Logic – Aristotle, Boole & Carroll: My Theory About Alice
My theory is that Lewis Carroll (1832 – 1898), author of Alice’s Adventures in Wonderland (1865), Through The Looking Glass (1871) and The Hunting of the Snark (1876), wrote his memorable silly stories to teach children serious forms of logic while keeping them entertained and engaged. Carroll studied and taught logic at Oxford and followed his famous fictions with The Game of Logic (1886) and Symbolic Logic (1897) to further teach and illustrate the same forms with silly examples.
In this overview, I present the remarkable fit between the ancient lessons of Aristotle (384 – 322 BCE) and the modern logic of Boole (1815 – 1864) with the events and characters of Carroll’s fantasies. Following this overview are my detailed explanations of Wonderland, the Looking Glass, and the Snark to show how the forms of thought Carroll taught make sense of his confusing fictions.
Aristotle’s Ten Categories & The Order of Alice’s Adventures
Aristotle is often called the father of logic, though Aristotle did not say himself that he invented the study of how we use words to reason or debate. Aristotle’s works on logic were collected into a larger work called the Organon (Tool or Instrument).
There are three basic ideas of Aristotle which I argue Carroll used to plot out the events and characters of his stories: the ten logical categories, the central idea of Aristotle’s first book on logic (the Categories), the four forms of proposition, the central idea of Aristotle’s second book on logic (On Interpretation), and balance between extremes, the central idea of Aristotle’s book on ethics (the Nicomachean Ethics).
John Duns Scotus (1266 – 1308) wrote a commentary on Aristotle’s Categories, and both he and William of Ockham (1285 – 1347), the two famous Neoplatonic philosophers and logicians of early Oxford, wrote commentaries on Aristotle’s On Interpretation and taught logic five centuries before Carroll did with his peers in their shadow.
Modern philosophers and logicians, including Kant, Mill, and Boole have ignored Aristotle’s ten types of being as antiquated and arbitrary, and it is not clear whether or not Aristotle intended the text to be a system or general discussion. In his Categories, Aristotle lists and explains ten categories of truth, ten types of things we say are true about things:
- Substance: the material being of this or that thing.
- Quantity: the number or amount of a thing.
- Quality: an aspect of a thing such as “good” or “green”.
- Relations: the interaction of a thing with others.
- Space: the place a thing is in and occupies.
- Time: the duration of a thing and the things it involves.
- Position: the situation of a thing with other things.
- State: the current status of a thing in terms of itself.
- Action: what a thing does to itself or other things.
- Passion: what moves a living thing to this or that action.
Aristotle discusses the highest and primary of the ten first, substance, the basis of truth and being itself, and proceeds only somewhat in his own stated order to illustrate many but not all of the ten, leaving the last few, including lowly passion, largely unillustrated. Aristotle says, “Of the rest, that is, time, place and state, they are so clear that I need say no more than I said at the very beginning – that a state is intended by terms such as being ‘shod’, ‘armed’ and the like, whereas place is intended by phrases like ‘in the Lyceum’ and so forth.”
If we turn Aristotle’s own list of ten categories backwards, starting from the lowest rather than the highest, like looking at a text backwards in a looking-glass mirror, we have: passion, action, state, position, time, space, relations, quality, quantity, and substance. This inverted list fits the order of events and characters Alice encounters in both of her adventures, with the two books mirroring each other, chapter by chapter, so well that it is well worth exploring, and so well that it can be argued, as I do here, that Carroll intended to teach children lessons about logic whether or not they knew how they were being trained during play.
|1: Passion||White Rabbit & Golden Key||Black Kitten & Looking Glass|
|2: Action||Mouse & Pool of Tears||Flowers, Red Queen|
|3: State||Dodo, Caucus Race||Train, Gnat, Fawn|
|4: Position||White Rabbit’s House, Puppy||Tweedle Dum & Dee, Crow|
|5: Time/Space||Caterpillar (Time)||White Queen (Time) & Sheep (Space)|
|6: Relations||Duchess (Relations) Cheshire Cat (Space)||Humpty Dumpty (Relations)|
|7: Quality||March Hare, Mad Hatter||Haigha, Hatta, Lion & Unicorn|
|8: Quantity||Queen of Hearts, Croquet||Red Knight, White Knight|
|9: Substance||Kind Duchess, Gryphon, Mock Turtle||Red & White Queens & Banquet|
|10: Substance||Lobster Quadrangle||Red Queen|
|11: Substance||King of Hearts, Trial||Black Kitten|
|12: End||Alice & Ending||Alice & Ending|
The first chapter of each of Alice’s adventures is about passion, with the White Rabbit and the Black Kitten. The second chapter is about action, with the swimming Mouse and the running Red Queen. The third chapter is about state, with the Dodo’s caucus race and the Gnat on the public train. The fourth chapter is about position, with Alice taking a position in the White Rabbit’s house and between Tweedle Dum and Dee while she’s in the Red King’s dream.
The fifth and sixth chapters of each work are about time, space and relations interwoven together such that the fifth chapter of each book is about time, with the Caterpillar of Wonderland and the White Queen of the Looking-Glass, the sixth chapter is about relations, with the Duchess and Humpty Dumpty, and space, the Cheshire Cat and the White Sheep, does not have its own chapter, its own space as space itself. Rather, in each book, space shares a chapter and space with others, with relations in Wonderland, the Cheshire Cat in the House of the Duchess, and with time in the Looking-Glass, with the White Queen turning into the Sheep.
The seventh chapter of each work is about quality, with the under-qualified Tea Party and the over-performing Lion & Unicorn, and with the Mad Hatter and Hare in both. The eighth chapter is about quantity, with the playing cards of the Queen of Hearts and endless inventions of the White Knight. The remaining chapters of each work are about substance, or lack thereof, with the insubstantial lies of the Mock Turtle leading to the King of Heart’s Trial over tarts, and the test about sums leading to the Queens’ Banquet.
I now use Wonderland to teach the order of Aristotle’s categories and my students say it works as a mnemonic device, one Carroll possibly intended to leave to us for use. If so, Carroll fleshes out the categories that Aristotle doesn’t, and does so with memorable characters that children and adults remember and use. Following this overview is my detailed argument and evidence for my extensive claim.
Aristotle’s Ten Categories & The Ten Who Hunt The Snark
In The Hunting of the Snark there are ten surreal characters who have no names other than their professions, jobs which all mysteriously start with the letter B: the Bellman, Boots, Bonnet-Maker, Barrister, Broker, Billiard-Maker, Banker, Beaver, Butcher and Baker, as listed in the beginning. In his Game of Logic, Carroll used buns, babies, beetles and battledores (an early badminton racket) as his examples of individual things, also known as beings, and he likewise used baked, beautiful, black and broken as his examples of attributes, also called qualities.
In Carroll’s introduction he sarcastically says Alice’s adventures show that as an author he is incapable of nonsense (though they instead show how capable of nonsense he is), and that this new brief but instructive poem, “includes precise arithmetic truth and natural history,” words that apply to Aristotle’s Categories, as well as Aristotle’s work as a whole.
If Aristotle’s ten types of being fit the ten who work jobs that start with B, and are thus b-ing, each in their own way on the hunt, it is certainly possible that the Hunting of the Snark works like a logic puzzle, like the very sort that Carroll designed using tables in his work on logic just after the Snark, and we can solve the puzzle by assigning each character an Aristotelian category. There is evidence from cover to cover, including the images on the front and back covers, that the Bellman, the first character of the story, the captain of the ship and the leader of the adventurous hunt, is time. Guiliano (1996) pointed out the Bellman looks like Father Time and carries a school bell, announcing the beginning and end of lessons.
Using evidence from the Snark and its interconnections with Alice’s adventures, I argue the best candidate for each category and the solution to the puzzle is: the Bellman is time, the Boots is place, the Maker of Bonnets and Hoods, for birth and death, is position, the Barrister who dreams of a pig on trial is relations, the Broker who values the goods is quality, the Billiard-Maker who chalks his own nose is action, the Banker who writes a check for England is state, the Beaver who paces on the deck and knits lace is passion, the Butcher who carves things up, dresses too formally and teaches the Beaver sums is quantity, and the Baker who leaves everything on the beach, wears many layers, bakes brides cake, doesn’t lie, forgets his specific name and fades away, vanishing without a trace in the end is substance, which Carroll believed is mortal, passing with time between the positions of birth and death, like all human ambitions, but unlike the human soul, which Carroll believed was capable of immortality.
Aristotle’s Four Forms of Proposition & The Rulers of Alice’s Two Dreams
In Aristotle’s On Interpretation, the second book of the Organon, Aristotle presents four forms of proposition, four ways we say things are true or not: we say that things are positive and include all, such as All apples are round, positive and include some, such as Some apples are round, negative and exclude all, such as No apples are round, and negative and exclude some, such as Some apples are not round. This pair of pairs form what logicians called the Square of Opposition centuries ago, with two propositions positive, two negative, and overlapping with this, two universal, and two partial.
|Square of Opposition||All – Universal||Some – Particular|
|Positive||All apples are round.||Some apples are round.|
|Negative||No apples are round.||Some apples are not round.|
Aristotle did not systematize his ten categories to Kant or Boole’s liking, and his categories have been largely ignored by logicians since, but Aristotle’s four forms of proposition are the simple, systematic form central to the study of logic for Aristotle, Al Farabi, Avicenna, Averroes, Aquinas, Scotus, Ockham, Leibniz, Kant, Boole, De Morgan, Carroll and many others for centuries in an Abrahamic/Greco-Roman family of overlapping cultures of talking about debate and truth that led to modern formal logic and the Boolean algebra found in telegraph, telephone and computer systems.
Just as Carroll could have imagined Aristotle’s categories as a series of events and characters in his Wonderland to engage and exercise the minds of children, and followed the same series in the sequel, it is possible Carroll imagined Aristotle’s four forms of proposition as the four royal court characters that rule Alice’s dreams to engage children with Aristotle’s four forms of proposition, the four corners of the Square of Opposition, central to ancient Aristotelian logic, modern Boolean logic, and Carroll’s own work and lessons on logic.
In Wonderland, the White Rabbit is characterized as overly inclusive and particular, who worries about the needs of his superiors, and orders Alice into his house to look after his own things, the Duchess is overly exclusive and particular, ignoring the needs of her cook, punishing the cries of her baby, and moralizing about who and what is best or worst, the Queen of Hearts is overly exclusive and general, ordering the complete subtraction of anyone who displeases her in the slightest from her garden and existence, and the King of Hearts is overly inclusive and general, who includes all conflicting testimony and evidence in his court whether or not it matters or makes sense and hates to cross-examine anyone.
The two male royal characters, the Rabbit and King, are overly inclusive, the two female characters, the Duchess and Queen, are overly exclusive. The Rabbit and Duchess, each with their own particular house, the first two royal characters, are overly particular and partial, but not entirely, and the Queen and King of Hearts, the second two royal characters, are overly general and universal, without degree. None of the four are wise, as all four serve as excessive examples Alice finds foolish rejects in the end, waking from her dream. Alice makes her way from the overly particular in the first half of the book to overly general in the second half, and from overly inclusive in the beginning, to and through overly exclusive in the middle, to overly inclusive again in the end, fed up with the entire cast of characters and royal court.
|Aristotle’s Four Forms of Proposition||Wonderland||Looking Glass||Character|
|Positive & Universal – Includes All||King of Hearts||White Queen||Inclusive|
|Negative & Universal – Excludes All||Queen of Hearts||Red Queen||Exclusive|
|Positive & Particular – Includes Some||White Rabbit||White King||Inclusive|
|Negative & Particular – Excludes Some||Duchess||Red King||Exclusive|
Similarly, in the Looking Glass, the Red Queen is overly exclusive and universal, like the Queen of Hearts, who says all ways are hers, which implies none at all are Alice’s, contradicts Alice entirely, and shows Alice the distance she must travel, the Red King is overly exclusive and particular, including Alice in his dream, but silently without interacting with her as she dreams of him, such that neither is entirely real to the other, the White Queen is overly inclusive and general, remembering time both ways and impossible things before breakfast, accepting Alice’s help, turning into a sheep and dashing the full distance of the board, and the White King is overly inclusive and particular, including most but not all of his horses and men to help Humpty Dumpty and nervously encouraging both sides of the battle between the Lion and Unicorn. Again, none are wise, and Alice wakes from her dream frustrated.
If Carroll embodied Aristotle’s four forms of proposition as the royal characters who rule Wonderland and then repeated this in the sequel, much as I argue he did with Aristotle’s ten categories, he did not follow the same order chapter by chapter in both books as he seems to have done with the categories. In both books, each royal character gets their own chapter, and each chapter of a royal character almost always has at least one chapters between it and any other, with one exception, but the chapters do not line up together, and Alice does not encounter them in the same order. In the sequel, she does not work from particular to general and from inclusive through exclusive to inclusive again, but rather the opposite, from exclusive to inclusive, and from general to particular twice. Alice starts on the side of the board with the Red Queen and King, overly exclusive, works her way to the end of the other side of the board past the White Queen and King, to sit between the White and Red Queen at the end, a ruler and queen herself of her own court banquet.
White Daisy Chains & Red Falling Leaves: Aristotle’s Complexions & Boolean Algebra
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.
Timid White Rabbit
White Roses Painted
Brash Queen of Hearts
Roses Painted Red
|Looking Glass||White Kitten
White Childlike Queen
White Knight protects Alice
Black Kitten – Red Queen
Red Governess Queen
Red King ignores Alice
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.
Aristotle, Boole, Time & Space
Carroll said many critical things about Aristotle and logicians who follow him in his early undergraduate essays and late works on logic. As a student at Oxford, Carroll wrote, “we can scarcely call (Aristotle) a philosopher,” and “we cannot admire him too much,” as Aristotle paid no attention to the immaterial beyond the material, to the eternal and Christian beyond time and space.
If Carroll incorporated Aristotle’s ten categories into his stories, the strange, disturbing Caterpillar and Cheshire Cat, time and space, who teach Alice what she needs to know to work her way through Wonderland, are helpful and enabling, but neither are presented as noble, both beneath but beyond the royal characters and their courts, as if both, like Diogenes, are superior, but can’t be bothered to conform to the spaces or relations of the rulers and rules. Alice tells us a cat can look upon a king, and Wittgenstein said a dog can’t expect someone to arrive on Thursday, as far as we know, because, like a rabbit without a watch, neither cat nor dog use words such as Thursday or pay any attention to royal positions, having no words for these things, and thus not sharing our world, logic, thinking and concepts in these ways with us.
If Aristotle paid too much attention to time and space for Carroll’s liking, Boole and modern formal logicians, who seek universal, eternal forms that are absolutely true in all times and spaces seem to have paid too little. The Mad Hatter attempts to teach Alice universal, formal lessons in logic as he ignores time and space and wears a formal hat that isn’t his own, but he is rude and has problems interacting with others that he does not have with formal abstractions. Just as Humpty Dumpty is an immature, narcissistic fool, but successfully translates some of the Jabberwocky poem for Alice, the Mad Hatter is a decent logician, but he is also a self-centered fool, who cannot function in good relations with others. The Mad Hatter has offended time, and the party ignores the circular space of their table and what this could mean for their future if things circle back around, and they are left with their own mess.
For more explanation that proceeds chapter by chapter through the fictional fantasies of Lewis Carroll and how they fit with ancient and modern forms of logic, please read my detailed explanations of Wonderland, the Looking Glass, and the Snark.