Thought Itself

In his Categories, Aristotle uses white and red as his examples of passive qualities, and says, “All colors, like white and black, are qualities as well and passive…  We give them that name from the fact that they spring from affections or passions.  There are numerous changes of color that clearly arise from affections.  When men are ashamed, then they blush; when alarmed, they turn pale and so on.  So much is this really the case that, I think, when a man is by nature disposed towards shame or alarm as arising from a certain concomitance of bodily elements in him, we may not unfairly conclude that he takes on the corresponding color.” (9b 10-20)  Aristotle says that there are temporary passing states of character, which are different from enduring dispositions.  He uses the example of anger as a temporary passionate state and anger as an enduring condition of madness, like the enduring rudeness and foolishness of the Mad Tea Party, who are unchanging in time.

In his On Interpretation, Aristotle uses white as his example of a quality that can be affirmed or denied with the four forms of proposition, and to illustrate what he means by contradiction, a central topic: “When the subject of two propositions is one and the same but the affirmative proposition clearly indicates that the subject is taken universally, then negative proposition, that the subject is not taken universally, I call contradictorily opposed.  Examples are, ‘Every man is white,’ ‘Not every man is white’ and the like, or, again, we have, ‘Some men are white,’ to which,  ‘No man is white’ is opposed in the manner of which I am speaking.” (17b 15-20)

Boole and Carroll, like many today, use the word white to mean European in ethnicity, a more permanent condition of complexion and color than Aristotle’s passive, temporary states of health, just as Hindu upper-caste Brahmins have identified the color white with their higher caste and complexion, the color black to refer to those of lower caste, and the British have referred to Indians as blacks in general, as they and Americans do Africans.  When Aristotle says Socrates was a white man as a central example of logic, he likely means that Socrates is old or sick, and certainly didn’t mean that Socrates was white in ethnicity, as are Germanic tribes, as ancient Greeks did not use the word this way.

Boole follows Aristotle’s example and uses white as a basic example of a class of things that are similar, saying, “Thus, if x alone stands for ‘white things,’ and y for ‘sheep,’ let xy stand for ‘white sheep…’”  but Boole proceeds to use white in the way of enduring complexion and ethnicity, which isn’t Aristotle but overlaps with his more temporary use of this color as character, when Boole, attempting to ground Aristotelian logic in algebraic mathematical expressions, expresses, “European men and European women,” as z(x + y), with z as European, x as men, and y as women, “All men except Asiatics,” as x – y, with x as men and y as Asiatics, and “White men, except white Asiatics,” as z(x – y), possibly referring to the Brahmins of India. (II.11)

Carroll uses the colors white and red several times in Wonderland and the Looking Glass as Aristotle does, to signify affections and passions as states of character, the extremes of too weak, pale and white, and the extreme of too brash, flush and red.  Carroll also pairs these affections with the positions of childhood and adulthood, passionate subject and reasoning ruler, several times in both books.  In the beginning of Wonderland, Alice thinks of making a white daisy chain, falls asleep and follows the White Rabbit, and in the end of Wonderland Alice disrupts the King of Heart’s trial, wakes up and sweeps falling red leaves from her face that she mistook for the playing cards rising up against her.  When she reaches the overly general in the garden of the Queen of Hearts, the first thing she sees is white roses painted red, and the Queen grows red in the face as she demands Alice’s execution.  Many of Carroll’s favorite poets spoke of the purity of childhood, and in the Looking Glass the White Queen is characterized as a carefree child, with Alice pinning her shawl for her, and the Red Queen is characterized as a strict governess.

Logicians who follow Aristotle, like Boole and Carroll, have taught that the propositions All A is B and No A is B contradict each other, and can’t both be true at the same time in the same way, just as Some A is B and No A is B contradict each other.  The two universal propositions at the top of the Square of Opposition contradict each other, such that, as Boole and Carroll both explain, the propositions All men are white and No men are white can’t both be true at the same time.  The positive universal proposition also contradicts the negative particular, and the negative universal contradicts the positive particular, from corner to diagonally opposite corner, such that if All men are white then it is contradictory to assert Some men are not white, and if No men are white it is contradictory to assert Some men are.

All of this was central and basic to the work of Boole, Carroll, and then Venn, who drew his famous circular diagrams to teach these Aristotelean lessons visually to students of all majors in an introductory logic course.  If Circle A is entirely inside Circle B, such that we can say All A is B, then it can’t be that Circle A is entirely or partly outside of B, so we can’t say without contradicting ourselves that No A is B, nor that Some A is not B, unless something changes.  Similarly, if Circle A is entirely outside Circle B, such that we can say No A is B, then it can’t be that Circle A is entirely or partly inside of B, so we can’t say without contradiction that All A is B nor that Some A is B.  In this way Venn visually presented the system of Aristotle’s four forms of proposition and syllogistic argumentation.

In The Laws of Thought, Boole speculates that if we were a species that split things into threes rather than twos, with trichotomies rather than dichotomies, the laws of human thought would be completely different.  When I was a small child, I listened to a Schoolhouse Rock record about multiplication tables, and the song about multiples of twelve told me as an amazed child that just as these twelve-fingered aliens would have an eleventh and twelfth finger, they would have a tenth and eleventh digit, much as a finger is a digit used for counting, a single symbol for ten and eleven, rather than our two, just as we have a single symbol that stands for the quantities of eight and nine, which aliens with eight fingers might represent with two symbols.  If these two alien species somehow came to the same numeral symbols as much of humanity did in the convergence of Indian, Islamic and European mathematics, the twelve-fingered would represent our “10” and “11” as single symbols we don’t use at all and would represent twelve as “10”, with our two symbols, and likewise the eight fingered aliens, with four on each hand.

If we did not have words intertwined with things, feelings and thoughts, we would not have the thoughts that we have, and if we were not dichotomous beings, we would not have the Square of Opposition, nor words that form pairs of opposites such as all and none, some and some not, without.  As Boole points out, we could be trichotomous beings that feel things are good, bad and zerblat, which is neither good nor bad, and not neutral, as it is opposed to both and its own thing.  Because we are creatures of dichotomy, all things are made up of the classes of men and not men together, and Boole says, “a class whose members are at the same time men and not men does not exist… it is impossible for the same individual to be at the same time a man and not a man,” and follows with the Aristotelian example Animals are either rational or irrational.

Lewis Carroll’s conjunctive White Rabbit is quite human and beast, and so, according to Aristotle, is impossibly a rational and irrational animal in the same individual, overly some and some, too inclusive of opposites to be real, and so is imaginary and fantastic.  Boole says we use the conjunctive words and and or permissively and strictly, equivalent to the combination of classes when permissive and the exclusive choice between classes when strict, and permissive and strict reflect the two colors of complexion Aristotle mentions.  We say x and y and x or y to mean what is both x and y when permissive and mean what is either x, or y, but not both, what is called an exclusive or by later logicians.  We are even told, in the opening of Wonderland, that a White Rabbit with pink eyes runs by, with red and white mixed together as some and some, in the eye of the Rabbit.

The White Rabbit is a strange sort of addition problem, a kind of conjunction, the adding of human reason to beast, which Aristotle argues is what we ourselves essentially are, and so is impossible in the case of a rabbit, who lacks what makes us human.  At the end of Alice’s adventures we find ourselves with Alice between the excessively inclusive and exclusive White and Red Queens, and they test her on whether or not she can do sums.  John Stuart Mill, whose work on logic Carroll owned, and who is said to be the most influential philosopher in Britain as Carroll studied logic and wrote Wonderland, argued that we learn logic and math through everyday practices of gathering and dividing objects, not from internal rules of logic.  The Queen of Hearts’ game of croquet similarly lacks rules and turns, like logic in real life, and Alice is tested in gathering and dividing things in everyday life, which she considers odd to call sums.

After praising Aristotle and laying out his examples of white sheep, men and Asiatics, Boole says that his work is designed to prove two positions: “First, That the operations of the mind… are subject to general laws.  Secondly, That those laws are mathematical in their form, and that they are actually developed in the essential laws of human language.” (III.11)  Whether or not Carroll believed this, Wonderland seems to supply counterexamples that contradict Aristotle, as well as Boole, as the characters who rule Wonderland, and later the Looking Glass, contradict Alice continuously, and hardly rule a coherent empire based on common purpose and form.  The Queen of Hearts’ croquet game, which doesn’t seem to have regular rules or turns to Alice, portrays the human world, British politics and history as a highly illogical affair, and Carroll mocks the insanity of politics and history throughout his fictions and works on logic.

Carroll owned several works by John Stuart Mill about logic and several other subjects, including the subjection of women, and Mill wrote: “Now I cannot wonder that so much stress should be laid on the circumstances of inconceivableness, when there is such ample experience to show that our capacity or incapacity of conceiving a thing has very little to do with the possibility of the thing in itself, but is in truth very much an affair of accident, and depends on the past history and habits of our minds.”  Whether or not we live in a regal, logical, law-abiding universe with Boole or a chaotic world of bloody, unruly politics like Wonderland, Carroll presents human logic, rules and authority as good and bad, as enabling but abusive, absurd authority figures who are logically operative, explaining their positions to Alice with their own logics and purposes, but in ways that are fantastic and absurd to Alice and us, embodying ideal, impossible extremes which may not be able to exist but Carroll can create with his imagination, imagining and then creating the impossible, as Mill suggests we do.  Carroll hopes that Alice and all of us keep the light of childhood alive in our adult selves, remaining creative and imaginative, as absurd examples can be highly instructive, as well as memorable.