Sunday, February 28, 2010

Logic Lecture March 1: Aristotle and the Analytics

BCC Logic
Eric Gerlach


The last time, we talked about the Square of Opposition. 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.

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).

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.

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.