Logic – Fallacies & Puzzles

The word fallacy comes from the Latin fallacia, an act of deception or illusion, much as Edgar Allan Poe’s detective Dupin does when he dupes and the Hoopoe bird does with its head feathers.  Fallacies are common types of mistakes we make when we use logic, debate and reason, either those we make ourselves or those others make that deceive us.  The subject of fallacies is only sometimes brought up near the end of a formal logic course and it doesn’t often seem to fit together with the logical forms of truth tables and proofs, which is why it is sometimes ignored and sometimes tacked on and out of place, with no bridge between true forms and false errors.

We have already looked at the fallacies of Gautama’s Nyaya school of logic and debate in India and those of Aristotle’s Peripatetic school in Greece, as well as paradoxes and puzzles Hui Shi and Gongsun Long’s School of Names used in China and the jokes found throughout Lewis Carroll’s Alice books that depend on the same fallacies of reasoning.  Now we’ll look at these again and fit them together to form a bigger picture of common ways that reasoning and debate with words, things, feelings and memories does and doesn’t work.

Gautama & The Nyaya Sutra On Fallacies

Gautama said that when the four sources of knowledge, perception, testimony, comparison and inference, are properly in their places there is justifiable truth and certainty, which means that errors and fallacies come about when one or many of these four are out of place, either the perception of the object, the feelings and testimony of the subject, the ways the object, knower and situation is like and unlike comparable situations, and concepts drawn through inference.  The Nyaya catalog problems that each source of knowledge shows us in experience, with perception being primary, the ultimate form of knowledge, certainty and satisfaction.

Perception can be faulty, and the Nyaya give us examples of confusing smoke and dust, a rope with a snake, and hot earth with the mirage of water.  As Vatsyayana argues, agreeing much with later Wittgenstein, thought can be mistaken without using words, as in these examples. Hui Shi’s 14th paradox is a dog can be a sheep, and a dog can be taken for a sheep, particularly in the dark, as Gautama said a man can be mistaken for a pole in the dark, so a dog can be a sheep in the mistaken perceptions, judgements and mind of the beholder even if nowhere else, much as Hui Shi said a wheel doesn’t much touch the ground at any given time, but does indeed in one particular place alone.

The Nyaya say we use words in three ways, referring to individuals, forms and classes, such that the word cow refers to this or that cow, to anything insofar as it is like a cow at all, what we call speaking metaphorically, and to the group of all cows as a class, and Vatsyayana says that many mistakes, doubts and fallacies spring from this triple-use of words.  When I say Cows are nice, I could mean some specific cows but not the whole group, or refer to anything that is cow-like in a particular way metaphorically, whether or not they are in the literal class, or that all cows are nice as a universal or general group, with or without exceptions.

The Nyaya Sutra gives three types of quibbling, which correspond to the three uses of words, quibbling over thing, quibbling over form, and quibbling over class.  Vatsyayana give us examples for each.  For quibbling over words used for things, he says someone could say they had a new (nava) blanket, and someone else could misunderstand the individual word and think they claimed they had nine (also nava) blankets, and point out the false fallacy.

When the Mouse tells Alice and Birds of Wonderland he had not (gotten to the fifth bend of his tail-shaped poem), Alice confuses the word with knot, says she will help to undo it, and the Mouse leaves offended, insisting that Alice is talking nonsense, not because the Mouse sees Alice is mistaken about his word, though she is not quibbling or disagreeing with him about his poem, but because not cannot be undone, as a negative can’t be negated, which is nonsense.  Remember that Carroll, like other mathematicians and logicians of his day, was opposed to the use of negative and impossible numbers, arguing they are nonsense.

Also in Wonderland, the Duchess mistakes Alice talking about the earth’s axis, the geometric line around which the world rotates, for axes, and orders the cook to chop off her head, but the cook stirs the soup and disobeys.  The Duchess, like Humpty Dumpty in the Looking Glass, is exclusively holding power and her own interpretation in her position, which is why she won’t listen to Alice, misunderstands her, orders her execution and is ignored and then opposed by the cook, her servant and underling.

For quibbling over form, Vatsyayana says someone could say, The stands cry out, and someone else could foolishly say Stands can’t cry, as they don’t have feelings, but they are wrong because they fail to understand their opponent is speaking metaphorically and are wrong to take what they say literally.  This does create more than problems with metaphor, as a dead cow does have the form of a cow, as a former cow, but it is not a living thing, as Gautama, Aristotle and their followers likely would classify a cow, as an animal.  Similarly, a three-legged cow has the form of a cow, and is a living cow, but doesn’t have four legs, only three, which is following much but not all of what some would say is the essential form of the cow.

For quibbling over class, Vatsyayana says someone could say, All Brahmins are educated, possibly to construct an example of a Nyaya syllogism, and someone else could wrongly object, Not all Brahmins are educated, because some are only three years old, and just learning to talk!  We could similarly get controversial about whether or not a three-legged cow counts as a full cow or only partly a cow if we define it as a four-legged mammal.  In Wonderland, the Executioner and King argue about whether or not the floating head of the Cheshire Cat can be beheaded, and the King, quite inclusively but ignorantly, says if anything has a head it can be beheaded, separated from its body, but as the Executioner correctly points out there are some things that have heads but no bodies, such as the Cheshire Cat in his current heady state, and things that don’t have bodies can’t have their heads separated from their bodies, whether or not they have a head or several.

For the other fallacies the Nyaya Sutra of Gautama and his school consider, we need to have more than the simple coherence, validity or soundness of an individual argument, as we need the social context of debate between points of view, which an individual can include in a single argument hypothetically, entertaining the possibility by themselves, much as we saw with entering conditionals into proofs which are simply larger conditionals, bars within bars, and dreams within dreams.  Some call these informal fallacies, as opposed to the forms of quibbling and other fallacies that depend on structural mistakes and misinterpretations rather than problems with the situation and context, much as many call the study of logic outside of strict forms informal as well.  It is serious business, but also sometimes casual.

The Nyaya say that silence, incoherent speech, repetition, overlooking other’s mistakes, and sharing the fault are all fallacies, but none of these are mistakes that an individual would encounter with themselves in the ways they do with others outside themselves.  If we are silent, or speak incoherently to ourselves, we are hardly upset or disappointed as we are when we expect something else out of someone else, much as Wittgenstein says we don’t watch our own faces or hands to guess what we will do next as we do with others.  Similarly we can’t help but share the fault with ourselves when we are at fault, so it is hardly a compounded error, a mistake on top of another mistake to point out a fault in our own logic, even if we can be accused of not seeing all of the error in our own position.

Aristotle & The Sophistical Refutations On Fallacies

In Aristotle’s Sophistical Refutations, he lists several fallacies that can be fit together with the forms of quibbling in Gautama’s Nyaya Sutra.  Aristotle argues that fallacies are false because they appear to be real reasoning but are not, much as we say a false front is hollow inside.  Sometimes we give evidence and premises that seems true, but is false, and sometimes we build arguments that seem valid, but are not, and if they are based also on false premises, would be both invalid and unsound, bad in form and content.  We can also give valid arguments that are sound, composed of true premises, that are irrelevant, beyond the structure of the argument itself. Aristotle says that we should study the false arguments of others and find their faults so that we will not be as foolish with ourselves in our own private reasoning and research.

For Aristotle, fallacies were not simply tacked on at the end of studying logic, but concluding the studies of logical forms with the false forms of those specific forms as best as could be done, using examples.  He groups equivocation, amphibology and form of expression apart from accent, composition and division, saying the first are due to double meanings and the second group not due to double meanings, and puts these two groups together as fallacies of language.  All the rest are labeled outside of language, not because they do not involve language, but because Aristotle thinks they are due primarily to causes other than double meanings or other problems with language.  Scholars and logicians do not agree entirely with Aristotle’s classifications or explanations.

The Fallacy of Equivocation is confusing a word that means one thing with meaning another.  This is much like the Nyaya’s quibbling over term, one word meaning two different things.  Aristotle uses the example of drawing the Barbara syllogism, Things that must be are good, and evil must be, therefore evil is good as false, relying on the double meaning of must to mean both what we need and want to survive and what has to be the case whether we want it or not.  It is interesting that Aristotle doesn’t use our English word must, which works better for translating this example than need or has to, though they also come close to Aristotle’s double meaning.

Aristotle gives us another useful example of a sick man getting better, such that we can say The sick man is healthy, meaning the man who was sick is now healthy, pointing to a now healthy man with our words the sick man, such that it appears to sound as if we are saying the same man is both sick and healthy at the same time, the sort of thing Heraclitus would say and Aristotle explicitly condemns.  Of course, Heraclitus might say that a man with an eye problem and an excellent liver is sick and healthy not at different times but in different parts, a problem Aristotle ignores.

The Fallacy of Amphibology is confusing a sentence that means one thing with meaning another, like equivocation, but for whole statements.  This turns on the arrangements of the words meaning more than one thing. For example, at the Mad Tea Party in Wonderland, Alice tells the Mad Hatter that she hasn’t had any tea yet, so she can’t have more tea than she already has, and the Hatter presumes to tell her that she misspoke, that she means to say that she can’t have any less, as it is easy to have more when you haven’t had any.

The joke relies on the words have more meaning different things at different times in the changing situation, as Alice is concerned that the Hatter thinks she already had some, but he is not concerned with time or context or changing places, so he ignores the context and argues he is technically correct about something Alice had no interest in saying.  Why would she inform the Hatter that she can’t have less than none, which is certainly true, but irrelevant, if she wants tea, rather than simply ask for some, or more?

The Fallacy of Form of Expression is for Aristotle a categorical mistake that arises from language treating all things similarly that are of different categories.  Aristotle illustrates this with an example about gambling dice, arguing that if we say What we have and then don’t have we lose, and if you had ten dice and lost one, you had ten and then don’t, then you lost ten dice, this would be confusing the quantity ten with a substance.  Aristotle was critical of both his teacher Plato and Pythagoras for speaking of ideas and numbers as if they are real in themselves without substantive existence.  We lost one, not ten dice, but we did go from having ten to not having ten, as if we lost the ten as a solid object rather than temporary state, but of course, Aristotle would say quantity and state are separate categories themselves, and should not be confused with each other.

The Mad Hatter of Wonderland makes a serious categorical error, according to Aristotle, when he tells Alice that time itself is not an it, but a him, and you don’t want to offend time, or time won’t move or do anything for you, which treats time like a living substance.  The Mad Hatter may well be referring to the Caterpillar, whom Alice has already met, who sits around smoking a hookah, asking Alice who she is, doing nothing for her and giving her little to nothing to go on, other than his mushroom.  It is also arguable that Wonderland opens with Alice making a categorical error when she confuses her desire to be like her sister with the White Rabbit, who embodies her desire as a substance, and isn’t actually real.

The other fallacies of language Aristotle lists seem to be types of amphibology, which lends itself to equivocation, to understanding more than one thing confusingly with the same term.  There are many ways we misunderstand words and their arrangements within arguments, just as we misunderstand arguments and their arrangements in life as a whole.  The Fallacy of Accent is misunderstanding a sentence based on what word is accented, or thought to be accented.  As shown with the various ways the statement I think she should have got the job can be accented, the sentence serves to clarify different questions or issues, such as who thought she should, or that she should, as opposed to someone else, or she should have the job, as opposed to something else, etcetera.

The Fallacy of Composition is wrongly attributing the property of a part to a greater whole.  Aristotle says that five is made of two added to three, but this doesn’t mean that five is both even, like two, and odd, like three, and that a good man could be a terrible carpenter, but this doesn’t mean he is both good and bad in the same way together, as this leads to contradiction.  Aristotle says, bringing amphibology confusingly into it, that if we say What we should know we should learn, and it is good to know evil, so it is good to learn evil, we are similarly making an error of composition, as we are saying learning what evil is includes learning how to do it also.

My favorite examples involve what we can call Wittgenstein’s oven, found in his Lectures and Conversations, which is actually notes on his mid-period seminar lectures by two of his graduate students.  If a human being is carbon based, that doesn’t mean we can cook them down to carbon to understand how they essentially behave, nor can we study water to understand how we behave, even though we are three fifths water ourselves, as we are complex situations made out of carbon, water and many other elements situated together, such that our properties and acts arise out of the whole and not specific elements apart from the others.

The converse Fallacy of Division is wrongly attributing the property of the greater whole to a particular part, the fallacy of composition in reverse.  We can provide examples by inverting the examples for composition. If five is odd, and two and three together make five, this doesn’t mean that two is odd, and if I know how to play chess, this doesn’t mean that the carbon or water in me knows how to play chess itself.  A biologist once wrote that our genes don’t care about our own individual survival in the long run, so why should we care about acting as our genes often make us act? We are much more, as individuals, than our mere DNA, otherwise we can send our DNA to school and claim attendance, or at least partial credit.

Bigotry and prejudice are types of fallacious composition and division.  If I say, “He is a Hindu, and he is a jerk, so all Hindus are jerks”, I have committed the fallacy of composition, judging the group by what is thought of the individual.  Likewise, if I say, “All atheists are immoral, and she is an atheist, so she is immoral”, I have committed the fallacy of division, judging the individual by what is thought of the group.  The Scottish empiricist David Hume argued that all of our beliefs are prejudices and mere habits, not absolute certainties, which would make composition and division, attributing properties too widely beyond the particular and too narrowly within the particular, continuous problems in talking, reasoning and thinking.

Aristotle follows his six fallacies of language with six he says are not based in language but involve it and cause problems in verbal, logical reasoning, and he also says that these twelve fallacies are the only types of errors that are possible, exhausting the set, which is odd, as his categories are ten in number, the errors are twelve, and while Kant said there has been no logic in his time improving on Aristotle, modern philosophers including Kant fail to follow Aristotle’s typologies, while striving for complete typologies of their own.  Wittgenstein would say there can be no complete typology of thought, and if so there probably isn’t a complete typology or zoology of logical problems, but we should, like Aristotle, Kant and Wittgenstein, examine the problems that arise between our minds, words and the world, even if it is endless.

The six fallacies Aristotle catalogs as not based in language are, like their language based brethren, confusingly interrelated and overlapping in nature, and they involve mistaking the place of a statement or argument in the situation as a whole.  The Fallacy of Irrelevant Conclusion, missing the point, is also known as a red herring.  Some say the expression comes from the hunting practice of dragging a fish behind a horse to see if hunting dogs would be easily thrown off the true prey’s trail by the trail of the fish, which distinguishes the truly trained and accomplished hunter.  Some have said the expression arose in the 1920s just after the Russian Revolution when a company spraying red dye into sardine cans to make them look fresh sometimes made one sardine dark red, starting rumors that Communists were secretly poisoning Americans.

Unfortunately for this theory,  Doyle wrote The Red-Headed League in 1891, a mystery case for his famous Sherlock Holmes to solve, and the story is structured by the fallacy of the red herring, which is a terrible pun, as the red-headed are also red-haired.  The red-headed man who is told there is a special well-paying job for him to do several hours each day away from his shop because he has a great head of red hair and a quirky noble left a fortune, possibly a Nigerian prince, that must be spent helping red-haired gentlemen such as himself, is being distracted from the fact that professional thieves are tunneling from his basement beneath the bank across the street every day while he is gone, and his red hair has nothing to do with the position of his shop, but rather with getting him out of the shop each day for hours.  The man is simply red-haired himself, and the thieves are red-herring him one day at a time.

There are also good but unfortunate examples of red herrings that psychologists and informal logicians have found in divorce court proceedings, as often when arguing over money, children and other important things the conversation becomes an exchange of accusations and debates about details in situations that either show what the problem truly is, with the other parent, or can be explained away due to temporary circumstance that particular day, a back and forth psychologists have called internal versus external attribution, which resembles the fallacies of division and composition, attributing the problem to something particular within the individual or to something general outside the individual.

One particular branch of brutal red herrings found in divorce courts and sadly elsewhere as well is the personal attack, which can involve internal or external attribution, or prejudice, or not.  In one way, this is opposite what modern logicians call the fallacy of appeal to authority, as it degrades the individual and the argument rather than supports it communally by relying on tradition and power structures.  For example, You cannot believe a word my opponent says, because she is a communist, Mormon, atheist, aquatic bird, etc, or His scientific theory is questionable, because he is a gambler.

Like with any red herring, it can be debatable whether or not a person’s character has any relationship to their argument, but a genuine and fallacious personal attack occurs when there is an unjustifiable appeal to fear based on the distrust of the group a person belongs to or a particular trait of the person.  Sometimes, it is prejudice to tell someone they shouldn’t speak to a police office, because they are a police officer, and sometimes not, according to several lawyers consulted.

Aristotle lists the Fallacy of Begging the Question, which similarly is stating something questionable, but saying something that requires further evidence rather than stating something that isn’t important.  This fallacy could be inclusive and overlapping with many, such as the red herring previously. If I tell you not to talk to someone because they are a cop, this begs the question as to whether or not I am acting as your personal lawyer judiciously.

The Fallacy of False Cause is misunderstanding something prior as cause when it isn’t, such as thinking the moon causes people to go to sleep.  Conversely, there is the Fallacy of Affirming the Consequent, assuming something prior is a cause when it isn’t, but it often is, such as thinking that if drugs can make people crazy, then all crazy people must be on drugs.  This is much like Tweedle Dum and Tweedle Dee from the Looking Glass, who tell Alice, If you think we’re wax-works you ought to pay, and, Contrariwise, if you think we’re alive, you ought to speak, as they are neither wax-works nor completely real people, so Alice should neither pay nor does she have to speak if she doesn’t choose to.

Finally, Aristotle says there is the Fallacy of the Multiple Question, asking a question that suggests things that either haven’t been stated or aren’t true, also known as a leading question.  The famous unsavory example is, Have you stopped beating your spouse and/or kids?  In a sense, Dum and Dee’s questions are somewhat loaded between the two, as they presume and expect something from Alice either way, unjustly.  It could be argued that Gongsun Long’s white horse as a horse, but also not a horse, is a loaded statement, that could be turned into a loaded question, Is a white horse a horse?  If someone answers one way, or the other, but not also both, in line with Nagarjuna and Wittgenstein’s duck-rabbit, they could be contradicted with the opposite, so the question is loaded as a trick that dupes us easily.

10th & Final Assignment: Create your own examples of each of the following fallacies, using an imaginary example involving one or more individuals: equivocation, form of expression, composition, division, red herring, false cause and affirming the consequent.

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