What are the differences between fuzzy, intuitionistic, and paraconsistent logic? Which one is considered the most useful and why?
11.06.2025 05:30
Fads, I submit, are the very negation of reason. To be swept up in every new craze that comes along, to surrender yourself to idiocy just because everybody else is doing it - this, to me, is the acme of mindlessness. Not, however, to Petey.
"Not that I know of. Why?"
Cool was I and logical. Keen, calculating, perspicacious, acute, and astute - I was all of these. My brain was as powerful as a dynamo, as precise as a chemist's scales, as penetrating as a scalpel. And - think of it! - I was only eighteen.
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"I beg your pardon," said I.
"You see, my dear, these things aren't so hard. All you have to do is concentrate. Think - examine - evaluate. Come now, let's review everything we have learned."
My brain, that precision instrument, slipped into high gear. "Anything?" I asked, looking at him narrowly.
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He flung the coat from him. "Never," he said stoutly.
She scratched her pretty, empty head. "I'm all confused," she admitted.
"Why not?" I demanded.
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"Logic," I said, clearing my throat, "is the science of thinking. Before we can think correctly, we must first learn to recognize the common fallacies of logic. These we will take up tonight."
Beautiful she was. She was not yet of pin-up proportions, but I felt sure that time would supply the lack. She already had the makings.
It is not often that one so young has such a giant intellect. Take, for example, Petey Bellows, my roommate at the university. Same age, same background, but dumb as an oak. A nice enough fellow, you understand, but nothing upstairs. Emotional type. Unstable. Impressionable. Worst of all, a faddist.
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I went back to my room with a heavy heart. I had gravely underestimated the size of my task. This girl's lack of information was terrifying. Nor would it be enough merely to supply her with information. First she had to be taught to think. This loomed as a project of no small dimensions, and at first I was tempted to give her back to Petey. But then I got to thinking about her abundant physical charms and about the way she entered a room and the way she handled a knife and fork, and I decided to make an effort. I went about it, as in all things, systematically. I gave her a course in logic. It happened that I, as a law student, was taking a course in logic myself, so I had all the facts at my fingertips.
"That's right."
"Polly," I said gently, "the argument is a fallacy. Exercise is good is an unqualified generalization. For instance, if you have heart disease, exercise is bad, not good. Many people are ordered by their doctors not to exercise. You must qualify the generalization. You must say exercise is usually good, or exercise is good for most people. Otherwise you have committed a Dicto Simpliciter. Do you see?"
"I certainly can," declared Polly. "He's got a racoon coat."
"You don't understand," he interrupted impatiently. "It's the thing to do. Don't you want to be in the swim?"
Gracious she was. By gracious I mean full of graces. She had an erectness of carraige, an ease of bearing, a poise that clearly indicated the best of breeding. At table her manners were exquisite. I had seen her at the Kozy Kampus Korner eating the specialty of the house - a sandwich that contained scraps of pot roast, gravy, chopped nuts, and a dipper of sauerkraut - without even getting her fingers moist.
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"Polly," I said sharply, "it's a fallacy. Eula Becker doesn't cause the rain. She has no connection with the rain. You are guilty of Post Hoc if you blame Eula Becker."
I sighed. "No, Polly, I'm not mad."
"Holy Toledo!" said Peter reverently. He plunged his hands into the raccoon coat and then his face. "Holy Toledo!" he repeated fifteen or twenty times.
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"What's Polly to me, or me to Polly?"
I chuckled with amusement. The dear child had learned her lessons well. "My dear," I said, patting her head in a tolerant manner, "five dates is plenty. After all, you don't have to eat a whole cake to know that it's good."
Heartened by the knowledge that Polly was not altogether a cretin, I began a long, patient review of all I had told her. Over and over and over again I cited instances, pointed out flaws, kept hammering away without letup. It was like digging a tunnel. At first everything was work, sweat, and darkness. I had no idea when I would reach the light, or even if I would. But I persisted. I pounded and clawed and scraped, and finally I was rewarded. I saw a chink of light. And then the chink got bigger and the sun came pouring in and all was bright. Five grueling nights this took, but it was worth it. I had made a logician out of Polly; I had taught her to think. My job was done. She was worthy of me at last. She was a fit wife for me, a proper hostess for my many mansions, a suitable mother for my well-heeled children.
"Dicto Simpliciter means an argument based on an unqualified generalization. For example: Exercise is good. Therefore everybody should exercise."
I stroked my chin thoughtfully. It so happened that I knew where to get my hands on a raccoon coat. My father had had one in his undergraduate days; it lay now in a trunk in the attic back home. It also happened that Petey had something I wanted. He didn't have it exactly, but at least he had first rights on it. I refer to his girl, Polly Espy.
"False Analogy," said Polly promptly. "I'm not a cake. I'm a girl."
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"Yeah," she said thoughtfully. "Well, then I guess He can't make the stone."
He swallowed. "It's a deal," he said and shook my hand.
"Then tell me some more fallacies."
"Listen," he said, clutching my arm eagerly, "while you're home, you couldn't get some money from your old man, could you, and lend it to me so I can buy a raccoon coat?"
"Wow-dow!" she cried, clapping her hands delightedly.
"But He can do anything," I reminded her.
"But if He can do anything, He can lift the stone," I pointed out.
I nodded with satisfaction. "In other words, if you were out of the picture, the field would be open. Is that right?"
"Polly," I said testily, "the argument is all wrong. Doctors, lawyers, and carpenters aren't taking a test to see how much they have learned, but students are. The situations are altogether different, and you can't make an analogy between them."
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"Because this afternoon I promised Petey Bellows that I would go steady with him."
"Polly," I said to her when I picked her up on our next date, "tonight we are going over to the Knoll and talk."
I reeled back, overcome with the infamy of it. After he promised, after he made a deal, after he shook my hand! "That rat!" I shrieked, kicking up great chuncks of turf. "You can't go with him, Polly. He's a liar. He's a cheat. He's a rat."
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I chuckled with somewhat less amusement. The dear child had learned her lesson perhaps too well. I decided to change tactics. Obviously the best approach was a simple, strong, direct declaration of love. I paused for a moment while my massive brain chose the proper words. Then I began: "Polly, I love you. You are the whole world to me, and the moon and the stars and the constellations of outer space. Please, my darling, say that you will go steady with me, for if you will not, life will be meaningless. I will languish. I will refuse my meals. I will wander the face of the earth, a shambling, hollow-eyed hulk." There, I thought, folding my arms, that ought to do it.
I dashed perspiration from my brow. "Polly," I croaked, "You mustn't take all these things so literally. I mean this is just classroom stuff. You know that the things you learn in school don't have anything to do with life."
"Dicto Simpliciter," she said, wagging her finger at me playfully.
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"Right!" I cried exultantly. "One hundred percent right. It's not fair. The first man has poisoned the well before anybody could drink from it. He has hamstrung his opponent before he could even start... Polly, I'm proud of you."
"Is there," I asked, "any other man for whom she has a particular fondness?"
"I want a raccoon coat," he wailed.
"Home for the weekend." I threw a few things into the bag.
"It isn't as though I was in love with Polly," he said thickly. "Or going steady or anything like that."
"Yes, let's," she chirped, blinking her eyes happily.
"No," I said truthfully.
"Hasty Generalization," she repeated. "How can you say that we are well matched on the basis of only five dates?"
"I guess so. What are you getting at?"
"It will be better if you stop tugging at my sleeve," I told her, and when she desisted, I continued. "Next we take up a fallacy called Hasty Generalization. Listen carefully: You can't speak French. I can't speak French. Petey Bellows can't speak French. I must therefore conclude that nobody at the University of Minnesota can speak French."
"Poisoning the Well," said Polly, "and stop shouting. I think shouting must be a fallacy too."
We went to the Knoll, the campus trysting place, and we sat down under an oak, and she looked at me expectantly: "What are we going to talk about?" she asked.
"I will not," she replied.
"True, true," said Polly, nodding her head. "Did you see the movie? Oh, it just knocked me out. That Walter Pidgeon is so dreamy. I mean he fractures me."
"I agree," said Polly earnestly. "I mean exercise is wonderful. I mean it builds the body and everything."
I ground my teeth. I was not Pygmalion; I was Frankenstein, and my monster had me by the throat. Frantically I fought back the tide of panic surging through me. At all costs I had to keep cool.
"Oo, terrif," she replied. One thing I will say for this girl: You would go far to find another so agreeable.
"Polly," I said when we next sat beneath our oak, "tonight we will not discuss fallacies."
"Know any more fallacies?" she asked breathlessly. "This is more fun than dancing even."
I winced, but went bravely on. "First let us examine the fallacy called Dicto Simpliciter."
Seated under the oak the next evening I said, "Our first fallacy tonight is called Ad Misericordiam."
He complied. The coat bunched high over his ears and dropped all the way down to his shoe tops. He looked like a mound of dead raccoons. "Fits fine," he said happily.
She thought this over for a minute and decided she liked it. "Magnif," she said.
"Really?" said Polly, amazed. "Nobody?"
"Aw, gee," she said, disappointed.
I frowned, but plunged ahead. "Here's an example of Contradictory Premises: If God can do anything, can He make a stone so heavy that He won't be able to lift it?"
"It's just been a casual kick - just a few laughs, that's all."
"I should have known it," he cried, pounding his temples. "I should have known they'd come back when the Charleston came back. Like a fool I spent all my money for textbooks, and now I can't get a raccoon coat."
"Ad Misericordiam," said Polly.
It must not be thought that I was without love for this girl. Quite the contrary. Just as Pygmalion loved the perfect woman he had fashioned, so I loved mine. The time had come to change our relationship from academic to romantic.
"I know somebody just like that," she exclaimed. "A girl back home - Eula Becker, her name is. It never fails. Every single time we take her on a picnic ---"
"Well, I do," he declared. "I'd give anything for a raccoon coat. Anything!"
"All the Big Men on Campus are wearing them. Where've you been?"
"Where are you going?" asked Petey.
"You did."
"Not a thing," said I.
"Hypothesis Contrary to Fact," she said instantly.
Intelligent she was not. In fact, she veered in the opposite direction. But I believed that under my guidance she would smarten up. At any rate, it was worth a try. It is, after all, easier to make a beautiful dumb girl smart than to make an ugly smart girl beautiful.
"Yes, it's awful," I agreed, "but it's no argument. The man never answered the boss's question about his qualifications. Instead he appealed to the boss's sympathy. He committed the fallacy of Ad Misericordiam.
"I'll never do it again," she promised contritely. "Are you mad at me?"
"All right. Let's try Contradictory Premises."
"Raccoon," he mumbled thickly.
"If you can forget Mr. Pidgeon for a moment," I said coldly, "I would like to point out that the statement is a fallacy. Maybe Madame Curie would have discovered radium at some later date. Maybe somebody else would have discovered it. Maybe any number of things would have happened. You can't start with a hypothesis that is not true and then draw any supportable conclusions from it."
I watched her closely as she knit her creamy brow in concentration. Suddenly a glimmer of intelligence - the first I had seen - came into her eyes. "It's not fair," she said with indignation. "It's not a bit fair. What chance has the second man got if the first man calls him a liar before he even begins talking?"
I handed her a handkerchief and tried to keep from screaming while she wiped her eyes. "Next," I said in a carefully controlled tone, "we will discuss False Analogy. Here is an example: Students should be allowed to look at their textbooks during examinations. After all, surgeons have X rays to guide them during an operation, lawyers have briefs to guide them during a trial, carpenters have blueprints to guide them when they are building a house. Why, then, shouldn't students be allowed to look at their textbooks during an examination?"
"You're darn right," she said with a vigorous nod.
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With an immense effort of will, I modulated my voice. "All right," I said. "You're a logician. Let's look at this thing logically. How could you choose Petey Bellows over me? Look at me - a brilliant student, a tremendous intellectual, a man with an assured future. Look at Petey - a knothead, a jitterbug, a guy who'll never know where his next meal is coming from. Can you give me one logical reason why you should go steady with Petey Bellows?"
I hid my exasperation. "Polly, it's a fallacy. The generalization is reached too hastily. There are too few instances to support such a conclusion."
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"They ought to put Walter Pidgeon in more pictures," said Polly. "I hardly ever see him anymore."
"Listen closely," I said. "A man applies for a job. When the boss asks him what his qualifications are, he replies that he has a wife and six children at home, the wife is a helpless cripple, the children have nothing to eat, no clothes to wear, no shoes on their feet, there are no beds in the house, no coal in the cellar, and winter is coming."
"Would you like it?" I asked.
She quivered with delight.
"Well, Polly," I said, forcing a smile, "you certainly have learned your fallacies."
"And who taught them to you, Polly?"
I shrugged. "Okay. If you don't want the be in the swim, I guess it's your business."
"Petey, why? Look at it rationally. Raccoon coats are unsanitary. They shed. They smell bad. They weigh too much. They're unsightly. They ---"
"Can you mean," I said incredulously, "that people are actually wearing raccoon coats again?"
"I think she's a keen kid," he replied, "but I don't know if you'd call it love. Why?"
"In the library," I said, naming a place not frequented by Big Men on Campus.
"Of course you are. Because when the premises of an argument contradict each other, there can be no argument. If there is an irresitible force, there can be no immovable object. If there is an immovable object, there can be no irresistible force. Get it?"
"Of course," she replied promptly.
"Listen: If Madame Curie had not happened to leave a photographic plate in a drawer with a chunk of pitchblende, the world today would not know about radium."
"Your girl," I said, mincing no words.
"Fire away," she said with an airy wave of her hand.
I sat down in a chair and pretended to read a book, but out of the corner of my eye I kept watching Petey. He was a torn man. First he looked at the coat with the expression of a waif at a bakery window. Then he turned away and set his jaw resolutely. Then he looked back at the coat, with even more longing in his face. Then he turned away, but with not so much resolution this time. Back and forth his head swiveled, desire waxing, resolution waning. Finally, he didn't turn away at all; he just stood and stared with mad lust at the coat.
"That's right," I murmured.
"Look," I said to Petey when I got back Monday morning. I threw open the suitcase and revealed the huge, hairy, gamy object that my father had worn in his Stutz Bearcat in 1925.
"Raccoon?" I said, pausing in my flight.
"Nothing, nothing," I said innocently, and took my suitcase out of the closet.
One afternoon I found Petey lying on his bed with an expression of such distress on his face that I immediately diagnosed appendicitis. "Don't move," I said. "Don't take a laxative. I'll get a doctor."
That did it. I leaped to my feet, bellowing like a bull. "Will you or will you not go steady with me?"
I had long coveted Polly Espy. Let me emphasize that my desire for this young woman was not emotional in nature. She was, to be sure, a girl who excited the emotions, but I was not one to let my heart rule my head. I wanted Polly for a shrewdly calculated, entirely cerebral reason. I was a freshman in law school. In a few years I would be out in practice. I was well aware of the importance of the right kind of wife in furthering a lawyer's career. The successful lawyers I had observed were, almost without exception, married to beautiful, gracious, intelligent women. With one omission, Polly fitted these specifications perfectly.
"I still think it's a good idea," said Polly.
"No," she confessed. "But this is marvy. Do more! Do more!"
"Logic."
"Have you got a handkerchief?" she blubbered.
I deposited her at the girl's dormitory, where she assured me that she had had a perfectly terrif evening, and I went glumly home to my room. Petey lay snoring in his bed, the raccoon coat huddled like a great hairy beast at his feet. For a moment I considered waking him and telling him that he could have his girl back. It seemed clear that my project was doomed to failure. The girl simply had a logic-proof head. But then I reconsidered. I had wasted one evening; I might as well waste another. Who knew? Maybe somewhere in the extinct crater of her mind a few embers still smoldered. Maybe somehow I could fan them into flame. Admittedly it was not a prospect fraught with hope, but I decided to give it one more try.
"Two men are having a debate. The first one gets up and says, 'My opponent is a notorious liar. You can't believe a word that he is going to say'... Now, Polly, think. Think hard. What's wrong?"
"Sounds yummy," was Polly's reaction.
I consulted my watch. "I think we'd better call it a night. I'll take you home now, and you go over all the things you've learned. We'll have another session tomorrow night."
"How cute!" she gurgled.
"Pshaw," she murmured, blushing with pleasure.
"That's right. So you do owe me something, don't you, my dear? If I hadn't come along you would never have learned about fallacies."
I fought off a wave of despair. I was getting nowhere with this girl, absolutely nowhere. Still, I am nothing if not persistent. I continued. "Next comes Post Hoc. Listen to this: Let's not take Bill on our picnic. Everytime we take him out with us, it rains."
"Polly?" he asked in a horrified whisper. "You want Polly?"
"Do you," I asked, "have any kind of formal arrangement with her? I mean are you going steady or anything like that?"
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I had my first date with Polly the following evening. This was in the nature of a survey; I wanted to find out just how much work I had to do to get her mind up to the standard I required. I took her first to dinner. "Gee, that was a delish dinner," she said as we left the restaurant. Then I took her to a movie. "Gee, that was a marvy movie," she said as we left the theater. And then I took her home. "Gee, I had a sensaysh time," she said as she bade me goodnight.
"No. We see each other quite a bit, but we both have other dates. Why?"
He leaped from the bed and paced the room. "I've got to have a raccoon coat," he said passionately. "I've got to!"
"Hasty Generalization," said Polly brightly.
"Oh yes!" he cried, clutching the greasy pelt to him. Then a canny look came into his eyes. "What do you want for it?"
"My dear," I said, favoring her with a smile, "we have now spent five evenings together. We have gotten along splendidly. It is clear that we are well matched."
A tear rolled down each of Polly's pink cheeks. "Oh, this is awful," she sobbed.
"Tell me some more of this keen stuff," she said eagerly.
"Petey," I said, "are you in love with Polly Espy?"
"I may do better than that," I said with a mysterious wink and closed my bag and left.
"By all means," she urged, batting her eyelashes eagerly.
"Nuts," I muttered. Doggedly I pressed on. "Next we'll try Hypothesis Contrary to Fact."
"Is it a deal?" I asked, extending my hand.
"Try on the coat," said I.
"There now," she said enthusiastically, "is the most marvy idea I've heard in years."
"Anything," he affirmed in ringing tones.
I perceived that his trouble was not physical, but mental. "Why do you want a raccoon coat?"
One more chance, I decided. But just one more. There is a limit to what flesh and blood can bear. "The next fallacy is called Poisoning the Well."
Do you understand?"