; uses tentacles for close vision too and for manipulation, of course; extended, they mean he’s at ease; retracted, on guard or nervous; sharply retracted, disgusted; greeting—”

Just then, one of them swept across my face like a sweet-smelling feather duster and I said, “Illy, man, it’s been a lot of sleeps,” and brushed my fingers across his muzzle. It still took a little self-control not to hug him, and I did reach a little cluckingly for his lopped tentacle, but he wafted it away from me and the little voice-box belted to his side squeaked, “Naughty, naughty. Papa will fix his little old self. Greta girl, ever bandaged even a Terra octopus?”

I had, an intelligent one from around a quarter billion a.d., but I didn’t tell him so. I stood and let him talk to the palm of my hand with one of his tentacles—I don’t savvy feather-talk but it feels good, though I’ve often wondered who taught him English—and watched him use a couple others to whisk a sort of Lunan band-aid out of his pouch and cap his wound with it.

Meanwhile, the satyr knelt over the bronze chest, which was decorated with little death’s heads and crosses with hoops at the top and swastikas, but looking much older than Nazi, and the satyr said to Sid, “Quick thinkin, Gov, when ya saw the Door comin in high n soffened up gravty unner it, but cud I hav sum hep now?”

Sid touched the Minor Maintainer and we all got very light and my stomach did a flip-flop while the satyr piled on the chest the clothes and weapons that Illy had been carrying and pranced off with it all and carefully put it down at the end of the bar. I decided the satyr’s English instructor must have been quite a character, too. Wish I’d met him—her—it.

Sid thought to ask Illy if he wanted Moon-normal gravity in one sector, but my boy likes to mix, and being such a lightweight, Earth-normal gravity doesn’t bother him. As he said to me once, “Would Jovian gravity bother a beetle, Greta girl?”


I ASKED Illy about the satyr and he squeaked that his name was Sevensee and that he’d never met him before this operation. I knew the satyrs were from a billion years in the future, just as the Loonies were from a billion in the past, and I thought—Kreesed us!—but it must have been a real big or emergency-like operation to have the Spiders using those two for it, with two billion years between them—a time-difference that gives you a feeling of awe for a second, you know.

I started to ask Illy about it, but just then Beau came scampering back from the bar with a big red-and-black earthenware goblet of wine—we try to keep a variety of drinking tools in stock so folks will feel more at home. Kaby grabbed it from him and drained most of it in one swallow and then smashed it on the floor. She does things like that, though Sid’s tried to teach her better. Then she stared at what she was thinking about until the whites showed all around her eyes and her lips pulled way back from her teeth and she looked a lot less human than the two ETs, just like a fury. Only a time traveler knows how like the wild murals and engravings of them some of the ancients can look.

My hair stood up at the screech she let out. She smashed a fist into the divan and cried, “Goddess! Must I see Crete destroyed, revived, and now destroyed again? It is too much for your servant.”

Personally, I thought she could stand anything.

There was a rush of questions at what she said about Crete—I asked one of them, for the news certainly frightened me—but she shot up her arm straight for silence and took a deep breath and began.

“In the balance hung the battle. Rowing like black centipedes, the Dorian hulls bore down on our outnumbered ships. On the bright beach, masked by rocks, Sevensee and I stood by the needle gun, ready to give the black hulls silent wounds. Beside us was Ilhilihis, suited as a sea monster. But then … then …”

Then I saw she wasn’t altogether the iron babe, for her voice broke and she started to shake and to sob rackingly, although her face was still a mask of rage, and she threw up the wine. Sid stepped in and made her stop, which I think he’d been wanting to do all along.


CHAPTER 5

Whenever I take up a newspaper and read it, I fancy I see ghosts creeping between the lines. There must be ghosts all over the world. They must be as countless as the grains of the sands, it seems to me.

—Ibsen

SID INSISTS ON GHOSTGIRLS

MY Elizabethan boy friend put his fists on his hips and laid down the law to us as if we were a lot of nervous children who’d been playing too hard.

“Look you, masters, this is a Recuperation Station and I am running it as such. A plague of all operations! I care not if the frame of things disjoints and the whole Change World goes to ruin, but you, warrior maid, are going to rest and drink more wine slowly before you tell your tale and your colleagues are going to be properly companioned. No questions, anyone. Beau, and you love us, give us a lively tune.”

Kaby relaxed a little and let him put his hand carefully against her back in token of support and she said grudgingly, “All right, Fat Belly.”

Then, so help me, to the tune of the Muskrat Ramble, which I’d taught Beau, we got girls for those two ETs and everybody properly paired up.

Right here I want to point out that a lot of the things they say in the Change World about Recuperation Stations simply aren’t so—and anyway they always leave out nine-tenths of it. The Soldiers that come through the Door are looking for a good time, sure, but they’re hurt real bad too, every one of them, deep down in their minds and hearts, if not always in their bodies or so you can see it right away.

Believe me, a temporal operation is no joke, and to start with, there isn’t one person in a hundred who can endure to be cut from his lifeline and become a really wide-awake Doubleganger—a Demon, that is—let alone a Soldier. What does a badly hurt and mixed-up creature need who’s been fighting hard? One individual to look out for him and feel for him and patch him up, and it helps if the one is of the opposite sex—that’s something that goes beyond species.

There’s your basis for the Place and the wild way it goes about its work, and also for most other Recuperation Stations or Entertainment Spots. The name Entertainer can be misleading, but I like it. She’s got to be a lot more than a good party girl—or boy—though she’s got to be that too. She’s got to be a nurse and a psychologist and an actress and a mother and a practical ethnologist and a lot of things with longer names—and a reliable friend.


NONE of us are all those things perfectly or even near it. We just try. But when the call comes, Entertainers have to forget grudges and gripes and envies and jealousies—and remember, they’re lively people with sharp emotions—because there isn’t any time then for anything but help and don’t ask who!

And, deep inside her, a good Entertainer doesn’t care who. Take the way it shaped up this time. It was pretty clear to me I ought to shift to Illy, although I wasn’t quite easy in my mind about leaving Erich, because the Lunan was a long time from home and, after all, Erich was among anthropoids. Ilhilihis needed someone who was simpatico.

I like Illy and not just because he is a sort of tall cross between a spider monkey and a persian cat—though that is a handsome combo when you come to think of it. I like him for himself. So when he came in all lopped and shaky after a mean operation, I was the right person to look out for him. Now I’ve made my little speech and know-nothings in the Change World can go on making their bum jokes. But I ask you, how could an arrangement between Illy and me be anything but Platonic?

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Write up on Piers Anthony’s A Spell for Chameleon  https://ddcomics.org/2026/06/21/write-up-on-piers-anthonys-a-spell-for-chameleon/ https://ddcomics.org/2026/06/21/write-up-on-piers-anthonys-a-spell-for-chameleon/#respond Sun, 21 Jun 2026 21:06:19 +0000 https://ddcomics.org/?p=6900 Introduction: born Oxford, England: 6 August 1934 Piers Anthony is an American author of science fiction and fantasy as novels, novellas, short stories, collections and most of his works are available as eBooks. His fictional work lies somewhere between space opera and high fantasy. He has also written some semi-autobiographical non-fiction. He born was in […]

The post Write up on Piers Anthony’s A Spell for Chameleon  appeared first on DayDreamin’ Comics.

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Introduction:

born Oxford, England: 6 August 1934

Piers Anthony is an American author of science fiction and fantasy as novels, novellas, short stories, collections and most of his works are available as eBooks. His fictional work lies somewhere between space opera and high fantasy. He has also written some semi-autobiographical non-fiction.

He born was in England and educated in the USA, he became a US citizen in 1958.

His first published work was the short story Possible to Rue (1963) in Fantastic, following which he often appeared in magazines during the 1960’s before focusing on longer works. His early work is collected in Anthonology (1985).

His first novel was Chthon (1967), concerning a vast underground prison. There was a sequel, Phthor (1975). His second novel was the extremely long space opera, Macroscope (1969).

Works of note include the Incarnations of Immortality series, beginning with On a Pale Horse (1983); the post-holocaust Battle Circle series, beginning with Sos the Rope (1968); and The Cluster series, beginning with Cluster (1977).

He is well-known for his long-running novel series set in the fictional realm of Xanth which continues to be written. The 1st novel in the series was A Spell for Chameleon (1977), the 47th novel was Apoca Lips (2023). The next novel will be Three Novel Nymphs and he promises even more to come.

He is another of the authors whose work has frequently appeared in the queries in our SFF Chronicles Book Search forum, an indication that it is remembered with some fondness.

A Spell for Chameleon by Piers Anthony.

Significance of the Story
In the land of Xanth it is the law that everyone must have a magical talent.  But Bink doesn’t have one, and he is approaching his 25th birthday, when he must demonstrate a magical talent or be exiled to Mundania.  He decides to go to Good Magician Humfrey, the Magician of Information, to ask whether he has a talent that he doesn’t know about.

The trip takes many days, and he meets various people and encounters various hazards along the way.

He meets Chester, a male centaur, who tells Bink to leave centaur territory.  But Bink stubbornly insists he is on a right of way, and they get into a fight.  A female centaur called Cherie rescues Bink, and then carries him part of the way.

Then he encounters the wide Gap Chasm.  Wynne, a beautiful but stupid girl, shows him the way down into it.  But then he has to run for his life from the Gap dragon.

Later he meets Iris, the Sorceress of Illusion.  She proposes to use her powers to make it seem he has a talent. Then he could become the King, and she, the power behind the throne.  But Bink rejects this offer.

Bink finds an injured soldier, Crombie, in a ditch, and heals him with magic water from the Spring of Life.  They travel on together.  They meet a plain-looking girl called Dee.  But Dee leaves them due to some of the comments made by the woman-hating Crombie.

Bink arrives at Magician Humfrey’s castle.  Humfrey determines that Bink has a strong magical talent, but there seems to be some magic preventing it from being identified.

Bink returns home, and then goes on trial before the King, to demonstrate his magical talent.  The King, who is known as the Storm King, is old; he previously had the power to generate storms, but now can hardly raise a gust of wind.  He does not accept the note from Magician Humfrey, and sentences Bink to exile.

Xanth is surrounded by a magical Shield which prevents living things from crossing. This is to prevent the waves of invasion of people from Mundania, which has resulted in violence and bloodshed in times past. A temporary gap is made in the Shield for Bink to leave Xanth. But outside the Shield, Bink is captured by soldiers under the command of Trent.  Trent, known as the Evil Magician, had tried to take the throne of Xanth 20 years previously, but had been defeated and exiled.  He has the magical talent of transforming any living thing into any other living thing.  He has now gathered an army, and is attempting to enter Xanth and take the throne again.  He wants Bink to tell him where the stone controlling the Shield is, so he can destroy it with a catapult, but Bink refuses.

Shortly afterwards, the soldiers capture another exile from Xanth, an ugly but smart woman called Fanchon.  Bink and Fanchon are imprisoned together.  But they manage to escape.  Somehow Bink, Fanchon and Trent end up back in Xanth, but without Trent’s men.  The three decide on a truce, while they help each other travel through the hazards of the Xanth wilderness.

They find an abandoned castle, Castle Roogna, which had been the King’s Castle many generations ago.  While staying there, Bink notices that Fanchon is becoming more attractive; in fact she looks a lot like Dee, whom he had previously met.  Fanchon confesses that her real name is Chameleon; over the period of a month, she changes from Wynne (beautiful and stupid), to Dee (average in both appearance and intelligence) and Fanchon (ugly and smart) and then back via Dee to Wynne.  The only way she can get rid of her magical nature is by living in Mundania, where she would always have the appearance of Dee.  Meanwhile, Bink is starting to think that Trent is not quite as evil as his reputation.

After some time in the castle, they travel on.

They discover a large gathering of creatures of all kinds.  The creatures have been brought together by Herman the Hermit, a centaur with the magic talent of summoning will-o’-the-wisps.  (Centaurs regard magic as obscene, and for this reason Herman had been exiled from his kind.)  Herman had brought the creatures together to eradicate a wiggle swarm.  Wiggles are small wormlike creatures that hover in the air, and then shoot off in a straight line, boring holes through everything in their path, which can kill any creature in their way.  The other creatures are killing the wiggles by crushing them between rocks, crunching them with their teeth, stomping them etc.  Trent, Bink and Chameleon join in.  Finally they defeat the wiggles, but many creatures, including Herman, have lost their lives.

Trent, Bink and Chameleon travel on.  But Trent still intends to take the throne, and Bink wants to stop him.  Bink challenges Trent to combat, in an attempt to prevent him taking the throne.

Literature Review

Bink is a likable young man, with a rare handicap. His home is the North Village of Xanth, a land of magic, where everyone and everything has at least a minor magical capability; however, Bink has none. Although he has physical strength, Bink is treated badly by his peers and lacks self-esteem. He does have the affection of Sabrina, a beautiful girl he wants to marry, and the love of his parents, Roland and Bianca. As the story opens, Bink is near his home, watching a chameleon change its appearance and ultimately lose its life to a moth hawk, which foreshadows several events to come. After the chameleon is gone, Bink discusses his problem with Sabrina, who urges him to seek the advice of the Good Magician Humphrey. Although Bink is reluctant to seek Humphrey’s help because the magician’s price is one year of servitude, he relents, as his twenty-fifth birthday is only a month away. If he is unable to demonstrate a magical talent by then, Bink will be exiled from Xanth, the only home he has ever known.

On the long journey to Good Magician Humphrey, Bink overcomes many obstacles. He is bedeviled by some creatures and plants, but aided, protected or taught by others. Bink’s first adversary is Chester, a mean centaur. Cherie, a friendly centaur, comes to his aid and helps Bink travel closer to Humphrey’s castle. As they travel, she educates him about the violent history of how humans came to Xanth. She tells Bink that magic may not be as valuable as most people think it is. Before leaving him, Cherie also tells Bink about a centaur named Herman the Hermit who was banished from her herd for an act of obscenity, but refuses to elaborate further.

Bink’s next obstacle is the Gap, a huge trench across Xanth that is filled with dragons and strange creatures. As night is falling, he agrees to stand in for a farmer at a rape trial in exchange for a night’s shelter, before trying to cross the Gap. At the rape trial the next morning, both the charge and the victim are treated dismissively, but Bink has fulfilled the requirement and, when he leaves, is provided with the service of a gorgeous guide named Wynne. Wynne may be the woman who was raped, although he is not certain. Bink quickly realizes that, unfortunately, her intelligence leaves much to be desired. When a dragon appears from the Gap, he sends her away and ends up trapped in a cave in the Gap. Bink escapes the dragon by allowing a shade, named Donald, to possess him so that he can give the location of a silver tree to his impoverished wife and child, many miles distant. Bink is offered the wealth of the silver tree as a reward for helping Donald, but refuses the gift.

After traveling further, Bink falls into a sea and is saved by the Sorceress Iris, a mistress of illusion, who offers him anything to help her take Xanth’s throne. Bink refuses and fights her illusions using the strength of his mind before escaping. As the journey continues Bink finds a badly wounded soldier, named Corporal Crombie, whom he saves using water from the Spring of Life. The Spring’s water can heal any injury or illness and helps Bink to grow a finger that he lost in a childhood accident. He saves a little water to be used later. Crombie’s magical talent is that he can discern danger from any direction and he becomes Bink’s bodyguard and guide. When he senses danger ahead, he urges Bink to go toward it because he says danger must always be faced. They find Dee, a very average-looking girl, whom Bink immediately likes. Crombie tells Bink that she is the danger, but Bink insists that she is not. The men ask Dee to travel with them, as she is heading for Humphrey too. The travelers are forced to take shelter from a magical hailstorm in a tangler tree, but Crombie’s opinion that all women are mean and manipulative, angers Dee. She goes off into the storm, with Bink and Crombie behind. They ask her to resume traveling with them, but she refuses. Crombie takes Bink close to the castle of Good Magician Humphrey and amiably leaves him.

Humphrey is unable to discover Bink’s talent but insists that he does have one. He gives Bink a note to give to the King at his trial, attesting to this fact and providing safe passage through the forest so he can quickly return to the North Village. At Bink’s trial in the North Village, the King disregards the note and refuses Bink’s gift of water from the Spring of Life. Bink is exiled and passes through the magical Shield separating Xanth from Mundania. He misses his parents, but realizes that his relationship with Sabrina was superficial and that he is better off without her. Soon, soldiers attack him. They take him to Trent, an evil magician, who twenty years ago had tried to overthrow the King. Trent still wants the throne and has amassed an army to help him attain it, but he needs the location of the Shield to do it. He has developed a magical suppression elixir that will render the Shield harmless, and tries to threaten Bink into revealing the Shield’s location. Bink resists, but a woman named Fanchon comes through the Shield from Xanth and appears before Trent. Fanchon is very ugly and very smart. When both are imprisoned in a pit, she tells Bink that she chose to leave Xanth because Humphrey told her that her ugliness, which is magical, might be reversed if she went to Mundania. She develops a plan to escape from Trent and steal the magical suppression elixir, but Bink, Fanchon, and Trent end up back in Xanth after a dramatic sea battle. If discovered, Bink and Trent will be executed and Fanchon is still ugly. To escape the wilderness, they join forces and declare a truce.

Bink, Trent, and Fanchon take shelter in Castle Roonga, where Fanchon reveals her identity as Chameleon, a woman who undergoes three drastic changes in appearance and intelligence level every month. She is average-looking Dee, beautiful, stupid Wynne, and ugly, smart Fanchon for a period each month. Her hope was to travel to Mundania, where she could revert to the average looks and intelligence of Dee forever. Fanchon also admits that she followed Bink because she knew his good nature, from having met him as both Dee and Wynne. Castle Roonga was once the stronghold of dead King Roonga, the Magician King of the Fourth Wave, a smart, just ruler. It maintains its own level of magic and protects itself with zombies and other mechanisms. After researching in Castle Roonga’s library, Trent announces that Xanth must open its borders to Mundanians or perish because of its isolation. He feels the pull of the throne again but the characters agree that their best option for survival is to work together and maintain their truce until they approach civilization. Trent promises the Castle that he will return to it to rule Xanth with its help.

After leaving Castle Roonga, Chameleon, Bink and Trent come upon a great battle, involving all of the creatures of Xanth, united against a swarm of wiggles, which are the most dangerous creatures in Xanth. They meet Herman the Hermit, who confesses that his obscenity was to use magic, something forbidden among the centaurs. The battle ends when Bink is transformed into a fiery salamander to set the wiggle swarm on fire. He saves Chameleon from a fiery death, but ends up inside the ring of fire. Herman the Hermit saves him, but is gravely injured and asks for a quick death from Trent. Trent obliges, praising the centaur’s courage.

At the edge of civilization, Bink and Chameleon agree to part company with Trent, but the Sorceress Iris appears. She coerces Trent into helping her take the throne. Bink is infuriated by Trent’s decision and the men agree to a duel in the forest. As Bink battles Trent, the magician changes Chameleon into a winged doe, yet, he finds himself unable to transform Bink. Through the duel, Bink’s magical talent is finally revealed: he cannot be harmed by magic. When his life is in true peril, Trent’s most powerful transformation spells are useless against him. When Trent tries to kill him physically with a sword, Chameleon comes between the men and is gravely wounded. The Sorceress Iris urges Trent to kill them both, but he refuses, saying the he will not kill a man who has saved his life or the woman who loves him. He quickly changes Bink into a phoenix to fly off to Good Magician Humphrey for help in saving Chameleon.

Although Sorceress Iris hounds Bink on the way, he reaches Humphrey and they are soon en route to the North Village to obtain the help of Bink’s father, Roland, a stunner. They also learn that the old King has died. When they return, Roland stuns Trent and Chameleon is healed. They all return to the North Village for Trent’s trial. Trent is exonerated on the conditions that he marries the Sorceress Iris to control her and assume the throne. At the coronation, Bink again meets many of his old acquaintances. Trent appoints him as the Official Researcher of Xanth, but urges him to keep his magical talent private, as he explores the roots of Xanth’s magic. Bink asks Chameleon to marry him, as the book closes.

The title of the book refers to Chameleon’s quest for a spell that will allow her to have a single form, and therefore a stable identity. Her arrival in Bink’s life is foreshadowed by his opening observations of the chameleon on the rock, which changes into a stingray beetle, stench puffer, fiery salamander and basilisk, before being killed and carried off by a moth hawk. The lizard’s guises are a comparison to the woman, Chameleon, who can never have real happiness while she is constantly changing. Initially, Bink interprets the death of the lizard as an omen that his life will be in danger, but it is actually about how he saves Chameleon by giving her the “spell” of true, unconditional, love.

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Write up on René Barjavel ‘s The Ice People (La Nuit des temps) https://ddcomics.org/2026/06/21/write-up-on-rene-barjavel-s-the-ice-people-la-nuit-des-temps/ https://ddcomics.org/2026/06/21/write-up-on-rene-barjavel-s-the-ice-people-la-nuit-des-temps/#respond Sun, 21 Jun 2026 21:00:41 +0000 https://ddcomics.org/?p=6897 Background of the Study Introduction René Barjavel is the author who may have been first to come up with the “grandfather paradox” in time-travel, and that’s pretty informative of what he wrote. We often recall France’s great science fiction leader, Jules Verne, while overlooking his waves of contemporaries, much less authors like Barjavel who penned […]

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Background of the Study

Introduction

René Barjavel is the author who may have been first to come up with the “grandfather paradox” in time-travel, and that’s pretty informative of what he wrote. We often recall France’s great science fiction leader, Jules Verne, while overlooking his waves of contemporaries, much less authors like Barjavel who penned SF novels in the 1940s into the 1980s. Many of his works remain untranslated, though several were printed in English in the late ’60s and early ’70s. One of his earliest novels, Ravage, was translated by Damon Knight as Ashes, Ashes; another, La nuit des temps, became The Ice People.

The grandfather paradox shows that traveling back in time is impossible. French journalist Rene Barjavel, the creator of the grandfather paradox, wrote the novel Le Voyageur Imprudent in 1944. In it, he writes of a man going back in time to kill his own grandfather at a date before his parents were born to illustrate his point.

According to Barjavel, it would be impossible for a man to go back in time and kill his own grandfather because then the man himself, the killer, would not exist to commit the crime. Once his grandfather is dead, he can never father the killer’s mom or dad, who then in turn can never create the killer himself. Thus, time travel is impossible because any change you make to history would have rippling consequences that would change the very fabric of your life as well as human history. In essence, the only way you or I exist right now is because everything in the past happened exactly the way it did. So by going back and changing something, you’re simultaneously preventing yourself from existing.

The grandfather paradox has become a mainstay in modern philosophy and physics. Some argue that time travel is indeed possible, because once a man goes back in time, he creates a parallel universe that operates separately from the one he left. Some also argue that even if Barjavel is correct, his point only leads to the conclusion that going back in time is impossible, but that time travel could still be possible into the futu

Will Time Travel ever be possible? Rene Barjavel was a French journalist and science fiction writer who spent a lot of his time thinking about time travel. In 1943, Barjavel asked what would happen if a man went back in time, to a date before his parents were born, and killed his own grandfather? With no grandfather, one of the man’s parents would never have been born – and therefore the man himself would never have existed. So there would be nobody to go back in time and kill the grandfather in the first place. Or the last place – depending on how you look at it. The Grandfather Paradox has been a mainstay of philosophy, physics and the entire Back to the Future Trilogy. Some people have tried to defend time travel, with arguments like the Parallel Universe Resolution – in which the changes made by the traveler create a new, separate history, branching off from the existing one. But the Grandfather Paradox prevails. Although the paradox only suggests that traveling backwards in time is impossible – it doesn’t say anything about going the other way…

                                                               Literature Review

A group of French explorers in Antarctica make a wondrous discovery—a large golden sphere buried deep under the ice. Joined by a multitude of foreigners to become international band of scientists and technicians, the expedition begins its delve into the ice and uncovers one miracle after another. The prize: two human beings, one male and one female, cryogenically frozen from some 900,000 years in the past. After thawing out the woman, Elea, she tells a pitiable story of a human utopia from before known history, of her love won and lost under the looming shroud of apocalyptic war, where great technologies have become humanity’s greatest asset (as well as its potential destruction). Meanwhile, agent provocateurs and spies have infiltrated the scientific base; steaming south are American carrier-groups and a Soviet sub-flotilla. As the story of the past’s final war is retold, the last war of the present may be playing out as well…

It’s pretty clear that the novel was intended as an anti-war novel, and while it doesn’t specify that theme in the “contemporary” timeline—the Soviets, Americans, and Europeans tend to grudgingly work together—the flashbacks to the time before are rife with metaphor. It takes Vietnam War-era turmoil and transplants it to the end times: as nuclear weapons sanitize the moon’s tropical jungles into ash and rubble, riot police and student protesters clash beneath giant monitors urging calm as it’s assured war will be averted. It’s epic scenery, pure sensawunda for me, and some of the book’s more effective scenes. Each side has their own weapon guaranteeing mutually assured destruction, and their petty politics and ideologies have led them astray. It’s firm-handed but poignant, definitely a product of its time.

The Ice People felt oddly old-fashioned to me, bringing to mind the ideological utopias, future histories, and suspended animation plots of the 1930s (Wells’ The Shape of Things to Come for one, his When the Sleeper Awakes for another, Stapledon’s Last and First Men for a third). Not surprisingly, Barjavel was influenced by H. Rider Haggard’s When the World Shook, with which is shares the suspended-animation trope. In terms of writing and translation, it’s perfectly modern, with very serviceable prose and some decent (if cardboard) characters, though Barjavel does tend to ramble a bit and venture off on asides—such as visiting one particular French family, standing in as civilian everymen. The themes are all very contemporary for the 1970s, dominated by Cold War politics—nods to the Non-Aligned Movement and France’s tenuous “neutral” role between the greater powers. No, it’s mostly that the novel is a utopian future-history in reverse—the utopian far-past is now just a dream, televised live from the Antarctic as a warning to a world in the grips of Cold War.

The discovery of a signal, thousands of feet below the Antarctic ice, turns a small French expedition into a billion dollar investment broadcasted on live television worldwide. What they uncover will change the history of mankind forever. Encased within a golden orb amid a metropolis, a man and a woman lie in cryogenic stasis – preserved with temperatures that modern science cannot replicate. The ruins are dated to be over nine-hundred thousand years old. This book is a romantic science fiction, and focuses on mankind’s repeated mistakes through the course of history, hinting at our fate in the future through the downfall of a people that thrived nearly a million years ago.
The Ice People does contain eroticism. It is mild, yes, but still present, as it is an important part of the plot-line.

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Write up on Pierre Bordage (REVISION 2) https://ddcomics.org/2026/06/21/write-up-on-pierre-bordage-revison-2/ https://ddcomics.org/2026/06/21/write-up-on-pierre-bordage-revison-2/#respond Sun, 21 Jun 2026 20:49:54 +0000 https://ddcomics.org/?p=6891 Translated in French to English:                                       Introduction Background of the Study Pierre Bordage was born in January 1955 in La Réorthe, Vendée, France. After uneventful years at school, nine years of karate and a few banjo lessons, he discovered writing during a workshop at Nantes University in 1975…A few trips to the Orient and some […]

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Translated in French to English:                                      

Introduction

Background of the Study

Pierre Bordage was born in January 1955 in La Réorthe, Vendée, France. After uneventful years at school, nine years of karate and a few banjo lessons, he discovered writing during a workshop at Nantes University in 1975…
A few trips to the Orient and some experience as a bookseller later, he settled down in the Gers in 1985 and wrote Les Guerriers du silence (The Warriors of Silence). Not until 1992 did he meet his first editors, at Vaugirard, having returned to Paris in the meantime to work as a sports reporter. He was offered the Rohel series. Back in 1992 L’Atalante received the manuscript of Les Guerriers du silence and published it in March 1993. This space opera with epic dimensions where hundreds of worlds clash, powerfully based on our myths and collective imagination, had immediate public success, which in no way diminished over the three years during the publication of the trilogy. Recognition by both amateurs and professionals came when he was awarded the Prize for l’Imaginaire and the Julia Verlanger Prize for Les Guerriers du Silence in 1994 and the Cosmos 2000 Prize for La Citadelle Hyponéros (The Citadel of Hyperneros) in 1996.
In the summer of 1999, he left to live in the United States for two years with his family before returning to his native region in France.

With L’Enjomineur in 2004, Pierre Bordage renders homage to this region by enhancing historical facts with fantastical elements. This trilogy was awarded the Imaginales Prize in 2007.

Today Pierre Bordage lives in the French South-West. During several years he served as President of the International Science Fiction Festival Les Utopiales.

Pierre Bordage was born in January 1955 in La Réorthe, Vendée, France. After uneventful years at school, nine years of karate and a few banjo lessons, he discovered writing during a workshop at Nantes University in 1975…

Literature Review

Some one hundred worlds make up the Confederation of Naflin, including the sumptuous and refined Syracusa. However, in the shadow of the ruling family, the mysterious Scaythes of Hyponeros, from a distant world, gifted with disturbing psychic powers, are plotting a gigantic plot of which the establishment of a dictatorship over the Confederation is only one step. Who could stand in their way? The warrior monks of the Absourate Order? Or should we take into account this obscure employee of a travel company who drowns his boredom in alcohol on the planet Deux-Saisons? Because his life changes the day a beautiful Syracusan woman, hunted, passes through the door of his agency… Few French science fiction novels are animated by a truly epic breath. Pierre Bordage, from his first attempt, delivers to us with Les guerriers du silence the first part of an authentic Space opera.

 Who, then, can stand up against them? The monk-warriors of the Absourate order? Or will they have to rely on an obscure employee in a travel agency, who is drowning his sorrows on the planet Two-Seasons? Because his life is turned upside-down the day a beautiful Syracusan woman who is being trailed, turns up on his doorstep.

The Confederation of Naflin is a vast political entity encompassing hundreds of worlds, with varied customs and religions, and diverse modes of government (although generally authoritarian and aristocratic), but which have managed to agree on a sort of “minimum regime”, consisting of a council of the main heads of state and adherence to a few common economic principles – in the best interests of each party.

The Confederation of Naflin also has military forces: the Interlice, the Confederal Army, a sort of interplanetary super-police force, and the Absourate Order, made up of warrior monks tasked with protecting the integrity of the Confederation. Within the latter, Syracusa is a powerful planet that dictates its clothing styles to the nobility of the entire known universe, and one of whose princes commands the Interlice.

The main oddity of the Syracuse aristocracy: the mysterious Scaythes of Hyponeros, non-humans from an unknown planet, who have managed to make themselves indispensable, have entered its ranks. The Scaythes first sold their services to the nobles as telepathic spies, then as bodyguards preventing… any telepathic espionage!

The final stage of their takeover of Syracusan society, one of their own, Pamynx, has been appointed Grand Constable, a key position in the government. With the full complicity of the Church of Kreuz, with its expansionist and totalitarian aims, the Scaythians will begin to brutally take control of the entire Confederation.

“The thoroughness and breadth of the description combine with the precision of the psychological portraits and the spiritual dimension of the characters and worlds crossed to make these WARRIORS OF SILENCE a success that is all the more exceptional since Pierre Bordage is signing his first novel here.” (Joëlle Wintrebert)

Tixu Oty was a lowly CILT employee, stuck on a jungle planet far from civilization, when he was thrust into adventure by the sudden arrival of a stunning young woman seeking a transfer. Despite CILT’s draconian regulations, Tixu allowed her to transfer without paying for the trip—and he was the last person to understand her reasons. Rescued by a native jungle sorcerer, Tixu found himself chasing after the young woman, always narrowly escaping the worst dangers, as if a grander destiny truly awaited him.

The young girl? She is Aphykit, daughter of one of the last three sages who possessed the secret of Indic mental science.

Barrofil the Twenty-Fourth is an abominable old man with a silhouette full of angles and edges, supreme leader of the Church of the Kreuz who crushes the former Confederation under his cruel rule, slowly burning on the torture of the Fiery Cross every opponent, every recalcitrant nobleman, every priest of another religion.

Pamynx is the Scaythe designated to lead the plot, he will later be replaced by Harkot, the superior link in the evolution of the Scaythe race (humanoid in appearance, they seem to be quite different in fact from what their appearance would suggest — some innuendos made me think of a race of fungi).

     In the beginning was a federation, a game of power regulated by laws and treaties… Then Hyponeros arrived, gently, imperceptibly, so humble, so servile. Represented by strange beings, the Scaythes, Hyponeros is at the service of humanity, Hyponeros is at the service of power. Then, the Church of Kreuz and the Kingdom of Siracusa joined forces to enslave worlds, to invade, destroy, and rule. The great Ang’Empire replaced the Confederation of Naflin, crushed everything in its path, and the Kreuzian Church launched its holy war and imposed its spiritual supremacy with “slow-burning fire crosses,” torture devices born from the darkest imagination. To counter this power grab, no one could rise up, not even the Absurate knights could resist the terrible Scaythes of Hyponeros and their weapon of death… Yet, somewhere in the universe, a resistance is slowly forming, through intelligence and purity, research and spirituality… The last bastions of refusal, these famous Warriors of Silence, are not necessarily those we think they are, but they all share a unique quality: innocence, because, it seems, only innocence can conquer nothingness.

     In this marvelous saga, Bordage takes us on a journey through humanity, drawing on both the great classics of science fiction and some delightful surprises of his own, playfully engaging with our old planet, its dreams and civilizations, to forge another that is an exaggerated caricature of them. Open any volume of this saga and you won’t be able to put it down; Bordage’s stories seep into us like dreams, like the myth of rebirth and freedom he sings to us with his words. Every human being is a part of the myth, just as the battle of the Warriors of Silence against the Hyponeros is a wonderful metaphor for a fight each of us should wage against the ease of nothingness. Essential reading

In the beginning was a federation, a game of power regulated by laws and treaties… Then Hyponeros arrived, gently, imperceptibly, so humble, so servile. Represented by strange beings, the Scaythes, Hyponeros is at the service of humanity, Hyponeros is at the service of power. Then, the Church of Kreuz and the Kingdom of Siracusa joined forces to enslave worlds, to invade, destroy, and rule. The great Ang’Empire replaced the Confederation of Naflin, crushed everything in its path, and the Kreuzian Church launched its holy war and imposed its spiritual supremacy with “slow-burning fire crosses,” torture devices born from the darkest imagination. To counter this power grab, no one could rise up, not even the Absurate knights could resist the terrible Scaythes of Hyponeros and their weapon of death… Yet, somewhere in the universe, a resistance is slowly forming, through intelligence and purity, research and spirituality… The last bastions of refusal, these famous Warriors of Silence, are not necessarily those we think they are, but they all share a unique quality: innocence, because, it seems, only innocence can conquer nothingness.

Themes

The Warriors of Silence addresses different themes on human nature, both creative and destructive, both sovereign and easily subjected to something else (the Scaythes). The work criticizes not religions, but dogmatism and the misuse of religious teachings for the benefit of the seizure of power by a minority.

The specter of a possible nuclear catastrophe (due to a war on Earth, an accident on Ut-Gen) is evoked in the background as a warning about the risks of progress, but the main illustration of this theme is the creation of the Scaythes from robots abandoned by humans.

Characters

The 12 Warriors of Silence

• Aphykit Alexu / Naia Phykit: daughter of Sri Alexu, a master of Inddic science. She flees during her father’s assassination and benefits from the help of Tixu Oty, whom she initiates into Inddic power, before becoming his wife.

• Yelle: the daughter of Tixu and Aphykit, she has the power to sense the advance of the threatening void and which she calls the “blouf”. It also seems that she benefits from a certain prescience. • Aphykit Alexu / Naia Phykit: daughter of Sri Alexu, a master of Inddic science. She flees during her father’s assassination and benefits from the help of Tixu Oty, whom she initiates into Inddic power, before becoming his wife.

• Shari Rampouline / The Mahdi Shari of the Hymlyas: one of the last inhabitants of the Earth, he is initiated by a mysterious hermit, the madman of the mountains. He learns to make stones fly then meets Tixu and Aphykit. His search for wisdom pushes him to undertake a long wandering between the worlds, during which he meets Oniki Kay. It is he who discovers the Indic annals.

• Jek At-Skin: born on the radioactive planet Ut-Gen, he discovers his powers as a human-source by calming a tribe of desert hyenas, which earns him the nickname “Prince of Hyenas”. Could he be the chosen one of the Jersalémines?

• San Francisco: Formerly prince of the American tribe of Jer Salem, he was condemned to exile for heresy. He became a space pirate under the name of San Frisco as second in command of the Viduc Papironda, and met Jek during his journey when the latter wanted to leave Ut-Gen. Thanks to the word of the Abyn Elian, he can make himself invisible for a few seconds. He masters bladed combat.

• Phoenix: San Francisco’s fiancée, originally from the same world as the latter. Thanks to the word of the Abyn Elian, she can also make herself invisible for a few seconds.

• Oniki Kay: Member of a female corporation on Ephren, this very agile thûta is a real organ acrobat, she is outlawed for having loved Shari. Exiled to an island, she gives birth to Tau Phraïm there.

• Tau Phraïm: the son of Shari and Oniki, he knows how to speak to coral snakes and move in the same way as them: silently, quickly and skillfully. Despite his very young age, he seems to benefit from a precocious maturity, even more than Yelle.

• Fracist Bogh: educated by the Kreuzian Church, he becomes its muffi (grand master) under the name of Barofill the Twenty-fifth. It is the post-mortem revelations of his predecessor Barofill the Twenty-fourth that persuade him to side with the warriors of silence. He learns the inddic graphemes of protection and healing, which allows him to escape mental inquisitions and to heal Ghë.

• Ghë: a passenger on the El Guazer, a convoy of ships of Earth exiles, she is revealed to be one of the twelve chosen ones during a trance. She will be the only survivor of the convoy. She masters telepathy.

• Whu Phan-Li: the last knight of the Absourate Order, converted into a child trafficker and brought back to the right path by a clairvoyant. Like any veteran knight of the Absourate Order, he kills with a scream and masters close combat.

Other important characters

The warriors of silence take up such a place in the story that it is difficult to see other important characters emerge. However, we can mention:

• Constable Pamynx, a Scaythe of Hyponeros who is the very first character that we discover in the story and a key piece of the first stage of the plan of the masters Germes.

• Menati Ang, lord of Syracusa then emperor of the galaxy, manipulated by the Scaythes of Hyponeros.

• Seneschal Harkot, a Scaythe deliberately “unfinished” by the Hyponeros in order to simulate human feelings.

• Marti de Kervaleur, a Syracusan nobleman on the run who ends up stranded on the Free City of Space before meeting Jek and San Frisco.

• Muffi Barofill the Twenty-fourth, high pontiff of the church of Kreuz. Other important characters

The warriors of silence take up such a place in the story that it is difficult to see other important characters emerge. However, we can mention:

Literature Sample

CHAPTER 1

No one knows how the Scaythes of Hyponeros managed to secure so much influence on the planet Bella Syracusa, the Queen of the Arts.
Or how they infiltrated the entourage of the Ang family, the dynasty that had ruled uninterrupted for 15 standard centuries.
Or how they progressively got hold of key positions within the Empire. Or how they managed to make themselves indispensable by creating the functions of thought detector and protector. Or how, feared because of their extraordinary mental abilities, they gradually created a reign of terror.
Who were they?
No one knew anything about Hyponeros, or had even heard of this distant world, so distant that it may only have existed in people’s imaginations. But, it turned out that one of its offspring, Pamynx, was given the supreme dignity of being named Chancellor, an honor which had, up until then, been reserved for the sons of Syracusa’s leading families.
This event took place during the reign of Lord Arghetti Ang. At the time, few were offended by it.
What had become of the proud Syracusans of the days of the conquest?
Were they empty shells, shadows, or just puppets of illusion?
Woe to he by whom the offence cometh.

Excerpt from an apocryphal mental text, received during his wanderings by Messaodyne Jhu-Piet, a Syracusan poet of the first post-Ang period. Some scholars think it may have come from stray thoughts of Naia Phytik, of Syracusan origin herself.


 
Pamynx the Chancellor, his face shrouded in the hood of his blue acaba, appeared from the darkness and joined the Lord Ranti Ang and his young protege Spergus, who were awaiting him, with their thought protectors, on the stationary gravitational platform.
“If my lordship would be so kind as to follow me,” said Pamynx, bowing.
“And none too soon,” scolded Ranti Ang. “Are you coming, Spergus?”
With their thought protectors following them like shadows, they stepped into a dark narrow tunnel. They soon came to a heavy wooden door that was incredibly ancient, blocked by thick metallic bars. After a short while, which seemed interminable to Spergus, the bars slid along their rails, which were sealed inside the walls of the tunnel. The damp, close air made the young Osgorite feel uncomfortable. He had the unpleasant feeling that the mold in the rank air was penetrating every pore of his skin.
The door opened onto a wide balcony, lit by two floating light-bubbles, where a small group of men were waiting, their faces hidden behind white masks. Three crossed silver triangles glimmered on the stiff breastplates of their gray uniforms.
Ranti Ang looked at Pamynx with wrath in his eyes.
“You are the high protector of the law, Chancellor! You are therefore aware that Pritivian mercenaries are forbidden to set foot on Syracusa!”
The restrained impatience that pervaded his words showed that he was on the brink of losing control.
“At least do me the honor of answering! Was it really necessary, for the public good, to retain these adventurers?”
“You will understand why they are here in good time, my Lord,” answered Pamynx in a dispassionate tone of voice.
The balcony overhung a huge empty round chamber; in the middle stood a figure, draped in the folds of a jet-black acaba.
“This place is sinister, my Lord!”
Spergus suppressed a shiver. The spectacle of this ghostly figure, standing as still as a statue on the floor below, dimly lit by underground water lamps, exuded a venom of anxiety in his young, impressionable mind. The smell of death wafted through the close air.
“Is that one of your students that you have told me about, Chancellor?” asked Ranti Ang.
Pamynx nodded in agreement.
“May I not see his face?”
“Not for the moment, your highness. But this is not out of a lack of respect for you. The hood of his acaba will cover his head during the experiment, to prevent our thoughts from focussing on his image, which could weaken his psychic potential.”
“Good gracious! And he really possesses this… this power that you have told me about?”
Pamynx did not reply to Ranti Ang’s mocking disbelief. He removed a tiny ring of golden optalium from within the folds of his acaba, and rang it with a rock crystal. A part of the wall slid away, as if changed by the lingering sound, and let in a flood of harsh light.
Three new figures were seen entering the room: two Pritivian mercenaries and a man whose coarse canvas clothes gave off a stench that was almost that of an animal. His simian face was ashen with fear.
Ranti Ang’s face showed a faint expression of disgust. “It looks like a Mikat.”
“A Mikat from the satellite Julius, your highness,” confirmed Pamynx. “He was put on the index and declared raskatta. I thought that… for our experiment…”
“From what I see, or should I say from what I hear, you are trying to vindicate yourself again, Chancellor!” said Ranti Ang, mockingly. “In fact, don’t you spend most of your time trying to vindicate yourself? For everything and especially for nothing!”
Spergus’ bright laughter punctuated the Lord of Syracusa’s comments.
“The Kreuzian Church considers that the Mikats are endowed with souls,” argued the Chancellor. “However, the…”
Ranti Ang cut him off curtly.
“Unfortunately for you, Sir, I am not Arghetti Ang but his elder son. My father thought he was doing the right thing when he appointed you to this position of great responsibility, and so be it. But if I must respect his choice, as he made me promise, nothing requires me to give my esteem to the beneficiary of his choice! Do be so kind as to not bring the Church of Kreuz into your sordid schemes! After all, isn’t this Mikat one of my subjects? Isn’t it up to me, and me alone, to decide if his life should be sacrificed for the common good?”
Pamynx held his resentment hidden behind the impassiveness of his face, and bowed ceremoniously. His day of revenge would soon come. This perspective helped him remain patient in spite of this constant harassment, these daily humiliations.
While this was going on, the two Pritivian mercenaries dragged the terrified Mikat a few feet away from the motionless black acaba.
“Spergus?” Ranti Ang’s voice was suddenly more gentle. “Would it please you to know what this Mikat is thinking about, at this very instant?”
“That… would greatly amuse me, my lord.” mumbled the young Osgorite.
A vague smile showed on his painted lips. He tried to hide the intense fear that this gloomy vault aroused in him.
Pamynx was annoyed by Spergus’ presence. Lord Ranti Ang thought it was a good idea to have his young protege present to witness the key experiment that was about to take place. But it was dangerous to bring affective elements into this first public trial, which required a psychically neutral environment.
“Well, what are you waiting for to reveal to our dear Spergus what is going through the Mikat’s mind? If something is going through it, of course! Is it fear that is causing this horrendous stench?”
Pamynx stared at the Mikat, whose greasy black hair was cut in the traditional manner of the Mikatun of Julius: very short, straight on the neck, and shaved on the sides. Under his protruding eyebrows, the poor man’s bulging eyes flitted back and forth around the chamber like crazy butterflies. From the balcony to the dark threatening figure; from the dark figure to the two Pritivian mercenaries, anonymous behind their white masks.
“His skin is all black!” whispered Spergus.
“That is because he works outdoors each day that Kreuz gives us with his kindness, under the rays of the fire-star Ahkit,” explained Ranti Ang.
The disgust that Spergus felt, induced by this creature from another world and another time, welled up within him like nausea. But he could not take his eyes off that thick neck, those strong arms, wide hands, and stubby fingers with their dust-encrusted nails.
Spergus’ wild uncontrolled thoughts perturbed his concentration and interfered with Pamynx’s mental investigation. The two protectors assigned to Spergus’ security turned out to be incapable of holding back the reckless torrents emanating from his mind. The Chancellor decided to not let anything show – it would be the wrong time to cast doubts on the Scaythes’ efficiency.
Pamynx was, like the thought protectors, a Scaythe from Hyponeros, a paritole, and his origin could bring up the question of the constitutional immunity that his high rank was supposed to confer on him. The great Arghetti Ang had had to stifle the anger of the Syracusan dignitaries to impose him as the Chancellor, and his position was becoming increasingly insecure as time went by, and as the memories of the current ruler’s father faded away.
But for now, Pamynx needed Ranti Ang’s support: this would guarantee the capital needed for the structure of the Great Project; for the fulfillment of the tremendous secret task he had been given by his masters, the Master Embryos of the Hyponeriate. He would soon have a chance to wipe the grin off the Lord of Syracusa’s face.
“We are still waiting, sir. Could it be that you have lost your so-called powers in a room in one of the brothels of Salaun? Yet, you are sexless, are you not?”
Spergus’s mischievous laugh broke out a second time.
“Fear paralyses the Mikat’s mental potential,” the Chancellor finally declared. “He is incapable of formulating the slightest coherent thought. I can tell you, however, that he is trying to recall the face and the body of a woman from Mikatun. Probably his own wife…”
“What an extraordinary discovery!” chuckled Ranti Ang. “You don’t need to be learned in the sciences of the mind to figure out that he is thinking of his wife!”
“Why do you say that, my Lord?” Spergus asked naively. The Lord of Syracusa let out a little sarcastic laugh.
“Before Julius was annexed to Syracusa, these animals, the Mikats, did not marry, and the women of the tribe belonged to all the men of the rural communities. For the last two centuries, the law and the church have required them to take just one spouse. This is the first law of the moral-genetic code governing the satellites. That is why, Chancellor, you are not revealing any wonders by stating that this sub-human is thinking of his wife!”
Impassive, Pamynx ignored Ranti Ang’s mockery and went on: “I also see the faces of some children. Three boys and two girls…”
Subjugated by the importance of the people that were watching him from the balcony, terrorized by the Chancellor’s words, which were the faithful transcription of the few images that were going through his mind, the Mikat let out a scream like a hunted animal and fell on his knees on the cold tiled floor.
“He has a very crude brain,” added Pamynx flatly. “If his brain were as simple as you are suggesting, what would be the value of this experiment applied to superior intelligences? We don’t have to bother with this muddle of cheap witchcraft to subdue the Mikatun of Julius! Our ancestors have already taken care of that without violating the precepts of our holy Church!”
Suddenly, Pamynx realized how delicate his situation was. Occupied by so many different things, he had not paid attention to the rumors that suggested that he had fallen into disgrace. He did not need to slip into Ranti Ang’s mind – a sacrilegious action, which could be punished by death – to understand the deadly intentions that his tone of voice implied. The Chancellor had underestimated the importance of the conspiracy that had been orchestrated against him by Tist of Argolon, the renowned bard of the Syracusan tradition. Even though he had intercepted some thoughts about the underground actions of his Syracusan rival, Pamynx had not deemed it worth his getting involved, thinking that the quality of his relations with the great Arghetti Ang and the length of his service puthim above all of these palace schemes. In fact, his behavior was irresponsible, unworthy of a higher level Scaythe, of a superior transceiver. This carelessness could compromise the Great Project, the universal plan that had been prepared over centuries by the Master Embryos of Hyponeros. He now realized that he had much less room to maneuver. The future of the entire project now rested on the success of this one experiment.
“Well, sir, this is no time for daydreaming!”
“My students will not be operational right away,” argued the Chancellor. “This demonstration is only designed to show you the current state of their progress. After this is finished, you will realize that the budget that has been allocated to mental research, which has been disparaged by so many of your counselors, has not been squandered in vain. In the future, we will continue our experiments on complex, refined brains and forge ahead until the technique is fully mastered.”
“What has this Mikat done to be put on the index and declared a raskatta?” Spergus’ airy voice was a striking contrast to the rich metallic sound of the Chancellor’s voice.
“For goodness sake, Chancellor! Answer his question!”
Ranti Ang’s increasing irritation was slowly breaking through the fragile barrier of his mental control. He was having a terrible time complying with the rigorous code of sycophantic emotion, which was followed at the court of Syracusa. Pamynx remained calm and found that his noble interlocutor’s anger gave him a new source of motivation.
“May I please request that you be patient for just a moment, my Lord? The data about raskattas from your territory has been entrusted to the Scaythe Markyat, who is the archiver of justice. It will just take a moment for me to enter into contact with him…”
“Hurry up! We would like to return to daylight soon. We feel like rats wallowing in a squalid sewer!”
Heavy greenish eyelids, furrowed with dark veinlets, fell over Pamynx’s uniformly yellow eyes. The hood of his acaba hung on his shoulders, uncovering a deformed face, a long bald head, and rough cracked skin. He looked like one of the monsters from the Osgorite legends, at least the idea that Spergus had of them. A chill went up his spine. The crimson circle of the Round Rouque Moon cut through the haze of his memories. For a brief moment, he was carried away to Osgos, the industrial mother, the largest of Syracusa’s satellites. He was running, naked and free, among the dried grass and the scalding stones of the abandoned gardens, chased by happy, noisy brown shapes that danced in the waves of heat. He breathed in the heavy smells of budding bucanas, and the heady sap of fruit fountains.
All of a sudden, he felt cramped in his bodstocking, the Syracusans’ usual undergarment, the second skin that covered them from head to foot. His mauve head cover and its light-band held his hair, his forehead, his cheeks and his chin all tightly together. His two braided blond locks of hair, the only extravagance allowed, stuck out under the edge, near his temples, and framed his effeminate face.
Spergus’ skin begged to feel the fervent caresses of the Round Rouque Moon. Getting back his self-control, he angrily fought off the melancholy that was coming over him. He was not allowed to have regrets: he, the son of humble Osgorite merchants, who was treated more considerately than the great courtiers, than the descendants of the old, illustrious Syracusan families. Even though this preference sometime became a heavy burden; even though he had to put up with the looks and the wounding words of Lady Sibrit, Ranti Ang’s wife; even though he was hardly comfortable among the never-ending schemes and intrigues of the court; even though he was never allowed to go any place without his thought protectors, hidden in the red and white acabas of the Royal Protection Corps, those ever-present shadows, silent and intriguing: he tried to push the nostalgic memories of his youth mercilessly from his mind. He accepted the obligations and the annoyances of the court for the love of his Lord. For the love of the absolute master of the most famous of all the planets of the Naflin Federation, for the love of this century-old man with such extraordinarily delicate features, whose eyes were limpid blue, whose blue gray locks of hair lay on the shimmering cloth of his hood. For the love of a man who was the living expression of nobility, of grace, of refinement, the cardinal virtues of the Syracusan etiquette and tradition.
The Mikat was in convulsions. The rhythmic banging of his knees on the tiles broke the silence that had become oppressive.
“He is a follower of the religions of the index,” said Pamynx suddenly, turning toward Spergus.
Spergus shuddered in surprise. He could not stand looking into the sharp impenetrable eyes of the Chancellor. He was terrified of the Scaythes’ telepathic powers, and particularly those of Pamynx. An instinctive reflex forced him to turn away, to seek the reassuring presence of his thought protectors.
“Those centers of abomination!” scolded Ranti Ang. “They should be destroyed once and for all!”
The Lord of Syracusa’s slender fingers, covered with rings of white optalium, were nervously twisting the silvery lock of hair which ran along the black edge of his hood. This tick was a forewarning that he was about to lose his control.
“This Mikat is a member of the Gudurayam heresy,” specified Pamynx. “He adores the effigy of Gudur, a false prophet that was burned on the crucifire three hundred standard years ago. He is venerated now like a martyr.”
“Animals! Stupid fanatics that do not hesitate to sacrifice humans!”
“And where do they hide?” asked Spergus; this information seemed to captivate him.
This question had the unexpected consequence of defusing Ranti Ang’s anger.
“Imagine, my friend, that some of them are found even on Syracusa! In the mountains of Taheu’ing and in Mesgomia, countries that are very difficult to get to and where it is not easy to clear them out. All the same, it is on Julius that the Gudurayam heresy is the most present, even though the number of his followers has been greatly reduced since reprisals have been stepped up and crucifires have been used more regularly.”
“Two details, if you will allow me to say so, my Lord,” added the Chancellor. “The first is that the parents of this Mikat were burned on a crucifire during your father, Arghetti Ang’s visit to Julius. The second, more picturesque, is that the person who turned him in is none other than his own wife, the one whose memory he is recalling at this very instant. And all this for the measly sum of one hundred Julian Keulis, the equivalent of a handful of standard units. This insignificant amount of money turned out to be more attractive than the love of her husband!”
The hint of a smile came across Ranti Ang’s face. The Mikat, lying on the floor prostrated, was hit head-on by the force of Pamynx’s words, harrowed by this final, hideous revelation. He stopped trembling. Large tears rolled down his unshaven cheeks.
“But… but he is crying! Do you see, my Lord? He’s crying!”
“Yes, my friend, he is crying!” mocked Ranti Ang. “He does not, like you or I, have a means of controlling his thoughts. This is how some creatures show their emotions, as unbelievable as that may seem!”
Spergus was leaning over the solid guardrail that ran along the edge of the balcony. His eyes were wide open, he was trying to look more closely at the shiny rivulets which flowed from the Mikat’s eyes.
In response to a discreet sign from the Chancellor, the Scaythe in the black acaba came closer to the prostrated body. Deep within his hood, Spergus got a quick glance of two flaming red embers, full of energy. Two evil stars in a pitch black sky.
“We are ready, my Lord.”
“Ready? But for what?”
The Mikat, very worried, picked his head up. Seeing the rough, black cloth coming closer, so close that it was brushing against his skin, his eyes opened wide in terror. His arms and legs shook violently.
“This is a great wonderful deed!” said Ranti Ang ironically. “Don’t tell me that you have prepared this grandiose presentation with the only goal of terrorizing a bumpkin!”
“If my Lord would please have a little bit of patience…”
The Chancellor’s mind was infiltrated by a pernicious doubt, a slow poison that he could not keep under check. But he had carefully chosen Harkot, the Scaythe doing the experiment, among a hundred handpicked postulants, all of them gifted with extraordinary mental capacities. He himself had overseen the selected student’s training, had carried out animal testing, and then the testing on the manimals of Getablan. However, he had not yet had the time to start working on complex minds, higher up on the evolutionary scale. There was therefore a chance that this experiment would fail. But Pamynx would not be allowed a single failure. He regretted this haste, which was not his usual way of doing things, but the race between his many critics and his few partisans had made it inevitable.
A plaintive gurgling escaped from the Mikat’s throat. Trickles of drool flowed from the corners of his mouth and dripped onto his slightly protruding chin.
“If you will please now remain totally silent,” whispered the Chancellor, who noted with relief the first signs of the Scaythe’s mental actions.
The Mikat’s convulsions got progressively further and further apart. His breathing turned into panting, then wheezing. Instinctively, he raised his large hands to his neck. Then, in a desperate jump, he tried to grab hold of the black acaba, but his curled up fingers only grabbed the air. There was a death rattle, a final spasm, and he fell, motionless on the floor.
The room was shrouded in a mortal silence. It was Spergus, who was still leaning over the guardrail, who broke the silence.
“What what happened to the Mikat? He’s not moving!”
“He… is… dead,” answered Pamynx, separating his words very carefully in order to highlight their terrible simplicity.
“Dead?”
“Dead, my Lord.”
“How is this possible?”
The Chancellor, who had now recovered his serenity, took a perverse pleasure baiting the curiosity of his listeners. He paused for a long while before answering.
“This Mikat was killed merely by the will of Harkot, our Scaythe experimenter. You have just witnessed the first mental execution, my Lord.”
He said these words with an indifferent tone of voice, as if he was talking about a banal, trivial incident. The Scaythe in the black acaba made a slight bow, to which Ranti Ang answered with a brief nod of his head.
“Do you think you can lead us to believe something that ridiculous, Chancellor?”
“Belief is not allowed in my laboratory, my Lord. I leave that to our holy Church. As a scientist, the only thing that convinces me is certainty. Harkot has just imploded this guinea pig’s brain, so to speak.”
“Do you mean that he can kill from a distance with his thoughts?” mumbled Spergus weakly.
“As long as this distance is not too far. At least for now. Interference from other thoughts may reduce, even staunch the efficiency of the mental intentions of death. But let us say that Harkot has effectively, to use your words, killed at a distance, without the help of a weapon. Right now, of course, this process is only effective on very simple types of brains, such as that of this Mikat. However, we have no worries about soon being operational with more evolved brains. And even those that are very highly evolved.”
The Chancellor’s self-confidence had come back to him. In spite of the thought protectors, those black and white wraiths whose job was to maintain psychic screens, he picked up some raw fragments of feelings from Ranti Ang, and he did not detect the slightest hint of resentment. The perspectives that had been opened by this extraordinary experiment, which had just been carried out under his eyes, were filling the Lord of Syracusa’s mind completely.
“And do all Scaythes have this ability?”
“Only those who have above average mental faculties.”
“This… this is witchcraft!” cried Ranti Ang.
He uttered this accusation without any conviction, as if he had already guessed the answer.
“You have nothing to fear from the Muffi of the Kreuzian Church, my Lord. These techniques are, I repeat, scientific, developed by our physicists specialized in the field of subtle waves, and not by some village witch or wizard. Witchcraft is a synonym of obscure, subjective practices. It is the exact opposite of our technology, which remains objective, provable and verifiable. In addition, if you so wish, my Lord, our scientists would be delighted to give you a more detailed explanation of the mental mechanisms used by our students. It is therefore out of the question” – and the Chancellor’s tone of voice here was very firm – “that our holy Church class the future mental killers on the index. It goes without saying that we would not have presented this new technique to you if it was found to go against Kreuzian principles.”
Pamynx was not taking too many risks in betting that the clergy would support him: Barrofill the Twenty-Fourth, the Muffi of the Kreuzian Church, had been informed about what was brewing in the Chancellor’s secret laboratory a long time ago.
“I would like you to tell us more about this technique, sir,” suggested Spergus.
“Oh, I am afraid that this might bore you,” answered Pamynx, who did not mind getting a small amount of revenge by being begged to continue.
“Go on, Chancellor, please grant our dear Spergus’ request,” interrupted Ranti Ang, in a wily tone of voice.
Even though he avoided showing it, Pamynx was jubilating. His lack of foresight could have fatal consequences for the realization of the Project, but he had managed to turn the situation around, as could be seen by Ranti Ang’s change of attitude and tone of voice. He had just won what he needed most: time. In addition, he now held the courtier Tist of Argolon and his accomplices in the palm of his hand, and this perspective filled him with boundless joy.
“These techniques come from a forgotten science that dates back thousands of years before Naflin. The only ancient science that ever really examined the potentials of the mind: Inddic science. We have found traces of it on Terra Mater, a very tiny planet in a solar system on the edge of the Milky Way. It also seems, as astonishing as this may be, that Inddic science originated on Terra Mater. To sum up briefly, two Scaythe ethnologists learned totally by accident that the religious hymns of a tribe of Terra Mater, the Amerynes, were sung in an Inddic dialect, even though this vernacular language had not been spoken for six thousand standard years. Our ethnologists went to Terra Mater, where they discovered a strange phenomenon: these hymns seemed to have geoclimactic repercussions on the environment, and they could cause seasonal upheavals, such as sudden blizzards in summer. When they collated their observations, they discovered the unbelievable properties of certain Inddic sounds, which are called uctras or antras.
“Good heavens, get to the point!” exclaimed Ranti Ang who had noticed that Spergus was no longer paying attention.
He, too, was in a hurry to escape from the macabre atmosphere of this cellar.
“I’m getting to the point, my Lord. It was necessary to give you some context in order to help you and Master Spergus understand a little bit more clearly. We quickly realized that the Amerynes were using very specific sounds for ritual animal sacrifices or for punishments given to those who broke the law. A concrete example: adultery. The guilty party, or both parties together, were tied together in the middle of a sacred circle. Four Amphanes, the Ameryne priests, would sit at the four cardinal points, singing the death chant, a succession of uctras, which would end up causing irreparable brain damage and bring about death in a few minutes. But one of our physicists recently discovered that these same uctras proved to be more effective, more powerful when they are given out at a subtle level.”
Spergus was once again paying unflagging attention to the Chancellor’s explanations.
“We based our work on the following theory: the destructive power of the Inddic uctras depends on the quality of the silence in which they are used. Little by little, the Amerynes forgot this basic principle. Instead of internalizing the uctras, they exteriorized them by chanting them and, because of that, reduced their power. One of the essential qualities of the Scaythes of Hyponeros is that they can attain levels of inner silence that no other living creatures in the universe can reach. Excited, superficial minds would not be able to use these uctras correctly. However, our students were trained in the greatest of secret, which is what called for the unpleasant but necessary presence of the Pritivian mercenaries, and they have managed to master them by stabilizing calm states of mind. They first tried them out on embryonic brains, then on mammals, then on the manimals of Getablan, and finally on this Mikat. By the way, I beg you to please clear up the concerns of some Kreuzian missionaries from the satellite Getablan. We had to…”
“Already having problems with the Church, Chancellor?” interrupted Ranti Ang. “I thought these experiments were kept totally secret! I imagine, in fact, that if the other member states of the Federation learn that you have been using the services from the mercenaries from Pritiv, we will not have any more credibility during the next Asma on Issigor.”
“The five-year assembly will not take place, as planned, on the planet Issigor.”
“How? And why?”
The Chancellor’s yellow eyes locked on those of Spergus.
“I will explain that to you later, my Lord. In private. May I continue? In order to have enough guinea pigs, we had to promise the missionaries that we would return these manimals unharmed. But…”
“A white lie, but a lie, Chancellor!” declaimed Ranti Ang, making fun of the bombastic tone of the people of the church.
“I thought that for the good of…”
“Don’t think anymore, if you please! The noble goal of these experiments was to serve science, was it not? And the fact that a few manimals have disappeared as a result of it does not shock my Kreuzian convictions. I will take care of all that with the Muffi Barrofill. Am I not, after all, his appointed protector and personal friend? But are you absolutely sure that no one else has found out about your experiments?”
“Absolutely sure. The only person who could impede us has been banished from Syracusa. By you, my Lord.”
“By me?”
“I am sure that you still remember the trial of Sri Mitsu, the Mustah.
“Sri Mitsu? What does he have to do with this?”
Even though Ranti Ang was using all the resources of his mental control to let nothing come through, he clearly loathed recalling this memory.
“Quite a bit, my Lord,” answered Pamynx; who could almost feel this discomfort – he knew exactly where it was coming from. “Inddic science had come through space and time, and there are three great masters who are still alive: Sri Mitsu is one of them.”
“If this were so, we would have known!” protested Ranti Ang. “Sri Mitsu has always refused mental protection: our inquisitors could read his thoughts as easily as they could read a light-book!”
“The exceptional psychic capabilities he had developed through practicing Inddic science exempted him from protection, my Lord. That, and the fact that he belonged to the brotherhood of Smellas, could have proved to have disastrous consequences for our projects. For that reason, and only for that reason, I insisted to you and to his Holiness the Muffi that he be tried in a sensational public forum. The accusations against him, unnatural sexual practices, were just a pretext, as I am sure you understood. He had to be removed. Fortunately, everything went as planned: his aura as a Smella, his influence on the other member states, his overall good reputation, all these things were turned against him during the trial and he was condemned to perpetual banishment.”
“Why have you hidden these true reasons from me, sir? Do you have such little esteem for me?” Ranti Ang’s voice was bitter. Pamynx refrained from showing the contempt that he had for the Lord of Syracusa. He thought he was superficial, frivolous, fickle, incapable of handling the heritage that had been left to him by the great Arghetti Ang. Behind the scenes, the Chancellor constantly worked for a more expeditious succession than that which was a part of Syracusan tradition.
“I did not wish to overload your already busy schedule, my Lord.” “Who are the other two masters of this Inddic science?” asked Spergus. “You said a minute ago that there were three of them and we have only heard the name of one.”
“Another Syracusan: Sri Alexu, a very discreet man that we never see in the court. But he lives right here, near Venicia. He is not involved in State affairs. He is only known to have two interests: his daughter, a young beauty named Aphykit, and flowers. He is under constant surveillance.”
“And the third one?” Spergus’ insistence bothered the Chancellor. Had he underestimated the role of the Syracusan Lord’s protege? Perhaps this disarming naivete hid some precise calculated intentions.
“Seqoram the Mahdi.” Ranti Ang gave an exclamation of surprise, an uncalled-for, indecent showing of his emotions, contrary to the code of sycophantic emotion.
“Good Lord! Do you realize what you are saying, Chancellor?”
“Why? What is it? What has he done?”
“The Grand Master of the Absurate Order. But don’t worry yourself, Spergus: we have compromised the Absurate knights and we have made sure to lead them down the wrong paths. And we go over their reports with a fine-toothed comb.”
“Perhaps! However, attacking the Absurate Order is attacking the very foundations of the Naflin Federation!” objected Ranti Ang. “The knighthood has devoted itself to the study of the arts of war for centuries. No lord, however powerful he may be, would have the recklessness to defy it! Have you lost your mind, Chancellor?”
“The Order knows nothing of the weapon that we are preparing, my Lord.”
Pamynx froze suddenly in a solemn attitude.
“My Lord, the time has finally come to carry out your father’s visionary dream. All of the conditions are right: the federal army, the Interlice, is under the command of your brother Menati, and this until the next five-year Asma. We are making sure that this will take place on Syracusa and not on Issigor. In accordance with our advice, Menati has managed to bring the

senior officers around to our cause with promises of titles and territorial concessions. Pritivian mercenaries are prepared to grant us unequivocal support, because they long to battle with the Absurate Order that their founders, the knights who broke away from the Order, came from. The Kreuzian Church is expanding thanks to the indefatigable activity of missionaries in the farthest corners of the Federation. Its crucifires and mental inquisitors are already a very useful repressive apparatus. There was only one thing we needed, my Lord, and this thing is what you have just seen materialized in front of your eyes.”
He stopped talking and watched the effects of his words on those in front of him. Spergus, his mouth hanging open, his eyes wide, looked like a holographic mannequin from the pre-Naflin museums. The only thing that made him look alive were his two blond locks of hair, moving lightly in the air. This young and exuberant boy, who was a victim of his curiosity and of Ranti Ang’s feelings, already knew too much. Whatever part he was playing, whether it had two sides or one, he was a danger. His wheel of fate, the rota individua of the Kreuzians, would soon stop turning.
As for the lord of Syracusa, he was rubbing his lips absentmindedly with his right index finger. His blue eyes wandered over to the body of the Mikat and the black acaba of his torturer. Fleeting bright sparkles came from the ephemeral gems, set by dozens in the long scarlet cape that covered his white bodstocking.
“We must now act very quickly,” continued the Chancellor. “We must definitively eliminate Sri Mitsu, who is still dangerous in spite of being in exile. The Pritivian mercenaries will take care of that. We must also eliminate Sri Alexu and his daughter. They don’t look dangerous but this is probably just to deceive us. You must use your discretionary power, my Lord, to obtain additional credits which will allow us to perfect our technology of mental execution. Then the Absurate Order must be attacked and destroyed, as well as the obsolete relic of the Federation, the final traces of Inddic civilization. In order to ensure that this is so, the Amerynes of Terra Mater should also be reduced to silence.”
“Do you realize, Chancellor, that if this genocide – because you are suggesting genocide – if this got out, we would be under a direct menace from the Absurate knights!” exclaimed Ranti Ang. “And it will get out, because the main member states have eyes and ears all around!”
“We need to learn that the Order is no longer an insurmountable obstacle. Our chances of success rest on speed and precision, on the element of surprise. All we need now is your formal agreement, my Lord. It is up to you to now become the first ruler of a post-Naflin empire.”
As he said this, he was thinking that Ranti Ang would never have this privilege. In the fifth stage of the Great Project, the masters of Hyponeros had planned for the Naflin Federation to be broken up and for power to be taken by a wise tyrant, a unifier. A man of a much different caliber than the current lord of Syracusa.
The four Scaythe thought protectors had slackened their watchfulness. The light from their half-closed eyes, coming from within the darkness of their red and white hoods, was less intense. They were violating the first law of the treaty of the Honorable Code of Protection: At all times day and night, I will be a zealous guardian of the mind of my Lord, because he alone has the right to follow the flow of his thoughts.
Pamynx noticed this inattention. He could have slipped for just a second into Ranti Ang’s mind, which was momentarily unscreened. He preferred to wait for his fellow Hyponerians to realize their unforgivable negligence. Today, the Chancellor would ask for no additional heads to fall. The most important ones would soon be rolling at his feet, and this perspective was more than enough to make him happy.
“My Lord, I would like to discuss the next steps of our undertaking,” he said softly, as if he did not want to awaken Ranti Ang from his daydreaming too suddenly. “Young Spergus should be allowed to avoid this tiresome chore. Send him someplace which is more in accordance with the concerns of his young age.”
Without waiting for Ranti Ang to answer and without paying any attention to the deadly look from Spergus, he walked off into the dark underground corridor with a firm step.
 

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Write up on Serbian history & Mythology(Revision 2) https://ddcomics.org/2026/06/21/write-up-on-serbian-history-mythologyrevision-2/ https://ddcomics.org/2026/06/21/write-up-on-serbian-history-mythologyrevision-2/#respond Sun, 21 Jun 2026 20:41:13 +0000 https://ddcomics.org/?p=6889 Significance of the Study Write up on Serbian history & Mythology Chapter 1 The South Slavs have a long tradition of belief in protective domestic spirits and in malevolent demons of the field, forest and water.(1) Such mythological creatures were prevalent among all Slavic peoples and are part of the common Indo-European heritage.(2) Whereas most […]

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Significance of the Study

Write up on Serbian history & Mythology

Chapter 1

The South Slavs have a long tradition of belief in protective domestic spirits and in malevolent

demons of the field, forest and water.(1) Such mythological creatures were prevalent among all Slavic

peoples and are part of the common Indo-European heritage.(2) Whereas most beliefs of this type receded

among the East and West Slavs by the end of the nineteenth century, they were maintained in many areas of

the Balkans until the beginning of the Second World War.(3) Ethnographic fieldwork conducted in the

1960s-1980s has shown that many farmers and stockbreeders in the more remote villages (of former

Yugoslavia) have not abandoned their traditional beliefs. For example, the protector housesnake,(4)

mischievous forest and dangerous water spirits, and many lesser mythological beings have been reported in

several South Slavic territories in the last forty years. Many traditional domestic rituals have their origin in

the conviction that the family ancestor’s spirit resides under the threshold or near the open hearth and, if

properly cared for, will ensure happiness and good fortune for the family. In Russia that spirit was manifest

in the domovoj, “house spirit,” but as this name itself was taboo, he was referred to in euphemisms such as

ded or deduška, “grandfather,” and xozjain “master.” Offerings of food, especially bread and salt, the

traditional symbols of hospitality, were routinely left for the domovoj at night before the family retired.

The corresponding belief in a protector-ancestor spirit in the South Slavic territories saw the male

founder of the family incarnate in a housesnake which was euphemized most commonly in Serbo-Croatian

as čúvarica “protector,” čúvarkuća “house protector,” and kúćarica “household one,” and in Bulgarian

stópanin or stópan “master of the house”; compare English “stoop,” i.e., the threshold, the resting place of

this spirit.(5)

The South Slavs’ perception of mythological beings was based on a dualistic view which

incorporated both positive and negative features. This was true of snakes as well. Most snakes were

considered incarnations of demons living in the Underworld (Donji svet) and were to be killed on sight,

poisonous or not. Yet the white snake rarely seen, but thought to live under the threshold, in the foundation,

or near the house was considered to embody the spirit of the family’s first male ancestor.

In Serbia the protector snake was sometimes believed to live in the foundation wall near the

threshold, but in most areas it was said to dwell in, behind, or under the hearth. Likewise, both locations

were thought to be the dwelling place of the domovoj in pre-Revolutionary Russia, and each has been the

site of domestic rituals in all parts of the Slavic world. The family’s albino protector snake not only caught

SEEFA Journal 2001, Vol. VI No. 1

4

mice and kept other, more dangerous snakes from the house, but was considered the source of good fortune

and well-being.

Representing a positive force, it was most often visualized as white. The white snake was called by

many names, the most common being (f.) sretna zmija “snake of (good) fortune” and (m.) zmijski

kralj/knez, “snake king/prince.” Incorporated into the Christian belief system, it was thought in some areas

to be sent by God Himself. Care was taken to protect the čuvarica, for it was believed that if it were killed,

the master or another person in the household would soon die. As a mediator between the Underworld,

Nature, and humans, this snake was thought to understand speech, to be able to teach man about medicinal

herbs, and to induce fertility in wives and female livestock. Many legends associate man and snake, and

metamorphosis of one into the other is common in South Slavic folklore.

Belief in a protector snake spirit has almost disappeared. Yet, as recently as the early 1970s

Macedonian field researchers examined evidence of a snake cult in the Skopje suburb of Orman, which is

known for the veneration of snakes, a custom surviving among the older generation. Many people still

believe in the power of the zmija-sajbija, or proprietor snake, and they are convinced that if the snake

should leave its dwelling place permanently, misfortune will come to the family of the house in which it

lives. Many Orman residents collectively celebrate both the coming of spring and the return of the snakes

from hibernation on March 22 (Orthodox), a day which is officially called Denot na proletta “Spring Day,”

but which retains its religious name, Četirieset mačenici, or “Forty Martyrs.” In 1969, after a new church

was built at the foot of Zmijarnik (“Snake Hill”), hundreds of villagers celebrated this holiday by bringing

bits of their clothing and laying them on the ground in the hope that the returning snakes might touch them

and thereby ensure good health and fortune for the owners. Belief in the magic power of this ritual imitation

of the snake’s shedding of its skin, i.e., a symbolic renewal of life, was strong enough that relatives brought

the clothing of the sick to the sacred hill. It is believed that only those whose clothes are “blessed” by a

returning snake will be cured or have their wish come true.

Post-World War II field research in Bosnia, Croatia, Serbia, and Macedonia has demonstrated that

mythological beings remain integrated into the belief system of a notable portion of the (mostly) rural

populations. For example, ethnographic research conducted in the 1950-1960s in the Bilo-gora region some

seventy kilometers east of Zagreb (including the sizeable towns of Bjelovar, Koprivnica, Virovitica,

Grubišno Polje, and Križevci) disclosed several tenacious beliefs stemming from pre-Christian

demonology. A prime example is the male spirit called ved, who lives in the forest (his name is derived

from the Old Church Slavic verb věděti “to know, to have secret knowledge”). Védi (pl.) were thought to be

as tall as a house and physically similar to men, but covered with hair. (Compare the Russian lešii who were

usually perceived as old men of human size and whose primary activity was frightening lone wanderers in

the forest.) The Croatian vedi were thought to inhabit the forest, long considered a transitional zone

between our world and the “other world”.(6) There they lived in sociable groups or even in their own cities.

5

Their singing and talking could be “heard” from far away. According to local tradition, there were both

good and bad vedi. The latter usually avoided humans and stayed in the forest; hence, they were called

šúmski vedi, or “forest vedi.” If a person strayed into the forest and came upon one or more of the latter he

could be tortured, starved, or beaten before being released.

Conversely, good vedi could be counted on for help. Each family or household had its own ved for

protection against hail, floods, and other dangerous natural phenomena. Thus the dual perception of vedi

incorporated both the positive features of the domestic protector spirit, and negative, demonic qualities

associated with forces from the Underworld. Several expressions survived in the Bilo-gora area which attest

to the folk belief in this combination forest and domestic spirit. A common exclamation in the Kajkavian

dialect of this region was Daj, Bože, da nam naši vedi pomognejo! (“God, let our vedi help us!”) Another

expression suggests that the borderline between good and bad vedi was less than clear: while a family

depended on its own spirit for help, that same ved could be induced to harm an unliked neighbor’s family

and crops. This may be illustrated by the expression: Bože ljubljeni, daj da nam naši vedi pomorejo, a da

nam njihovi vedi ne nahudijo! (“Dear God, let our vedi help us and don’t let their vedi harm us”). It was

believed that after one pronounced such a prayer, the spirit would come quickly to the person’s aid. Belief in

the vedi has generally receded by now, but many of the older generation in Bilo-gora still spoke of the them

in the late 1960s.

A second male spirit, the vodénjak or vódeni čóvjek “water man,” has proven more durable in the

Bilo-gora region, as elsewhere in former Yugoslavia. The vodenjak is thought to live in whirlpools or other

places where there is deep water. When someone drowns it is usually said that the vodenjak has claimed his

victim, or has taken him to his realm deep below the surface. Reports as to the physical appearance of this

dangerous spirit are fairly consistent: when seen in the water, he is naked but his skin is green and entirely

covered with blue or green hair which makes him look like a submerged tree stump. Out of the water, he

looks like a man but is dressed in green and carries a stick for beating his victims and implementing magic

charms. He is thought to be very strong, so strong that a person can rarely escape his grasp. Explanations of

drownings as being caused by the vodenjak are very frequent in the Bilo-gora region (as elsewhere in

Southeastern Europe).

In Serbia, The water spirit is known as nečástivi, “evil one” or “devil,” especially in northeast

Serbia, surrounding the town of Donji Milanovac; because of the danger of invoking his presence by calling

his name he is usually referred to as ónaj stári “that old man” or onaj máli “that little man.” His reported

size varies between fifty and one hundred centimeters, and he is similar in appearance to the common

European devil, i.e., he has a cloven hoof, horns and goat’s ears (usually disguised by a bathing cap or red

[Turkish] fez), and he is dressed in black or white, colors of chthonic gods. He is said to change his shape at

will, and often to take the form of a baby or relative calling someone to the river. It is thought unwise to

answer if one’s name is called three times at night, for it means that the nečastivi has set the time for one’s

SEEFA Journal 2001, Vol. VI No. 1

6

death. The various localities’ water spirits are thought to gather once a year on Devils’ Day (djavolji dan)

and, after their elder gives them instructions for the remainder of the year, they are said to celebrate by

dancing the kolo (round dance).

The Serbian villagers of this Danube region believe that it is dangerous to see one’s reflection in

the water: should this happen the spirit will try to claim that person as his victim. Danger lurks not only in

the water, but also nearby: if one falls asleep on a river bank or in a moored boat there is a chance that the

water spirit(s) will dance the kolo around the site; or one may awake, as reported in the case of three

fishermen, several kilometers downstream. Field researchers have recorded much “eyewitness” testimony

telling of such incidents, and these and similar explanations are especially common in July, the month in

which the nečastivi is thought to be most active.

The mythology surrounding the Serbian variant of the water spirit is well developed. For example,

as in the case of vampires, the water demon can only be “seen” by persons born on a Saturday. And sighting

of the nečastivi by a villager is a sign that death or misfortune is soon to strike. Yet, it is also believed that

certain women voluntarily fraternize with them. Such women are reputed to go out at night in order to visit

with the spirits. They are said to be naked and with hair unbraided but with their genital area demurely

covered by a pan or pillow (to protect them from the devil’s lust). It is said that if a woman has sexual

relations with one of them her husband will soon die and she herself will become infertile or her next child

will be born without a skeleton.

Many villagers in this region are convinced that some women give themselves freely to the water

spirits and thereby gain power over them. Such women are usually the village conjurers (vráčare/bájalice).

They take care to reinforce their power by whispering magic charms (básme,)(7) and by performing special

rituals involving nine grains of wheat (i.e., thrice three, the most important magic number, which increases

their power threefold), nine pieces of salt, garlic, and a special staff with which they strike the water. When

the spirit appears the conjurer tosses money as a tribute to him, or promises an animal or human sacrifice. If

this ritual is done properly, her wish will be fulfilled.

Conjurers in this region are often hired to enlist the aid of the water spirit in support of the

customer’s desire for success in fishing, hunting, and even catching thieves. Furthermore, it is thought

possible to sell one’s soul to the nečastivi to ensure successful fishing. As in the case of other such demons

and devils, the pact guarantees only temporary success; eventually, the spirit will claim what is his. When a

drowned man is found, a proverbial expression confirms this belief: Došao ðavo po svoje. (“The devil came

for his own”).

A third male creature, and one considerably less dangerous than the water spirits, is the Croatian

vrag or “devil,” who can be met in a forest, a meadow, or on a path, but is most often encountered at

crossroads (the traditional haunting place of witches as well). This minor demon dares not enter a church,

yet he is often reported to have been in the vicinity of nearby cemeteries. In Croatian Bilo-gora he is

7

described as physically similar to a man, but bearing the familiar distinguishing marks of traditional

European devils: a cloven hoof, horns, and a tail.

It is thought that the vrag is strong enough to stop a pair of running horses; however, in combat

with humans he is neither crafty nor agile. Rather he is thought to be a relatively simple and trusting

creature who can be deceived easily. For example, the vrag likes to fight with priests and women, but he

always loses the contest. He is afraid of the cross, holy water and rosaries, and his constant goal is to turn

believers from the “true faith,” to win over souls by “registering them in his book.”

The ved, vodenjak/nečastivi and vrag are examples of male mythological beings which are

supernatural extensions of, or mediators from the Underworld.

Yet there is a fourth male creature, one who originates from the soul of a deceased human: the

vampire. Belief in vampires is still strong in certain areas of Southeastern Europe and the Mediterranean

basin, but for our purposes is most concentrated today in Macedonia. Common Croatian-Serbian terms for

vampire are vúkodlak “wolf-hair,” vámpir, and ténac “shade” (= ghost) or “werewolf;” in Macedonian he is

called vóper or vópir (cf. Russian upýr’, Polish upiór), gróbnik “graveman,” and tálasam, a term derived

from Arabic, which has a second meaning in Macedonian: (probably from Turkish) “evil house spirit.”

According to reports from Macedonia, when seen, the vampire may look like a person but, as he has no

skeleton, his body is filled only with blood. His voice can be heard and his red eyes seen, but he has no

shadow. He can also change his shape freely; he is most often seen as a cat, rabbit, rooster, or in fact any

living creature.

Like other South and East European vampires, the Macedonian voper attempts to suck blood or at

least to frighten his victim to death. However, when less violently inclined, he merely makes noises in the

house, stirs up the ashes in the hearth, or starts fires (!). Not a wholly negative creature, he is commonly

believed in Macedonia to return to his former place of work and even continue to live with his widow.

Explanations of the origin of this phenomenon are generally traditional, the most common being that an

animal jumped over the deceased as the body lay in state. For this reason, no corpse is left unguarded at

night: a “wake” is held for three days and nights. Should it happen that a cat, chicken, rooster or dog jumps

over the body, the offending animal (thought to be the devil in disguise) must be made to jump back over it

in the reverse direction so as to prevent the creation of a vampire from the Devil’s theft of the deceased’s

soul.

Other reasons given for the origin of Macedonian vampires are that a person died during the

“unbaptized days,” that is, in the period between Christmas and Epiphany (January 6) when Christ was said

to have been baptized; died a violent death and was not mourned properly by ritual lament; died with an

unfulfilled wish; or died on the spot where a star fell. In addition, there is another explanation which has

been given: a vampire is “a corpse which has been urinated on by the devil (mrtovec što go pomočal

ģavolot).

SEEFA Journal 2001, Vol. VI No. 1

8

To summarize, many of these spirits support a dual function: the family’s protector spirit, whether

snake or progenitor incarnate, is counted on for health, fertility, and prosperity. The family’s adopted ved is

likewise helpful, but those of neighbors may be detrimental to one’s well-being. The Christianity-inspired

vrag is a generally mischievous sprite, and one seen most often in a humorous light; still, the best defenses

are caution, one’s own superior wits, and a firm belief in the principles of the Church. But the pre-Christian,

purely mythological vodenjak, nečastivi, and especially the semi-human vukodlak, vampir or voper, are

dangerous spirits to be avoided if at all possible. The water spirits are demons lurking below to catch the

unaware, yet even they can be the source of power for certain villagers (ordinary women as well as female

conjurers); one must always be careful in and near the water.

There are yet other mythological beings in traditional South Slavic folklore which have become for

the most part only figures in legends and folktales. In addition to the male creatures discussed in this paper,

there are those such as the vúčji pástir “wolf shepherd” thought to control the countryside; drékovi, souls of

deceased soldiers; drekavci, souls of children who died during the nekršteni dani; and the Christian ánđeo i

đávo, angel and devil. Note: There are a number of female spirits including personified diseases such as

Ćúma or Kúga, the plague, and Karakónđula, an old hag who is thought to ride drunken men at night.(8)

There are likewise many meteorological and animal spirits, e.g., ala, the summer hail demon; duga, the

rainbow thought to drink from rivers, lakes, and oceans; Zmaj, the serpent which spreads fire everywhere

and is visualized as a bolt of lightning; or even Psoglava, a dog-headed monster with iron teeth who lives in

a dark cave in a land where there is no sun, but who comes to our world to seek out victims for gnawing. No

longer believed in as are the personified male and female spirits, these may still be named as the cause of

otherwise inexplicable phenomena in many parts of former (and present) Yugoslavia and constitute a

significant portion of the body of folk belief in the mythology of the South Slavs.(9)

NOTES:

1. Research for this study was conducted at the Zagreb Institut za etnologiju i folkloristiku

(formerly Zavod za istraživanje folklora) under the auspices of a Senior Fulbright Research grant

(sponsored by the Council for International Exchange of Scholars) Consultations were held with folklorists

and ethnographers in the Slovene and Serbian academies of sciences and arts, as well as with those in

museums and universities in Ljubljana, Zagreb, and Belgrade.

A somewhat expanded version, “Mythological Beings in South Slavic Folklore,” was read May 6,

2000, at the Balkan and South Slavic Conference, held at the University of Kansas, Lawrence.

2. For a comprehensive survey of traditional Slavic mythology, see A Afanas’ev, Poètičeskija

vozzrenija slavjan na prirodu, I-III (Moscow: 1865-69; reprinted in Slavistic Printings and Reprintings

214/1-3 [The Hague-Paris: Mouton, 1970]). More specifically for the East Slavs, consult S.A. Tokarev,

9

Religioznye verovanija vostočnoslavjanskih narodov XIX-načala XX v. (Moscow: Akademija Nauk, 1957).

See also the more recent Enciklopedičeskij Slovar’: Slavjanskaja mifologija (Moskva: Ellis Lak, 1995).

For discussion of Balkan mythology, Špiro Kulišić, Stara slovenska religija u svijetlu novijih

istraživanja posebno balkanoloških. Sarajevo: ANBiH, Dijela, knj. LVI. Centar za balkanološka

ispitivanja, knj. 3, 1979.

3. For a general introduction to South Slavic folk belief, see E Schneeweiss, Serbokroatische

Volkskunde. Erster Teil: Der volkstümliche Glaube (Second edition, Berlin: Walter de Gruyter and Co.,

1961), pp. 3-33. For a similar introduction to Bulgarian folk belief, see Christo Vakarelski, Bulgarische

Volkskunde. Zweiter Teil: Geistige Kultur (Berlin: Walter de Gruyter and Co., 1968), pp. 207-47.

Specialized information concerning almost all aspects of Yugoslav mythology can be found in Veselin

Čajkanović, Mit i religija u Srba. Izabrane studije (Beograd: Srpska književna zadruga, kolo LXVI, knj.

443, 1973). For a more recent treatment of the mythological spirits and demons discussed in this chapter,

see Slobodan Zečević, Mitska bića srpskih predanja (Beograd: Vuk Karadžić, 1981).

For traditional Bulgarian mythology, see Mikhail Arnaudov, Očerci po B”lgarskija folklor vol. 2

(Sofija: B”lgarski pisatel, 1969).

4. zmija, the usual word for snake, is fem sg. but refers to the male progenitor of the family. As it is

related to zmaj, “serpent” which is masculine, it fits into the category of male mythological spirits.

5. Please note that the accent marks are not those used by traditional linguists for Croatian/Serbian-

-rather they are merely to indicate the stressed syllable for those interested in South Slavic folklore

6. Similarly, rivers are considered fixed boundaries between “this” world and “that” or “the other”

world

7. For discussion and analysis of charms and rituals used by vračare, bajalice, and their basme, see

Joseph L Conrad, “Magic Charms and Healing Rituals in Contemporary Yugoslavia,” Southeastern Europe

10 (1983):99-120. See also my “Bulgarian Magic Charms: Ritual, Form, and Content,” Slavic and East

European Journal 31, 4 (1987):548-62; and “Slovene Oral Incantations: Topics, Texts, and Rituals,”

Slovene Studies 12/1 (1990):55-66. For comparison, see Conrad, “Russian Ritual Incantations: Tradition,

Diversity, and Continuity,” SEEJ 33, 3 (1989):422-444.

For a solid collection of South Slavic charms, see Ljubinko Radenković, Narodne Basme i Bajanja

(Udruženi izdavači: Niš: “Gradina,” Priština: “Jedinstvo,” Kraguevac: “Svetlost,” 1982). See also his recent

Narodna Bajanja kod Južnih Slovena (Beograd: Balkanološki Institut SANU, and “Prosveta,” 1996).

8. For information concerning more important female demons, see Conrad, “Female Spirits Among

the South Slavs,” SEEFA Journal vol 5, no. 2 (Fall 2000):27-34.

9. For commentary on these lesser demons in Northeastern Serbia, see Slobodan Zečević,

“Narodna verova

SEEFA Journal 2001, Vol. VI No. 1

Paganism and Religion

There is available but slender material concerning the pre-Christian

history of the Southern-Slavonic races, and their worship of Nature has

not been adequately studied. Immediately after the Slavonic immigration

into the Balkan Peninsula during the seventh and eighth centuries,

Christianity, which was already deeply rooted in the Byzantines, easily

destroyed the ancient faith. The last survivors of paganism lived

in the western part of the peninsula, in the regions round the river

Neretva, and these were converted to Christianity during the reign of

Basil I. A number of Croatians had been converted to Christianity as

early even as the seventh century, and had established an episcopate

at Agram (Zagreb). In the course of some thousand years Graeco-Oriental

myths and legends, ancient Illyrian and Roman propaganda and Christian

legends and apocryphal writings exercised so great an influence upon

the ancient religions of the Southern-Slavonic peoples that it is

impossible to unravel from the tangled skein of such evidence as is

available a purely Southern-Slavonic mythology.

The God Peroon

Of Peroon, the Russian God of Thunder, by whom the Russian pagans

used to swear in their treaties and conventions concluded with the

Byzantines during the tenth century, only a few insignificant traces

remain. There is a village named ‘Peroon’ near Spalato; a small number

of persons in Montenegro bear the name; [13] and it is preserved

also in the name of a plant, ‘Peroonika’ (iris), which is dedicated

to the god. There is hardly a cottage-garden in the Serbian villages

where one does not see the iris growing by the side of the house-leek

(Tchuvar-Koutchye). The Serbians say that the god lives still in the

person of St. Elias (Elijah), and Serbian peasants believe that this

saint possesses the power of controlling lightning and thunder. They

also believe that St. Elias has a sister ‘Ognyena Maria’ (Mary the

Fiery One), who frequently acts as his counsellor.

The God Volos

From the Russian God of Cattle, ‘Volos,’ the city ‘Veless’ has obtained

its name; also a village in the western part of Serbia, and there

is a small village on the lower Danube called ‘Velessnitza.’ But

the closest derivative appears in the Serbian word ‘Vo,’ or ‘Voll’

(in the singular) ‘Volovi’ (in the plural) which means ‘Ox.’

The Sun God

Other phenomena of Nature were also personified and venerated as gods.

The Sun god, ‘Daybog’ (in Russian ‘Daszbog,’ meaning literally ‘Give,

O God!’), whose idols are found in the group of idols in Kief, and

whose name reappears as a proper name of persons in Russia, Moldavia

and Poland, is to the Serbians the personification of sunshine,

life, prosperity and, indeed, of everything good. But there have

been found no remains of idols representing the god ‘Daybog’ among

the Southern-Slavonic nations, as with the Russians, who made figures

of him in wood, with head of silver and moustache of gold.

The Veele

The Serbian legends preserve to this day interesting traces of the

worship of those pagan gods and of minor deities–which still occupy

a considerable place in the national superstition. The “nymphai”

and “potami” mentioned by the Greek historian Procope, as inferior

female divinities inhabiting groves, forests, fountains, springs

or lakes, seem to have been retained in the Serbian popular Veela

(or Vila–in the singular; Veele or Vile–in the plural). There

are several fountains called “Vilin Izvor” in Montenegro (e.g. on

Mount Kom), as also in the district of Rudnik in Serbia. During

the Renaissance the Serbian poets of Ragusa and other cities of

Dalmatia made frequent reference to the nymphs, dryads, and oreads

beloved by them as “veele.” The Serbian bards or troubadours from

the early fourteenth century to our day have ever glorified and sung

of the veele, describing them as very beautiful and eternally young,

robed in the whitest and finest gauze, with shimmering golden hair

flowing down over snow-white bosoms. Veele were said to have the most

sweet voices and were sometimes armed with bows and arrows. Their

melodious songs were often heard on the borders of the lakes or in

the meadows hidden deep in the forests, or on high mountain-peaks

beyond the clouds. They also loved to dance, and their rings are

called ‘Vrzino (or Vilino) Kollo.’ In Mount Kom in Montenegro,

there is one of these rings which measures about twenty metres

across and is called ‘Vilino Kollo.’ The Treaty of Berlin mentions

another situated between Vranya and Kuestandil, through which ran

the Serbo-Bulgarian frontier. When veele were dancing nobody dare

disturb them, for they could be very hostile to men. Like the Greek

nymphs, veele could also be amicably disposed; and on occasions they

assisted the heroes. They could become the sisters of men and of women,

and could even marry and have off-springs. But they were not by any

means invulnerable. Prince Marko, the favourite hero of the Serbians,

was endowed with superhuman strength by a veela who also presented

him with a most wonderful courser, ‘Sharatz,’ which was, indeed,

almost human. A veela also became his possestrima (Spiritual sister,

or ‘sister-in-God’) and when Marko was in urgent need of help, she

would descend from the clouds and assist him. But she refused to aid

him if he fought in duels on Sundays. On one occasion [14] Marko all

but slew the Veela Raviyoyla who wounded his pobratim (brother-in-God)

Voivode Milosh. The veele were wise in the use of herbs, and knew

the properties of every flower and berry, therefore Raviyoyla could

heal the wounds of Milosh, and his pierced heart was “sounder than

ever before.” They believed in God and St. John, and abhorred the

Turk. The veele also possessed the power of clairvoyance, and Prince

Marko’s ‘sister-in-God’ prophesied his death and that of Sharatz. [15]

Veele had power to control tempests and other phenomena of nature; they

could change themselves into snakes or swans. When they were offended

they could be very cruel; they could kill or take away the senses of

any who threatened them with violence; they would lead men into deep

waters or raze in a night magnificent buildings and fortresses. [16]

To veele was attributed also the power of deciding the destiny of

newly born children. On the seventh night after the birth of a child

the Serbian peasant woman watches carefully for the Oossood, a veela

who will pronounce the destiny of her infant, and it is the mother

only who can hear the voice of the fairy.

Predestination and Immortality

The Serbians believe firmly in predestination, and they say that

“there is no death without the appointed day” (Nema smrti bez soodyena

dana). They believe universally in the immortality of the soul,

of which even otherwise inanimate objects, such as forests, lakes,

mountains, sometimes partake. After the death of a man, the soul delays

its departure to the higher or lower spheres until the expiration of

a certain period (usually forty days), during which time it floats in

the air, and can perhaps enter into the body of some animal or insect.

Good and Evil Spirits

Spirits are usually good; in Montenegro the people believe that each

house has its Guardian-Spirit, whom they call syen or syenovik. Such

syens can enter into the body of a man, a dog, a snake, or even a

hen. In the like manner every forest, lake, and mountain has each

its syen, which is called by a Turkish word djin. So, for example,

the djin of the mountain Riyetchki Kom, near the northern side of

the lake of Scutari, does not allow passers-by to touch a branch or a

leaf in the perpetually green woods on the mountain side, and if any

traveller should gather as much as a flower or a leaf he is instantly

pursued by a dense fog and perceives miraculous and terrifying visions

in the air. The Albanians dread similar spirits of the woods in the

region round Lurya, where they do not dare touch even the dry branches

of fallen firs and larches. This recalls the worship of sacred bushes

common among the ancient Lithuanians.

Besides the good spirits there appear evil spirits (byess), demons,

and devils (dyavo), whom the Christians considered as pagan gods,

and other evil spirits (zli doossi) too, who exist in the bodies of

dead or of living men. These last are called vookodlaks or Vlkodlaks

(i.e. vook, meaning ‘wolf,’ and dlaka, meaning ‘hair’), and, according

to the popular belief, they cause solar and lunar eclipses. This

recalls the old Norse belief that the sun and moon were continually

pursued by hungry wolves, a similar attempt to explain the same natural

phenomena. Even to-day Serbian peasants believe that eclipses of the

sun and moon are caused by their becoming the prey of a hungry dragon,

who tries to swallow them. In other parts of Serbia it is generally

believed that such dragons are female beings. These mischievous

and very powerful creatures are credited with the destruction of

cornfields and vineyards, for they are responsible for the havoc

wrought by the hail-carrying clouds. When the peasants observe a

partial eclipse of the moon or the sun, believing that a hailstorm is

imminent, they gather in the village streets, and all–men, women,

and children–beat pots and pans together, fire pistols, and ring

bells in order to frighten away the threatening monster.

In Montenegro, Herzegovina, and Bocca Cattaro the people believe that

the soul of a sleeping man is wafted by the winds to the summit of a

mountain, and, when a number of such has assembled, they become fierce

giants who uproot trees to use as clubs and hurl rocks and stones at

one another. Their hissing and groans are heard especially during the

nights in spring and autumn. Those struggling crowds are not composed

merely of human souls, but include the spirits of many animals, such as

oxen, dogs, and even cocks, but oxen especially join in the struggles.

Witches

Female evil spirits are generally called veshtitze (singular,

veshtitza, derived obviously from the ancient Bohemian word ved, which

means ‘to know’), and are supposed to be old women possessed by an evil

spirit, irreconcilably hostile to men, to other women, and most of all

to children. They correspond more or less to the English conception of

‘witches.’ When an old woman goes to sleep, her soul leaves her body

and wanders about till it enters the body of a hen or, more frequently,

that of a black moth. Flying about, it enters those houses where there

are a number of children, for its favourite food is the heart of an

infant. From time to time veshtitze meet to take their supper together

in the branches of some tree. An old woman having the attributes of

a witch may join such meetings after having complied with the rules

prescribed by the experienced veshtitze, and this is usually done by

pronouncing certain stereotyped phrases. The peasants endeavour to

discover such creatures, and, if they succeed in finding out a witch,

a jury is hastily formed and is given full power to sentence her to

death. One of the most certain methods used to discover whether the

object of suspicion is really a witch or not, is to throw the victim

into the water, for if she floats she is surely a witch. In this case

she is usually burnt to death. This test was not unknown in England.

Vampires

The belief in the existence of vampires is universal throughout the

Balkans, and indeed it is not uncommon in certain parts of western

Europe. Some assert that this superstition must be connected with

the belief generally held in the Orthodox Church that the bodies of

those who have died while under excommunication by the Church are

incorruptible, and such bodies, being taken possession of by evil

spirits, appear before men in lonely places and murder them. In

Montenegro vampires are called lampirs or tenatz, and it is thought

that they suck the blood of sleeping men, and also of cattle and other

animals, returning to their graves after their nocturnal excursions

changed into mice. In order to discover the grave where the vampire

is, the Montenegrins take out a black horse, without blemish, and

lead it to the cemetery. The suspected corpse is dug up, pierced

with stakes and burnt. The authorities, of course, are opposed to

such superstitious practices, but some communities have threatened

to abandon their dwellings, and thus leave whole villages deserted,

unless allowed to ensure their safety in their own way. The code of

the Emperor Doushan the Powerful provides that a village in which

bodies of dead persons have been exhumed and burnt shall be punished

as severely as if a murder had been committed; and that a resnik, that

is, the priest who officiates at a ceremony of that kind, shall be

anathematized. Militchevitch, a famous Serbian ethnographist, relates

an incident where a resnik, as late as the beginning of the nineteenth

century, read prayers out of the apocrypha of Peroon when an exorcism

was required. The revolting custom has been completely suppressed in

Serbia. In Montenegro the Archbishop Peter II. endeavoured to uproot

it, but without entire success. In Bosnia, Istria and Bulgaria it

is also sometimes heard of. The belief in vampires is a superstition

widely spread throughout Roumania, Albania and Greece. [17]

Nature Worship

Even in our own day there are traces of sun and moon worship, and

many Serbian and Bulgarian poems celebrate the marriage of the sun

and the moon, and sing Danitza (the morning star) and Sedmoro Bratye

(‘The Seven Brothers’–evidently The Pleiades). [18] Every man has

his own star, which appears in the firmament at the moment of his

birth and is extinguished when he dies. Fire and lightning are also

worshipped. It is common belief that the earth rests on water, that

the water reposes on a fire and that that fire again is upon another

fire, which is called Zmayevska Vatra (‘Fire of the Dragons’).

Similarly the worship of animals has been preserved to our times. The

Serbians consider the bear to be no less than a man who has been

punished and turned into an animal. This they believe because the

bear can walk upright as a man does. The Montenegrins consider the

jackal (canis aureus) a semi-human being, because its howls at night

sound like the wails of a child. The roedeer (capreolus caprea) is

supposed to be guarded by veele, and therefore she so often escapes

the hunter. In some parts of Serbia and throughout Montenegro it is

a sin to kill a fox, or a bee.

The worship of certain snakes is common throughout the Balkans. In

Montenegro the people believe that a black snake lives in a hole

under every house, and if anybody should kill it, the head of the

house is sure to die. Certain water-snakes with fiery heads were also

considered of the same importance as the evil dragons (or hydra) who,

at one time, threatened ships sailing on the Lake of Scutari. One

of these hydras is still supposed to live in the Lake of Rikavatz,

in the deserted mountains of Eastern Montenegro, from the bottom of

which the hidden monster rises out of the water from time to time, and

returns heralded by great peals of thunder and flashes of lightning.

But the Southern Slavs do not represent the dragon as the Hellenes

did, that is to say as a monster in the form of a huge lizard or

serpent, with crested head, wings and great strong claws, for they

know this outward form is merely used as a misleading mask. In his

true character a dragon is a handsome youth, possessing superhuman

strength and courage, and he is usually represented as in love with

some beautiful princess or empress. [19]

Enchanters

Among celebrants of the various pagan rites, there is mention of

tcharobnitzi (enchanters), who are known to have lived also in

Russia, where, during the eleventh century, they sapped the new

Christianity. The Slavonic translation of the Gospel recognized

by the Church in the ninth century applies the name ‘tcharobnitzi’

to the three Holy Kings.

To this same category belong the resnitzi who, as is apparent in

the Emperor Doushan’s Code referred to previously, used to burn the

bodies of the dead. Resnik, which appears as a proper name in Serbia,

Bosnia and Croatia, means, according to all evidence, “the one who

is searching for truth.”

Sacrificial Rites

From translations of the Greek legends of the saints, the exact

terminology of the sacrificial ceremonies and the places where they

had been made is well known. Procopius mentions oxen as the animals

generally offered for sacrifice, but we find that calves, goats,

and sheep, in addition to oxen, were used by the Polapic Slavs and

Lithuanians, and that, according to Byzantine authorities, the Russians

used even birds as well. In Montenegro, on the occasion of raising a

new building, a ram or a cock is usually slaughtered in order that a

corner-stone may be besprinkled with its blood, and, at the ceremony of

inaugurating a new fountain, a goat is killed. Tradition tells of how

Prince Ivan Tzrnoyevitch once shot in front of a cavern an uncommonly

big wild goat that, being quite wet, shook water from its coat so that

instantly a river began to flow thence. This stream is called even

now the River of Tzrnoyevitch. The story reminds one of the goats’

horns and bodies of goats which are seen on the altar dedicated to

the Illyrian god, Bind, near a fountain in the province of Yapod.

It is a fact that Russians and Polapic Slavs used to offer human

sacrifices. Mention of such sacrifices among the Southern Slavs

is found only in the cycle of myths relating to certain buildings,

which, it was superstitiously believed, could be completed only if a

living human being were buried or immured. Such legends exist among

the Serbians and Montenegrins concerning the building of the fortress

Skadar (Scutari) and the bridge near Vishegrad; with the Bulgarians

in reference to building the fort Lidga-Hyssar, near Plovdiv, and the

Kadi-Koepri (Turkish for ‘the bridge of the judge’) on the river Struma;

and again among modern Greeks in their history of the bridge on the

river Arta, and the Roumanians of the church ‘Curtea de Ardyesh.’ It

seems very likely that certain enigmatic bas-reliefs, representing

oval human faces with just the eyes, nose and mouth, which are found

concealed under the cemented surface of the walls of old buildings

have some connexion with the sacrificial practice referred to. There

are three such heads in the fortress of Prince Dyouragy Brankovitch

at Smederevo (Semendria), not far from Belgrade, on the inner side of

the middle donjon fronting the Danube, and two others in the monastery

Rila on the exterior wall close to the Doupitchka Kapiya.

Funeral Customs

During the siege of Constantinople in the year 626, the Southern Slavs

burnt the bodies of their dead. The Russians did the same during the

battles near Silistria, 971, and subsequently commemorative services

were held in all parts of Russia, and the remains of the dead were

buried.

The Slavs of north Russia used to keep the ashes of the dead in a

small vessel, which they would place on a pillar by the side of a

public road; that custom persisted with the Vyatitchs of southern

Russia as late as 1100.

These funeral customs have been retained longest by the Lithuanians;

the last recorded instance of a pagan burial was when Keystut,

brother of the Grand Duke Olgerd, was interred in the year 1382,

that is to say, he was burnt together with his horses and arms,

falcons and hounds.

There are in existence upright stones, mostly heavy slabs of stone,

many of them broken, or square blocks and even columns, which

were called in the Middle Ages kami, or bileg, and now stetyak or

mramor. Such stones are to be found in large numbers close together;

for example, there are over 6000 in the province of Vlassenitza,

and some 22,000 in the whole of Herzegovina; some can be seen also in

Dalmatia, for instance, in Kanovli, and in Montenegro, at Nikshitch;

in Serbia, however, they are found only in Podrigne. These stones

are usually decorated with figures, which appear to be primitive

imitations of the work of Roman sculptors: arcades on columns, plant

designs, trees, swords and shields, figures of warriors carrying

their bows, horsemen, deer, bears, wild-boars, and falcons; there

are also oblong representations of male and female figures dancing

together and playing games.

The symbol of the Cross indicates the presence of

Christianity. Inscriptions appear only after the eleventh century. But

many tombstones plainly had their origin in the Middle Ages. Some

tombs, situated far from villages, are described by man’s personal

name in the chronicles relating to the demarcations of territories,

for example, Bolestino Groblye (the cemetery of Bolestino) near Ipek;

Druzetin Grob (the tomb of Druzet). In Konavla, near Ragusa, there

was in the year 1420 a certain point where important cross-roads met,

known as ‘Obugonov Grob.’ Even in our day there is a tombstone here

without inscription, called ‘Obugagn Greb.’ It is the grave of the

Governor Obuganitch, a descendant of the family of Lyoubibratitch,

famous in the fourteenth century.

Classic and Mediaeval Influence

When paganism had disappeared, the Southern-Slavonic legends received

many elements from the Greeks and Romans. There are references to the

Emperors Trajan and Diocletian as well as to mythical personages. In

the Balkans, Trajan is often confused with the Greek king Midas. In

the year 1433 Chevalier Bertrandon de la Broquiere heard from the

Greeks at Trajanople that this city had been built by the Emperor

Trajan, who had goat’s ears. The historian Tzetzes also mentions

that emperor’s goat’s ears otia tragou. In Serbian legends the

Emperor Trajan seems also to be confused with Daedalus, for he is

given war-wings in addition to the ears.

To the cycle of mediaeval myths we owe also the djins (giants) who dwelt

in caverns, and who are known by the Turkish name div–originally

Persian. Notable of the divs were those having only one eye–who

may be called a variety of cyclops–mentioned also in Bulgarian,

Croatian and Slovenian mythology. On the shores of the river Moratcha,

in Montenegro, there is a meadow called ‘Psoglavlya Livada’ with a

cavern in which such creatures are said to have lived at one time

Chapter 2: Turks Presence

In 1169 a dynasty destined to rule Serbia for more than two centuries

(1169-1372) within ever-changing political boundaries, was founded

by the celebrated Grand Djoupan Stephan Nemanya (1169-1196) who was

created Duke (grand djoupan) of Serbia by the Byzantine Emperor after

he had instigated a revolution, the result of which was favourable

to his pretensions. By his bravery and wisdom he succeeded not only

in uniting under his rule the provinces held by his predecessors,

but also in adding those which never had been Serbian before, and he

placed Ban Koulin, an ally, upon the throne of Bosnia. Furthermore he

strengthened the orthodox religion in his state by building numerous

churches and monasteries, and by banishing the heretic Bogoumils. [7]

Feeling the weakness of advanced age, and wishing to give fresh proof

of his religious faith to his people, the aged Nemanya abdicated in

1196, in favour of his able second son Stevan, and withdrew into a

monastery. On his accession in the year 1217 Stevan assumed the title

of King of Serbia.

When the crusaders vanquished Constantinople, Sava, Stevan’s youngest

brother, obtained from the Greek patriarch the autonomy of the Serbian

Church (1219), and became the first Serbian archbishop.

Stevan was succeeded by his son Radoslav (1223-1233), who was dethroned

by his brother Vladislav (1233-1242), who was removed from the throne

by his third brother Ourosh the Great (1242-1276). Ourosh increased his

territory and established the reputation of Serbia abroad. In his turn,

he was dethroned by his son Dragoutin (1276-1281), who, owing to the

failure of a campaign against the Greeks, retired from the throne in

favour of a younger brother Miloutin (1281-1321), reserving, however,

for himself a province in the north of the State. Soon afterward

Dragoutin received from his mother-in-law, the queen of Hungary,

the lands between the Rivers Danube Sava and Drina, and assumed the

title of King of Sirmia. Dragoutin, while still alive, yielded his

throne and a part of his lands to Miloutin, and another part remained

under the suzerainty of the King of Hungary. Miloutin is considered

one of the most remarkable descendants of Nemanya. After his death the

usual discord obtained concerning the succession to the throne. Order

was re-established by Miloutin’s son, Stevan Detchanski (1321-1331),

who defeated the Bulgarians in the famous battle of Velbouzd, and

brought the whole of Bulgaria under his sway. Bulgaria remained a

province of Serbia until the Ottoman hordes overpowered both.

Doushan the Powerful

Stevan Detchanski was dethroned by his son Doushan the Powerful

(1331-1355), the most notable and most glorious of all Serbian

sovereigns. He aimed to establish his rule over the entire Balkan

Peninsula, and having succeeded in overpowering nearly the whole of

the Byzantine Empire, except Constantinople, he proclaimed himself,

in agreement with the Vlastela (Assembly of Nobles), Tsar of

Serbia. He elevated the Serbian archbishopric to the dignity of the

patriarchate. He subdued the whole of Albania and a part of Greece,

while Bulgaria obeyed him almost as a vassal state. His premature death

(some historians assert that he was poisoned by his own ministers)

did not permit him to realize the whole of his great plan for Serbia,

and under the rule of his younger son Ourosh (1355-1371) nearly all

his magnificent work was undone owing to the incessant and insatiable

greed of the powerful nobles, who thus paved the way for the Ottoman

invasion.

Among those who rebelled against the new Tsar was King

Voukashin. Together with his brother and other lords, he held almost

independently the whole territory adjoining Prizrend to the south of

the mountain Shar. [8]

King Voukashin and his brother were defeated in a battle with the

Turks on the banks of the River Maritza (1371), and all Serbian lands

to the south of Skoplye (Ueskueb) were occupied by the Turks.

The Royal Prince Marko

The same year Tsar Ourosh died, and Marko, the eldest son of King

Voukashin, the national hero of whom we shall hear much in this book,

proclaimed himself King of the Serbians, but the Vlastela and the

clergy did not recognize his accession. They elected (A.D. 1371) Knez

[9] (later Tsar) Lazar, a relative of Tsar Doushan the Powerful, to be

the ruler of Serbia, and Marko, from his principality of Prilip, as a

vassal of the Sultan, aided the Turks in their campaigns against the

Christians. In the year 1399 he met his death in the battle of Rovina,

in Roumania, and he is said to have pronounced these memorable words:

“May God grant the victory to the Christians, even if I have to perish

amongst the first!” The Serbian people, as we shall see, believe that

he did not die, but lives even to-day.

Knez Lazar ruled from 1371 to 1389, and during his reign he made

an alliance with Ban [10] Tvrtko of Bosnia against the Turks. Ban

Tvrtko proclaimed himself King of Bosnia, and endeavoured to extend

his power in Hungary, whilst Knez Lazar, with the help of a number

of Serbian princes, prepared for a great war against the Turks. But

Sultan Amourath, informed of Lazar’s intentions, suddenly attacked

the Serbians on June 15 1389, on the field of Kossovo. The battle

was furious on both sides, and at noon the position of the Serbians

promised ultimate success to their arms.

The Treachery of Brankovitch.

There was, however, treachery in the Serbian camp. Vook (Wolf)

Brankovitch, one of the great lords, to whom was entrusted one wing

of the Serbian army, had long been jealous of his sovereign. Some

historians state that he had arranged with Sultan Amourath to betray

his master, in return for the promise of the imperial crown of

Serbia, subject to the Sultan’s overlordship. At a critical moment

in the battle, the traitor turned his horse and fled from the field,

followed by 12,000 of his troops, who believed this to be a stratagem

intended to deceive the Turks. This was a great blow to the Serbians,

and when, later in the day the Turks were reinforced by fresh

troops under the command of the Sultan’s son, Bajazet, the Turkish

victory was complete. Knez Lazar was taken prisoner and beheaded,

and the Sultan himself perished by the hand of a Serbian voivode,

[11] Milosh Obilitch.

Notwithstanding the disaster, in which Brankovitch also perished, the

Serbian state did not succumb to the Turks, thanks to the wisdom and

bravery of Lazar’s son, Stevan Lazarevitch (1389-1427). His nephew,

Dyourady Brankovitch (1427-1456), also fought heroically, but was

compelled, inch by inch, to cede his state to the Turks.

The Final Success of the Turks

After the death of Dyourady the Serbian nobles could not agree

concerning his successor, and in the disorder that ensued the Turks

were able to complete their conquest of Serbia, which they finally

achieved by 1459. Their statesmen now set themselves the task of

inducing the Serbian peasantry in Bosnia, by promises of future

prosperity, to take the oath of allegiance to the Sultan, and in

this they were successful during the reign of the King of Bosnia,

Stevan Tomashevitch, who endeavoured in vain to secure help from the

Pope. The subjugation of Bosnia was an accomplished fact by 1463, and

Herzegovina followed by 1482. An Albanian chief of Serbian origin,

George Kastriotovitch-Skander-Beg (1443-1468), successfully fought,

with great heroism, for the liberty of Albania. Eventually, however,

the Turks made themselves master of the country as well as of all

Serbian lands, with the exception of Montenegro, which they never

could subdue, owing partly to the incomparable heroism of the bravest

Serbians–who objected to live under Turkish rule–and partly to the

mountainous nature of the country. Many noble Serbian families found

a safe refuge in that land of the free; many more went to Ragusa as

well as to the Christian Princes of Valahia and Moldavia. The cruel

and tyrannous nature of Turkish rule forced thousands of families to

emigrate to Hungary, and the descendants of these people may be found

to-day in Batchka, Banat, Sirmia and Croatia. Those who remained

in Serbia were either forced to embrace Islam or to live as raya

(slaves), for the Turkish spahis (land-lords) not only oppressed the

Christian population, but confiscated the land hitherto belonging to

the natives of the soil.

The Miseries of Turkish Rule

We should be lengthening this retrospect unduly if we were to describe

in full the miserable position of the vanquished Christians, and so

we must conclude by giving merely an outline of the modern period.

When it happens that a certain thing, or state of things, becomes

too sharp, or acute, a change of some sort must necessarily take

place. As the Turkish atrocities reached their culmination at the

end of the XVIIth century, the Serbians, following the example of

their brothers in Hungary and Montenegro, gathered around a leader

who was sent apparently by Providence to save them from the shameful

oppression of their Asiatic lords. That leader, a gifted Serbian,

George Petrovitch–designated by the Turks Karageorge (‘Black

George’)–gathered around him other Serbian notables, and a general

insurrection occurred in 1804. The Serbians fought successfully,

and established the independence of that part of Serbia comprised

in the pashalik of Belgrade and some neighbouring territory. This

was accomplished only by dint of great sacrifices and through the

characteristic courage of Serbian warriors, and it was fated to endure

for less than ten years.

Serbia again Subjugated

When Europe (and more particularly Russia) was engaged in the war

against Napoleon, the Turks found in the pre-occupation of the Great

Powers the opportunity to retrieve their losses and Serbia was again

subjugated in 1813. George Petrovitch and other Serbian leaders left

the country to seek aid, first in Austria, and later in Russia. In

their absence, Milosh Obrenovitch, one of Karageorge Petrovitch’s

lieutenants, made a fresh attempt to liberate the Serbian people

from the Turkish yoke, and in 1815 was successful in re-establishing

the autonomy of the Belgrade pashalik. During the progress of his

operations, George Petrovitch returned to Serbia and was cruelly

assassinated by order of Milosh who then proclaimed himself hereditary

prince and was approved as such by the Sublime Porte in October

1815. Milosh was a great opponent of Russian policy and he incurred the

hostility of that power and was forced to abdicate in 1839 in favour of

his son Michel (Serbian ‘Mihaylo’). Michel was an excellent diplomat,

and had previously incorporated within the independent state of Serbia

several districts without shedding blood. He was succeeded by Alexandre

Karageorgevitch (1842-1860) son of Karageorge Petrovitch. Under the

prudent rule of that prince, Serbia obtained some of the features

of a modern constitution and a foundation was laid for further and

rapid development. But an unfortunate foreign policy, the corruption

existing among the high dignitaries of the state and especially the

treachery of Milosh’s apparent friends, who hoped to supplant him,

forced that enlightened prince to abandon the throne and to leave his

country. The Skoupshtina (National Assembly) restored Milosh but the

same year the prince died and was succeeded once again by his son

Michel (1860-1868). At the assassination of this prince his young

cousin, Milan (1868-1889), ruled with the aid, during his minority,

of three regents, in conformity of a Constitution voted in 1869.

The principal events during the rule of Milan were: the war against

Turkey (1876-1878) and the annexation of four new districts; the

acknowledgment of Serbian independence by the famous Treaty of Berlin;

the proclamation of Serbia as a kingdom in 1882; the unfortunate war

against Bulgaria, which was instigated by Austria, and the promulgation

of a new Constitution, which, slightly modified, is still in force.

After the abdication of King Milan, his unworthy son, Alexander,

ascended the throne. Despite the vigorous advices of his friends and

the severe admonishments of his personal friend M. Chedo Miyatovich,

he married his former mistress, Draga Mashin, under whose influence

he entered upon a period of tyranny almost Neronian in type. He went

so far as to endeavour to abolish the Constitution, thus completely

alienating his people and playing into the hands of his personal

enemies, who finally murdered him (1903).

King Peter I

The Skoupshtina now elected the son of Alexander Karageorgevitch,

the present King Peter I Karageorgevitch, whose glorious rule will

be marked with golden letters in modern Serbian history, for it is to

him that Christendom owes the formation of the league whereby the Turk

was all but driven from Europe in 1913. But, alas! the Serbians have

only about one-half of their lands free, the rest of their brethren

being still under the foreign yoke.

Brief as is this retrospect it will suffice to show the circumstances

and conditions from which sprung the Serbian national poetry with

which we shall be largely concerned in the following pages. The

legends have their roots in disasters due as much to the self-seeking

of Serbian leaders as to foreign oppressors; but national calamities

have not repressed the passionate striving of a high-souled people

for freedom, and these dearly loved hero tales of the Balkans express

the ideals which have inspired the Serbian race in its long agony, and

which will continue to sustain the common people in whatever further

disappointments they may be fated to suffer ere they gain the place

among the great nations which their persistence and suffering must

surely win in the end.

General Characteristics

The Serbians inhabiting the present kingdom of Serbia, having been

mixed with the ancient indigenous population of the Balkan Peninsula,

have not conserved their true national type. They have mostly brown

visages and dark hair; very rarely are blonde or other complexions

to be seen. Boshnyaks (Serbians inhabiting Bosnia) are considered

to be the most typical Serbians, they having most strongly retained

the national characteristics of the pure Southern-Slavonic race. The

average Serbian has a rather lively temperament; he is highly sensitive

and very emotional. His enthusiasm is quickly roused, but most emotions

with him are, as a rule, of short duration. However, he is extremely

active and sometimes persistent. Truly patriotic, he is always ready

to sacrifice his life and property for national interests, which he

understands particularly well, thanks to his intimate knowledge of the

ancient history of his people, transmitted to him from generation to

generation through the pleasing medium of popular epic poetry composed

in very simple decasyllabic blank verse–entirely Serbian in its

origin. He is extremely courageous and always ready for war. Although

patriarchal and conservative in everything national, he is ready

and willing to accept new ideas. But he has remained behind other

countries in agricultural and industrial pursuits. Very submissive in

his Zadrooga [12] and obedient to his superiors, he is often despotic

when elevated to power. The history of all the Southern Slavs pictures

a series of violations, depositions, political upheavals, achieved

sometimes by the most cruel means and acts of treachery; all mainly

due to the innate and hitherto inexpugnable faults characteristic of

the race, such as jealousy and an inordinate desire for power. These

faults, of course, have been most apparent in the nobles, hence the

decay of the ancient aristocracy throughout the Balkans.

.

Significance of the Study

Chapter 3 : Moor Presence in Serbia

PRINCE MARKO AND A MOORISH CHIEFTAIN

A great and powerful Moorish chieftain had built for himself a

magnificent castle, rising to the height of twenty storeys. The place

he had chosen for the castle was by the sea, and when it was quite

completed he had panes of the most beautiful glass put in for windows;

he hung all the rooms and halls with the richest silks and velvets and

then soliloquized thus: “O my koula, [30] why have I erected thee? for

there is no one but I who is there to tread, with gentle footsteps,

upon these fine rugs, and behold from these windows the blue and

shining sea. I have no mother, no sister, and I have not yet found a

wife. But I will assuredly go at once and seek the Sultan’s daughter

in marriage. The Sultan must either give me his daughter or meet me

in single combat.” As soon as the Moor, gazing at his castle, had

uttered these words, he wrote a most emphatic letter to the Sultan at

Istamboul, [31] the contents of which ran thus: “O Sire, I have built

a beautiful castle near the shore of the azure sea, but as yet it has

no mistress, for I have no wife. I ask thee, therefore, to bestow upon

me thy beloved daughter! In truth, I demand this; for if thou dost

not give thy daughter to me, then prepare thyself at once to meet me

face to face with thy sword. To this fight I now challenge thee!”

The letter reached the Sultan and he read it through. Immediately he

sought for one who would accept the challenge in his stead, promising

untold gold to the knight who would show himself willing to meet the

Moor. Many a bold man went forth to fight the Moor, but not one ever

returned to Istamboul.

Alas! the Sultan soon found himself in a most embarrassing position

for all his best fighters had lost their lives at the hand of the

haughty Moor. But even this misfortune was not the worst. The Moor

prepared himself in all his splendour, not omitting his finest sabre;

then he proceeded to saddle his steed Bedevia, securely fastening the

seven belts and put on her a golden curb. On one side of the saddle

he fastened his tent, and this he balanced on the other side with

his heaviest club. He sprang like lightning on to his charger, and

holding before him, defiantly, his sharpest lance, he rode straight

to Istamboul.

The instant he reached the walls of the fort, he spread his tent,

struck his lance well into the earth, bound his Bedevia to the lance

and forthwith imposed on the inhabitants a daily tax, consisting of:

one sheep, one batch of white loaves, one keg of pure brandy, two

barrels of red wine, and a beautiful maiden. Each maiden, after being

his slave and attending on him for twenty-four hours, he would sell

in Talia for large sums of money. This imposition went on for three

months, for none could stop it. But even yet there was a greater evil

to be met.

The Entrance of the Moor

The inhabitants of Istamboul were terrorized one day when the haughty

Moor mounted upon his dashing steed entered the city. He went to the

Palace, and cried loudly: “Lo! Sultan, wilt thou now, once and for

ever, give me thy daughter?” As he received no answer he struck the

walls of the Palace with his club so violently that the shattered

glass poured down from the windows like rain. When the Sultan saw

that the Moor might easily destroy the Palace and even the whole

city in this way, he was greatly alarmed, for he knew that there was

no alternative open to him in this horrible predicament but to give

up his only daughter. Although overwhelmed with shame, therefore,

he promised to do this. Pleased with his success, the Moor asked for

fifteen days’ delay before his marriage took place that he might go

back to his castle and make the necessary preparations.

When the Sultan’s daughter heard of her father’s desperate resolution,

she shrieked and exclaimed bitterly: “Alas! Behold my sorrow, O

almighty Allah! For whom have I been taught to prize my beauty? For a

Moor? Can it be true that a Moor shall imprint a kiss upon my visage?”

The Sultana’s Dream

That night the Sultana had a strange dream, in which the figure of

a man appeared before her, saying: “There is within the Empire of

Serbia a vast plain Kossovo; in that plain there is a city Prilip;

and in that city dwells the Royal Prince Marko who is known among

all men as a truly great hero.”

And the man went on to advise the Sultana to send, without delay, a

message to Prince Marko and beg him to become her son-in-God, and at

the same time to offer him immense fortune, for he was without doubt

the only one living likely to vanquish the terrible Moor and save

her daughter from a shameful fate. The next morning she sped to the

Sultan’s apartments and told him of her dream. The Sultan immediately

wrote a firman [32] and sent it to Prince Marko at Prilip, beseeching

him to journey with all speed to Istamboul and accept the challenge of

the Moor, and if he should succeed in saving the Princess the Sultan

would give him three tovars [33] of pure golden ducats.

When Marko read the firman, he said to the Sultan’s young courier,

a native of Tartary: “In the name of God go back, thou Sultan’s

messenger, and greet thy master–my father-in-God–tell him that I dare

not face the Moor. Do we not, all of us, know that he is invincible? If

he should cleave my head asunder, of what avail would three tovars,

or three thousand tovars, of gold be to me?”

The young Tartar brought back Marko’s answer which caused the

Sultana so much grief, that she determined to send a letter to him

herself, once more beseeching him to accept the challenge and this

time increasing the reward to five tovars of pure gold. But Marko,

though generally so chivalrous and courteous to all women, remained

inexorable, replying that he would not meet the Moor in combat even

if he were to be presented with all the treasure the Sultan possessed;

for he did not dare.

The Princess appeals to Marko

When the broken-hearted bride heard that this answer had come from

Marko she sprang to her feet, took a pen and some paper, struck her

rosy cheek with the pen and with her own blood traced the following:

“Hail, my dear brother-in-God, O, thou Royal Prince Marko! Be a true

brother to me! May God and Saint John be our witnesses! I implore thee,

do not suffer me to become the wife of the Moor! I promise thee seven

tovars of pure gold, seven boshtchaluks, which have been neither woven

nor spun, but are embroidered with pure gold. Moreover, I shall give

thee a golden plate decorated with a golden snake, whose raised head

is holding in its mouth a priceless gem, from which is shed a light

of such brilliance, that by it alone you can see at the darkest hour

of midnight as well as you can at noon. In addition to these I shall

present thee with a finely tempered sabre; this sabre has three hilts,

all of pure gold, and in each of them is set a precious stone. The

sabre alone is worth three cities. I shall affix to this weapon the

Sultan’s seal so that the Grand Vizir may never put thee to death

without first receiving his Majesty’s special command.”

When he had read this missive, Marko reflected thus: “Alas! O my dear

sister-in-God! It would be but to my great misfortune if I came to

serve thee, and to my still greater misfortune if I stayed away. For,

although I fear neither the Sultan nor the Sultana, I do in all truth

fear God and Saint John, by whom thou hast adjured me! Therefore I

now resolve to come and, if necessary, to face certain death!”

Marko prepares to succour the Princess

Having sent away the Princess’ messenger without telling him what he

had resolved to do, Marko entered his castle and put on his cloak and

a cap, made of wolves’ skins; next he girded on his sabre, selected

his most piercing lance, and went to the stables. For greater safety

he fastened the seven belts under the saddle of his Sharatz with his

own hands; he then attached a leathern bottle filled with red wine on

one side of his saddle and his weightiest war-club on the other. Now

he was ready and threw himself upon Sharatz and rode off to Istamboul.

Upon reaching his destination he did not go to pay his respects either

to the Sultan or to the Grand Vizir, but quietly took up his abode in

a new inn. That same evening, soon after sun-set, he led his horse to

a lake near by to be refreshed: but to his master’s surprise Sharatz

would not even taste the water, but kept turning his head first to

the right, then to the left, till Marko noticed the approach of a

Turkish maiden covered with a long gold-embroidered veil. When she

reached the edge of the water she bowed profoundly toward the lake

and said aloud: “God bless thee, O beauteous green lake! God bless

thee, for thou art to be my home for ever more! Within thy bosom am I

henceforth to dwell; I am now to die, O beauteous lake; rather would

I choose such a fate than become the bride of the cruel Moor!”

Marko greets the Princess

Marko went nearer to the maiden and spoke thus: “O, thou unhappy

Turkish maiden! What is thy trouble? What is it that has made thee

wish to drown thyself?”

She answered: “Leave me in peace, thou ugly dervish, [34] why dost

thou ask me, when there is nought that thou canst do to help me?”

Then the maiden related the story of her coming marriage with the

Moorish chieftain, of the messages sent to Marko, and finally she

bitterly cursed that Prince for the hardness of his heart.

Thereupon Marko said: “O, curse me not, dear sister-in-God! Marko is

here and is now speaking to thee himself!”

Hearing these words the maiden turned toward the famous knight,

embraced him and earnestly pleaded: “For God’s sake, O my brother

Marko! Suffer not the Moor to wed me!”

Marko was greatly affected, and declared: “O dear sister-in-God! I

swear that so long as my head remains upon my shoulders, I shall never

let the Moor have thee! Do not tell others that thou hast seen me

here, but request the Sultan and thy mother to have supper prepared

and sent to the inn for me, and, above all things, beg them to send

me plenty of wine. Meanwhile I shall await the Moor’s coming at the

inn. When the Moor arrives at the Palace, thy parents should welcome

him graciously, and they should go so far as to yield thee to him in

order to avoid a quarrel. And I know exactly the spot where I shall

be able to rescue thee, if it may so please the true God, and if my

customary good luck, and my strength, do not desert me.”

The Prince returned to the inn, and the maiden hastened back to

the Palace.

When the Sultan and the Sultana knew that Marko had come to their aid,

they were much comforted, and immediately ordered a sumptuous repast

to be sent to him, especially good red wine in abundance.

Now all the shops in Istamboul were closed, and there was silence

everywhere as Marko sat drinking the delicious wine in peace. The

landlord of the inn came presently to close his doors and windows,

and, questioned by Marko as to why the citizens were all shutting

up their dwellings so early that day, he answered: “By my faith,

you are indeed a stranger here! The Moorish chieftain has asked

for our Sultan’s daughter in marriage, and as, to our shame, she is

to be yielded to him, he is coming to the Palace to fetch her this

day. Therefore, owing to our terror of the Moor, we are forced to close

our shops.” But Marko did not allow the man to close the door of the

inn, for he wished to see the Moor and his gorgeous train pass by.

The Moor in Istamboul

At that very moment, as they were speaking, Marko could hear from

the city the clangour caused by the Moorish chieftain and his black

followers, numbering at least five hundred, and all in glittering

armour. The Moor had roused his Bedevia, and she trotted in such

a lively manner that the stones, which she threw up with her hoofs,

whizzed through the air in all directions, and broke windows and doors

in all the shops she passed! When the cavalcade came up to the inn,

the Moor thought: “Allah! I am struck with wonder and astonishment! The

windows and doors of all the shops and houses throughout the entire

city of Istamboul are closed from the great fear the people have of

me, except, I see, the doors of this inn. There must either be nobody

within, or if there is anybody inside, he is assuredly a great fool;

or perhaps he is a stranger, and has not yet been told how terrible

I am.” The Moor and his retinue passed that night in tents before

the Palace.

Next morning the Sultan himself presented his daughter to the Moorish

chieftain, together with all the wedding gifts, which were known to

weigh twelve tovars. As the wedding procession passed the inn where

Marko waited, the Moor again noticed the open door, but this time he

urged Bedevia right up to it to see who might be there.

Sharatz and Bedevia

Marko was seated at his ease in the most comfortable room the inn

could boast, leisurely drinking his favourite red wine; he was not

drinking from an ordinary goblet, but from a bowl which held twelve

litres; and each time he filled the bowl he would drink only one

half of its contents, giving, according to his habit, the other half

to his Sharatz. The Moor was on the point of attacking Marko, when

Sharatz barred his way and kicked viciously at Bedevia. The Moor,

meeting such unexpected resistance, promptly turned to rejoin the

procession. Then Marko rose to his feet, and, turning his cloak

and cap inside out, so that to the first glance of those who saw

him he presented the terrifying appearance of a wolf, inspected his

weapons and Sharatz’s belts carefully, and dashed on his charger after

the procession. He felled horsemen right and left, till he reached

the dever and the second witness, and killed them both. The Moorish

chieftain was immediately told of the stranger who had forced his way

into the midst of the procession, and of those whom he had killed, also

that he did not look like other knights, being clad in wolves’ skins.

Marko and the Moor

The Moor astride his Bedevia, wheeled round and addressed Marko thus:

“Ill fortune is indeed overtaking thee to-day, O stranger! Thou must

have been driven here by Satan to disturb my guests and even kill my

dever and second witness; thou must be either a fool, knowing nothing

of to-day’s events, or thou must be extremely fierce and hast gone

mad; but maybe thou art merely tired of life? By my faith, I shall

draw in the reins of my Bedevia, and shall spring over thy body seven

times; then shall I strike off thy head!” Thereupon Marko answered:

“Cease these lies, O Moor! If God, and my usual luck, do but attend me

now, thou shalt not even spring near to me; still less can I imagine

thee carrying out thy intention of springing over my body!” But,

behold! The Moor drew in his Bedevia, spurred her violently forward

and indeed he would have sprung over Marko, had not Sharatz been

the well-trained fighter that he was, and in a trice he reared so

as to receive the adversary against his forefeet and swiftly bit

off Bedevia’s right ear, from which blood gushed forth profusely

and streamed down over her neck and chest. In this way Marko and

the Moor struggled for four hours. Neither would give way, and when

finally the Moor saw that Marko was overpowering him, he wheeled

his steed Bedevia round and fled along the main street of Istamboul,

Marko after him. But the Moor’s Bedevia was swift as a veela of the

forest, and would certainly have escaped from Sharatz if Marko had

not suddenly recollected his club, and flung it after his adversary,

striking him between his shoulders. The Moor fell from his horse and

the Prince severed his head from his body. Next he captured Bedevia,

returned to the street where he had left the bride, and found, to his

astonishment, that she with her twelve tovars of presents, was alone,

awaiting him, for all the wedding-guests and the retinue of the Moorish

chieftain had fled at full gallop. Marko escorted the Princess back

to the Sultan, and cast the head of the Moorish chieftain at his feet.

The hero now took his leave and started at once on his journey back

to Prilip, and the following morning he received the seven tovars of

gold which had been promised to him, the many precious gifts which

the Princess had described, and last of all a message thanking him

for the marvellous deeds he had done, and telling him that the vast

stores of gold belonging to his father-in-God, the Sultan, would for

ever be at his disposal.

The Spread of Christianity

When the pagan Slavs occupied the Roman provinces, the Christian

region was limited to parts of the Byzantine provinces. In Dalmatia

after the fall of Salona, the archbishopric of Salona was transferred

to Spalato (Splyet), but in the papal bulls of the ninth century it

continued always to be styled Salonitana ecclesia, and it claimed

jurisdiction over the entire lands as far as the Danube.

According to Constantine Porphyrogenete, the Serbians adopted the

Christian faith at two different periods, first during the reign of

the Emperor Heraclius, who had requested the Pope to send a number of

priests to convert those peoples to the Christian faith. It is well

known, however, that the Slavs in Dalmatia even during the reign of

Pope John IV (640-642) remained pagans. No doubt Christianity spread

gradually from the Roman cities of Dalmatia to the various Slav

provinces. The Croatians already belonged to the Roman Church at the

time when its priests were converting the Serbians to Christianity

between the years 642 and 731, i.e., after the death of Pope John IV

and before Leon of Isauria had broken off his relations with Rome.

The second conversion of those of the Southern Slavs who had remained

pagans was effected, about 879, by the Emperor Basil I.

At first the Christian faith spread amongst the Southern Slavs only

superficially, because the people could not understand Latin prayers

and ecclesiastical books. It took root much more firmly and rapidly

when the ancient Slavonic language was used in the church services.

Owing to the differences arising over icons and the form their worship

should take, enthusiasm for the conversion of the pagans by the Latin

Church considerably lessened. In the Byzantine provinces, however,

there was no need for a special effort to be made to the people,

for the Slavs came in constant contact with the Greek Christians,

whose beliefs they adopted spontaneously.

From the Slavonic appellations of places appearing in certain official

lists, one can see that new episcopates were established exclusively

for the Slavs by the Greek Church. The bishops conducted their

services in Greek, but the priests and monks, who were born Slavs,

preached and instructed the people in their own languages. Thus they

prepared the ground for the great Slav apostles.

The Slav apostles of Salonica, Cyrillos and his elder brother

Methodius, were very learned men and philosophers. The principal of

the two, Cyrillos, was a priest and the librarian of the Patriarchate;

in addition he was a professor of philosophy in the University of

the Imperial Palace at Constantinople, and he was much esteemed on

account of his ecclesiastical erudition. Their great work began in 862

with the mission to the Emperor Michel III., with which the Moravian

Princes Rastislav and Svetopluk entrusted them.

The Moravians were already converted to Christianity, but they wished

to have teachers among them acquainted with the Slav language. Before

the brothers started on their journey, Cyrillos composed the Slav

alphabet and translated the Gospel.

Thus the Serbians obtained these Holy Books written in a language

familiar to them, and the doctrines of the great Master gradually,

but steadily, ousted the old, primitive religion which had taken

the form of pure Naturalism. But the worship of Nature could not

completely disappear, and has not, even to our day, vanished from the

popular creed of the Balkans. The folk-lore of those nations embodies

an abundance of religious and superstitious sentiment and rites handed

down from pre-Christian times, for after many years’ struggle paganism

was only partially abolished by the ritual of the Latin and afterwards

of the Greek Christian Church, to which all Serbians, including the

natives of Montenegro, Macedonia and parts of Bosnia, belong.

Superstition

The foundations of the Christian faith were never laid properly in

the Balkans owing to the lack of cultured priests, and this reason,

and the fact that the people love to cling to their old traditions,

probably accounts for religion having never taken a very deep hold on

them. Even to this day superstition is often stronger than religion,

or sometimes replaces it altogether. The whole daily life of the

Southern Slav is interwoven with all kinds of superstition. He is

superstitious about the manner in which he rises in the morning and

as to what he sees first; for instance, if he sees a monk, he is sure

to have an unfortunate day; when he builds a house, a ‘lucky spot’

must be found for its foundation. At night he is superstitious about

the way he lies down; he listens to hear if the cocks crow in time,

and if the dogs bark much, and how they are barking. He pays great

attention to the moment when thunder is first heard, what kind of

rain falls, how the stars shine–whether or not they shine at all,

and looks anxiously to see if the moon has a halo, and if the sun

shines through a cloud. All these things are portents and omens to

his superstitious mind, and they play a considerable part in all his

actions. When he intends to join a hunting expedition, for example,

he decides from them whether there will be game or not; he believes

that he is sure to shoot something if his wife, or sister (or any

other good-natured person) jumps over his gun before he calls up his

dogs. Especially there are numberless superstitions connected with

husbandry, for some of which fairly plausible explanations could be

given;  for others, however, explanations are hopelessly unavailing,

and the reasons for their origin are totally forgotten. Nevertheless,

all superstitions are zealously observed because, the people say,

“it is well to do so,” or “our ancestors always did so and were happy,

why should we not do the same?”

The planting of fruit-trees and the growing of fruit must be aided by

charms, and numerous feasts are organized to secure a fruitful year,

or to prevent floods, hail, drought, frost, and other disasters. But

undoubtedly the greatest number of superstitions exist regarding the

daily customs, most of which refer to birth, marriage and death. Charms

are used to discover a future bridegroom or bride; to make a young man

fall in love with a maid or vice versa; also, if it seems desirable,

to make them hate each other. Sorcery is resorted to to ensure the

fulfilment of the bride’s wishes with regard to children; their

number and sex are decided upon, their health is ensured in advance,

favourable conditions are arranged for their appearance. Death can

come, it is believed, only when the Archangel Michael removes a soul

from its body, and that can only happen on the appointed day.

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Zodiac World: Arboreal the Enchantress (ORIGINS) https://ddcomics.org/2026/06/21/arboreal-the-enchantress-origins/ https://ddcomics.org/2026/06/21/arboreal-the-enchantress-origins/#respond Sun, 21 Jun 2026 10:12:29 +0000 https://ddcomics.org/?p=3084 In the Realm of Scorpio, a great conflict is being plotted. Since the beginning of time in the Zodiac World it had been war, non Zodiacs versus Zodiacs, who would rule the planet. In the crossfire some Zodiac sold their soul to work with non Zodiac they were called Diwers. In the midst of darkness, […]

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In the Realm of Scorpio, a great conflict is being plotted. Since the beginning of time in the Zodiac World it had been war, non Zodiacs versus Zodiacs, who would rule the planet. In the crossfire some Zodiac sold their soul to work with non Zodiac they were called Diwers. In the midst of darkness, in the bizarre land of the resurrected Scorpio; Garanzana de loco stands in the caves of Alchemist waiting for word of his brother Kacarkus the monkey shifter; his messengers the Black Dogs of Kuman teleports to the dark one with a weight on their shoulder.
“Glamadour. What is the word of my brother Kacurkus?”
“Dark one. He has been slain. The Enforcer. She beheaded him with the Umtus and burned his body. I’m sorry.”
Garanzana de loco’s back is turned away from his servants with his down.
“Bring Oachi Kabouche.”
“Yes, Master”
Dark One. You have called me. I am here to serve you.
Oachi Kabouche as you know Kacurkus is dead. Now I am the last of my kind; the last Kuman.
Yes, Doris was slain by Galloc Muh. Za was killed by Tayillah. What about Haflo?
Haflo. He is weak. He stays in that jungle almost brain dead; after the dead of his mate; Kath. Now Haflo stays in the jungle he calls Kath. That is why I bring you here. As you know Kath was slain by the rogue Libra priestess magicians. Those women go by a different code than Tayillah. They deal in mystic sorcery. That is why you come into play. I know Za thought you great black magic. In these caves I want you to start a clan of black magic magicians.
“What about the Bleeks? If Modin ever found out?”
“The Alchemist cave is too much out of the reach of the Scorpio capital. They would never know.”
As you wish dark one
On my part; I will create the first wave of vengeance against the Libra and the Zodiac. This night I will create another shadow; she will be unlike the Kuman of my world; she will not shift. She will be powerful; and run a new mock against this Zodiac World; until one day I regain my powers to fight Polyganda myself.
Write this down Oachi Kabouchi. For we will call this the Rage War.
Out of his Semen and Burning Black Fire -Garranzana de loco created his daughter a tool of revenge to fullfil prophecies of the Kuman . Garranzana de loco called her La Luc , she looked as of man kind Zodiac creature , she was formed of no animal .
ARBOREAL THE ENCHANTRESS (ORIGINS)
Atiwa this is your time to meet Motiz Bleek the king every Scorpio woman dreams of meeting such power, such a handsome man, and maybe just maybe he will make you queen of the Realm of Scorpio.
Mother I heard he loves women from beyond the Eboke Mountain?
Motiz Bleek loves all women he has numerous mistresses he has chosen you to come to Modin Bleek because he heard Atiwa the Duchess has luxurious red hair that extends to down your torso.
I will make you proud mother I will please him and be the best mistress I can be and the Duchess beyond the Eboke Mountain will be queen of the Realm of Scorpio.
It took me days to travel to Modin Bleek I was so nervous, and I was so young accompanied by the Kings guards I traveled over the purple mountains of Ebooke and on to the white sands of the Realm. Before I left the Ebooke I talked to Scorpio whores on how to please a man, for I was innocent and did not know anything about men after all Motiz had been with hundreds of women. There was a place we passed called Cauldron there were caves that stretched on white sands you would hear screams there and black smoke. One of the kings’ guards gripped my arm
“There are witches in those caves the king has banded this sorcery from Modin Bleek.”
I felt a great power in those caves and I wanted it more than I wanted to be queen.
As I entered the capital the guardians of the door Manix yelled out
“Introducing the Duchess from beyond the Eboke Mountains Atiwa.”
Every one bowed down in the hall of Modin Bleek’s palace then I saw him Motiz Bleek he was the tallest and most powerful Scorpio in the Realm. My mother was right I would be pleased he was so handsome. He had black hair his eyes were green and then sometimes black. His voice trembled the halls of the palace.
“ Welcome Duchess Atiwa I have asked your mother for you to stay in my palace for a week because I have heard of your great beauty. I am pleased. I have servants here that will take you to your room.”
I was so shy my head was down and I just said
“Thank you my king.”
“Dutchess Atiwa you are to stay in Modin Bleek things you must know, the Motiz Bleek wants you only inside the walls of the Capital City if he to consider you a Queen of all Scorpio.
There was a Big War with Scorpio and Motiz Bleek; they recently fought a Kuman Kacurkus who introduced the races to Black Magic in the Realm of Scorpio although the Kuman was defeated and killed by Motiz Bleek, there are still witches in the region outside the Realm of Scorpio that practice Black Magic. Atiwa you are here for the Bleek Family to comfort him and be a companion, abide by his rules and never travel beyond the Ebooke Mountains.”
Said the Handmaid of Atiwa and the Motiz family servant.
“I hope that I can satisfy Motiz and live up to his expectations emotionally and sexually that is what he sent for me to be a potential wife.”
Stated The Dutchess Atiwa.
“Seducing shouldn’t be necessary Atiwa, Motiz has plenty of whores to fill his den, he needs a wife, just be yourself this is your first experience with a man. He’s looking for innocence and Royality like yourself.”
Said the Handmaid of Atiwa and the Motiz family servant.
II.
Motiz Bleek has his way with me on any desired night ; I don’t feel special because Motiz Bleek has his way with other Concubines but heirs no sons nor daughters to his royal kingdom. I was bored sitting in my chambers locked away to only be told to bathe and wait for Motiz Bleek to come in my chamber and have his way with me.
Finally I expressed to my hand maid that I wanted to walk the countryside and get out of the dreaded chambers of the North Realm Palace Modin Bleek.
It was so many procedures and approvals for myself to even leave my chambers so imagine I made a huge fuzz and especially out of walking the countryside.
Atwia you must understand that the Ultimate has enemies we can’t just let his mistresses go unattended for walks among the countryside.
I’m bored. Why am I hear to be a whore. I am the Dutchess beyond the Ebooke Mountains, the King doesn’t even hold a conversation with me. He has his way with me like the other concubine whores. I was told that I was hear for something like love or marriage
Women in the Realm of Scorpio don’t demand things especially conversation with their sutters or to break rules like going outside of the palace for a walk in the countryside because they are bored.
Atiwa the Dutchess beyond the Ebooke Mountain screamed the whole half of the palace could hear “ Bring me the King , the Ultimate I want to walk the Countryside I demand it
9 foot Half Scorpio / Half Man the Ultimate King of all Scorpio in the Realm came to Atiwa the Dutches Beyond the Ebooke Mountains chamber asked
Have you lost your mind child? Why should I give in the demands of a woman. I should beat you for causing such a commotion. You demand nothing you’re a woman.
I want my needs meet Motiz , I want a conversation from you if you plan on marrying me or I’m going back home Beyond the Ebooke Mountains.

“Very well. I will satisfy your needs anything you desire Atiwa- it’s just not safe. We are at a Civil War and a huge unrest outside of Modin Bleek ,Kumans lead Black Magic worshippers to disrupt my law . It isn’t just my law it’s Polyganda’s laws.”
“ I honor what you say my Lord but we were free to walk Beyond the Ebooke Mountains. I just want to see the countryside.”
“You will need bodyguards, if I allow it. Royal Blood are the main targets our bourgeois society the poor and weak that eat bread, corn and water worship Kumans in the Cave and plan to overthrow my Laws and take away Royal Bourgeoise Society. You’re such a young thing and a woman. You don’t understand.”

“Yes, I am fresh. You are the First Scorpio male I had ever made love with. I do understand the Politics. I was convinced to marry you or be a concubine in hopes of Political Changes when or if I am Queen. I believe I can make a change. The change comes from a woman or women in the Realm of Scorpio voicing their opinions Publicly and Politically. I prepared myself for you to slap me for saying such against your laws.”

“The Bourgeois Society in Modin Bleek eat Fish and Bread because we can afford such luxurious things and we drape our Palaces in Gold. We Worship Polyganda only. We do not worship Kuman Demons this is the words of Galloc Muh the prophet. We feed off fish but No Pisces. These are the words of Galloc Muh. The Fish give us prosperity and enhance our Magical ways . These Commoners who worship Kumans you think they deserve to eat of Fish?”

“Just maybe the commoners feel unequal because they are meant to eat of Bread and Corn and Water, Fish and magic are only meant for the Bourgeois Society? Where does it say that in the Laws of Galloc Muh the Prophet?”
“Magic casting should not be harnessed for everyone ‘s usage. Your still a child . You don’t understand my Laws . You must abide by and respect my laws as my Queen. If I choose to marry you Atiwa.”

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Write up on “Bram Stoker’s” The Jewel of Seven Stars  https://ddcomics.org/2026/06/13/write-up-on-bram-stokers-the-jewel-of-seven-stars/ https://ddcomics.org/2026/06/13/write-up-on-bram-stokers-the-jewel-of-seven-stars/#respond Sat, 13 Jun 2026 16:55:37 +0000 https://ddcomics.org/?p=6871             Background of the Study       Abraham (Bram) Stoker was born November 8, 1847 in Dublin, Ireland. His father was a civil servant and his mother was a charity worker and writer. Stoker was a sickly child and spent a lot of time in bed. Growing up his mother told him a lot of horror stories […]

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            Background of the Study      

Abraham (Bram) Stoker was born November 8, 1847 in Dublin, Ireland. His father was a civil servant and his mother was a charity worker and writer. Stoker was a sickly child and spent a lot of time in bed. Growing up his mother told him a lot of horror stories which may have influenced his later writings. In 1864 Stoker entered Trinity College Dublin. While attending college he began working as an Irish civil servant. He also worked part time as a free lance journalist and drama critic. In 1876 he met Henry Irving, a famous actor, and they soon became friends. Not long after that, Stoker met and fell in love with an aspiring actress named Florence Balcombe. In 1878 Stoker accepted a job working in London as Irving’s personal secretary. According to an announcement in the December 5, 1878 issue of The Freeman’s Journal: and Daily Commercial Advertiser Stoker and Balcombe were married on December 4, 1878 at St. Anne’s Parish Church, Dublin, by the Rev. Charles W. Benson. On December 9, Stoker and his new wife moved to England to join Irving. His first book “The Duties of Clerks of Petty Sessions in Ireland” though written while he was still in Dublin, was published in 1879. On December 30, 1879 Stoker and his wife had their only child, a son Noel. While in England Stoker also wrote several novels and short stories. His first book of fiction, “Under the Sunset,” was published in 1881. Although best known for “Dracula”, Stoker wrote eighteen books before his death in 1912. He died of exhaustion at the age of 64.        

The Jewel of Seven Stars 

was Bram Stoker’s eighth novel. This novel, along with The Lair of The White Worm, is one of his most famous after Dracula. The novel is a horror story about attempt to resurrect an Egyptian queen. It was first published in the UK in 1903 by William Heinemann, London. The book itself is a 337 page hardcover with a red cloth cover stamped on the spine and front panel in blind, back and gold. In the same year, this book was also published by Heinemann as a part of their Heinemann’s Colonial Library series (No. 276). This edition is a hardcover with a decorated cloth cover [1]. The Jewel of Seven Stars was first published in the US in 1904 by Harper & Brothers Publishers, New York and London. This edition is a 311 page hardcover with a dark blue cloth cover stamped on the spine and front panel in light green and silver. In the same year it was also published in the US by W. R. Caldwell & Co, New York, as a part of The International Adventure Library (Three Owls Edition) series. This edition is a 310 page hardcover with a red cloth cover stamped on the spine and front panel in blind, black and gold.

Literature Review

The Jewel of Seven Stars is relatively unknown, which is surprising — it is one of the earlier stories about the reanimation of an ancient Egyptian mummy, and it is quite a thrilling tale!  It also uses state-of-the-art science of the time to bolster the story — with rather amusing results.

When Malcolm Ross arrives at the Trelawny house, he finds a scene in near chaos.  Abel has been found in his room of Egyptian antiquities, lying on the floor near his safe with his wrist horribly mauled by some animal.  A Doctor Winchester has already been called to tend to the patient, and Ross quickly calls a sharp police officer, Sergeant Daw, to assist in the investigation.  Abel Trelawny is in a cataleptic state unfamiliar to medical science, but seems to have anticipated trouble: he has left explicit written instructions that he is not to be moved from his Egyptian room, and furthermore he is to be guarded at all times by no less than one man and one woman.

The latter directive turns out to be prophetic: the following night, even with people in the room, Abel is attacked again by an unseen force.  Ross and the others realize they are guarding against a deadly and likely supernatural threat — and have no real defense against it.  More troubling, Margaret Trelawny seems to gradually be falling under the influence of this same threat, as her personality shifts subtly but unmistakably.

The arrival of Eugene Corbeck, a colleague of Abel’s freshly returned from Egypt, sheds some light on the situation and raises the stakes considerably.  With Corbeck’s help, Trelawny has spent years of his life seeking out the tomb of the ancient Egyptian queen and sorceress Tera, and nearly all of Tera’s funerary possessions are in Abel’s room — including the queen’s mummy.    Corbeck has brought the final pieces with him, and with everything together the group plans a dangerous experiment that may prove the existence of the supernatural — or lead to their destruction.

The focus and true heroine of this novel is Tera, who, in life, was Queen of the Egypts (Upper and Lower), daughter of Antef, Monarch of the North and the South, Daughter of the Sun and Queen of the Diadems. Even in her day she was the original emancipated woman because she claimed all the privileges of Kingship and masculinity and had power to compel the gods but never gave up her femininity even to the end of this story. According to the story she lived during the Theban Dynasty which was the 11th and at her birth a great aerolite fell, from whose heart was finally extracted that Jewel of Seven Stars which she regarded as the talisman of her life. It’s description is given to be that of a rare ruby with seven stars of which each star had seven points.

                                                Significance of Study

Set in the early twentieth century, the story opens when the first-person narrator, a barrister named Malcolm Ross, is summoned to the house of a young woman he recently met and is attracted to named Margaret Trelawny. She found her father, Abel Trelawny, unconscious on the floor of his study with seven scratches on his arm. The study is filled with Ancient Egyptian artifacts and Ross immediately notices the strong “mummy smell,” referring to the odor of bitumen and other substances the ancient Egyptians used to embalm their dead. Margaret has a letter of instructions from her father should such a strange occurrence take place. He is not to be removed from the room and a man and a woman must stand watch over him at all times. At this point, Malcolm and Margaret, along with a team consisting of Mr. Trelawny’s personal physician, Dr. Winchester, and his nurse watch over the Egyptologist. Over the coming days, strange things happen. Margaret’s cat, Silvio, tries to attack a cat mummy in the room. One night while Malcolm is on watch, he falls asleep. When he awakens, he discovers Mr. Trelawny on the floor and one of the knives in his collection has been moved. At first Silvio is suspected, but they realize the cat could not have pulled a grown man from his resting place on the study’s couch much less moved the knife. After this, Malcolm takes care to wear a respirator to avoid succumbing to the mummy smell. Malcolm also compares notes with Dr. Winchester and Police Superintendent Dolan. It begins to seem that the only person who could be responsible is none other than Margaret Trelawny, but there seems no good reason she would cause the strange occurrences.

At last, an old acquaintance of Mr. Trelawny named Eugene Corbeck arrives. He claims that he’d been tasked with searching out seven special lamps. However, soon after arriving in England, the lamps have vanished. The next day, the lamps are discovered in the Trelawny house. Corbeck gives us some backstory. It turns out that he was Mr. Trelawny’s partner in exploring Egypt. They came across a tomb of Queen Tera, in the Valley of the Sorcerers. Queen Tera had herself mummified with all her organs in tact and believed she had found a way to return to her body in the distant future. The key to the revival is, of course, the titular jewel of the seven stars, which serves as a map for the correct placement of the seven lamps. While Corbeck is telling his story, Abel Trelawny revives. To Malcolm Ross’s delight, now that we’re about two-thirds of the way through the novel, he gives his blessing for Ross to formally court Margaret. Once that’s out of the way, Mr. Trelawny, Corbeck, Ross, Margaret, and Dr. Winchester put plans in motion to see if they can bring Queen Tera back from the dead.

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Write up on Tech Geek History: LISP vs Python (Logical /Functional Programming) Revision 2 https://ddcomics.org/2026/06/13/write-up-on-tech-geek-history-lisp-vs-python-logical-functional-programming-revision-2/ https://ddcomics.org/2026/06/13/write-up-on-tech-geek-history-lisp-vs-python-logical-functional-programming-revision-2/#respond Sat, 13 Jun 2026 07:32:34 +0000 https://ddcomics.org/?p=6835 Write up on Tech Geek history : LISP vs Python Searches Significance of the Study Introduction Python was created by Guido van Rossum and first released on February 20, 1991. While the word “python” might bring to mind a large snake, the language’s name actually comes from the classic BBC comedy series Monty Python’s Flying Circus. What makes […]

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Write up on Tech Geek history : LISP vs Python Searches

Significance of the Study

Introduction

Python was created by Guido van Rossum and first released on February 20, 1991. While the word “python” might bring to mind a large snake, the language’s name actually comes from the classic BBC comedy series Monty Python’s Flying Circus.

What makes Python unique is that it began as the vision and work of a single person. Unlike most languages born in big tech companies, Python was created by Guido – driven by a simple goal: to make programming more intuitive and enjoyable.

From that idea, a global community has grown. Thousands of developers, educators, scientists, and enthusiasts continue to shape Python, expanding its reach into AI, data science, education, and beyond.

Even though Python has many of the features of Lisp, it is instructive to look at the original Lisp evaluation mechanism. At the heart of the Lisp language is a recursive interplay between the evaluation of expressions and application of functions. If you look at the code, there is an apply() function, and an eval() function. The interplay of these two functions results in a very elegant piece of code.

Common Lisp: there are similar patterns than in Python, but we can escape them. We can use macros, be concise and do what we want. We can have the decorator syntax with the cl-annot library, and any other by writing our reader macros (they can bring triply-quoted docstrings, string interpolation, infix notation, C syntax…). It’s not only macros though. The polymorphism of the object system (or generic dispatch) helps, and Lisp’s “moldability” in a whole allows us to refactor code exactly how we want, to build a “Domain Specific Language” to express what we want. Other language features than macros help here, like closures or multiple values (which are different, and safer for refactoring, than returning a tuple).

            Chapter 1: LISP Searches vs Python Searches

LISP Searches

A collection (in other words, a list) of assertions is called a database. Given a database , we can write functions to answer questions such as, “What color is block B2?” or “What blocks support block B1?”

To answer these questions, we will use a function called a pattern matcher to search the database for us. For example, to find out the color of block B2, we use the pattern (B2 COLOR?).

> (fetch’(b2 color ?))

 ((B2 COLOR RED))

To find which blocks support B1, we use the pattern (? SUPPORTS B1):

> (fetch'(? supports b1))

 ((B2 SUPPORTS B1) (B3 SUPPORTS B1))

FETCH returns those assertions from the database that match a given pattern. It should be apparent from the preceding examples that a pattern is a triple, like an assertion, with some of its elements replaced by question marks.

Structures are programmer-defined Lisp objects with an arbitrary number of named components. Structure types automatically become part of the Lisp type hierarchy. The DEFSTRUCT macro defines new structures and specifies the names and default values of their components.

 For example, we can define a structure called STARSHIP like this:

 (defstruct starship (name nil) (speed 0)

(condition ‘green) (shields ‘down))

https://www.cs.cmu.edu/~dst/LispBook/lisp-book-figures.pdf

This DEFSTRUCT form defines a new type of object called a STARSHIP whose components are called NAME, SPEED, CONDITION, and SHIELDS. STARSHIP becomes part of the system type hierarchy and can be referenced by such functions as TYPEP and TYPE-OF.

To introduce graph search programming in Lisp, we next represent and solve the farmer, wolf, goat, and cabbage problem: A farmer with his wolf, goat, and cabbage come to the edge of a river they wish to cross. There is a boat at the river’s edge, but, of course, only the farmer can row it. The boat also can carry only two things (including the rower) at a time. If the wolf is ever left alone with the goat, the wolf will eat the goat; similarly, if the goat is left alone with the cabbage, the goat will eat the cabbage. Devise a sequence of crossings of the river so that all four characters arrive safely on the other side of the river.

 The Lisp version searches the same space and has structural similarities to the Prolog solution; however, it differs in ways that reflect Lisp’s imperative/functional orientation. The Lisp solution searches the state space in a depth-first fashion using a list of visited states to avoid loops. The heart of the program is a set of functions that define states of the world as an abstract data type. These functions hide the internals of state representation from higher-level components of the program. States are represented as lists of four elements, where each element denotes the location of the farmer, wolf, goat, or cabbage, respectively.

Thus, (e w e w) represents the state in which the farmer (the first element) and the goat (the third element) are on the east bank and the wolf and cabbage are on the west. The basic functions defining the state data type will be a constructor, make-state, which takes as arguments the locations of the farmer, wolf, goat, and cabbage and returns a state, and four access functions, farmer-side, wolf-side, goatside, and cabbage-side, which take a state and return the location of an individual. These functions are defined:

(defun make-state (f w g c) (list f w g c))

(defun farmer-side (state) (nth 0 state))

(defun wolf-side (state) (nth 1 state))

 (defun goat-side (state) (nth 2 state))

 (defun cabbage-side (state) (nth 3 state))

The rest of the program is built on these state access and construction functions. In particular, they are used to implement the four possible actions the farmer may take: rowing across the river alone or with either of the wolf, goat, or cabbage. Each move uses the access functions to tear a state apart into its components. A function called opposite (to be defined shortly) determines the new location of the individuals that cross the river, and make-state reassembles

these into the new state.

 For example, the function farmer-takes-self may be defined:

(defun farmer-takes-self (state)

 (make-state (opposite (farmer-side state))

 (wolf-side state)

(goat-side state)

 (cabbage-side state)))

Note that farmer-takes-self returns the new state, regardless of whether it is safe or not. A state is unsafe if the farmer has left the goat alone with the cabbage or left the wolf alone with the goat. The program must find a solution path that does not contain any unsafe states. Although this “safe” check may be done at a number of different stages in the execution of the program, our approach is to perform it in the move functions. This is implemented by using a function called safe, which we also define shortly. safe has the following behavior:

> (safe ‘(w w w w)) ;safe state, return unchanged (w w w w) > (safe ‘(e w w e)) ;wolf eats goat, return nil nil > (safe ‘(w w e e)) ;

What is Search? Search is a process of finding a value in a list of values. In other words, searching is the process of locating given value position in a list of values. Linear Search Algorithm (Sequential Search Algorithm)

• Linear search algorithm finds given element in a list of elements with O(n) time complexity where n is total number of elements in the list.

 • This search process starts comparing of search element with the first element in the list. • If both are matching then results with element found otherwise search element is compared with next element in the list.

• If both are matched, then the result is “element found”. Otherwise, repeat the same with the next element in the list until search element is compared with last element in the list.

• if that last element also doesn’t match, then the result is “Element not found in the list”. That means, the search element is compared with element by element in the list. Linear search is implemented using following steps…

Step 1: Read the search element from the user

Step 2: Compare, the search element with the first element in the list.

Step 3: If both are matching, then display “Given element found!!!” and terminate the function

 Step 4: If both are not matching, then compare search element with the next element in the list.

Step 5: Repeat steps 3 and 4 until the search element is compared with the last element in the list.

Step 6: If the last element in the list is also doesn’t match, then display “Element not found!!!” and terminate the function

General overview Scheme is a functional programming language Scheme is a small derivative of LISP: LISt Processing Dynamic typing and dynamic scooping Scheme introduced static scooping

 · Data Objects

§ An expression is either an atom or a list

§ An atom is a string of characters

Definitions

· Functional programming languages were originally developed specifically to handle symbolic computation and listprocessing applications.

 · In FPLs the programmer is concerned only with functionality, not with memory-related variable storage and assignment sequences.

· FPL can be categorized into two types; ü PURE functional languages, which support only the functional paradigm (Haskell), and ü Impure functional languages that can also be used for writing imperative-style programs (LISP).

A lambda abstraction is rather similar to a function definition in a conventionallanguage, such as C: Inc( x ) int x; {retum( x + 1 );} The formal parameter of the lambda abstraction corresponds to the formal parameter of the function, and the body of the abstraction is an expression rather than a sequence of commands. However, functions in conventional languages must have a name(such as Inc), whereas lambda abstractions are ‘anonymous’functions.

The bodyof a lambdaabstraction extends as far to the right as possible, so that in the expression (Ax. + x 1) 4 the body of the Ax abstraction is (+ x 1), not just +. As usual, we may add extra bracketsto clarify, thus (ax.(+ x 1)) 4 When a lambdaabstraction appears in isolation we may write it without any brackets: dAx.+ x1

Applications

· AI is the main application domain for functional programming, covering topics such as: expert systems knowledge representation machine learning natural language processing modelling speech and vision A. Bellaachia Page: 4 o In terms of symbolic computation, functional programming languages have also proven useful in some editing environments (EMACS) and some mathematical software (particularly calculus)

· Lisp and its derivatives are still the dominant functional languages (we will consider one of the simpler derivatives, Scheme, in some detail).

We define a lambda expression to be an expression in the lambda calculus,

Definition 1 (λ-terms)

We begin by positing the existence of an infinite set of atomic variables, x, y, z, . . ..

We then define the set Λ of λ-terms as follows:

 • every variable is a λ-term,

• if M and N are λ-terms, then apply(M, N) is a λ-term, and

• if x is a variable, and M is a λ-term, then lambda(x, M) is a λ-term

https://www.classes.cs.uchicago.edu/archive/2004/spring/15300-1/docs/lambda-intro.pdf

Syntax of a lambda special form.

A lambda special form has the following syntax: (lambda (x1 x2 … xn) b1 b2 … bk) where:

• each xi is an expression denoting a Lisp symbol;

• each bi is a Lisp expression of any kind;

• n ≥ 0; and • k ≥ 1.

But real-world programming languages, like Lisp, have facilities that are not obviously present in the pure λ-calculus: they have primitive datatypes for things like the integers, they have ways to create composite types (tuples, records, vectors, etc.), and they have control structures such as conditionals and recursion. It is not immediately whether or not analogs to these facilities can be found in the λ-calculus.

Chapter 2

They Called It LISP for a Reason: List Processing

Lists play an important role in Lisp–for reasons both historical and practical. Historically, lists were Lisp’s original composite data type, though it has been decades since they were its only such data type. These days, a Common Lisp programmer is as likely to use a vector, a hash table, or a user-defined class or structure as to use a list.

Practically speaking, lists remain in the language because they’re an excellent solution to certain problems. One such problem–how to represent code as data in order to support code-transforming and code-generating macros–is particular to Lisp, which may explain why other languages don’t feel the lack of Lisp-style lists. More generally, lists are an excellent data structure for representing any kind of heterogeneous and/or hierarchical data. They’re also quite lightweight and support a functional style of programming that’s another important part of Lisp’s heritage.

Thus, you need to understand lists on their own terms; as you gain a better understanding of how lists work, you’ll be in a better position to appreciate when you should and shouldn’t use them.

“There Is No List”

Spoon Boy: Do not try and bend the list. That’s impossible. Instead . . . only try to realize the truth.

Neo: What truth?

Spoon Boy: There is no list.

Neo: There is no list?

Spoon Boy: Then you’ll see that it is not the list that bends; it is only yourself.1

The key to understanding lists is to understand that they’re largely an illusion built on top of objects that are instances of a more primitive data type. Those simpler objects are pairs of values called cons cells, after the function CONS used to create them.

CONS takes two arguments and returns a new cons cell containing the two values.2 These values can be references to any kind of object. Unless the second value is NIL or another cons cell, a cons is printed as the two values in parentheses separated by a dot, a so-called dotted pair.

(cons 1 2) ==> (1 . 2)

The two values in a cons cell are called the CAR and the CDR after the names of the functions used to access them. At the dawn of time, these names were mnemonic, at least to the folks implementing the first Lisp on an IBM 704. But even then they were just lifted from the assembly mnemonics used to implement the operations. However, it’s not all bad that these names are somewhat meaningless–when considering individual cons cells, it’s best to think of them simply as an arbitrary pair of values without any particular semantics. Thus:

(car (cons 1 2)) ==> 1

(cdr (cons 1 2)) ==> 2

Both CAR and CDR are also SETFable places–given an existing cons cell, it’s possible to assign a new value to either of its values.3

(defparameter *cons* (cons 1 2))

*cons*                 ==> (1 . 2)

(setf (car *cons*) 10) ==> 10

*cons*                 ==> (10 . 2)

(setf (cdr *cons*) 20) ==> 20

*cons*                 ==> (10 . 20)

Because the values in a cons cell can be references to any kind of object, you can build larger structures out of cons cells by linking them together. Lists are built by linking together cons cells in a chain. The elements of the list are held in the CARs of the cons cells while the links to subsequent cons cells are held in the CDRs. The last cell in the chain has a CDR of NIL, which–as I mentioned in Chapter 4–represents the empty list as well as the boolean value false.

This arrangement is by no means unique to Lisp; it’s called a singly linked list. However, few languages outside the Lisp family provide such extensive support for this humble data type.

So when I say a particular value is a list, what I really mean is it’s either NIL or a reference to a cons cell. The CAR of the cons cell is the first item of the list, and the CDR is a reference to another list, that is, another cons cell or NIL, containing the remaining elements. The Lisp printer understands this convention and prints such chains of cons cells as parenthesized lists rather than as dotted pairs.

(cons 1 nil)                   ==> (1)

(cons 1 (cons 2 nil))          ==> (1 2)

(cons 1 (cons 2 (cons 3 nil))) ==> (1 2 3)

When talking about structures built out of cons cells, a few diagrams can be a big help. Box-and-arrow diagrams represent cons cells as a pair of boxes like this:

The box on the left represents the CAR, and the box on the right is the CDR. The values stored in a particular cons cell are either drawn in the appropriate box or represented by an arrow from the box to a representation of the referenced value.4 For instance, the list (1 2 3), which consists of three cons cells linked together by their CDRs, would be diagrammed like this:

However, most of the time you work with lists you won’t have to deal with individual cons cells–the functions that create and manipulate lists take care of that for you. For example, the LIST function builds a cons cells under the covers for you and links them together; the following LIST expressions are equivalent to the previous CONS expressions:

(list 1)     ==> (1)

(list 1 2)   ==> (1 2)

(list 1 2 3) ==> (1 2 3)

Similarly, when you’re thinking in terms of lists, you don’t have to use the meaningless names CAR and CDRFIRST and REST are synonyms for CAR and CDR that you should use when you’re dealing with cons cells as lists.

(defparameter *list* (list 1 2 3 4))

(first *list*)        ==> 1

(rest *list*)         ==> (2 3 4)

(first (rest *list*)) ==> 2

Because cons cells can hold any kind of values, so can lists. And a single list can hold objects of different types.

(list “foo” (list 1 2) 10) ==> (“foo” (1 2) 10)

The structure of that list would look like this:

Because lists can have other lists as elements, you can also use them to represent trees of arbitrary depth and complexity. As such, they make excellent representations for any heterogeneous, hierarchical data. Lisp-based XML processors, for instance, usually represent XML documents internally as lists. Another obvious example of tree-structured data is Lisp code itself. In Chapters 30 and 31 you’ll write an HTML generation library that uses lists of lists to represent the HTML to be generated. I’ll talk more next chapter about using cons cells to represent other data structures.

Common Lisp provides quite a large library of functions for manipulating lists. In the sections “List-Manipulation Functions” and “Mapping,” you’ll look at some of the more important of these functions. However, they will be easier to understand in the context of a few ideas borrowed from functional programming.

Functional Programming and Lists

The essence of functional programming is that programs are built entirely of functions with no side effects that compute their results based solely on the values of their arguments. The advantage of the functional style is that it makes programs easier to understand. Eliminating side effects eliminates almost all possibilities for action at a distance. And since the result of a function is determined only by the values of its arguments, its behavior is easier to understand and test. For instance, when you see an expression such as (+ 3 4), you know the result is uniquely determined by the definition of the + function and the values 3 and 4. You don’t have to worry about what may have happened earlier in the execution of the program since there’s nothing that can change the result of evaluating that expression.

Functions that deal with numbers are naturally functional since numbers are immutable. A list, on the other hand, can be mutated, as you’ve just seen, by SETFing the CARs and CDRs of the cons cells that make up its backbone. However, lists can be treated as a functional data type if you consider their value to be determined by the elements they contain. Thus, any list of the form (1 2 3 4) is functionally equivalent to any other list containing those four values, regardless of what cons cells are actually used to represent the list. And any function that takes a list as an argument and returns a value based solely on the contents of the list can likewise be considered functional. For instance, the REVERSE sequence function, given the list (1 2 3 4), always returns a list (4 3 2 1). Different calls to REVERSE with functionally equivalent lists as the argument will return functionally equivalent result lists. Another aspect of functional programming, which I’ll discuss in the section “Mapping,” is the use of higher-order functions: functions that treat other functions as data, taking them as arguments or returning them as results.

Most of Common Lisp’s list-manipulation functions are written in a functional style. I’ll discuss later how to mix functional and other coding styles, but first you should understand a few subtleties of the functional style as applied to lists.

The reason most list functions are written functionally is it allows them to return results that share cons cells with their arguments. To take a concrete example, the function APPEND takes any number of list arguments and returns a new list containing the elements of all its arguments. For instance:

(append (list 1 2) (list 3 4)) ==> (1 2 3 4)

From a functional point of view, APPEND‘s job is to return the list (1 2 3 4) without modifying any of the cons cells in the lists (1 2) and (3 4). One obvious way to achieve that goal is to create a completely new list consisting of four new cons cells. However, that’s more work than is necessary. Instead, APPEND actually makes only two new cons cells to hold the values 1 and 2, linking them together and pointing the CDR of the second cons cell at the head of the last argument, the list (3 4). It then returns the cons cell containing the 1. None of the original cons cells has been modified, and the result is indeed the list (1 2 3 4). The only wrinkle is that the list returned by APPEND shares some cons cells with the list (3 4). The resulting structure looks like this:

In general, APPEND must copy all but its last argument, but it can always return a result that shares structure with the last argument.

Other functions take similar advantage of lists’ ability to share structure. Some, like APPEND, are specified to always return results that share structure in a particular way. Others are simply allowed to return shared structure at the discretion of the implementation.

“Destructive” Operations

If Common Lisp were a purely functional language, that would be the end of the story. However, because it’s possible to modify a cons cell after it has been created by SETFing its CAR or CDR, you need to think a bit about how side effects and structure sharing mix.

Because of Lisp’s functional heritage, operations that modify existing objects are called destructive–in functional programming, changing an object’s state “destroys” it since it no longer represents the same value. However, using the same term to describe all state-modifying operations leads to a certain amount of confusion since there are two very different kinds of destructive operations, for-side-effect operations and recycling operations.5

For-side-effect operations are those used specifically for their side effects. All uses of SETF are destructive in this sense, as are functions that use SETF under the covers to change the state of an existing object such as VECTOR-PUSH or VECTOR-POP. But it’s a bit unfair to describe these operations as destructive–they’re not intended to be used in code written in a functional style, so they shouldn’t be described using functional terminology. However, if you mix nonfunctional, for-side-effect operations with functions that return structure-sharing results, then you need to be careful not to inadvertently modify the shared structure. For instance, consider these three definitions:

(defparameter *list-1* (list 1 2))

(defparameter *list-2* (list 3 4))

(defparameter *list-3* (append *list-1* *list-2*))

After evaluating these forms, you have three lists, but *list-3* and *list-2* share structure just like the lists in the previous diagram.

*list-1*                  ==> (1 2)

*list-2*                  ==> (3 4)

*list-3*                  ==> (1 2 3 4)

Now consider what happens when you modify *list-2*.

(setf (first *list-2*) 0) ==> 0

*list-2*                  ==> (0 4)     ; as expected

*list-3*                  ==> (1 2 0 4) ; maybe not what you wanted

The change to *list-2* also changes *list-3* because of the shared structure: the first cons cell in *list-2* is also the third cons cell in *list-3*. SETFing the FIRST of *list-2* changes the value in the CAR of that cons cell, affecting both lists.

On the other hand, the other kind of destructive operations, recycling operations, are intended to be used in functional code. They use side effects only as an optimization. In particular, they reuse certain cons cells from their arguments when building their result. However, unlike functions such as APPEND that reuse cons cells by including them, unmodified, in the list they return, recycling functions reuse cons cells as raw material, modifying the CAR and CDR as necessary to build the desired result. Thus, recycling functions can be used safely only when the original lists aren’t going to be needed after the call to the recycling function.

To see how a recycling function works, let’s compare REVERSE, the nondestructive function that returns a reversed version of a sequence, to NREVERSE, a recycling version of the same function. Because REVERSE doesn’t modify its argument, it must allocate a new cons cell for each element in the list being reversed. But suppose you write something like this:

(setf *list* (reverse *list*))

By assigning the result of REVERSE back to *list*, you’ve removed the reference to the original value of *list*. Assuming the cons cells in the original list aren’t referenced anywhere else, they’re now eligible to be garbage collected. However, in many Lisp implementations it’d be more efficient to immediately reuse the existing cons cells rather than allocating new ones and letting the old ones become garbage.

NREVERSE allows you to do exactly that. The N stands for non-consing, meaning it doesn’t need to allocate any new cons cells. The exact side effects of NREVERSE are intentionally not specified–it’s allowed to modify any CAR or CDR of any cons cell in the list–but a typical implementation might walk down the list changing the CDR of each cons cell to point to the previous cons cell, eventually returning the cons cell that was previously the last cons cell in the old list and is now the head of the reversed list. No new cons cells need to be allocated, and no garbage is created.

Most recycling functions, like NREVERSE, have nondestructive counterparts that compute the same result. In general, the recycling functions have names that are the same as their non-destructive counterparts except with a leading N. However, not all do, including several of the more commonly used recycling functions such as NCONC, the recycling version of APPEND, and DELETEDELETE-IFDELETE-IF-NOT, and DELETE-DUPLICATES, the recycling versions of the REMOVE family of sequence functions.

In general, you use recycling functions in the same way you use their nondestructive counterparts except it’s safe to use them only when you know the arguments aren’t going to be used after the function returns. The side effects of most recycling functions aren’t specified tightly enough to be relied upon.

However, the waters are further muddied by a handful of recycling functions with specified side effects that can be relied upon. They are NCONC, the recycling version of APPEND, and NSUBSTITUTE and its -IF and -IF-NOT variants, the recycling versions of the sequence functions SUBSTITUTE and friends.

Like APPENDNCONC returns a concatenation of its list arguments, but it builds its result in the following way: for each nonempty list it’s passed, NCONC sets the CDR of the list’s last cons cell to point to the first cons cell of the next nonempty list. It then returns the first list, which is now the head of the spliced-together result. Thus:

(defparameter *x* (list 1 2 3))

(nconc *x* (list 4 5 6)) ==> (1 2 3 4 5 6)

*x* ==> (1 2 3 4 5 6)

NSUBSTITUTE and variants can be relied on to walk down the list structure of the list argument and to SETF the CARs of any cons cells holding the old value to the new value and to otherwise leave the list intact. It then returns the original list, which now has the same value as would’ve been computed by SUBSTITUTE6

The key thing to remember about NCONC and NSUBSTITUTE is that they’re the exceptions to the rule that you can’t rely on the side effects of recycling functions. It’s perfectly acceptable–and arguably good style–to ignore the reliability of their side effects and use them, like any other recycling function, only for the value they return.

Combining Recycling with Shared Structure

Although you can use recycling functions whenever the arguments to the recycling function won’t be used after the function call, it’s worth noting that each recycling function is a loaded gun pointed footward: if you accidentally use a recycling function on an argument that is used later, you’re liable to lose some toes.

To make matters worse, shared structure and recycling functions tend to work at cross-purposes. Nondestructive list functions return lists that share structure under the assumption that cons cells are never modified, but recycling functions work by violating that assumption. Or, put another way, sharing structure is based on the premise that you don’t care exactly what cons cells make up a list while using recycling functions requires that you know exactly what cons cells are referenced from where.

In practice, recycling functions tend to be used in a few idiomatic ways. By far the most common recycling idiom is to build up a list to be returned from a function by “consing” onto the front of a list, usually by PUSHing elements onto a list stored in a local variable and then returning the result of NREVERSEing it.7

This is an efficient way to build a list because each PUSH has to create only one cons cell and modify a local variable and the NREVERSE just has to zip down the list reassigning the CDRs. Because the list is created entirely within the function, there’s no danger any code outside the function has a reference to any of its cons cells. Here’s a function that uses this idiom to build a list of the first n numbers, starting at zero:8

(defun upto (max)

  (let ((result nil))

    (dotimes (i max)

      (push i result))

    (nreverse result)))

(upto 10) ==> (0 1 2 3 4 5 6 7 8 9)

The next most common recycling idiom9 is to immediately reassign the value returned by the recycling function back to the place containing the potentially recycled value. For instance, you’ll often see expressions like the following, using DELETE, the recycling version of REMOVE:

(setf foo (delete nil foo))

This sets the value of foo to its old value except with all the NILs removed. However, even this idiom must be used with some care–if foo shares structure with lists referenced elsewhere, using DELETE instead of REMOVE can destroy the structure of those other lists. For example, consider the two lists *list-2* and *list-3* from earlier that share their last two cons cells.

*list-2* ==> (0 4)

*list-3* ==> (1 2 0 4)

You can delete 4 from *list-3* like this:

(setf *list-3* (delete 4 *list-3*)) ==> (1 2 0)

However, DELETE will likely perform the necessary deletion by setting the CDR of the third cons cell to NIL, disconnecting the fourth cons cell, the one holding the 4, from the list. Because the third cons cell of *list-3* is also the first cons cell in *list-2*, the following modifies *list-2* as well:

*list-2* ==> (0)

If you had used REMOVE instead of DELETE, it would’ve built a list containing the values 1, 2, and 0, creating new cons cells as necessary rather than modifying any of the cons cells in *list-3*. In that case, *list-2* wouldn’t have been affected.

The PUSH/NREVERSE and SETF/DELETE idioms probably account for 80 percent of the uses of recycling functions. Other uses are possible but require keeping careful track of which functions return shared structure and which do not.

In general, when manipulating lists, it’s best to write your own code in a functional style–your functions should depend only on the contents of their list arguments and shouldn’t modify them. Following that rule will, of course, rule out using any destructive functions, recycling or otherwise. Once you have your code working, if profiling shows you need to optimize, you can replace nondestructive list operations with their recycling counterparts but only if you’re certain the argument lists aren’t referenced from anywhere else.

One last gotcha to watch out for is that the sorting functions SORTSTABLE-SORT, and MERGE mentioned in Chapter 11 are also recycling functions when applied to lists.10 However, these functions don’t have nondestructive counterparts, so if you need to sort a list without destroying it, you need to pass the sorting function a copy made with COPY-LIST. In either case you need to be sure to save the result of the sorting function because the original argument is likely to be in tatters. For instance:

CL-USER> (defparameter *list* (list 4 3 2 1))

*LIST*

CL-USER> (sort *list* #'<)

(1 2 3 4)                      ; looks good

CL-USER> *list*

(4)                            ; whoops!

List-Manipulation Functions

With that background out of the way, you’re ready to look at the library of functions Common Lisp provides for manipulating lists.

You’ve already seen the basic functions for getting at the elements of a list: FIRST and REST. Although you can get at any element of a list by combining enough calls to REST (to move down the list) with a FIRST (to extract the element), that can be a bit tedious. So Common Lisp provides functions named for the other ordinals from SECOND to TENTH that return the appropriate element. More generally, the function NTH takes two arguments, an index and a list, and returns the nth (zero-based) element of the list. Similarly, NTHCDR takes an index and a list and returns the result of calling CDR n times. (Thus, (nthcdr 0 …) simply returns the original list, and (nthcdr 1 …) is equivalent to REST.) Note, however, that none of these functions is any more efficient, in terms of work done by the computer, than the equivalent combinations of FIRSTs and RESTs–there’s no way to get to the nth element of a list without following n CDR references.11

The 28 composite CAR/CDR functions are another family of functions you may see used from time to time. Each function is named by placing a sequence of up to four As and Ds between a C and R, with each A representing a call to CAR and each D a call to CDR. Thus:

(caar list) === (car (car list))

(cadr list) === (car (cdr list))

(cadadr list) === (car (cdr (car (cdr list))))

Note, however, that many of these functions make sense only when applied to lists that contain other lists. For instance, CAAR extracts the CAR of the CAR of the list it’s given; thus, the list it’s passed must contain another list as its first element. In other words, these are really functions on trees rather than lists:

(caar (list 1 2 3))                  ==> error

(caar (list (list 1 2) 3))           ==> 1

(cadr (list (list 1 2) (list 3 4)))  ==> (3 4)

(caadr (list (list 1 2) (list 3 4))) ==> 3

These functions aren’t used as often now as in the old days. And even the most die-hard old-school Lisp hackers tend to avoid the longer combinations. However, they’re used quite a bit in older Lisp code, so it’s worth at least understanding how they work.12

The FIRSTTENTH and CARCADR, and so on, functions can also be used as SETFable places if you’re using lists nonfunctionally.

Tablesummarizes some other list functions that I won’t cover in detail.

Table . Other List Functions

FunctionDescription
LASTReturns the last cons cell in a list. With an integer, argument returns the last n cons cells.
BUTLASTReturns a copy of the list, excluding the last cons cell. With an integer argument, excludes the last n cells.
NBUTLASTThe recycling version of BUTLAST; may modify and return the argument list but has no reliable side effects.
LDIFFReturns a copy of a list up to a given cons cell.
TAILPReturns true if a given object is a cons cell that’s part of the structure of a list.
LIST*Builds a list to hold all but the last of its arguments and then makes the last argument the CDR of the last cell in the list. In other words, a cross between LIST and APPEND.
MAKE-LISTBuilds an n item list. The initial elements of the list are NIL or the value specified with the :initial-element keyword argument.
REVAPPENDCombination of REVERSE and APPEND; reverses first argument as with REVERSE and then appends the second argument.
NRECONCRecycling version of REVAPPEND; reverses first argument as if by NREVERSE and then appends the second argument. No reliable side effects.
CONSPPredicate to test whether an object is a cons cell.
ATOMPredicate to test whether an object is not a cons cell.
LISTPPredicate to test whether an object is either a cons cell or NIL.
NULLPredicate to test whether an object is NIL. Functionally equivalent to NOT but stylistically preferable when testing for an empty list as opposed to boolean false.

Mapping

Another important aspect of the functional style is the use of higher-order functions, functions that take other functions as arguments or return functions as values. You saw several examples of higher-order functions, such as MAP, in the previous chapter. Although MAP can be used with both lists and vectors (that is, with any kind of sequence), Common Lisp also provides six mapping functions specifically for lists. The differences between the six functions have to do with how they build up their result and whether they apply the function to the elements of the list or to the cons cells of the list structure.

MAPCAR is the function most like MAP. Because it always returns a list, it doesn’t require the result-type argument MAP does. Instead, its first argument is the function to apply, and subsequent arguments are the lists whose elements will provide the arguments to the function. Otherwise, it behaves like MAP: the function is applied to successive elements of the list arguments, taking one element from each list per application of the function. The results of each function call are collected into a new list. For example:

(mapcar #'(lambda (x) (* 2 x)) (list 1 2 3)) ==> (2 4 6)

(mapcar #’+ (list 1 2 3) (list 10 20 30)) ==> (11 22 33)

MAPLIST is just like MAPCAR except instead of passing the elements of the list to the function, it passes the actual cons cells.13 Thus, the function has access not only to the value of each element of the list (via the CAR of the cons cell) but also to the rest of the list (via the CDR).

MAPCAN and MAPCON work like MAPCAR and MAPLIST except for the way they build up their result. While MAPCAR and MAPLIST build a completely new list to hold the results of the function calls, MAPCAN and MAPCON build their result by splicing together the results–which must be lists–as if by NCONC. Thus, each function invocation can provide any number of elements to be included in the result.14 MAPCAN, like MAPCAR, passes the elements of the list to the mapped function while MAPCON, like MAPLIST, passes the cons cells.

Finally, the functions MAPC and MAPL are control constructs disguised as functions–they simply return their first list argument, so they’re useful only when the side effects of the mapped function do something interesting. MAPC is the cousin of MAPCAR and MAPCAN while MAPL is in the MAPLIST/MAPCON family.

Other Structures

While cons cells and lists are typically considered to be synonymous, that’s not quite right–as I mentioned earlier, you can use lists of lists to represent trees. Just as the functions discussed in this chapter allow you to treat structures built out of cons cells as lists, other functions allow you to use cons cells to represent trees, sets, and two kinds of key/value maps. I’ll discuss some of those functions in the next chapter.


1Adapted from The Matrix (http://us.imdb.com/Quotes?0133093)

2CONS was originally short for the verb construct.

3When the place given to SETF is a CAR or CDR, it expands into a call to the function RPLACA or RPLACD; some old-school Lispers–the same ones who still use SETQ–will still use RPLACA and RPLACD directly, but modern style is to use SETF of CAR or CDR.

4Typically, simple objects such as numbers are drawn within the appropriate box, and more complex objects will be drawn outside the box with an arrow from the box indicating the reference. This actually corresponds well with how many Common Lisp implementations work–although all objects are conceptually stored by reference, certain simple immutable objects can be stored directly in a cons cell.

5The phrase for-side-effect is used in the language standard, but recycling is my own invention; most Lisp literature simply uses the term destructive for both kinds of operations, leading to the confusion I’m trying to dispel.

6The string functions NSTRING-CAPITALIZENSTRING-DOWNCASE, and NSTRING-UPCASE are similar–they return the same results as their N-less counterparts but are specified to modify their string argument in place.

7For example, in an examination of all uses of recycling functions in the Common Lisp Open Code Collection (CLOCC), a diverse set of libraries written by various authors, instances of the PUSH/NREVERSE idiom accounted for nearly half of all uses of recycling functions.

8There are, of course, other ways to do this same thing. The extended LOOP macro, for instance, makes it particularly easy and likely generates code that’s even more efficient than the PUSHNREVERSE version.

9This idiom accounts for 30 percent of uses of recycling in the CLOCC code base.

10SORT and STABLE-SORT can be used as for-side-effect operations on vectors, but since they still return the sorted vector, you should ignore that fact and use them for return values for the sake of consistency.

11NTH is roughly equivalent to the sequence function ELT but works only with lists. Also, confusingly, NTH takes the index as the first argument, the opposite of ELT. Another difference is that ELT will signal an error if you try to access an element at an index greater than or equal to the length of the list, but NTH will return NIL.

12In particular, they used to be used to extract the various parts of expressions passed to macros before the invention of destructuring parameter lists. For example, you could take apart the following expression:

(when (> x 10) (print x))

Like this:

;; the condition

(cadr ‘(when (> x 10) (print x))) ==> (> X 10)

;; the body, as a list

(cddr ‘(when (> x 10) (print x))) ==> ((PRINT X))

13Thus, MAPLIST is the more primitive of the two functions–if you had only MAPLIST, you could build MAPCAR on top of it, but you couldn’t build MAPLIST on top of MAPCAR.

14In Lisp dialects that didn’t have filtering functions like REMOVE, the idiomatic way to filter a list was with MAPCAN.

(mapcan #'(lambda (x) (if (= x 10) nil (list x)))  list) === (remove 10 list)

Python Linear Search

Searching is when we find something in a data structure. We frequently search for strings in things like web pages, PDFs, documents, etc., but we can also search through other data structures, like lists, dictionaries, etc. Depending on how our data is organized, we can search in different ways. For unorganized data, we usually have to do a linear search, which is the first type of search we will discuss. If our data is organized in some way, we can do more efficient searches. If our data is in a strict order, we can perform a binary search, which is the second type of search we will look at.:

 Linear Searching Lecture 10: Linear Searching The most straightforward type of search is the linear search. We traverse the data structure (e.g., a string’s characters, or a list) until we find the result. How would we do a linear search on a list, like this? Let’s say we are searching for 15

. lst = [12, 4, 9, 18, 53, 82, 15, 99, 98, 14, 11]

The most straightforward type of search is the linear search. We traverse the data structure (e.g., a string’s characters, or a list) until we find the result. How would we do a linear search on a list, like this? Let’s say we are searching for 15.

: Linear Searching lst = [12, 4, 9, 18, 53, 82, 15, 99, 98, 14, 11]

def linear_search(lst, value_to_find):

“”” Perform a linear search to find a value in the list :

param lst: a list :param value_to_find: the value we want to find

:return:

the index of the found element, or -1

 if the element does not exist in the list

 >>> linear_search([12, 4, 9, 18, 53, 82, 15, 99, 98, 14, 11], 15) 6

>>> linear_search([12, 4, 9, 18, 53, 82, 15, 99, 98, 14, 11], 42) -1

“”” for i, value in enumerate(lst):

if value == value_to_find:

 return i return –

Chapter 3

Python’s  Linear Search

Linear Search What is a Linear Search? Linear search is a method of finding elements within a list. It is also called a sequential search. It is the simplest searching algorithm because it searches the desired element in a sequential manner. It compares each and every element with the value that we are searching for. If both are matched, the element is found, and the algorithm returns the key’s index position. Concept of Linear Search Let’s understand the following steps to find the element key = 7 in the given list.

Step – 1: Start the search from the first element and Check key = 7 with each element of list x.

Linear Search Algorithm There is list of n elements and key value to be searched. Below is the linear search algorithm.

 1. LinearSearch(list, key)

2. for each item in the list

3. if item == value

4. return its index position

 5. return -1

Python Program Let’s understand the following Python implementation of the linear search algorithm.

Program 1.

def linear_

Search(list1, n, key): 2.

 3. # Searching list1 sequentially

 4. for i in range(0, n):

 5. if (list1[i] == key):

6. return i

7. return -1

8.

9.

10. list1 = [1 ,3, 5, 4, 7, 9]

11. key = 7

12.

13. n = len(list1)

14. res = linear_Search(list1, n, key)

 15. if(res == -1):

16. print(“Element not found”)

17. else:

18. print(“Element found at index: “, res)

 Output: Element found at index:

Explanation: In the above code, we have created a function linear_Search(), which takes three arguments – list1, length of the list, and number to search. We defined for loop and iterate each element and compare to the key value. If element is found, return the index else return -1 which means element is not present in the list.

11. Functional Programming Using Python

 · A list is a collection of objects.

· List constants are surrounded by square brakets and the elements in the list are separated by commas.

· Lists are “mutable” – we can change an element of a list using the index operator

 · Examples: myFriendsList = [ ‘Paul’, ‘Mary’, ‘Sally’ ] myNums = [1,2,3,4] myList = []

 · A list can element of another list:

>>> myList2 = [1, 2]

>>> myList3 = [‘a’, myList2, 50]

>>> myList2 [1, 2]

>>> myList3 [‘a’, [1, 2], 50]

 . Defining functions

· User-defined functions can be created through the use of the lambda operator as follows: (define functionname (lambda (functionparameters)

(expression) (expression) …

(expression) ))

NOTE: The value returned by the function is the value of the last expression in the list ü Example: For example, the function below calculates factorials:

>(define factorial (lambda (n) ( if (< n 3) n (* n (factorial (- n 1)))) )) ;

Value: factorial

 >(factorial 3) ; Value: 6

# Python lambda

x : lambda y : y

Rules of Our System

The Lambda Calculus asserts that any computational system can be implemented with a set of three simple rules:

  • You can define variables
  • You can define single-argument functions
  • You can call single-argument functions

That’s it. No numbers. No operators. No control flow. No data structures.

I find it fascinating that with these very minimal concepts, the Lambda Calculus asserts that we can create a fully functional computer! This is, of course, a very minimal explanation of the rules of the Lambda Calculus, and I invite you to consult the references above for more information and formal definitions!

x = lambda a : a + 10
print(x(5))

https://www.cs.umd.edu/class/spring2024/cmsc330-030X-040X/assets/notes/lambdacalc.pdf

Why Use Lambda Functions?

The power of lambda is better shown when you use them as an anonymous function inside another function.

Say you have a function definition that takes one argument, and that argument will be multiplied with an unknown number:

def myfunc(n):
  return lambda a : a * n

A Data structures are specialized objects for organizing data efficiently. There are many kinds, each with specific strengths and weaknesses, and different applications require different structures for optimal performance. For example, some data structures take a long time to build, but once built their data are quickly accessible. Others are built quickly, but are not as efficiently accessible. These strengths and weaknesses are determined by how the structure is implemented. Python has several built-in data structure classes, namely list, set, dict, and tuple. Being able to use these structures is important, but selecting the correct data structure to begin with is often what makes or breaks a good program. In this lab we create a structure that mimics the built-in list class, but that has a different underlying implementation. Thus our class will be better than a plain Python list for some tasks, but worse for others.

Linked Lists A linked list is a data structure that chains nodes together. Every linked list needs a reference to the first node in the chain, called the head. A reference to the last node in the chain, called the tail, is also often included. Each node instance in the list stores a piece of data, plus at least one reference to another node in the list. The nodes of a singly linked list have a single reference to the next node in the list (see Figure 1.1), while the nodes of a doubly linked list have two references: one for the previous node, and one for the next node in the list (see Figure 1.2). This allows for a doubly linked list to be traversed in both directions, whereas a singly linked list can only be traversed in one direction.

class Linked

ListNode(Node):

 “””A node class for doubly linked lists. Inherits from the ‘Node’ class. Contains references to the next and previous nodes in the linked list.

 “”” def __init__(self, data): “

“”Store ‘data’ in the ‘value’ attribute and initialize attributes for the next and previous nodes in the list.

“”” Node.__init__(self, data)

 # Use inheritance to set self.value

. self.next = None

 self.prev = None

Linked Lists:

A linked list is a collection of objects/nodes, where each node contains both the item stored in the list, as well as a “pointer” to the next node in the list.

Linked lists typically have three instance variables: a head pointer (self.head), a tail pointer (self.tail), and the number of nodes in the list (self.size)

In the above picture, the linked list has 5 nodes, and each node stores a simple string, as well as a pointer to the next node in the list. In class I often draw linked lists and nodes as follows:

              ——    ——   ——   ——   ——

self.head –> |rich|    |lisa|   |andy|   |zach|   |tia |<–self.tail

              ——    ——   ——   ——   ——

              |    |–> |    |–>|    |–>|    |–>|    |–>None

              ——    ——   ——   ——   ——

                             self.size = 5

Linked lists are often used to create other useful data structures, such as stacks and queues. They are also a good introduction to more advanced data structures, such as binary trees.

One big advantage of the linked list is not needing large chunks of consecutive memory. Each node in the list is just put in memory wherever there is space, and all nodes are “linked” together into one “list”.

Also, adding an item into the middle of the list does not require shifting all other elements down one slot. Simply reset the next “pointers” to include the new node.

Implementing A Linked List

We will implement a linked list using two classes:

  1. Node, which will store a data value (the list item) and a “pointer” to the next node in the list
  2. LinkedLlist, which will store pointers to the head Node and the tail Node, as well as the current size of the list.

the Node class:

Let’s write a Node class, where each node has two instance variables: one to hold the node data (call it “data”), and one to hold a pointer to another node (call it “next”). Later we will use these node objects to build a linked-list!

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Write up on Discrete Mathematics in an educational Curriculum for grade school and high school students (Part 1 & Part 2) https://ddcomics.org/2026/06/13/write-up-on-discrete-mathematics-in-an-educational-curriculum-for-grade-school-and-high-school-students-part-1-part-2/ https://ddcomics.org/2026/06/13/write-up-on-discrete-mathematics-in-an-educational-curriculum-for-grade-school-and-high-school-students-part-1-part-2/#respond Sat, 13 Jun 2026 06:24:13 +0000 https://ddcomics.org/?p=6828  Literature Review1.1 Discrete Mathematics used for Computer Science as a Fundamental basics to learn AlgorithmsThe Need for Computer ScienceThis is largely based on how exposed students are to computational thinking and computer science concepts. Additionally, educating students in computer science is beneficial for all students. With the digital age rising, there is a need to […]

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Literature Review
1.1 Discrete Mathematics used for Computer Science as a Fundamental basics to learn Algorithms
The Need for Computer Science
This is largely based on how exposed students are to computational thinking and computer science concepts. Additionally, educating students in computer science is beneficial for all students. With the digital age rising, there is a need to develop logical thinking and problem-solving which are all a part of learning computer science.
Computer Science Standards and Model Curriculum give students experiences that help them discover and take part in a world continually influenced by technology and to understand the role of computing
What is Discrete Mathematics?
Discrete Mathematics is an area of mathematics that most closely connects with the field of computer science. It is the study of mathematical structures that are countable or otherwise distinct and separable (as opposed to continuous quantities like in algebra or calculus).
What Is Discrete Mathematics?
Discrete Mathematics is a rapidly growing and increasingly used area of mathematics, with many practical and relevant applications.
• Because it is grounded in real-world problems, discrete mathematics lends itself easily to implementing the recommendations fo the National Council of Teachers of Mathematics (NCTM) standards. (The recently published Standards and Principles for School Mathematics notes that “As an active branch of contemporary mathematics that is widely used in business and industry, discrete mathematics should be an integral part of the school mathematics curriculum.”)
• Because many discrete math problems are simply stated and have few mathematical prerequisites, they can be introduced at all grade levels, even with children who are not yet fluent readers.
Discrete mathematics will make math concepts come alive for your students. It’s an excellent tool for improving reasoning and problem-solving skills, and is appropriate for students at all levels and of all abilities. Teachers have found that discrete mathematics offers a way of motivating unmotivated students while challenging talented students at the same time.
Because many discrete math problems are simply stated and have few mathematical prerequisites, they can be easily be introduced at the middle school grade level.
1.2 How Discrete Mathematics Can Help Students
The National Council of Teachers of Mathematics (NCTM) recommends that discrete mathematics be implemented into the curriculum as early as the seventh 4 grade, because it benefits students and helps maintain their interest in mathematics [5, p. 362]. But beyond what the NCTM recommends, is discrete mathematics really advantageous to students, and if so, why? Discussed below are some of the reasons. First, it keeps students interested in mathematics. It helps entice them to regularly attend and participate in class. When students are interested in the material, it is easier for them to learn and stay focused when presented with tough, complex problems. While solving problems in discrete mathematics can be complicated, the problems themselves can be easily understood. The students, therefore, are able to understand and work the problems, which gives them a much needed confidence [12, pp. 35-36].
 Many students are lost from mathematics forever during high school. Discrete mathematics is a great way to help these students stay interested and involved in mathematics. Second, discrete mathematics benefits students by allowing them to see the connections between the mathematics they are studying and the real world. “For example, we might be able to convince our students that calculus can be used to help civil engineers build better bridges, but the students still might not see how it really works. But in graph theory [and other areas of discrete mathematics], we can explain the applications, the students can see how they work, and they can actually see real problems” [7, p. 94]. Teachers need to help students dismiss the idea that there is nothing new left to discover in mathematics and help them to look beyond basic arithmetic computation.
 Discrete mathematics is the mathematical foundation of computer science and is “used extensively in business, industry, and government. For example, difference equations are an essential mathematical tool for high-technology engineering firms, and matrices are indispensable for computer graphics” [4, p. 75].
 Another benefit of discrete mathematics is that it enriches the traditional curriculum. It places more emphasis on teaching students to think mathematically and less emphasis on certain computational skills [1, p. 83]. Discrete mathematics lends itself to group work more easily than does traditional mathematics. It is also helpful to teachers because it gives them a new way of teaching elements in the curriculum, which may make the traditional concepts easier to teach and learn.
By using discrete mathematics to teach already existing elements in the curriculum, it can help to change the way students view mathematics altogether. Below are some of the ways that discrete mathematics complements topics already in the traditional secondary-school curriculum: · Algebraic skills are needed and reinforced throughout discrete mathematics. · In geometry, graph theory can be used to enrich the study of polygons and polyhedral. · Difference equations give use to the fascinating new geometry of fractals [4, p. 75]. The NCTM does not provide teachers with detailed guidelines. They only provide a set of goals and topics to cover in high school for discrete mathematics. Therefore, it is up to each individual teacher, or the mathematics department within each school, to decide how these topics should be implemented.

Unfortunately, many teachers are unfamiliar, and even uncomfortable, with discrete mathematics. Thus, it can be difficult for them to know how to incorporate these topics into the curriculum. Because discrete mathematics can be used to teach traditional elements in the curriculum, these topics can be covered in different ways throughout the school year, without having to set aside extra time to cover them. 6 While the implementation of discrete mathematics into the curriculum is not discussed here in detail, many references cited in this thesis give numerous ideas on how to do so. Some also give sample lessons and projects for different skill levels. Some excellent resources are: · DIMACS Series in Discrete Mathematics and Theoretical Computer Science 36 (eds. J.G. Rosenstein, D.S. Franzblau, and F.S. Roberts), American Mathematical Society; 1997. · Discrete Mathematics Across the Curriculum, K–12 (eds. M.J. Kenney, and C.R. Hirsch), NCTM, Inc, Reston, Virginia; 1991. · Mathematics Teacher, a monthly journal magazine.
https://repository.lsu.edu

These political battles over the mathematics curriculum resulted in discrete mathematics largely being ignored in these countries. Of the discrete mathematics topics specifed above, very few are part of the standard curriculum in most of the countries we are familiar with. Through personal communication, it appears that:
Combinatorics is included in the secondary curriculum of several countries, including Spain, US, England, Germany, Hungary, Brazil, Israel.
Connecting recursive patterns and sequences with algebraic formulas is taught, to some degree, in Spain, Germany, and the United States.
• Graph theory is included in Italy and some isolated state curriculum in the US. In England, students focusing on mathematics can take a special track in which they have extensive exposure to discrete mathematics. In many countries, it appears that the opportunities for dealing with discrete mathematics in schools, especially when it goes beyond combinatorics, are often only seen on the level of optional recreational mathematics (Colipan & Liendo, 2022; Gravier & OuvrierBufet, 2022; Greefrath et al., 2022). We wonder if one contributing issue for the lack of discrete topics taught in the schools may be that the term discrete mathematics’ is not well understood. Perhaps, each of the discrete topics mentioned above should be considered individually. For example, fair division algorithms and economic game theory are almost self-explanatory. While combinatorics sounds complicated, counting is clearly important. Instead of using the termgraph theory’ which may be misleading to some, we could talk about vertex-edge graphs or networks, which most people are familiar with. Iteration can be described in terms of simple recursive situations, such as repeatedly folding a piece of paper or compounding of interest, along with its accessibility through the use of spreadsheets. Therefore, it might be more productive to talk about the individual discrete topics than the discipline as a whole. This is also intended to describe the mentioned topic areas of discrete mathematics for school more clearly once again. We therefore shortly go into a little more detail on the most common discrete topics and how they may support mathematical competencies.



1.3  Potential benefits of discrete mathematics topics for mathematics education
We see the potential benefits of teaching discrete mathematical topics in three broad areas, and some of these benefits have been highlighted in existing literature. The first potential benefit is offered by the content, which is accessible and offers interesting and relevant topics for teaching and learning (Anderson et al., 2004; DeBellis & Rosenstein, 2004).
Potential benefits of discrete mathematics topics for mathematics education
The second potential benefit is the learning of mathematics and the acquisition of general mathematical competencies (Coenen et al., 2018; Vorhölter et al., 2019) including afect (Goldin, 2018) that influences the learning of mathematics and the third potential is the relevance of discrete mathematics for living in the modern world (Hart & Martin, 2018; Rosenstein, 2007). In this section, we will discuss general benefts and in Sect. 6, the benefts resulting from specifc discrete mathematics topics.
Accessible topics for teaching and learning As early as the end of the 1980s, there were calls to integrate discrete mathematics into teaching, not only in higher education but also in schools (Dolgos, 1990). Advocates of discrete mathematics have noted that problems in discrete mathematics are relatively accessible, in the sense that a student may be able to understand what a problem is asking or can explore a situation without needing a lot of prior mathematical experience (Anderson et al., 2004; Devaney, 2018; Ferrarello & Mammana, 2018; Rosenstein, 2007). This is often because the problems themselves do not require knowledge of technical definitions or specific mathematical knowledge, and students can exemplify and explore objects.
The discrete nature of the objects would seem to lend itself to this accessibility. This accessibility of discrete mathematics is something that seems to be agreed upon by many mathematicians, mathematics educators and mathematics education researchers, and it is often used as an argument for the importance of the inclusion of discrete mathematics in curriculum (Anderson et al., 2004; Burghes, 1995; Dolgos, 1990; Hart & Martin, 2018). While we note that this is an aspect of discrete mathematics that would benefit from more systematic study and research, this accessibility has come through in some research studies, but not as an explicit focus of the study. Some of the research on the teaching and learning of discrete mathematics with younger students highlights not only the accessibility of topics but that this accessibility can help students make sense of the current curriculum.
 For example, iterative problems and difference equations can help students learn algebra (Amit & Neria, 2008; Blanton & Kaput, 2005; Carraher et al., 2008; Radford, 2008; Rivera & Becker, 2008; Sandefur et al., 2018; Steele, 2008; Yeap & Kaur, 2008). Even very young students can reason about combinatorial problems in meaningful ways (de Beer et al., 2015; Maher et al., 2011). English (1991, 1993) reports on young children’s strategies as they engage with combinatorial problems. As another example, students can naturally `invent’ graph theory to solve a problem (Ferrarello & Mammana, 2018; Greefrath et al., 2022; van den Heuvel & Krabbendam, 1991). We can only wonder how much better would the students’ work be if they already knew some graph theory or had previous experience with counting or recursive problems?

For many topics in discrete mathematics, students ranging from young children to undergraduate students can be posed similar questions and have a reasonable chance at investigating the problem at their different levels. This results in self differentiating tasks that allow individual approaches to the problem (Ostkirchen & Greefrath, 2022). For example, how many 2-color towers can I make of height 5, can be extended to more complicated problems for older students by increasing the number of colors and the height of the tower. Maher et al. (2011) describe the use of combinatorial problems in such contexts among students in a longitudinal study. Students as young as the third grade can investigate recursive structures they build with toothpicks and stickers while high school students can develop recursive models of bacteria growth and the spread of epidemics, which is a simplified version of models studied by epidemiologists (Radford, 2008; Sandefur & Manaster, 2022; Yeap & Kaur, 2008).
https://link.springer.com/article/10.1007/s11858-022-01399-7
Chapter 2:
Write up on Discrete Mathematics Logic, Tree, and Lambda Equations could be utilized in Highschool Computer Science Curriculum
Discrete Mathematics
2.1
The key aspect to rooted trees — which is both their greatest advantage and greatest limitation — is that every node has one and only one path to the root. This behavior is inherited from free trees: as we noted, every node has only one path to every other.
Trees have a myriad of applications. Think of the files and folders on your hard drive: at the top is the root of the filesystem (perhaps “/” on Linux/Mac or “C:\\” on Windows) and underneath that are named folders. Each folder can contain files as well as other named folders, and so on down the hierarchy. The result is that each file has one, and only one, distinct path to it from the top of the filesystem. The file can be stored, and later retrieved, in exactly one way.
An “org chart” is like this: the CEO is at the top, then underneath her are the VP’s, the Directors, the Managers, and finally the rank-and-file employees. So is a military organization: the Commander in Chief directs generals, who command colonels, who command majors, who command captains, who command lieutenants, who command sergeants, who command privates.
The human body is even a rooted tree of sorts: it contains skeletal, cardiovascular, digestive, and other systems, each of which is comprised of organs, then tissues, then cells, molecules, and atoms. In fact, anything that has this sort of part-whole containment hierarchy is just asking to be represented as a tree.
In computer programming, the applications are too numerous to name. Compilers scan code and build a “parse tree” of its underlying meaning. HTML is a way of structuring plain text into a tree-like hierarchy of displayable elements. AI chess programs build trees representing their possible future moves and their opponent’s probable responses, in order to “see many moves ahead” and evaluate their best options. Object-oriented designs involve “inheritance hierarchies” of classes, each one specialized from a specific other. Etc. Other than a simple sequence (like an array), trees are probably the most common data structure in all of computer science.
Rooted tree terminology
root.
Figure 2-1

in our example. Note that unlike trees in the real world, computer science trees have their root at the top and grow down. Every tree has a root except the empty tree, which is the “tree” that has no nodes at all in it. (It’s kind of weird thinking of “nothing” as a tree, but it’s kind of like the empty set ∅∅, which is still a set.)
parent.
Every node except the root has one parent: the node immediately above it. D’s parent is C, C’s parent is B, F’s parent is A, and A has no parent.
child.
Some nodes have children, which are nodes connected directly below it. A’s children are F and B, C’s are D and E, B’s only child is C, and E has no children.
sibling.
A node with the same parent. E’s sibling is D, B’s is F, and none of the other nodes have siblings.
ancestor.
Your parent, grandparent, great-grandparent, etc., all the way back to the root. B’s only ancestor is A, while E’s ancestors are C, B, and A. Note that F is not C’s ancestor, even though it’s above it on the diagram: there’s no connection from C to F, except back through the root (which doesn’t count).
descendent.
Your children, grandchildren, great-grandchildren, etc., all the way the leaves. B’s descendents are C, D and E, while A’s are F, B, C, D, and E.
leaf.
A node with no children. F, D, and E are leaves. Note that in a (very) small tree, the root could itself be a leaf.
internal node.
Any node that’s not a leaf. A, B, and C are the internal nodes in our example.
depth (of a node).
A node’s depth is the distance (in number of nodes) from it to the root. The root itself has depth zero. In our example, B is of depth 1, E is of depth 3, and A is of depth 0.
height (of a tree).
A rooted tree’s height is the maximum depth of any of its nodes; i.e., the maximum distance from the root to any node. Our example has a height of 3, since the “deepest” nodes are D and E, each with a depth of 3. A tree with just one node is considered to have a height of 0. Bizarrely, but to be consistent, we’ll say that the empty tree has height -1! Strange, but what else could it be? To say it has height 0 seems inconsistent with a one-node tree also having height 0. At any rate, this won’t come up much.
level.
All the nodes with the same depth are considered on the same “level.” B and F are on level 1, and D and E are on level 3. Nodes on the same level are not necessarily siblings. If F had a child named G in the example diagram, then G and C would be on the same level (2), but would not be siblings because they have different parents. (We might call them “cousins” to continue the family analogy.)
subtree.
Finally, much of what gives trees their expressive power is their recursive nature. This means that a tree is made up of other (smaller) trees. Consider our example. It is a tree with a root of A. But the two children of A are each trees in their own right! F itself is a tree with only one node. B and its descendents make another tree with four nodes. We consider these two trees to be subtrees of the original tree. The notion of “root” shifts somewhat as we consider subtrees — A is the root of the original tree, but B is the root of the second subtree. When we consider B’s children, we see that there is yet another subtree, which is rooted at C. And so on. It’s easy to see that any subtree fulfills all the properties of trees, and so everything we’ve said above applies also to it.
Binary trees (BT’s)
The nodes in a rooted tree can have any number of children. There’s a special type of rooted tree, though, called a binary tree which we restrict by simply saying that each node can have at most two children. Furthermore, we’ll label each of these two children as the “left child” and “right child.” (Note that a particular node might well have only a left child, or only a right child, but it’s still important to know which direction that child is.)
The left half of is a binary tree, but the right half is not (C has three children). A larger binary tree (of height 4) is shown in  .
Traversing binary trees
There were two ways of traversing a graph: breadth-first, and depth-first. Curiously, there are three ways of traversing a tree: pre-orderpost-order, and in-order. All three begin at the root, and all three consider each of the root’s children as subtrees. The difference is in the order of visitation.
Figure 2-2
The node at the top of the tree, which is A

Rules in Prolog
Using rules, we can build a relationships among facts.
% Define what a mother, father, and a grandparent is
mother(M, C) :- parent(M, C), female(M).
father(F, C) :- parent(F, C), male(F).
grandparent(X, Z) :- parent(X, Y), parent(Y, Z).
The rule named mother defines the meaning of the mother relationship: If M is a parent of C, and if M is female, then M is a mother.
Similarly, the father rule defines the meaning of the father relationship: If F is a parent of C, and if F is male, then F is a father.
Lastly, the grandparent rule defines the meaning of the grandparent relationship: if X is a parent of Y, and Y is a parent of Z, then X is a grandparent of Z.
Rules in Prolog
Using rules, we can build a relationships among facts.
% Define what a mother, father, and a grandparent is
mother(M, C) :- parent(M, C), female(M).
father(F, C) :- parent(F, C), male(F).
grandparent(X, Z) :- parent(X, Y), parent(Y, Z).
The rule named mother defines the meaning of the mother relationship: If M is a parent of C, and if M is female, then M is a mother.
Similarly, the father rule defines the meaning of the father relationship: If F is a parent of C, and if F is male, then F is a father.
Lastly, the grandparent rule defines the meaning of the grandparent relationship: if X is a parent of Y, and Y is a parent of Z, then X is a grandparent of Z.
                                                Logical Programming
Logic is the discipline concerned with unassailably valid reasoning. By valid we mean that if we start with true statements and from them deduce new statements, following the given logical laws of deduction, we will always end up with new statements that are also true. Logics are thus systems of symbols and rules for manipulating them that have the property that syntactic deduction, involving mechanical manipulation of strings, is always a semantically valid operation, involving the truths of derived statements.
There is not just one logic. There are many. First-order predicate logic with equality is central to everyday mathematics. Propositional logic is equivalent to the language of Boolean expressions as found in conditional expressions in most programming languages. Temporal logics provide ways to reason about what statements remain true in evolving worlds. Dependent type theory is a logic, a richer form of predicate logic, in which propositions are formalized as types and proofs are essentially programs and data structures written in pure, typed, functional programming languages, and so can be type checked for correctness.
Logic is a pillar of computer science. It has been said that logic is to computer science as calculus is to natural science and engineering. As scientists and engineers use everyday mathematics to represent and reason about properties of physical things, so computer scientists use various logical languages to specify and reason about properties of programs, algorithms, the states of programs as they execute, problems to be solved by algorithms, and even about the real world in which software is meant to operate.
Propositional logic , essentially the language of Boolean expressions, is ubiquitous in programming. First-order predicate logic is widely used to reason about many issues that arise in computer science, from the complexity of algorithms to the correctness of programs. Hoare logic is a specialized extension of first-order predicate logic that is especially well suited to specifying how programs must behave and for showing that they do behave according to given logical specifications. Dependent type theory is the logic of modern proof assistant tools, including Lean (as well as Coq and Agda), which we will use in this class.
Dependent type theory and the tools that support it now play important roles in both the development of programming languages and in the production of trustworthy software. In lieu of testing of a given computer program to see if it works correctly on some inputs, one proves that it works correctly on all possible inputs. A tool then checks such proofs for correctness. Mathematicians are increasingly interested in the possibilities for automated checking of complex proofs as well.
At the heart of logic are the concepts of propositions and proofs. A proposition is an expression that we interpret as making a claim that some particular state of affairs holds in some particular domain of discourse (some world of interest). A proof is a compelling argument, in a precise form, that shows beyond any doubt that such a proposition is true. The existence of a valid proof of a proposition shows that it is true. In mathematical logic, the arbiter of truth is the existence of a proof. Proof implies truth; truth demands proof.
This first section of this course, on logic, provides a rigorous survey of forms of propositions and the construction and use of corresponding proofs in the language of predicate logic. You will learn how to write propositions in predicate logic, and how to both construct and use proofs of them in a logical reasoning system called natural deduction. As you will also quickly see, natural deduction is essentially a form of computation in a typed, pure functional programming language in which all programs terminate (there can be no infinite loops). To learn logic in this style is thus also learn a very important style of programming: functional programming. You will learn to compute with propositions and proofs.
 propositions need not be a true statement. Propositions only need to be declarative. Their truth value may be true or false. However, all propositions must have a particular truth value. The statement cannot be both true and false. The statement must be able to be interpreted as true or false.
From the previous definition and examples, propositions are therefore not questions, general statements, demands, or hypotheses. Propositions do not have any variables, quantifiers, or parameters (e.g. the words “some” or “any” typically do not appear). Consider now a few non-examples.
Examples of statements that are not propositions
Do you have a dog?
Let’s go!
Some coffee mug with a mermaid on it.
x+2=3
y=x2−1
Checkpoint
Are each of these propositions?
I am a dolphin.
Supercalifragilisticexpialidocious.
Jupiter is the 5th planet from the sun.
On Thursdays, van Gogh painted landscapes.
11+56∗3−819=9
Solution
Constructing Propositions
An entire proposition is often denoted by a single propositional variable. Propositional variables are typically among p,q,r,s,t,….
Using propositional variables
p:= “The sky is blue”
q:= “The sun rises from the west”
We also denote truth values in particular ways. “True” may be denoted by T. “False” may be denoted by F. When a proposition (or proposition variable) is known to always be true, we can replace it by T. When a proposition (or proposition variable) is known to always be false, we can replace it by F.
Connectives
We can combine propositions (and propositional variables) into compound propositions or propositional formulas. This is akin to compound sentences and other logical connectives in natural language.
In propositional logic, we have 5 main connectives. Each connective has a corresponding meaning in natural language as we will soon see.
Negation: ¬
Conjunction: ∧
Disjunction: ∨
Implication: →
Biconditional: ↔
Logical connectives are like arithmetic operators (+,−,×,÷).
Negation
The negation of a proposition results in a proposition with the opposite truth value. It is akin to adding “not” into a sentence, or starting a sentence with “it is not that case that…”.
Given a proposition p its negation is ¬p and has the following truth values.
Table 1.1 Negation truth table
p
¬p
F
T
T
F
Negation
Let p:= “the sky is blue”.
¬p is “the sky is not blue” or “it is not the case that the sky is blue”.
Let q:= “2+2=5”.
¬q is “2+2≠5.
Notice in these examples that negation does not necessarily make a proposition false. Rather, it makes the proposition have the opposite truth value.
Conjunction
The conjunction of two propositions is the logical “and” of the two propositions. The conjunction of two proposition is only true if both the propositions are individually true, otherwise the conjunction is false.
Given proposition p and q their conjunction is denoted p∧q and has the following truth values.
Table 1.2 Conjunction truth table
p
q
p∧q
F
F
F
F
T
F
T
F
F
T
T
T
Conjunction
Let p be “birds lay eggs” and q be “my eyes are blue”. p∧q is then “birds lay eggs and my eyes are blue”.
Disjunction
The disjunction of two propositions is the “or” of the two propositions. The disjunction is true if at least one of the propositions is individually true, otherwise the disjunction is false.
Given proposition p and q their disjunction is denoted p∨q and has the following truth values.
Table 1.3 Disjunction truth table
p
q
p∨q
F
F
F
F
T
T
T
F
T
T
T
T
Disjunction
Let p be “it is raining” and q be “I am wearing sunglasses”. p∨q is then “it is raining or I am wearing sunglasses”.
Notice that in this previous example, it is may be true that it is both raining and that I am wearing sunglasses. While that may be silly, p∨q is still true! In logic, we only require that at least one of the propositions in a disjunction is true. That means both are allowed to simultaneously be true.
Caution
In natural language, “or” is often interpreted as an exclusive or.
Language “or”
“You can have a cookie or a piece of cake.” Most people assume that this means you can have a cookie or a piece of cake, but not both.
In logic, “or” is not exclusive. You can have a cookie, a piece of cake, or both!
If you want logical exclusive or, we use the symbol ⊕. However, we will not use that in this course.

Checkpoint
What is the truth value of these compound propositions?
“The earth is round and the sky is blue.”
“Dogs or cats make great pets.”
“It is 20∘ Celsius outside and it is snowing.”
“Lemons are purple or grass is green”
Solution

Implication
Implication is one of the most challenging connectives to understand. Yet it is arguably the most important for creating logical arguments).
An implication is a conditional statement. For two propositions p and q, p→q is an implication which is read “if p, then q”. You can also say “p implies q”.
Implication
Let p be “it is raining” and q be “the ground is wet”. p→q can be read “if it is raining, then the ground is wet”.
In an implication p→q, the first proposition p is known as the hypothesisantecedent, or premise. The second proposition q is known as the conclusion or consequence.
Because an implication is a conditional, the truth value of the implication as a whole changes depending on the truth value of the premise. The following truth table summarizes the truth values of an implication.
Table 1.4 Implication truth table
p
q
p→q
F
F
T
F
T
T
T
F
F
T
T
T
An implication can be viewed as an obligation, a contract, or a commitment. The implication p→q is false (the contract is broken; the obligation is unmet) only when p is true and q is false.
There are several important observations from this truth table about logical implication.
If q is true, then p→q is always true.
If p is true and the implication correct (the obligation is upheld), then q can never be false.
“Falsity can imply anything.” If the hypothesis is false, then the implication is always true, regardless of the whether or not the conclusion is true.
Some of these observations may seem counter-intuitive at first. Let us clarify with some examples.
The truth value of implications
Let p be “that animal is a panda bear” and q be “that animal is black and white”. p→q can be read as “if that animal is a panda bear, then that animal is black and white”.
If p is true, and that animal is indeed a panda bear (and the implication is correct),
then it is also black and white. If q is true, and the animal is black and white, it might be a panda bear, but it might also be a cow.
From p→q, we can say that knowing the animal is a panda bear is sufficient to know that the animal is black and white.
Valid implications can be formed from completely unrelated propositions. Moreover, if you begin with a nonsensical hypothesis, then one can construct valid (but equally nonsensical) implications. Falsity implies anything.
Absurd but valid implications
“If pigs can fly, then I am the pope.”
“If 2+2=5, then lemons are purple.”
“If the sun is made of ice, then my father is Morgan Freeman”.
There are many equivalent ways to think about the implication p→q.
If p, then q
p implies q
q when p
q, if p
q whenever p
q follows from p
p is sufficient for q
q is necessary for p
Necessity and Sufficiency
An implication connects propositions by a necessary or sufficient condition. From p→q we get two relations:
p is sufficient for q
q is necessary for p
That is, “if sufficient condition, then necessary condition”.
Necessary and Sufficient
“If all birds have feathers, then a chicken is a type of bird.”
Knowing birds have feathers is sufficient information to conclude that a chicken is a type of bird. If a chicken is a type of bird, then chickens necessarily have feathers.
Fig. 1.1 Being in the inner circle is sufficient for being in the outer circle. Being in the outer circle is necessary for being in the inner circle.
Biconditional
For two propositions p and q, they can be connected by a biconditional as p↔q.
A biconditional is an double implication. A biconditional is true if both propositions have the same truth value. p↔q can be read as “p if and only if q”. A biconditional has the following truth table.
Table 1.5 Biconditional truth table
p
q
p↔q
F
F
T
F
T
F
T
F
F
T
T
T
The biconditional p↔q can be expressed in many ways:
“p if and only if q”
“if p then q, and if q then p”
“p is necessary and sufficient for q”
“p iff q”
Biconditional
Let p be “2 is an even number”. Let q be “4 is an even number”. p↔q is a biconditional and its truth value is true, since both p and q are true.
Tip (thinking in memes)
“The venn diagram is a circle” exactly means that the two subjects form a biconditional.

Checkpoint
What is the truth value of these compound propositions?
“The Earth is flat” → “Pigeons are robots”
“Bats have wings” → “Bats are birds”
“A square is a rectangle” ↔ “A square had four 90∘ interior angles”
“Spinach is green” ↔ “Penguins can fly”
Solution

Propositional Formulas
In the previous section we saw 5 different logical connectives: ¬, ∧, ∨, →, and ↔. Much like arithmetic formulas using addition, multiplication, division, etc., propositional formulas may use several connectives simultaneously.
Remember BEDMAS or PEDMAS? Now we have “PaNCo DIB” (“Panko Dib”)?
For logical connectives we have a similar order of precedence.
Parenthesis: always perform operations on expressions inside parentheses first.
Negation: apply negation to a proposition before binary connectives.
Conjunction: conjunction before disjunction
Disjunction: disjunction after conjunction, but before implication
Implication: → after ∧, ∨
Biconditional: ↔ after ∧,∨,→.
Logical order of precendence
p∨q→¬r   is the same as   (p∨q)→(¬r)
p∨¬q∧r   is the same as   p∨( (¬q) ∧r)
Propositional variables need not be associated with a particular proposition or truth value. A propositional variable could be just that: a variable. Replacing the variables in a propositional formula with a truth value is called a truth assignment.
Definition (truth assignment)
truth assignment is the assignment of a truth value (true or false) to a propositional variable. Equally, it is the replacement of a propositional variable with a truth value.
Much like logical connectives, propositional formulas will result in different truth values depending on the particular truth assignment on its consituent propositional variables. When at least one truth assignment exists so that a formula is true, that formula is said to be satisfiable.
Definition (satisfiable)
A propositional formula is satisfiable if its truth value can be true under some truth assignment. If every possible truth assignment makes the formula have false as its truth value, that formula is said to be unsatisfiable.
In order to determine the truth value of a propositional formula, and to determine if it is satisfiable, we can create a truth table.

Truth Tables
Truth tables are tools for determining the truth values of propositional formulas.
The table is separated into two sets of columns:
The first set of columns represent each proposition (or propositional variable) in a formula.
The second set of columns represents the sub-formulas and formulas whose truth values are to be determined.
There must be one row in the table for every possible combination of truth values of the propositional variables. For example, in a formula with two variables, the possible combinations are: (T,T),(T,F),(F,T),(F,F).
3-variable truth table
Let p,q,r be propositional variables. A truth table for the formula (p∧q)∨r is:
p
q
r
p∧q
(p∧q)∨r
F
F
F
F
F
F
F
T
F
T
F
T
F
F
F
F
T
T
F
T
T
F
F
F
F
T
F
T
F
T
T
T
F
T
T
T
T
T
T
T
Notice that every possible combination of truth values for p, q, and r is contained in this table. Since at least one choice of truth value for p, q, and r results in the formula being true, then this formula is satisfiable.
In a truth table, you begin by filling out the columns corresponding to each propositional variable. These columns represent every possible combination of truth values on those variables. Then, you add columns for each sub-formula, one at a time, building up to the final formula.
Consider the formula p∧q∧r ∨ ¬q∧r→p. By order of precendence, this is equal to ( (p∧q∧r) ∨ ((¬q)∧r) )→p This contains several sub-formulas which we can parse:
¬q
¬q∧r
p∧q
(p∧q)∧r
(p∧q∧r)∨(¬q∧r)
( (p∧q∧r)∨(¬q∧r) )→p
To be as explicit as possible, we could create a truth table with 3 + 6 = 9 columns (3 variables, 6 sub-formulas). But this is excessive. For example, we could directly compute (¬q∧r) and (p∧q∧r). This gives the following truth table.
A large truth table
A truth table for the propositional formula p∧q∧r ∨ ¬q∧r→p.
p
q
r
p∧q∧r
¬q∧r
(p∧q∧r)∨(¬q∧r)
(p∧q∧r)∨(¬q∧r)→p
F
F
F
F
F
F
T
F
F
T
F
T
T
F
F
T
F
F
F
F
T
F
T
T
F
F
F
T
T
F
F
F
F
F
T
T
F
T
F
T
T
T
T
T
F
F
F
F
T
T
T
T
T
F
T
T

Checkpoint
Construct a truth table
Give a truth table for the propositional formula p∧r→q∨¬r
Solution

Implication, Inverse, Converse, and Contrapositive
Now that we have seen propositional formulas and truth tables, let’s revisit implications. This connective has many related conditionals.
Consider the propositional formula p→q. Then, we have:
Converse: q→p
Inverse: ¬p→¬q
Contrapositive: ¬q→¬p
A conditional and its inverse
The proposition “if it is raining, then I wear a jacket” is a conditional statement. Its inverse is “if it is not raining I do not wear a jacket”.
Notice from this previous example than an implication and its inverse are not exactly the same. If the conditional “if it is raining, then I wear a jacket” is true, that is not the same as its inverse. Indeed, you might still wear a jacket even if its not raining. Maybe you’re just cold.
Important
An implication is not equivalent to its converse or inverse. However, it is equivalent to its contrapositive.

 Logical laws are similar to algebraic laws. For example, there is a logical law corresponding to the associative law of addition, a+(b+c)=(a+b)+c.a+(b+c)=(a+b)+c. In fact, associativity of both conjunction and disjunction are among the laws of logic. Notice that with one exception, the laws are paired in such a way that exchanging the symbols ∧,∧, ∨,∨, 1 and 0 for ∨,∨, ∧,∧, 0, and 1, respectively, in any law gives you a second law. For example, p∨0⇔pp∨0⇔p results in p∧1⇔p.p∧1⇔p. This is called a duality principle. For now, think of it as a way of remembering two laws for the price of one. We will leave it to the reader to verify a few of these laws with truth tables. However, the reader should be careful in applying duality to the conditional operator and implication since the dual involves taking the converse. For example, the dual of p∧q⇒pp∧q⇒p is p∨q⇐p,p∨q⇐p, which is usually written p⇒p∨q.p⇒p∨q.
Example : Verification of an Identity Law
The Identity Law can be verified with this truth table. The fact that (p∧1)↔p(p∧1)↔p is a tautology serves as a valid proof.
Table 3.4.13.4.1: Truth table to demonstrate the identity law for conjunction.
pp
11
p∧1p∧1
(p∧1)↔p(p∧1)↔p
00
11
00
11
11
11
11
11
Some of the logical laws in Table 3.4.33.4.3 might be less obvious to you. For any that you are not comfortable with, substitute actual propositions for the logical variables. For example, if pp is “John owns a pet store” and qq is “John likes pets,” the detachment law should make sense.
Table 3.4.23.4.2: Basic Logical Laws – Equivalences
Commutative Laws
p∨q⇔q∨pp∨q⇔q∨p
p∧q⇔q∧pp∧q⇔q∧p
Associative Laws
(p∨q)∨r⇔p∨(q∨r)(p∨q)∨r⇔p∨(q∨r)
(p∧q)∧r⇔p∧(q∧r)(p∧q)∧r⇔p∧(q∧r)
Distributive Laws
p∧(q∨r)⇔(p∧q)∨(p∧r)p∧(q∨r)⇔(p∧q)∨(p∧r)
p∨(q∧r)⇔(p∨q)∧(p∨r)p∨(q∧r)⇔(p∨q)∧(p∨r)
Identity Laws
p∨0⇔pp∨0⇔p
p∧1⇔pp∧1⇔p
Negation Laws
p∧¬p⇔0p∧¬p⇔0
p∨¬p⇔1p∨¬p⇔1
Idempotent Laws
p∨p⇔p∨p⇔
p∧p⇔pp∧p⇔p
Null Laws
p∧0⇔0p∧0⇔0
p∨1⇔1p∨1⇔1
Absorption Laws
p∧(p∨q)⇔pp∧(p∨q)⇔p
p∨(p∧q)⇔pp∨(p∧q)⇔p
DeMorgan’s Laws
¬(p∨q)⇔(¬p)∧(¬q)¬(p∨q)⇔(¬p)∧(¬q)
¬(p∧q)⇔(¬p)∨(¬q)¬(p∧q)⇔(¬p)∨(¬q)
Involution Laws
¬(¬p)⇔p¬(¬p)⇔p
Table 3.4.33.4.3: Basic Logical Laws – Common Implications and Equivalences
Detachment (AKA Modus Ponens)
(p→q)∧p⇒q(p→q)∧p⇒q
Indirect Reasoning (AKA Modus Tollens)
(p→q)∧¬q⇒¬p(p→q)∧¬q⇒¬p
Disjunctive Addition
p⇒(p∨q)p⇒(p∨q)
Conjunctive Simplification
(p∧q)⇒p(p∧q)⇒p and (p∧q)⇒q(p∧q)⇒q
Disjunctive Simplification
(p∨q)∧¬p⇒q(p∨q)∧¬p⇒q and (p∨q)∧¬q⇒p(p∨q)∧¬q⇒p
Chain Rule
(p→q)∧(q→r)⇒(p→r)(p→q)∧(q→r)⇒(p→r)
Conditional Equivalence
p→q⇔¬p∨qp→q⇔¬p∨q
Biconditional Equivalences
(p↔q)⇔(p→q)∧(q→p)⇔(p∧q)∨(¬p∧¬q)(p↔q)⇔(p→q)∧(q→p)⇔(p∧q)∨(¬p∧¬q)
Contrapositive
(p→q)⇔(¬q→¬p)
Lambda
Function terms, called lambda abstractions, are literal expressions that represent mathematical functions. Yes, you can and should now think of functions as being values, too. Function definitions are terms in predicate logic and in functional programming languages. As we will see later on, we can pass functions as arguments to other functions, and receive them as results. Functions that take functions as arguments or that return functions as results are called higher-order functions. We will get to this topic later on.
Consider the simple lambda expression, . It’s a term that represents a function that takes one argument, n, of type, nat. When applied to an actual parameter, or argument, it returns the value of that argument plus one.
Here’s the example in Lean. It first shows that the literal function expression reduces to the value that it represents directly. It then shows that this function can be applied to an actual parameter, i.e., an argument, 5, reducing to the value 6. Third, it shows that a literal function term can be bound to an identifier, here f, and finally that this makes it easier to write code to apply the function to an argument.
#reduce (λ n : ℕ, n + 1)
#eval (λ n : ℕ, n + 1) 5
def f := (λ n : ℕ, n + 1)
#eval f 5
Lambda calculus terms can be viewed as a kind of binary tree. A lambda calculus term consists of:
Variables, which we can think of as leaf nodes holding strings.
Applications, which we can think of as internal nodes.
Lambda abstractions, which we can think of as a special kind of internal node whose left child must be a variable. (Or as a internal node labeled with a variable with exactly one child.)
It’s slightly annoying to have to define a helper procedure add-three just so we can use it as the argument to every. We’re never going to use that procedure again, but we still have to come up with a name for it. We’d like a general way to say “here’s the function I want you to use” without having to give the procedure a name. In other words, we want a general-purpose procedure-generating procedure!
Lambda is the name of a special form that generates procedures. It takes some information about the function you want to create as arguments and it returns the procedure. It’ll be easier to explain the details after you see an example.
(define (add-three-to-each sent)
  (every (lambda (number) (+ number 3)) sent))
> (add-three-to-each ‘(1 9 9 2))
(4 12 12 5)
The first argument to every is, in effect, the same procedure as the one we called add-three earlier, but now we can use it without giving it a name. (Don’t make the mistake of thinking that lambda is the argument to every. The argument is the procedure returned by lambda.)
Perhaps you’re wondering whether “lambda” spells something backward. Actually, it’s the name of the Greek letter L, which looks like this: λ. It would probably be more sensible if lambda were named something like make-procedure, but the name lambda is traditional.[1]
Creating a procedure by using lambda is very much like creating one with define, as we’ve done up to this point, except that we don’t specify a name. When we create a procedure with define, we have to indicate the procedure’s name, the names of its arguments (i.e., the formal parameters), and the expression that it computes (its body). With lambda we still provide the last two of these three components.
As we said, lambda is a special form. This means, as you remember, that its arguments are not evaluated when you invoke it. The first argument is a sentence containing the formal parameters; the second argument is the body. What lambda returns is an unnamed procedure. You can invoke that procedure:
> ((lambda (a b) (+ (* 2 a) b)) 5 6)
16
> ((lambda (wd) (word (last wd) (first wd))) ‘impish)
It’s slightly annoying to have to define a helper procedure add-three just so we can use it as the argument to every. We’re never going to use that procedure again, but we still have to come up with a name for it. We’d like a general way to say “here’s the function I want you to use” without having to give the procedure a name. In other words, we want a general-purpose procedure-generating procedure!
Lambda is the name of a special form that generates procedures. It takes some information about the function you want to create as arguments and it returns the procedure. It’ll be easier to explain the details after you see an example.
(define (add-three-to-each sent)
  (every (lambda (number) (+ number 3)) sent))
> (add-three-to-each ‘(1 9 9 2))
(4 12 12 5)
The first argument to every is, in effect, the same procedure as the one we called add-three earlier, but now we can use it without giving it a name. (Don’t make the mistake of thinking that lambda is the argument to every. The argument is the procedure returned by lambda.)
Perhaps you’re wondering whether “lambda” spells something backward. Actually, it’s the name of the Greek letter L, which looks like this: λ. It would probably be more sensible if lambda were named something like make-procedure, but the name lambda is traditional.[1]
Creating a procedure by using lambda is very much like creating one with define, as we’ve done up to this point, except that we don’t specify a name. When we create a procedure with define, we have to indicate the procedure’s name, the names of its arguments (i.e., the formal parameters), and the expression that it computes (its body). With lambda we still provide the last two of these three components.
As we said, lambda is a special form. This means, as you remember, that its arguments are not evaluated when you invoke it. The first argument is a sentence containing the formal parameters; the second argument is the body. What lambda returns is an unnamed procedure. You can invoke that procedure:
> ((lambda (a b) (+ (* 2 a) b)) 5 6)
> ((lambda (wd) (word (last wd) (first wd))) ‘impish)
Searching algorithms are used to find a specified element within a data structure. For example, a searching algorithm could be used to find the name “Alan Turing” in an array of names. Numerous different searching algorithms exist, each of which is suited to a particular data structure of format of data. Different searching algorithms are used depending on each individual scenario. Binary Search The binary search algorithm can only be applied on sorted data and works by finding the middle element in a list of data before deciding which side of the data the desired element is to be found in. The unwanted half of the data is then discarded and the process repeated until the desired element is found or until it is known that the desired element doesn’t exist in the data.
 Pseudocode for binary search is shown below. A ​ = Array of data x ​ = Desired element low = 0  high = A.length -1  while low <= high:  mid = (low + high) / 2  if A[mid] == x:  return mid  else if
A[mid] > x:  high = mid -1  else:  low = mid + 1  endif  endwhile  return
 “Not found in data”  With each iteration of binary search, half of the input data is discarded, making the algorithm very efficient. In fact, the time complexity of binary search is O(log n)​ .
www.pmt.education Example 1 Find the location of “Dylan” in the data below.
 0  Alice
1 Bob
2  Charlie
3 Dylan
5 Ellie
 6  Franz
Gabbie Hugo Ingrid To start with, our values for high and low and 8 and 0 respectively. The first step is to find the middle position.
 In this case, it’s (0 + 8) / 2 = ​4​ . Next we inspect the data in position 4: Ellie. This is higher than the desired data and so we discard elements 4-8, setting high as 3. 0  Alice 1  2  3  4  5  6  7  8  Bob Charlie Dylan Ellie Franz Gabbie Hugo Ingrid Again, we calculate the value of the middle position.
This time it’s (0 + 3) / 2 = ​2 (We have to round to the nearest whole number). Inspecting the data at this position we find Charlie, which is lower than the. This breaks the condition of low being less than or equal to high, and so breaks the while loop in the pseudocode. The algorithm then returns “Not found in data” before terminating. Example 3 Look at how the algorithm finds the letter R in the first 20 characters of the alphabet. It’s clear from this example that the algorithm halves the remaining data to be searched with each iteration, gradually reducing the size of the problem to be solved. A B C D E F G H I J K L M N O P Q R S T  A B C D E F G H I J K L M N O P Q R S T  A B C D E F G H I J K L M N O P Q R S T  A B C D E F G H I J K L M N O P Q R S T 
.pmt.education
Linear Search Linear search is the most basic searching algorithm. You can think of it as going along a bookshelf one by one until you come across the book you’re looking for. Sometimes the algorithm gets lucky and finds the desired element almost immediately, while in other situations, the algorithm is incredibly inefficient.
 It’s time complexity is O(n)​ . There’s a lot of pot luck involved with linear search, but it’s incredibly easy to implement. Unlike binary search, linear search doesn’t require the data to be sorted. A ​ = Array of data x ​ = Desired element i = 0  while i < A.length:  if A[i] == x:  return i  else:  i = i + 1  endif  endwhile  return “Not found in data”  Example 1
 Find the position of Apple in the data. 0  Banana
1  Orange
2  3  4  Apple
Kiwi First we inspect position 0,
 and find Banana.
Not the element we’re after.
0  1  2  3  Mango
 4  Banana
Orange
 Apple
 Kiwi
Next we inspect position 1, finding Orange.
Again, not what we’re looking for. 0  1  2  3 
www.pmt.education Example 2 Look at how this algorithm finds the letter R in the first 20 characters of the alphabet and compare it to binary search above. It’s clear from this example that the algorithm is much less efficient than binary search, particularly when the search value is towards the end of the data to be searched.
A B C D E F G H I J K L M N O P Q R S T
 A B C D E F G H I J K L M N O P Q R S T
A B C D E F G H I J K L M N O P Q R S T
A B C D E F G H I J K L M N O P Q R S T
A B C D E F G H I J K L M N O P Q R S T
A B C D E F G H I J K L M N O P Q R S T
 A B C D E F G H I J K L M N O P Q R S T
 A B C D E F G H I J K L M N O P Q R S T
 A B C D E F G H I J K L M N O P Q R S T
A B C D E F G H I J K L M N O P Q R S T
A B C D E F G H I J K L M N O P Q R S T
 A B C D E F G H I J K L M N O P Q R S T
 A B C D E F G H I J K L M N O P Q R S T
A B C D E F G H I J K L M N O P Q R S T
 A B C D E F G H I J K L M N O P Q R S T
 A B C D E F G H I J K L M N O P Q R S T
A B C D E F G H I J K L M N O P Q R S T A B C D E F G H I J K L M N O P Q
Conclusion:
While we note that this is an aspect of discrete mathematics that would benefit from more systematic study and research, this accessibility has come through in some research studies, but not as an explicit focus of the study. Some of the research on the teaching and learning of discrete mathematics with younger students highlights not only the accessibility of topics but that this accessibility can help students make sense of the current curriculum. For example, iterative problems and difference equations can help students learn algebra (Amit & Neria, 2008; Blanton & Kaput, 2005; Carraher et al., 2008; Radford, 2008; Rivera & Becker, 2008; Sandefur et al., 2018; Steele, 2008; Yeap & Kaur, 2008). Even very young students can reason about combinatorial problems in meaningful ways (de Beer et al., 2015; Maher et al., 2011). English (1991, 1993) reports on young children’s strategies as they engage with combinatorial problems. As another example, students can naturally `invent’ graph theory to solve a problem (Ferrarello & Mammana, 2018; Greefrath et al., 2022; van den Heuvel & Krabbendam, 1991).
We can only wonder how much better would the students’ work be if they already knew some graph theory or had previous experience with counting or recursive problems? For many topics in discrete mathematics, students ranging from young children to undergraduate students can be posed similar questions and have a reasonable chance at Investigating the problem at their different levels.
This results in self differentiating tasks that allow individual approaches to the problem (Ostkirchen & Greefrath, 2022). For example, how many 2-color towers can I make of height 5, can be extended to more complicated problems for older students by increasing the number of colors and the height of the tower. Maher et al. (2011) describe the use of combinatorial problems in such contexts among students in a longitudinal study. Students as young as the third grade can investigate recursive structures they build with toothpicks and stickers while high school students can develop recursive models of bacteria growth and the spread of epidemics, which is a simplified version of models studied by epidemiologists (Radford, 2008; Sandefur & Manaster, 2022; Yeap & Kaur, 2008).
This highlights what we mean by accessibility – students can have access to mathematical topics and ideas regardless of background and prerequisite knowledge (DeBellis & Rosenstein, 2004; Ferrarello & Mammana, 2018). Although we have several examples of empirical research that implicitly demonstrates the accessibility of discrete mathematics and the value of this accessibility, we note that there is also more work to be done. There is a need for the field to focus on the issue of accessibility more systematically and explicitly.
Part II
Computational Thinking at first glance – What, why and how
1.1. Definition of Computational Thinking (CT) Is that a fact? There is still no universally accepted definition of “Computational Thinking”. The concept of computational thinking (CT) was first introduced by an educationalist Seymour Papert in 1967 talking about LOGO, the programming language he developed at MIT (Massachusetts Institute of Technology) to teach programming to children. He was convinced that the use of computers could foster formal thinking in children and, in particular, could allow children to autonomously “construct” their learning and thinking. The concept of CT was then revitalized in 2006 by a computer scientist Jeannette M. Wing who, in the article “Computational Thinking”, argued that it addresses the conundrum of machine intelligence by asking what machines do better than man and what man does better than machines. Wing argues that computational thinking is not simply a procedural coding activity, but is a basic conceptual skill that, along with reading, writing and arithmetic, should be taught to all children. It appears that computational thinking purports to be critical thinking in evaluating situations and an advanced problem-solving ability using computerized tools. If computer science is the science of what can be computerized and how to computerize it, however, computational thinking is not a skill unique to computer scientists. It allows problems to be solved, a system to be designed and human behaviour to be understood in everyday life, in an alternative way, through the fundamental concepts of information technology.
Key Skills for Computational Thinking There are four key skills in computational thinking:
 1 Decomposition
2 Pattern Recognition
3 Pattern Abstraction
4 Algorithm Design
 1) Decomposition Breaking down big and difficult problems into something much simpler. Often big problems are just many little problems put together. Decomposition is an important life skill to be relied on in the future when students and adults need to take on larger tasks. Students will learn ways to delegate group projects and build time management skills
 2) Pattern Recognition Sometimes when a problem is made up of many small bits, you will notice that these bits have something in common. If they do not, they could, however, have strong similarities with the pieces of another problem that has already been solved. If you are able to find these regularities, it will become a lot easier to identify the individual pieces: pattern recognition is simply looking for patterns in the puzzles and determining if any of the problems or solutions we encountered in the past may apply to a present situation. May, what we learned in the past, help us sort out the actual problem? If you have ever built a piece of IKEA furniture, you will understand the importance of patterns. While assembling an IKEA drawer unit, it is likely to take you much longer to assemble the first drawer than the fourth or fifth. When we repeat steps in our assembling process we learn how to solve the instructions more quickly and learn from our mistakes. The painstaking process of assembling that first part teaches us the skills to perform the process more efficiently in the future.
3) Pattern Abstraction Once you have located a pattern, it is possible to abstract (ignore) the details that differentiate the various things and use general techniques for finding solutions that work for more than one problem. Identifying the crucial information in a problem and disregarding the irrelevant information is one of the hardest parts of computational learning.
 4) Algorithm design When the solution is ready, it is possible to write it down so it can be executed step by step. This makes easier to obtain the expected results. Algorithm design is setting out the steps and rules needed to follow in order to achieve the same desired outcome every time.
Concepts of computational thinking Computational thinking is a cognitive or thought process involving logical reasoning by which problems are solved and artefacts, procedures and systems are better understood. It embraces:
● the ability to think algorithmically;
● the ability to think in terms of decomposition;
 ● the ability to think in generalisations, identifying and making use of patterns;
 ● the ability to think in abstractions, choosing good representations; and
 ● the ability to think in terms of evaluation. Computational thinking skills enable pupils to access parts of the Computing subject content. Importantly, they relate to thinking skills and problem solving across the whole curriculum and through life in general. Computational thinking can be applied to a wide range of artefacts including: systems, processes, objects, algorithms, problems, solutions, abstractions, and collections of data or information. In the following discussion of concepts, artefact refers to any of these. Logical reasoning Logical reasoning enables pupils to make sense of things by analysing and checking facts through thinking clearly and precisely. It allows pupils to draw on their own knowledge and internal models to make and verify predictions and to draw conclusions. It is used extensively by pupils when they test, debug, and correct algorithms. Logical reasoning is the novel application of the other computational thinking concepts to solve problems.
Design and technology pupils, designing a model of a truck, choose materials for different elements of a project. They are employing generalisation when they recognise that the properties of a material used in one situation make it suitable to use in another completely different context. Being able to divide the new project into different parts, requiring different materials, is an example of decomposition. The pupil is using logical reasoning to design a truck. Pupils use logical reasoning when learning about gravity using a weighted string suspended from the lid of a glass jar. Before tilting the jar, pupils can make predictions about the behaviour of the weighted string. They can then evaluate the results of their tests. They may be able to generalise the behaviour to other situations such as a crane. The novel use in understanding a property of gravity is logical reasoning. Logical reasoning is key in allowing pupils to debug their code. They can work with peers to evaluate each other’s code, to isolate bugs, and to suggest fixes. During this process, they may have opportunities to employ abstraction, evaluation, and algorithmic design. The novel use in correcting mistakes in code requires logical reasoning.
 Abstraction in the Computer Science Classroom
Although we have presented relevant characterisations of abstraction, it is still far from clear how an abstraction-oriented perspective could become part of the pedagogical practice. Already in the late 1990s, as a revision of the spreading object-first orientaTable 1
 PGK Hierarchy levels from Perrenet et al. (2005) and their mapping to K-5 settings according to Waite et al. (2018). PGK Level Definition K-5 name K-5 question Problem Algorithm perceived as a problem solving strategy problem “What is needed” Object (algorithm) Algorithm understood independently of any specific implementation design “What it should do” Program Algorithmic grasp of the program code “How it is done” Execution Focus on individual runs with specific inputs running the code “What it does” 630 C. Mirolo et al. tion, Machanick (1998) endorsed an abstraction-first instructional approach where the implementation of abstract data types is delayed as much as possible in order to stress an abstract view of the models.
Kramer remarked that abstraction per se is not the subject of any computing course, but that all computing courses “rely on or utilize abstraction to explain, model, specify, reason or solve problems,” so confirming that “abstraction is an essential aspect of computing, but that it must be taught indirectly through other topics” (Kramer, 2007, p. 41). In line with Kramer’s remark, Hazzan (2008) discussed abstraction as a soft idea, “that can be neither rigidly nor formally defined, nor is it possible to guide students as to its precise application.” And although “it is not a trivial matter,” like other soft ideas, abstraction should be taught in a computer science curriculum.
Then, a small number of educators have provided guidelines to teach abstraction at different instruction levels. Hence, this section briefly explores their approaches to foster and assess abstraction skills. 5.1. Teaching to Trigger Abstraction in Computer Science Often instructors aim to develop students’ abstraction skills indirectly, by devising particular learning trajectories that are supposed to foster higher-level thinking and require students to use abstraction to succeed. In a program development project, for example, they could assign refactoring tasks in which learners are asked to look for recurrent patterns of code and to re-organise the code by introducing meaningful procedural and/or data abstractions with the purpose of making the whole program easier to read, debug and modify. In the following paragraphs we will outline a selection of representative approaches to (an implicit) abstraction-oriented instruction. Pattern-oriented instruction. This approach has the aim of improving students’ competencies in algorithmic problem solving (Muller and Haberman, 2008).
An algorithm is indeed seen by these authors as a combination of plan patterns in Soloway’s sense (Soloway, 1986), resulting via sequencing, nesting or merging plans from a repertory of basic algorithmic patterns specifically designed for pedagogical purposes. In Muller and Haberman’s scenario, abstraction plays a crucial role in pattern recognition, chunking, and problem structure identification. Their approach relies on having an appropriate pattern repository, as well as on presenting carefully selected problems of gradually increasing difficulty; teachers should then discuss and compare different solutions to a given problem in terms of pattern composition. Additional guidelines for pattern-oriented instruction include:
(1) patterns should be abstracted from related examples or by generalising a simpler problem,
(2) patterns should be revisited in different contexts, in order to make the identification of common problem features easier, and (3) similarities, differences, and also possible misuses of patterns should be considered. According to Muller and Haberman, comparative studies appear to show that novices exposed to this approach develop enhanced problem solving abilities. Multiple representations perspective. Dealing with multiple representations of a given phenomenon can play a key role in the development of abstract concepts. Ac- cording to Ainsworth (2008), in particular, in order “to construct a deeper understanding of a domain,” if the learners “fail to relate representations, then processes like abstraction cannot occur. Moreover, although learners find it difficult to relate different forms of representations, if the representations are too similar, then abstraction is also unlikely to occur.” She then recommends that teachers should foster abstraction over multiple representations “by providing focused help and support on how to relate representations and giving learners sufficient time to master this process.” In this respect, Gautam et al. (2020) have recently proposed an interesting interdisciplinary approach to integrate science (namely, chemistry) and computational thinking in the curriculum. While abstraction is usually “presented as hierarchical” in terms of (i) extracting important features and ignoring unimportant ones, and (ii) finding commonalities across contexts, in their standpoint “abstraction in science” as well as in computing “requires students to move laterally across different representations of the concepts or actions.” In the reported study, the micro-level process of photosynthesis was modeled by a code snippet, and by discussing commonalities and differences between, e.g., a whiteboard and the code representation of the implied chemical reaction, “the instructor pushed students towards higher-level abstract thinking.” Moreover, they suggest to allow for friction emerging when the students explore different representations, in that it encourages to consider alternative views and “negotiate the elements with one another.” According to the authors, this approach “created meaningful accounts of science phenomenon and the science provided access to how computation embeds ideas.” Exploration of artefacts.
 A more recent pedagogical trend in programming education attempts to trigger abstraction through activities inspired by the use-modify-create framework. The idea is that the understanding of artefacts such as programs would gradually progress through three major stages, corresponding to
exploration via passive use (as a consumer),
(ii) experimentation of the internal machinery by modifying some features, and finally
(iii) creation of new, original artefacts to achieve specific goals. While discussing the use-modify-create approach, Lee et al. (2014) observe that abstraction, as well as other computational thinking abilities, are “not explicitly taught but rather [develop] through one’s impetus to create;” nevertheless, in this progression the abilities to modify and, later, to create imply the enhancement of learner’s abstraction skills.
This is a course on discrete mathematics as used in Computer Science. It’s only a one-semester course, so there are a lot of topics that it doesn’t cover or doesn’t cover in much depth. But the hope is that this will give you a foundation of skills that you can build on as you need to, and particularly to give you a bit of mathematical maturity—the basic understanding of what mathematics is and how mathematical definitions and proofs work.
So why do I need to learn all this nasty mathematics?
 Why you should know about mathematics, if you are interested in Computer Science: or, more specifically, why you should take CS202 or a comparable course:
• Computation is something that you can’t see and can’t touch, and yet (thanks to the efforts of generations of hardware engineers) it obeys strict, well-defined rules with astonishing accuracy over long periods of time.
• Computations are too big for you to comprehend all at once. Imagine printing out an execution trace that showed every operation a typical $500 desktop computer executed in one (1) second.
If you could read one operation per second, for eight hours every day, you would die of old age before you got halfway through. Now imagine letting the computer run overnight. So in order to understand computations, we need a language that allows us to reason about things we can’t see and can’t touch, that are too big
 
for us to understand, but that nonetheless follow strict, simple, well-defined rules. We’d like our reasoning to be consistent: any two people using the language should (barring errors) obtain the same conclusions from the same information. Computer scientists are good at inventing languages, so we could invent a new one for this particular purpose, but we don’t have to: the exact same problem has been vexing philosophers, theologians, and mathematicians for much longer than computers have been around, and they’ve had a lot of time to think about how to make such a language work.
 Philosophers and theologians are still working on the consistency part, but mathematicians (mostly) got it in the early 20th-century. Because the first virtue of a computer scientist is laziness, we are going to steal their code.
But isn’t math hard? Yes and no. The human brain is not really designed to do formal mathematical reasoning, which is why most mathematics was invented in the last few centuries and why even apparently simple things like learning how to count or add require years of training, usually done at an early age so the pain will be forgotten later. But mathematical reasoning is very close to legal reasoning, which we do seem to be very good at.1 There is very little structural difference between the two sentences: 1. If x is in S, then x + 1 is in S. 2. If x is of royal blood, then x’s child is of royal blood.
 But because the first is about boring numbers and the second is about fascinating social relationships and rules, most people have a much easier time deducing that to show somebody is royal we need to start with some known royal and follow a chain of descendants than they have deducing that to show that some number is in the set S we need to start with some known element of S and show that repeatedly adding 1 gets us to the number we want. And yet to a logician these are the same processes of reasoning. So why is statement
(1) trickier to think about than statement
 (2)? Part of the difference is familiarity—we are all taught from an early age what it means to be somebody’s child, to take on a particular social role, etc. For mathematical concepts, this familiarity comes with exposure and practice, just as with learning any other language. But part of the difference is that 1For a description of some classic experiments that demonstrate this, see http://
Foundations and logic Why: This is the assembly language of mathematics—the stuff at the bottom that everything else compiles to.
 • Propositional logic.
 • Predicate logic.
 • Axioms, theories, and models.
• Proofs.
• Induction and recursion
 
English Language Arts
To Critical Thinking, the critical person is something like a critical consumer of information; he or she is driven to seek reasons and evidence. Part of this is a matter of mastering certain skills of thought: learning to diagnose invalid forms of argument, knowing how to make and defend distinctions, and so on. Much of the literature in this area, especially early on, seemed to be devoted to lists and taxonomies of what a “critical thinker” should know and be able to do (Ennis 1962, 1980). More recently, however, various authors in this tradition have come to recognize that teaching content and skills is of minor import if learners do not also develop the dispositions or inclination to look at the world through a critical lens. By this, Critical Thinking means that the critical person has not only the capacity (the skills) to seek reasons, truth, and evidence, but also that he or she has the drive (disposition) to seek them. For instance, Ennis claims that a critical person not only should seek reasons and try to be well informed, but that he or she should have a tendency to do such things (Ennis 1987, 1996). Siegel criticizes Ennis somewhat for seeing dispositions simply as what animates the skills of critical thinking, because this fails to distinguish sufficiently the critical thinker from critical thinking. For Siegel, a cluster of dispositions (the “critical spirit”) is more like a deep-seated character trait, something like Scheffler’s notion of “a love of truth and a contempt of lying” (Siegel 1988; Scheffler 1991). It is part of critical thinking itself. Paul also stresses this distinction between skills and dispositions in his distinction between “weak-sense” and “strong-sense” critical thinking. For Paul, the “weak-sense” means that one has learned the skills and can demonstrate them when asked to do so; the “strong-sense” means that one has incorporated these skills into a way of living in which one’s own assumptions are re-examined and questioned as well. According to Paul, a critical thinker in the “strong sense” has a passionate drive for “clarity, accuracy, and fairmindedness” (Paul 1983, 23; see also Paul 1994). This dispositional view of critical thinking has real advantages over the skills-only view. But in important respects it is still limited. First, it is not clear exactly what is entailed by making such dispositions part of critical thinking. In our view it not only broadens the notion of criticality beyond mere “logicality,” but it necessarily requires a greater attention to institutional contexts and social relations than Critical Thinking authors have provided. Both the skills-based view and the skills-plus-dispositions view are still focused on the individual person. But it is only in the context
A second theme in the Critical Thinking literature has been the extent to which critical thinking can be characterized as a set of generalized abilities and dispositions, as opposed to content-specific abilities and dispositions that are learned and expressed differently in different areas of investigation. Can a general “Critical Thinking” course develop abilities and dispositions that will then be applied in any of a range of fields; or should such material be presented specifically in connection to the questions and content of particular fields of study? Is a scientist who is a critical thinker doing the same things as an historian who is a critical thinker? When each evaluates “good evidence,” are they truly thinking about problems in similar ways, or are the differences in interpretation and application dominant? This debate has set John McPeck, the chief advocate of content-specificity, in opposition to a number of other theorists in this area (Norris 1992; Talaska 1992). This issue relates not only to the question of how we might teach critical thinking, but also to how and whether one can test for a general facility in critical thinking (Ennis 1984). A third debate has addressed the question of the degree to which the standards of critical thinking, and the conception of rationality that underlies them, are culturally biased in favor of a particular masculine and/or Western mode of thinking, one that implicitly devalues other “ways of knowing.” Theories of education that stress the primary importance of logic, conceptual clarity, and rigorous adherence to scientific evidence have been challenged by various advocates of cultural and gender diversity who emphasize respect for alternative world views and styles of reasoning. Partly in response to such criticisms, Richard Paul has developed a conception of critical thinking that regards “sociocentrism” as itself a sign of flawed thinking (Paul 1994). Paul believes that, because critical thinking allows us to overcome the sway of our egocentric and sociocentric beliefs, it is “essential to our role as moral agents and as potential shapers of our own nature and destiny”(Paul 1990, 67). For Paul, and for some other Critical Thinking authors as well, part of the method of critical thinking involves fostering dialogue, in which thinking from the perspective of others is also relevant to the assessment of truth claims; a too-hasty imposition of one’s own standards of evidence might result not only in a premature rejection of credible alternative points of view, but might also have the effect of silencing the voices of those who (in the present context) need to be encouraged as much as possible to speak for themselves. In this respect, we see Paul introducing into the very definition of critical thinking some of the sorts of social and contextual factors that Critical Pedagogy writers have emphasized.
http://mediaeducation.org.mt/wp-content/uploads/2013/05/Critical-Thinking-and-Critical-Pedagogy.pdf
CHAPTER TWO BASIC CONCEPTS OF LOGIC
Chapter Overview Logic, as field of study, may be defined as the organized body of knowledge, or science that evaluates arguments. The aim of logic is to develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own. Argument is a systematic combination of two or more statements, which are classified as a premise or premises and conclusion. A premise refers to the statement, which is claimed to provide a logical support or evidence to the main point of the argument, which h known as conclusion. A conclusion is a statement, which is claimed to follow from the alleged evidence. Depending on the logical and real ability of the premise(s) to support the conclusion, an argument can be either a good argument or a bad argument. However, unlike all kinds of passages, including those that resemble arguments, all arguments purport to prove something. Arguments can generally be divided into deductive and inductive arguments. A deductive argument is an argument in which the premises are claimed to support the conclusion in such a way that it is impossible for the premises to be true and the conclusion false. On the other hand, an inductive argument is an argument in which the premises are claimed to support the conclusion in such a way that it is improbable that the premises be true and the conclusion false. The deductiveness or inductiveness of an argument can be determined by the particular indicator word it might use, the actual strength of the inferential relationship between its component statements, and its argumentative form or structure. A deductive argument can be evaluated by its validity and soundness. Likewise, an inductive argument can be evaluated by its strength and cogency. Depending on its actually ability to successfully maintain its inferential claim, a deductive argument can be either valid or invalid. That is, if the premise(s) of a certain deductive argument actually support its conclusion in such a way that it is impossible for the premises to be true and the conclusion false, then that particular deductive argument is valid. If, however, its premise(s) actually support its conclusion in such a
way that it is possible for the premises to be true and the conclusion false, then that particular deductive argument is invalid. Similarly, an inductive argument can be either strong or weak, depending on its actually ability to successfully maintain its inferential claim. That is, if the premise(s) of a certain inductive argument actually support its conclusion in such a way that it is improbable for the premises to be true and the conclusion false, then that particular inductive argument is strong. If, however, its premise(s) actually support its conclusion in such a way that it is probable for the premises to be true and the conclusion false, then that particular inductive argument is weak. Furthermore, depending on its actually ability to successfully maintain its inferential claim as well as its factual claim, a deductive argument can be either sound or unsound. That is, if a deductive argument actually maintained its inferential claim, (i.e., if it is valid), and its factual claim, (i.e., if all of its premises are true), then that particular deductive argument will be a sound argument. However, if it fails to maintain either of its claims, it will be an unsound argument. Likewise, depending on its actually ability to successfully maintain its inferential claim as well as its factual claim, an inductive argument can be either cogent or uncogent. That is, if an inductive argument actually maintained its inferential claim, (i.e., if it is strong), and its factual claim, (i.e., if all of its premises are probably true), then that particular inductive argument will be a cogent argument. However, if it fails to maintain either of its claims, it will be an uncogent argument. In this chapter, we will discuss logic and its basic concepts, the techniques of distinguishing arguments from non-argumentative passages, and the types of arguments.
Chapter Objectives: Dear learners, after the successful completion of this chapter, you will be able to:
Ø Understand the meaning and basic concepts of logic;
Ø Understand the meaning, components, and types of arguments; and
Ø Recognize the major techniques of recognizing and evaluating arguments
Lesson 1: Basic Concepts of Logic: Arguments, Premises and Conclusions Lesson Overview Logic is generally be defined as a philosophical science that evaluates arguments. An argument is a systematic combination of one or more than one statements, which are claimed to provide a logical support or evidence (i.e., premise(s) to another single statement which is claimed to follow logically from the alleged evidence (i.e., conclusion). An argument can be either good or bad argument, depending on the logical ability of its premise(s) to support its conclusion. The primary aim of logic is to develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own. The study of logic increases students‘ confidence to criticize the arguments of others and advance arguments of their own. In this lesson, we will discuss the meaning and basic concepts of logic: arguments, premises, and conclusions. Lesson Objectives: After the accomplishment of this lesson, you will be able to:
Ø Understand the meaning.
Ø Identify the subject matter of logic.
 Ø Understand the meaning of an argument.
Ø Identify the components of an argument.
 Ø Understand the meaning and nature of a premise.
Ø Comprehend the meaning and nature of a conclusion.
Ø Recognize the techniques of identifying the premises and conclusion of an argument.
Conditional Statements
A conditional statement is an ―if . . . then . . .‖ statement.
Example: If you study hard, then you will score „A‟ grade.
 Every conditional statement is made up of two component statements. The component statement immediately following the ―if‖ is called the antecedent (if-clause), and the one following the ―then‖ is called the consequent (then-clause). However, there is an occasion that the order of antecedent and consequent is reversed.
That is, when occasionally the word ‗‗then‘‘ is left out, the order of antecedent and consequent is reversed. For example if we left out ―then‖ from the above example the antecedent and consequent is reversed: You will score „A‟ grade if you study hard. In the above example, the antecedent is ―You study hard,‖ and the consequent is ―
You will score „A‟ grade.‖ In this example, there is a meaningful relationship between antecedent and consequent.
However, such a relationship need not exist for a statement to count as conditional. The statement ―If Getaneh Kebede is a singer, then Hawassa is in Mekelle‖ is just as much a conditional statement as that in the above example.
Conditional statements are not arguments, because they fail to meet the criteria given earlier.
In an argument, at least one statement must claim to present evidence, and there must be a claim that this evidence implies something. In a conditional statement, there is no claim that either the antecedent or the consequent presents evidence. In other words, there is no assertion that either the antecedent or the consequent is true. Rather, there is only the assertion that if the antecedent is true, then so is the consequent.
For example, the above example merely asserts that if you study hard, then you will score ‗A‘. It does not assert that you study hard.
Nor does it assert you scored ‗A‘. Of course, a conditional statement as a whole may present evidence because it asserts a relationship between statements. Yet when conditional statements are taken in this sense, there is still no argument, because there is then no separate claim that this evidence implies anything.
 Therefore, a single conditional statement is not an argument.
The fact that a statement begins with ―if‖ makes it the idea conditional and not a final reasonable assertion.
That is why also conditional statements are not evaluated as true or false without separately evaluating the antecedent and the consequent. They only claim that if the antecedent is true then so is the consequent. However, some conditional statements are similar to arguments in that they express the outcome of a reasoning process. As such, they may be said to have a certain inferential content. Consider the following example: If destroying a political competitor gives you joy, then you have a low sense of morality
https://wcu.edu.et/FirstYearModule/CRITICAL%20THINKING%20module.pdf
Discrete Mathematics
The Conditional Statement Before we give a formal definition of the conditional statement, we start with an example so we can understand when a conditional statement should be true. For the example, we need the following notation and terminology:
 Notation 1.1.
 If p and q are statements, the conditional of q by p is “if p then q” denoted p → q. We call p the hypothesis of the conditional and q the conclusion.
 Example 1.2. Consider the conditional statement, “if I am healthy, I will come to class.” To determine the truth value of this statement, we need to determine when this statement is false, so we consider the four different possibilities for the truth values of p and q.
Let p :=“I am healthy” and q :=“I will come to class”. We shall fill in the following table:
p q p → q T T T T F F F T T F F T
 • For case # 1, if I am healthy and I come to class, the conditional is clearly true.
 • For case # 2, if I am healthy, but I have decided to stay home and not go to class, the conditional is false – the hypothesis is satisfied, but the conclusion is not satisfied, so the statement cannot possibly be true.
• For case # 3, if I am not healthy, but I have come to class anyway though all the people sitting around may not be happy about it, the conditional statement has not been violated since the hypothesis does not hold i.e. the conditional statement is meaningless since the hypothesis is not true. Therefore, the conditional must be true.
 • Likewise, for case # 4, if I am not healthy, and I did not come to class, the conditional statement has not been violated since
the hypothesis does not hold. Therefore, the conditional is true. This example implies that a conditional statement is false only when the hypothesis is true and the conclusion is false. Though it is clear that a conditional statement is false only when the hypothesis is true and the conclusion is false, it is not clear why when the hypothesis is false, the conditional statement is always true. To try to explain why this is this case, we consider another example.
 Example 1.3.
 Consider the mathematical statement “if n is a perfect square, then n is not prime.” Clearly this is a true statement for any n, so it will be true when we substitute values in for n. Now substitute 3 for n: “if 3 is a perfect square, then 3 is not prime.” As remarked above, this conditional statement is still true yet its hypothesis and conclusion are both false. Similarly, if we substitute 6 into this statement, it becomes “if 6 is a perfect square, then 6 is not prime.” This conditional statement is true yet its hypothesis is false and its conclusion is true.
 We can now write down a formal definition for the conditional statement.
 Definition 1.4. If p and q are statements, the conditional of q by p is “if p then q” or “p implies q” denoted p → q. A conditional statement is false only when the hypothesis is false and the conclusion is true. The truth table for the conditional statement is as follows:
p q p → q
T T T T F F
F T T F F T
https://faculty.up.edu/wootton/discrete/section1.2.pdf
Conditional reasoning
Conditional reasoning is based on the construction “if 𝑝, then 𝑞” when the premise is true,
the conclusion will be true. However, this leaves open the question of what happens when
𝑝 is false, which means that, in this case, 𝑞 can logically be either true or false. Studies
are abundant about four main conditional inferences: modus ponens, modus tollens,
affirmation of the consequent and denial of the antecedent. Johnson-Laird and Byrne
(2002) discuss that, among all four conditional inferences mentioned in §§3.1.2, only
modus ponens and modus tollens are valid for the conditional interpretation. The following
is an example of modus ponens:
If it rains, then you get wet.
It rains.
Therefore, you get wet.
https://summit.sfu.ca/_flysystem/fedora/2022-08/input_data/22441/etd21791.pdf
Critics have claimed that mathematics taught in K-11 and K-12 is nothing more than memorizing the facts rather than computing the method of solving the given problem with a known concept of study.
Topic of study which includes discrete mathematics are Set Theory, Relations and Functions, Principles of Mathematical Induction, Permutation and Combinations, Mathematical Reasoning, Probability and some study about Matrices and Determinants.
Discrete Mathematics course is a core part of computer / information science & technology and it facilitates the study of applications in the field of computer science, especially in the areas of data structures, the theory of computer languages and the analysis of algorithms. In addition, this course also provides students with understanding of applications in engineering and the physical and life sciences, as well as in statistics and the social sciences.
 To introduce the student at the high-school level, if not earlier, to the topics and techniques of discrete methods and combinatorial reasoning. Whenever the structures from abstract algebra are required, only the basic theory needed for the application development. Further, the solution of the some applications contribute to the iterative procedures that lead to specific algorithms. The algorithmic approach and solution for the problems is fundamental in discrete mathematics.
Counting concept introduces the basic collection of counting techniques with few motivational examples such as paper folding example, Rubik’s cube problem etc. This provides, count visually distinguishable patterns (Binomial Theorem) for collection of objects with identifiable types of objects, each with several copies are available. Counting the number of distinct elements in a union of possibly non-disjoint sets (inclusion-exclusion formula). Probability theory conceptualize the foundational explanations (event, sample space, independence). Methods of determining the probabilities of events are introduced and the notion of equally likely outcomes are defined. The notion of a random variable is to create a variable whose value is determined by the outcome of a random experiment. Probability distribution described for a particular pattern and a collection of conditional probabilities into a different set of conditional probabilities (Bayes’s Theorem). On higher level, they should be aware of some meta-knowledge and heuristics, and be able to use them appropriately. They should be aware that there are many approaches to achieve the same goal but using the appropriate method of solving the problem and reach the desired result. Influenced by all the examples in discrete mathematics concepts, students shall know it is good to work systematically and in phases, virtually every time when it is possible. Educational targets describes above includes both practical usability and theoretical knowledge. These two aspect shall strength each other systematically, where every student of K-11 and K-12 grade classes efficiently.
 

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Write up on Discrete Mathematics in an educational Curriculum for grade school and high school students(Part 2) https://ddcomics.org/2026/06/13/write-up-on-discrete-mathematics-in-an-educational-curriculum-for-grade-school-and-high-school-studentspart-2/ https://ddcomics.org/2026/06/13/write-up-on-discrete-mathematics-in-an-educational-curriculum-for-grade-school-and-high-school-studentspart-2/#respond Sat, 13 Jun 2026 06:21:06 +0000 https://ddcomics.org/?p=6823 Part II 1.1. Definition of Computational Thinking (CT) Is that a fact? There is still no universally accepted definition of “Computational Thinking”. The concept of computational thinking (CT) was first introduced by an educationalist Seymour Papert in 1967 talking about LOGO, the programming language he developed at MIT (Massachusetts Institute of Technology) to teach programming […]

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Part II

  1. Computational Thinking at first glance – What, why and how

1.1. Definition of Computational Thinking (CT) Is that a fact? There is still no universally accepted definition of “Computational Thinking”. The concept of computational thinking (CT) was first introduced by an educationalist Seymour Papert in 1967 talking about LOGO, the programming language he developed at MIT (Massachusetts Institute of Technology) to teach programming to children. He was convinced that the use of computers could foster formal thinking in children and, in particular, could allow children to autonomously “construct” their learning and thinking. The concept of CT was then revitalized in 2006 by a computer scientist Jeannette M. Wing who, in the article “Computational Thinking”, argued that it addresses the conundrum of machine intelligence by asking what machines do better than man and what man does better than machines. Wing argues that computational thinking is not simply a procedural coding activity, but is a basic conceptual skill that, along with reading, writing and arithmetic, should be taught to all children. It appears that computational thinking purports to be critical thinking in evaluating situations and an advanced problem-solving ability using computerized tools. If computer science is the science of what can be computerized and how to computerize it, however, computational thinking is not a skill unique to computer scientists. It allows problems to be solved, a system to be designed and human behaviour to be understood in everyday life, in an alternative way, through the fundamental concepts of information technology.

Key Skills for Computational Thinking There are four key skills in computational thinking:

 1 Decomposition

2 Pattern Recognition

3 Pattern Abstraction

4 Algorithm Design

 1) Decomposition Breaking down big and difficult problems into something much simpler. Often big problems are just many little problems put together. Decomposition is an important life skill to be relied on in the future when students and adults need to take on larger tasks. Students will learn ways to delegate group projects and build time management skills

 2) Pattern Recognition Sometimes when a problem is made up of many small bits, you will notice that these bits have something in common. If they do not, they could, however, have strong similarities with the pieces of another problem that has already been solved. If you are able to find these regularities, it will become a lot easier to identify the individual pieces: pattern recognition is simply looking for patterns in the puzzles and determining if any of the problems or solutions we encountered in the past may apply to a present situation. May, what we learned in the past, help us sort out the actual problem? If you have ever built a piece of IKEA furniture, you will understand the importance of patterns. While assembling an IKEA drawer unit, it is likely to take you much longer to assemble the first drawer than the fourth or fifth. When we repeat steps in our assembling process we learn how to solve the instructions more quickly and learn from our mistakes. The painstaking process of assembling that first part teaches us the skills to perform the process more efficiently in the future.

3) Pattern Abstraction Once you have located a pattern, it is possible to abstract (ignore) the details that differentiate the various things and use general techniques for finding solutions that work for more than one problem. Identifying the crucial information in a problem and disregarding the irrelevant information is one of the hardest parts of computational learning.

 4) Algorithm design When the solution is ready, it is possible to write it down so it can be executed step by step. This makes easier to obtain the expected results. Algorithm design is setting out the steps and rules needed to follow in order to achieve the same desired outcome every time.

Concepts of computational thinking Computational thinking is a cognitive or thought process involving logical reasoning by which problems are solved and artefacts, procedures and systems are better understood. It embraces:

● the ability to think algorithmically;

● the ability to think in terms of decomposition;

 ● the ability to think in generalisations, identifying and making use of patterns;

 ● the ability to think in abstractions, choosing good representations; and

 ● the ability to think in terms of evaluation. Computational thinking skills enable pupils to access parts of the Computing subject content. Importantly, they relate to thinking skills and problem solving across the whole curriculum and through life in general. Computational thinking can be applied to a wide range of artefacts including: systems, processes, objects, algorithms, problems, solutions, abstractions, and collections of data or information. In the following discussion of concepts, artefact refers to any of these. Logical reasoning Logical reasoning enables pupils to make sense of things by analysing and checking facts through thinking clearly and precisely. It allows pupils to draw on their own knowledge and internal models to make and verify predictions and to draw conclusions. It is used extensively by pupils when they test, debug, and correct algorithms. Logical reasoning is the novel application of the other computational thinking concepts to solve problems.

Design and technology pupils, designing a model of a truck, choose materials for different elements of a project. They are employing generalisation when they recognise that the properties of a material used in one situation make it suitable to use in another completely different context. Being able to divide the new project into different parts, requiring different materials, is an example of decomposition. The pupil is using logical reasoning to design a truck. Pupils use logical reasoning when learning about gravity using a weighted string suspended from the lid of a glass jar. Before tilting the jar, pupils can make predictions about the behaviour of the weighted string. They can then evaluate the results of their tests. They may be able to generalise the behaviour to other situations such as a crane. The novel use in understanding a property of gravity is logical reasoning. Logical reasoning is key in allowing pupils to debug their code. They can work with peers to evaluate each other’s code, to isolate bugs, and to suggest fixes. During this process, they may have opportunities to employ abstraction, evaluation, and algorithmic design. The novel use in correcting mistakes in code requires logical reasoning.

 Abstraction in the Computer Science Classroom

Although we have presented relevant characterisations of abstraction, it is still far from clear how an abstraction-oriented perspective could become part of the pedagogical practice. Already in the late 1990s, as a revision of the spreading object-first orientaTable 1

 PGK Hierarchy levels from Perrenet et al. (2005) and their mapping to K-5 settings according to Waite et al. (2018). PGK Level Definition K-5 name K-5 question Problem Algorithm perceived as a problem solving strategy problem “What is needed” Object (algorithm) Algorithm understood independently of any specific implementation design “What it should do” Program Algorithmic grasp of the program code “How it is done” Execution Focus on individual runs with specific inputs running the code “What it does” 630 C. Mirolo et al. tion, Machanick (1998) endorsed an abstraction-first instructional approach where the implementation of abstract data types is delayed as much as possible in order to stress an abstract view of the models.

Kramer remarked that abstraction per se is not the subject of any computing course, but that all computing courses “rely on or utilize abstraction to explain, model, specify, reason or solve problems,” so confirming that “abstraction is an essential aspect of computing, but that it must be taught indirectly through other topics” (Kramer, 2007, p. 41). In line with Kramer’s remark, Hazzan (2008) discussed abstraction as a soft idea, “that can be neither rigidly nor formally defined, nor is it possible to guide students as to its precise application.” And although “it is not a trivial matter,” like other soft ideas, abstraction should be taught in a computer science curriculum.

Then, a small number of educators have provided guidelines to teach abstraction at different instruction levels. Hence, this section briefly explores their approaches to foster and assess abstraction skills. 5.1. Teaching to Trigger Abstraction in Computer Science Often instructors aim to develop students’ abstraction skills indirectly, by devising particular learning trajectories that are supposed to foster higher-level thinking and require students to use abstraction to succeed. In a program development project, for example, they could assign refactoring tasks in which learners are asked to look for recurrent patterns of code and to re-organise the code by introducing meaningful procedural and/or data abstractions with the purpose of making the whole program easier to read, debug and modify. In the following paragraphs we will outline a selection of representative approaches to (an implicit) abstraction-oriented instruction. Pattern-oriented instruction. This approach has the aim of improving students’ competencies in algorithmic problem solving (Muller and Haberman, 2008).

An algorithm is indeed seen by these authors as a combination of plan patterns in Soloway’s sense (Soloway, 1986), resulting via sequencing, nesting or merging plans from a repertory of basic algorithmic patterns specifically designed for pedagogical purposes. In Muller and Haberman’s scenario, abstraction plays a crucial role in pattern recognition, chunking, and problem structure identification. Their approach relies on having an appropriate pattern repository, as well as on presenting carefully selected problems of gradually increasing difficulty; teachers should then discuss and compare different solutions to a given problem in terms of pattern composition. Additional guidelines for pattern-oriented instruction include:

(1) patterns should be abstracted from related examples or by generalising a simpler problem,

(2) patterns should be revisited in different contexts, in order to make the identification of common problem features easier, and (3) similarities, differences, and also possible misuses of patterns should be considered. According to Muller and Haberman, comparative studies appear to show that novices exposed to this approach develop enhanced problem solving abilities. Multiple representations perspective. Dealing with multiple representations of a given phenomenon can play a key role in the development of abstract concepts. Ac- cording to Ainsworth (2008), in particular, in order “to construct a deeper understanding of a domain,” if the learners “fail to relate representations, then processes like abstraction cannot occur. Moreover, although learners find it difficult to relate different forms of representations, if the representations are too similar, then abstraction is also unlikely to occur.” She then recommends that teachers should foster abstraction over multiple representations “by providing focused help and support on how to relate representations and giving learners sufficient time to master this process.” In this respect, Gautam et al. (2020) have recently proposed an interesting interdisciplinary approach to integrate science (namely, chemistry) and computational thinking in the curriculum. While abstraction is usually “presented as hierarchical” in terms of (i) extracting important features and ignoring unimportant ones, and (ii) finding commonalities across contexts, in their standpoint “abstraction in science” as well as in computing “requires students to move laterally across different representations of the concepts or actions.” In the reported study, the micro-level process of photosynthesis was modeled by a code snippet, and by discussing commonalities and differences between, e.g., a whiteboard and the code representation of the implied chemical reaction, “the instructor pushed students towards higher-level abstract thinking.” Moreover, they suggest to allow for friction emerging when the students explore different representations, in that it encourages to consider alternative views and “negotiate the elements with one another.” According to the authors, this approach “created meaningful accounts of science phenomenon and the science provided access to how computation embeds ideas.” Exploration of artefacts.

 A more recent pedagogical trend in programming education attempts to trigger abstraction through activities inspired by the use-modify-create framework. The idea is that the understanding of artefacts such as programs would gradually progress through three major stages, corresponding to

  • exploration via passive use (as a consumer),
  • (ii) experimentation of the internal machinery by modifying some features, and finally
  • (iii) creation of new, original artefacts to achieve specific goals. While discussing the use-modify-create approach, Lee et al. (2014) observe that abstraction, as well as other computational thinking abilities, are “not explicitly taught but rather [develop] through one’s impetus to create;” nevertheless, in this progression the abilities to modify and, later, to create imply the enhancement of learner’s abstraction skills.

This is a course on discrete mathematics as used in Computer Science. It’s only a one-semester course, so there are a lot of topics that it doesn’t cover or doesn’t cover in much depth. But the hope is that this will give you a foundation of skills that you can build on as you need to, and particularly to give you a bit of mathematical maturity—the basic understanding of what mathematics is and how mathematical definitions and proofs work.

  1. So why do I need to learn all this nasty mathematics?

 Why you should know about mathematics, if you are interested in Computer Science: or, more specifically, why you should take CS202 or a comparable course:

• Computation is something that you can’t see and can’t touch, and yet (thanks to the efforts of generations of hardware engineers) it obeys strict, well-defined rules with astonishing accuracy over long periods of time.

• Computations are too big for you to comprehend all at once. Imagine printing out an execution trace that showed every operation a typical $500 desktop computer executed in one (1) second.

If you could read one operation per second, for eight hours every day, you would die of old age before you got halfway through. Now imagine letting the computer run overnight. So in order to understand computations, we need a language that allows us to reason about things we can’t see and can’t touch, that are too big

for us to understand, but that nonetheless follow strict, simple, well-defined rules. We’d like our reasoning to be consistent: any two people using the language should (barring errors) obtain the same conclusions from the same information. Computer scientists are good at inventing languages, so we could invent a new one for this particular purpose, but we don’t have to: the exact same problem has been vexing philosophers, theologians, and mathematicians for much longer than computers have been around, and they’ve had a lot of time to think about how to make such a language work.

 Philosophers and theologians are still working on the consistency part, but mathematicians (mostly) got it in the early 20th-century. Because the first virtue of a computer scientist is laziness, we are going to steal their code.

  1. But isn’t math hard? Yes and no. The human brain is not really designed to do formal mathematical reasoning, which is why most mathematics was invented in the last few centuries and why even apparently simple things like learning how to count or add require years of training, usually done at an early age so the pain will be forgotten later. But mathematical reasoning is very close to legal reasoning, which we do seem to be very good at.1 There is very little structural difference between the two sentences: 1. If x is in S, then x + 1 is in S. 2. If x is of royal blood, then x’s child is of royal blood.

 But because the first is about boring numbers and the second is about fascinating social relationships and rules, most people have a much easier time deducing that to show somebody is royal we need to start with some known royal and follow a chain of descendants than they have deducing that to show that some number is in the set S we need to start with some known element of S and show that repeatedly adding 1 gets us to the number we want. And yet to a logician these are the same processes of reasoning. So why is statement

(1) trickier to think about than statement

 (2)? Part of the difference is familiarity—we are all taught from an early age what it means to be somebody’s child, to take on a particular social role, etc. For mathematical concepts, this familiarity comes with exposure and practice, just as with learning any other language. But part of the difference is that 1For a description of some classic experiments that demonstrate this, see http://

Foundations and logic Why: This is the assembly language of mathematics—the stuff at the bottom that everything else compiles to.

 • Propositional logic.

 • Predicate logic.

 • Axioms, theories, and models.

• Proofs.

• Induction and recursion

English Language Arts

To Critical Thinking, the critical person is something like a critical consumer of information; he or she is driven to seek reasons and evidence. Part of this is a matter of mastering certain skills of thought: learning to diagnose invalid forms of argument, knowing how to make and defend distinctions, and so on. Much of the literature in this area, especially early on, seemed to be devoted to lists and taxonomies of what a “critical thinker” should know and be able to do (Ennis 1962, 1980). More recently, however, various authors in this tradition have come to recognize that teaching content and skills is of minor import if learners do not also develop the dispositions or inclination to look at the world through a critical lens. By this, Critical Thinking means that the critical person has not only the capacity (the skills) to seek reasons, truth, and evidence, but also that he or she has the drive (disposition) to seek them. For instance, Ennis claims that a critical person not only should seek reasons and try to be well informed, but that he or she should have a tendency to do such things (Ennis 1987, 1996). Siegel criticizes Ennis somewhat for seeing dispositions simply as what animates the skills of critical thinking, because this fails to distinguish sufficiently the critical thinker from critical thinking. For Siegel, a cluster of dispositions (the “critical spirit”) is more like a deep-seated character trait, something like Scheffler’s notion of “a love of truth and a contempt of lying” (Siegel 1988; Scheffler 1991). It is part of critical thinking itself. Paul also stresses this distinction between skills and dispositions in his distinction between “weak-sense” and “strong-sense” critical thinking. For Paul, the “weak-sense” means that one has learned the skills and can demonstrate them when asked to do so; the “strong-sense” means that one has incorporated these skills into a way of living in which one’s own assumptions are re-examined and questioned as well. According to Paul, a critical thinker in the “strong sense” has a passionate drive for “clarity, accuracy, and fairmindedness” (Paul 1983, 23; see also Paul 1994). This dispositional view of critical thinking has real advantages over the skills-only view. But in important respects it is still limited. First, it is not clear exactly what is entailed by making such dispositions part of critical thinking. In our view it not only broadens the notion of criticality beyond mere “logicality,” but it necessarily requires a greater attention to institutional contexts and social relations than Critical Thinking authors have provided. Both the skills-based view and the skills-plus-dispositions view are still focused on the individual person. But it is only in the context

A second theme in the Critical Thinking literature has been the extent to which critical thinking can be characterized as a set of generalized abilities and dispositions, as opposed to content-specific abilities and dispositions that are learned and expressed differently in different areas of investigation. Can a general “Critical Thinking” course develop abilities and dispositions that will then be applied in any of a range of fields; or should such material be presented specifically in connection to the questions and content of particular fields of study? Is a scientist who is a critical thinker doing the same things as an historian who is a critical thinker? When each evaluates “good evidence,” are they truly thinking about problems in similar ways, or are the differences in interpretation and application dominant? This debate has set John McPeck, the chief advocate of content-specificity, in opposition to a number of other theorists in this area (Norris 1992; Talaska 1992). This issue relates not only to the question of how we might teach critical thinking, but also to how and whether one can test for a general facility in critical thinking (Ennis 1984). A third debate has addressed the question of the degree to which the standards of critical thinking, and the conception of rationality that underlies them, are culturally biased in favor of a particular masculine and/or Western mode of thinking, one that implicitly devalues other “ways of knowing.” Theories of education that stress the primary importance of logic, conceptual clarity, and rigorous adherence to scientific evidence have been challenged by various advocates of cultural and gender diversity who emphasize respect for alternative world views and styles of reasoning. Partly in response to such criticisms, Richard Paul has developed a conception of critical thinking that regards “sociocentrism” as itself a sign of flawed thinking (Paul 1994). Paul believes that, because critical thinking allows us to overcome the sway of our egocentric and sociocentric beliefs, it is “essential to our role as moral agents and as potential shapers of our own nature and destiny”(Paul 1990, 67). For Paul, and for some other Critical Thinking authors as well, part of the method of critical thinking involves fostering dialogue, in which thinking from the perspective of others is also relevant to the assessment of truth claims; a too-hasty imposition of one’s own standards of evidence might result not only in a premature rejection of credible alternative points of view, but might also have the effect of silencing the voices of those who (in the present context) need to be encouraged as much as possible to speak for themselves. In this respect, we see Paul introducing into the very definition of critical thinking some of the sorts of social and contextual factors that Critical Pedagogy writers have emphasized.

http://mediaeducation.org.mt/wp-content/uploads/2013/05/Critical-Thinking-and-Critical-Pedagogy.pdf

CHAPTER TWO BASIC CONCEPTS OF LOGIC

Chapter Overview Logic, as field of study, may be defined as the organized body of knowledge, or science that evaluates arguments. The aim of logic is to develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own. Argument is a systematic combination of two or more statements, which are classified as a premise or premises and conclusion. A premise refers to the statement, which is claimed to provide a logical support or evidence to the main point of the argument, which h known as conclusion. A conclusion is a statement, which is claimed to follow from the alleged evidence. Depending on the logical and real ability of the premise(s) to support the conclusion, an argument can be either a good argument or a bad argument. However, unlike all kinds of passages, including those that resemble arguments, all arguments purport to prove something. Arguments can generally be divided into deductive and inductive arguments. A deductive argument is an argument in which the premises are claimed to support the conclusion in such a way that it is impossible for the premises to be true and the conclusion false. On the other hand, an inductive argument is an argument in which the premises are claimed to support the conclusion in such a way that it is improbable that the premises be true and the conclusion false. The deductiveness or inductiveness of an argument can be determined by the particular indicator word it might use, the actual strength of the inferential relationship between its component statements, and its argumentative form or structure. A deductive argument can be evaluated by its validity and soundness. Likewise, an inductive argument can be evaluated by its strength and cogency. Depending on its actually ability to successfully maintain its inferential claim, a deductive argument can be either valid or invalid. That is, if the premise(s) of a certain deductive argument actually support its conclusion in such a way that it is impossible for the premises to be true and the conclusion false, then that particular deductive argument is valid. If, however, its premise(s) actually support its conclusion in such a

way that it is possible for the premises to be true and the conclusion false, then that particular deductive argument is invalid. Similarly, an inductive argument can be either strong or weak, depending on its actually ability to successfully maintain its inferential claim. That is, if the premise(s) of a certain inductive argument actually support its conclusion in such a way that it is improbable for the premises to be true and the conclusion false, then that particular inductive argument is strong. If, however, its premise(s) actually support its conclusion in such a way that it is probable for the premises to be true and the conclusion false, then that particular inductive argument is weak. Furthermore, depending on its actually ability to successfully maintain its inferential claim as well as its factual claim, a deductive argument can be either sound or unsound. That is, if a deductive argument actually maintained its inferential claim, (i.e., if it is valid), and its factual claim, (i.e., if all of its premises are true), then that particular deductive argument will be a sound argument. However, if it fails to maintain either of its claims, it will be an unsound argument. Likewise, depending on its actually ability to successfully maintain its inferential claim as well as its factual claim, an inductive argument can be either cogent or uncogent. That is, if an inductive argument actually maintained its inferential claim, (i.e., if it is strong), and its factual claim, (i.e., if all of its premises are probably true), then that particular inductive argument will be a cogent argument. However, if it fails to maintain either of its claims, it will be an uncogent argument. In this chapter, we will discuss logic and its basic concepts, the techniques of distinguishing arguments from non-argumentative passages, and the types of arguments.

Chapter Objectives: Dear learners, after the successful completion of this chapter, you will be able to:

Ø Understand the meaning and basic concepts of logic;

Ø Understand the meaning, components, and types of arguments; and

Ø Recognize the major techniques of recognizing and evaluating arguments

Lesson 1: Basic Concepts of Logic: Arguments, Premises and Conclusions Lesson Overview Logic is generally be defined as a philosophical science that evaluates arguments. An argument is a systematic combination of one or more than one statements, which are claimed to provide a logical support or evidence (i.e., premise(s) to another single statement which is claimed to follow logically from the alleged evidence (i.e., conclusion). An argument can be either good or bad argument, depending on the logical ability of its premise(s) to support its conclusion. The primary aim of logic is to develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own. The study of logic increases students‘ confidence to criticize the arguments of others and advance arguments of their own. In this lesson, we will discuss the meaning and basic concepts of logic: arguments, premises, and conclusions. Lesson Objectives: After the accomplishment of this lesson, you will be able to:

Ø Understand the meaning.

Ø Identify the subject matter of logic.

 Ø Understand the meaning of an argument.

Ø Identify the components of an argument.

 Ø Understand the meaning and nature of a premise.

Ø Comprehend the meaning and nature of a conclusion.

Ø Recognize the techniques of identifying the premises and conclusion of an argument.

Conditional Statements

A conditional statement is an ―if . . . then . . .‖ statement.

Example: If you study hard, then you will score „A‟ grade.

 Every conditional statement is made up of two component statements. The component statement immediately following the ―if‖ is called the antecedent (if-clause), and the one following the ―then‖ is called the consequent (then-clause). However, there is an occasion that the order of antecedent and consequent is reversed.

That is, when occasionally the word ‗‗then‘‘ is left out, the order of antecedent and consequent is reversed. For example if we left out ―then‖ from the above example the antecedent and consequent is reversed: You will score „A‟ grade if you study hard. In the above example, the antecedent is ―You study hard,‖ and the consequent is ―

You will score „A‟ grade.‖ In this example, there is a meaningful relationship between antecedent and consequent.

However, such a relationship need not exist for a statement to count as conditional. The statement ―If Getaneh Kebede is a singer, then Hawassa is in Mekelle‖ is just as much a conditional statement as that in the above example.

Conditional statements are not arguments, because they fail to meet the criteria given earlier.

In an argument, at least one statement must claim to present evidence, and there must be a claim that this evidence implies something. In a conditional statement, there is no claim that either the antecedent or the consequent presents evidence. In other words, there is no assertion that either the antecedent or the consequent is true. Rather, there is only the assertion that if the antecedent is true, then so is the consequent.

For example, the above example merely asserts that if you study hard, then you will score ‗A‘. It does not assert that you study hard.

Nor does it assert you scored ‗A‘. Of course, a conditional statement as a whole may present evidence because it asserts a relationship between statements. Yet when conditional statements are taken in this sense, there is still no argument, because there is then no separate claim that this evidence implies anything.

 Therefore, a single conditional statement is not an argument.

The fact that a statement begins with ―if‖ makes it the idea conditional and not a final reasonable assertion.

That is why also conditional statements are not evaluated as true or false without separately evaluating the antecedent and the consequent. They only claim that if the antecedent is true then so is the consequent. However, some conditional statements are similar to arguments in that they express the outcome of a reasoning process. As such, they may be said to have a certain inferential content. Consider the following example: If destroying a political competitor gives you joy, then you have a low sense of morality

https://wcu.edu.et/FirstYearModule/CRITICAL%20THINKING%20module.pdf

Discrete Mathematics

  1. The Conditional Statement Before we give a formal definition of the conditional statement, we start with an example so we can understand when a conditional statement should be true. For the example, we need the following notation and terminology:
  2.  Notation 1.1.

 If p and q are statements, the conditional of q by p is “if p then q” denoted p → q. We call p the hypothesis of the conditional and q the conclusion.

 Example 1.2. Consider the conditional statement, “if I am healthy, I will come to class.” To determine the truth value of this statement, we need to determine when this statement is false, so we consider the four different possibilities for the truth values of p and q.

Let p :=“I am healthy” and q :=“I will come to class”. We shall fill in the following table:

p q p → q T T T T F F F T T F F T

 • For case # 1, if I am healthy and I come to class, the conditional is clearly true.

 • For case # 2, if I am healthy, but I have decided to stay home and not go to class, the conditional is false – the hypothesis is satisfied, but the conclusion is not satisfied, so the statement cannot possibly be true.

• For case # 3, if I am not healthy, but I have come to class anyway though all the people sitting around may not be happy about it, the conditional statement has not been violated since the hypothesis does not hold i.e. the conditional statement is meaningless since the hypothesis is not true. Therefore, the conditional must be true.

 • Likewise, for case # 4, if I am not healthy, and I did not come to class, the conditional statement has not been violated since

the hypothesis does not hold. Therefore, the conditional is true. This example implies that a conditional statement is false only when the hypothesis is true and the conclusion is false. Though it is clear that a conditional statement is false only when the hypothesis is true and the conclusion is false, it is not clear why when the hypothesis is false, the conditional statement is always true. To try to explain why this is this case, we consider another example.

 Example 1.3.

 Consider the mathematical statement “if n is a perfect square, then n is not prime.” Clearly this is a true statement for any n, so it will be true when we substitute values in for n. Now substitute 3 for n: “if 3 is a perfect square, then 3 is not prime.” As remarked above, this conditional statement is still true yet its hypothesis and conclusion are both false. Similarly, if we substitute 6 into this statement, it becomes “if 6 is a perfect square, then 6 is not prime.” This conditional statement is true yet its hypothesis is false and its conclusion is true.

 We can now write down a formal definition for the conditional statement.

 Definition 1.4. If p and q are statements, the conditional of q by p is “if p then q” or “p implies q” denoted p → q. A conditional statement is false only when the hypothesis is false and the conclusion is true. The truth table for the conditional statement is as follows:

p q p → q

T T T T F F

F T T F F T

https://faculty.up.edu/wootton/discrete/section1.2.pdf

Conditional reasoning

Conditional reasoning is based on the construction “if 𝑝, then 𝑞” when the premise is true,

the conclusion will be true. However, this leaves open the question of what happens when

𝑝 is false, which means that, in this case, 𝑞 can logically be either true or false. Studies

are abundant about four main conditional inferences: modus ponens, modus tollens,

affirmation of the consequent and denial of the antecedent. Johnson-Laird and Byrne

(2002) discuss that, among all four conditional inferences mentioned in §§3.1.2, only

modus ponens and modus tollens are valid for the conditional interpretation. The following

is an example of modus ponens:

If it rains, then you get wet.

It rains.

Therefore, you get wet.

https://summit.sfu.ca/_flysystem/fedora/2022-08/input_data/22441/etd21791.pdf

Critics have claimed that mathematics taught in K-11 and K-12 is nothing more than memorizing the facts rather than computing the method of solving the given problem with a known concept of study.

Topic of study which includes discrete mathematics are Set Theory, Relations and Functions, Principles of Mathematical Induction, Permutation and Combinations, Mathematical Reasoning, Probability and some study about Matrices and Determinants.

Discrete Mathematics course is a core part of computer / information science & technology and it facilitates the study of applications in the field of computer science, especially in the areas of data structures, the theory of computer languages and the analysis of algorithms. In addition, this course also provides students with understanding of applications in engineering and the physical and life sciences, as well as in statistics and the social sciences.

 To introduce the student at the high-school level, if not earlier, to the topics and techniques of discrete methods and combinatorial reasoning. Whenever the structures from abstract algebra are required, only the basic theory needed for the application development. Further, the solution of the some applications contribute to the iterative procedures that lead to specific algorithms. The algorithmic approach and solution for the problems is fundamental in discrete mathematics.

Counting concept introduces the basic collection of counting techniques with few motivational examples such as paper folding example, Rubik’s cube problem etc. This provides, count visually distinguishable patterns (Binomial Theorem) for collection of objects with identifiable types of objects, each with several copies are available. Counting the number of distinct elements in a union of possibly non-disjoint sets (inclusion-exclusion formula). Probability theory conceptualize the foundational explanations (event, sample space, independence). Methods of determining the probabilities of events are introduced and the notion of equally likely outcomes are defined. The notion of a random variable is to create a variable whose value is determined by the outcome of a random experiment. Probability distribution described for a particular pattern and a collection of conditional probabilities into a different set of conditional probabilities (Bayes’s Theorem). On higher level, they should be aware of some meta-knowledge and heuristics, and be able to use them appropriately. They should be aware that there are many approaches to achieve the same goal but using the appropriate method of solving the problem and reach the desired result. Influenced by all the examples in discrete mathematics concepts, students shall know it is good to work systematically and in phases, virtually every time when it is possible. Educational targets describes above includes both practical usability and theoretical knowledge. These two aspect shall strength each other systematically, where every student of K-11 and K-12 grade classes efficiently.

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