Logic: Aristotle & The Syllogisms

Aristotle (380-320 BCE) was not originally from Athens, but he traveled there to study with the great Athenian philosopher Plato.  After Plato’s death (350 BCE) Aristotle left Athens, and then was offered the position of tutor to Alexander the Great.  As Alexander went on his campaigns of conquest, Aristotle returned to Athens and founded a school like the Academy of his teacher Plato named the Lyceum.  Upon Alexander’s death, Aristotle found himself unwanted in Athens and fled, as he was not Athenian and the Athenians were not too pleased to have been conquered by Alexander in the first place.

Aristotle is often praised as the father of Logic, but as we have seen with the Indian logicians it is far more reasonable (and logical) to say that Aristotle developed theories of debate and seeking truth just as earlier thinkers from many different cultures had done.  He is also called the father of the scientific method, but this is quite unobservant of the similarities his theories have with the cosmology shared by much of the ancient world (including Egypt, India and Persia).  He was not, in fact, primarily interested in logic or debate, but rather in cosmology, particularly what we would call today psychology, physics and biology.  In Indian terms, Aristotle was more like Kanada than Gotama.  The book he was most famous for up through the middle ages in Islamic lands and Christian Europe was his On The Soul (a psychology and biology text), but this book is hard to find in print today.

There are four of his texts that concern logic and debate: the Categories, On Interpretation, Prior Analytics and Posterior Analytics.  This first lecture will cover the Categories and On Interpretation, and the second lecture will cover the Prior and Posterior Analytics.


The Categories

Starting with the Categories we notice that Aristotle believes in the power of speech.  This lines up with the Zoroastrian and Abrahamic traditions.  Order is spoken downward, from the eternal mind into the particular beings or substances of changing matter the same way that a leader issues orders to subordinates and the mind issues orders to the body and the limbs.  Thus, ‘predicated’, something being said of a subject, is a central concept.  Aristotle assumes that things have purposes according to their natures, and these purposes correspond to concepts that can be put into words.

Aristotle believes that we must observe nature and say what can be said of things, based on their similarities and differences (the same method of his teacher Plato and others).  He believes that what can be said that is necessary of the thing must be what truly is.  Thus, as example, he says that ‘animal’ can be said, can be predicated, of both man and ox, just as ‘man’ and ‘animal’ can be predicated of any human individual (male, that is).

A Genera or Genus is the family to which a thing belongs.  Today, we think of this as the concept.  This is paired with Species, the sub group of the family.  He says (80) that if you are giving an account of a particular tree, you would say more with the species of tree than you would of the genus ‘plant’.

Aristotle gives us 10 categories, which he claims are in no way composite, meaning they are completely separate categories from each other, ‘categorically separate’: Substance, quantity, quality, relation, place, time, position, state, action, and affection.  Note that we saw several of these in Kanada, and that ‘relation, place, time, position, state, and affection’ are all quite confusingly interrelated, even though Aristotle says that they are all categorically separate.
Substances ‘underlie’ everything else.  This has led some to call Aristotle a materialist, but this is not so.  He thought there were motions above (mover) , immobile matter (the moved) below, so the substances are material, the receiver of motions (like Plato’s forms).  Aristotle argues that substances simply are, while ‘white’ or ‘beautiful’ or ‘warm’ can be more or less so in a thing.

(81) “The distinctive mark of substance is that it can admit contrary qualities.  Thus, a color cannot be both white and black, nor can the same act be good and bad: this is true for any non substance, but substances can at different times be white and black…The same individual person is at one time white, at another black, at one time warm, at another cold, at one time good, at another bad.”  (Note that by “white person” Aristotle means a pale or elderly person and nothing more, certainly not race).

Thus ‘he is sitting’ can be true, then false, then true of a person.  Aristotle denies that things can possess contrary qualities (be both white and black) or be in contrary states (good and evil) at the same time.  This is a major positivist point that any skeptic, before and after Aristotle, would deny.  Heraclitus, a major opponent that Aristotle argues against, thought that things were good and bad by perspective and positioning and so things can have contrary qualities and be in contrary states at the same time.  Aristotle completely denies the possibility of this: “If, then, someone says that statements and opinions are capable of admitting contrary qualities, his contention is unsound”.  For Aristotle, a person cannot be healthy and sick at the same time, but it can be one at one time, then another at another time.  (What if you have a healthy liver, but an eye infection?  What if you then get a liver problem?)


On Interpretation

Aristotle says that we must limit our discussion to propositions, sentences true OR false exclusively.  He argues that prayers and other requests are neither true nor false because they do not guarantee whether they will be fulfilled or not.  An affirmation is a positive statement of something, a denial a negative statement.  Thus, ‘I exist’ is positive, whereas ‘I do not exist’ or ‘I doubt that I exist’ are negative.

The Square of Opposition is a central concept for the ideas that follow.  In fact, Aristotle does not mention the concept himself but medieval Europeans created the square as a way of organizing his ideas.  In the same way, Aristotle did not use algebraic notation but his syllogisms which we will focus on in the next lecture were put in symbolic forms by Muslims and Christians in the middle ages to make them simpler to digest.

A says there are universals and individuals.  This corresponds to the eternal sphere and the temporal sphere.  Universals are the forces and motions of the stars and planets above, individuals are the particular substances acted upon on and in the earth below.  We saw this before in Kanada and Gotama’s texts.  Cows is the universal group of all cows, while a particular cow is an individual.  Thus, ‘All people are pale’ and ‘No people are pale’ is a universal proposition, whereas ‘Some people are pale’ and ‘Some people are not pale’ are particular propositions.

Notice that ‘men are white’ becomes the central example used, and how this can be misperceived by later cultures.  Muslims and Christians mistook this to be speaking about themselves equally, and unfortunately Aquinas in the 1200s uses “Socrates was a white man” as a central example for teaching Aristotle’s On Interpretation.  Yes, Aquinas is thinking that he and Aristotle are “white men”, and that this refers to the European race.  Muslim logicians assumed that Aristotle was a “white” Asiatic man like themselves earlier, and Aquinas is getting his Aristotle from Persians like Al Farabi and Al Ghazali.
To come back to the subject, Aristotle tells us (and this is critical for understanding the Square of Opposition) that particular propositions can contradict each other but both be true, whereas universal propositions can not contradict each other and both be true.

Consider the four propositions, “I like all cows”, “I dislike all cows”, “I like some cows”, and “I dislike some cows”.  The first two are universal propositions, as they refer to the group of all cows, while the second two are particular propositions, as they only refer to some individual cows.  These propositions correspond to the four corners of a square.  The top is universal (All and None), the bottom is particular (Some and Some Not), the left is positive (All and Some), and the right is negative (None and Some Not).  If we want to put the four corners in an algebraic form with X and Y, they would be:  All X is Y, No X is Y, Some X is Y, Some X is not Y.

Aristotle notes that the cross corners cannot both be true, and neither can the top two corners (ALL and NONE).  Also, he argues that one or the other of the bottom must be true, but there are counter examples to this (like we saw with Nagarjuna’s neither Y or not Y).

At end of the reading, Aristotle gets himself into another problem given what he has said.  He argues that he has a problem with things that are neither y nor not y, with the example of neither good nor bad person.  Is ‘x is good’ the contradiction of ‘x is bad’ or ‘x is not good’?  If x is neither good nor bad, the first is false, the second false, but the third is true.  This is an interesting problem, as it shows two sets of contraries.  Good and bad are contraries, but they are BOTH contrary to neither good nor bad (neutral, like Swiss).  Aristotle says, strangely, that ‘not good’ is MORE REALLY FALSE than ‘bad’, the contrary quality.  This seems to reverse his position from before, where he was saying that if something is good, it can be bad at some other time, but cannot be bad, because he is now saying that it is more that it cannot be good.  He says, ‘it SEEMS more contrary’.  Aristotle has already said that prayers (also commands) are neither true nor false, as are statements about the future (‘There will be a sea-battle tomorrow’).  Are these, then, more the contradiction of truths?  Or are false statements more?  Now Aristotle seems to be saying that True means not not true, and not false is simply a byproduct, an accident, even though the duality of true and false is central to his system.

Aristotle has founded everything on two things, what is later called the PRINCIPLE OF BIVALENCE (a proposition must be either true or false, not neither true nor false) and the PRINCIPLE OF NONCONTRADICTION (a proposition must be true or false, not both).  We will see with later European thinkers that positivism tends to support these principles while skepticism denies them.


INCLUSIVE OR VS. EXCLUSIVE OR

The top and bottom of the square are a good way of introducing the concept of Exclusive OR and Inclusive OR which will become important for understanding how truth tables function for the second half of the course after the midterm.

There are two different ways that the conjunction OR can function: inclusively and exclusively.  Let us say you are at a buffet, and the sign says, “You can have eggs, toast, bacon, soup, or salad”.  At a buffet, you can have as much of any number of things as you want, so the OR in the sign is being used INCLUSIVELY here.  Now let us say someone is buying you a car, and says “You can have an A, a B or a C”.  Since you only get to have one item, the OR is being used EXCLUSIVELY by the generous person who is buying you a (single) car.  When you can have your choice of more than one, OR is used inclusively.  When you can have ONLY one choice, not more, OR is used exclusively.

Notice that the top of the Square of Opposition, the Universal and General side, functions like an exclusive OR because both “All X is Y” and “All X is not Y” cannot both be true.  If all trees are green, then it can’t be that all trees are not green and vice versa.  The bottom of the Square of Opposition, the particular and individual side, functions like an inclusive OR because both “Some X is Y” and “Some X is not Y” can’t both be false but one, the other or both can be true.  If some trees are green, it is possible that some trees are not green.

Notice also that positivist and categorical thinking (black & white) functions like the top of the square, and skeptical and relative thinking (grey between black & white) functions like the bottom of the square of opposition.  Aristotle, a positivist thinker, wants universal all or nothing truths to have necessary and certain knowledge, while skeptical thinkers (like Heraclitus from Greece who we will examine after Aristotle) want relative some and some not truths to have perspective and wisdom.


SWITCHING TERMS:

Aristotle notes that SOME and NONE can have terms switched, but not ALL.  If some X is Y, then some Y is X (ex: if some trees are green things, then some green things are trees, and if some people are secret agents, then some secret agents are people).  Likewise, if no X is Y then no Y is X (if no trees are happy, then no happy things are trees, and if no planets are friendly, then no friendly things are planets).  However, if all X is Y it is not necessarily the case that all Y is X (ex: if all humans are animals, it does not mean that all animals are humans, and if all trees can scream it does not follow that all things that can scream are trees).


DEMONSTRATION VS. DIALECTIC

Aristotle says that true science or knowledge starts from starts from first principles to deduce necessary conclusions, and he says that this originated in ancient Babylon (modern day Iraq).  While he includes consideration of some and some not, notice that nothing is known certainly about X if we only know that some X is Y or not Y.  To know something certain about X, we would have to know that all X is Y or no X is Y.

Aristotle describes the difference between DEMONSTRATION, which starts from certain principles to deduce and conclude additional certain principles (If A, B, and C, therefore D) and DIALECTIC, which argues back and forth about a thing to see which side is more certain (Is A B or not B, just like the Nyaya Form of Debate).  While his teacher Plato thought dialectic was the ultimate device for achieving certain knowledge (like Hegel, whose dialectics we will learn about in the second half of the course) Aristotle believed that demonstration is superior to dialectic even though he uses both throughout his texts.  While some have said this makes Aristotle the first scientist, he himself believes that it originated in Babylon and it is also true that scientific study and daily reasoning make constant use of both forms.


THE FOUR PERFECT FORMS OF SYLLOGISM:

Aristotle presents us with many forms of argument that can be used in debate, but he only believes that the first four require no additional conditions, outside inferences or evidence.  For this reason, Aristotle’s four “perfect” syllogisms were studied as the forms of logic up until Wittgenstein replaced them with truth tables.  Aristotle does not put it in the easiest way, so we will reorder this like the Nyaya proof to make it easier to digest and recreate by keeping the order of A-B, B-C therefore A-C.  Notice there is one for each of the four corners of the square.

BARBARA, the Positive Universal Syllogism:
If All A are B, and All B are C, then All A are C.
If all humans are animals, and all animals are alive, then all humans are alive.
In the Venn diagram form, if a circle A is entirely within a circle B, and this circle B is entirely in a third circle C, then circle A must be entirely inside circle C.

CELARENT, the Negative Universal Syllogism:
If All A are B, and No B are C, then No A are C.
If all humans are animals, and no animals are made of stone, then no humans are made of stone.
As a Venn diagram, if A is entirely within B, and no B is inside C, then no A can be inside C.

DARII, the Positive Particular Syllogism:
If Some A are B, and All B are C, then Some A are C.
If some animals are humans, and all humans are funny, then some animals are funny.
As a Venn diagram, if some A is inside B and all B is inside C then some A must be inside C.

FERIO, the Negative Particular Syllogism:
If Some A are B, and No B are C, then Some A are not C.
If some animals are humans, and no humans are reptiles, then some animals are not reptiles.
As a Venn diagram, if some A is in B and no B is in C then some of A is outside C.

Aristotle believed that you could derive pure knowledge from chaining these.  He argues in the text that since the Scythians have no vines, thus no grapes, thus no intoxication, thus no flute players.  He gives another example: If it is metal, then it will cut, Hatchets are made of metal, therefore hatchets will cut.  He argues, just like Gotama, that the eternal is uncreated and the temporal is created, and, just like Kanada, that lightning is fire that passes down into water which then rises with the fire from sun up into the clouds until it falls as rain.

In the 1600s, Sir Francis Bacon rejected the syllogism as fallible, just as Islamic scholars and scientists had before.  Aristotle’s forms thus became forms of logic but were too simple for science.  Consider that all metal things do not cut, nor do all knives, the butter knife being an example of something metal and a knife that does not cut.

Aristotle sometimes goes back on his earlier statements and gives us examples when things that are normally universal and certain can be conditional, can be different in certain situations and circumstances.  He says that it is never right to kill your father, but among the Triballi tribe, the gods sometimes demand it.  Since the gods are one’s super-parents and one’s obligations to them supersedes one’s obligations to ones parents, he says that the Triballi rightly sacrifice their fathers.  Notice that Aristotle believes that the polytheistic gods are real and that this is logical.

Interestingly Aristotle like the Nyaya provides us with defenses against syllogisms.  He says that in order to avoid having a syllogism drawn against one’s own argument, one should not let the opponent give the same term twice over.  This is an interesting place where arguing what is right blends with tactics and strategy for winning debates.  If one’s opponent argues that A is B, and B is C, therefore A is C, one should attack the twice used middle term (B that links A and C in the syllogisms) to attack the syllogism.  For example, if one’s opponent argues the war is American, what is American is good, therefore the war is good, one should argue that the war is only somewhat American or that only some of America is good because America is being used to link the war to the good.  We naturally know to do this in arguing, just like using the forms.

Funny enough, this makes Aristotle look like a skeptic, a person who argues that absolute knowledge is only some and some relative, and he calls skeptics “destroyers” and “no better than plants” in the text.  He says that we should conceal our syllogisms to prevent our opponent attacking our middle terms, which makes him sound like a sophist, someone who argues (like a lawyer) for a living and thus can’t be trusted with true science and knowledge above mere temporary, imperfect and uncertain opinions.