Some have claimed Lewis Carroll’s Alice in Wonderland and Through the Looking Glass are both works of nonsense, meant to amuse but not educate, but this is wrong. Carroll designed both books to illustrate forms from the history of logic with memorable, emotional and unreasonable characters. While Carroll mocked the work of Boole, De Morgan and others throughout the two tales, both also primarily serve to illustrate and teach central concepts of Aristotle’s work on Logic, specifically his categories and syllogisms, the forms of Logic that Carroll taught and studied for a living.
I actually had the chance to use Wonderland this morning to teach Aristotle’s categories to my Greek philosophy students, and one said that it served well to help her visualize and remember each category, as the examples draw on classic memories and are emotively meaningful. This demonstrates the texts are not useless nonsense or mere entertainment, but lesson plans in logic. My theory is that Carroll believed others would find this list of Aristotle’s categories reversed, but when no one noticed he began the sequel Through the Looking Glass with the idea of mirror-images, reversals and putting a text up to the mirror to show that he was inverting Aristotle’s classic text on logic, and going to use inversions and reversals with logic even more in the second story.
Alice’s first adventure in Wonderland illustrates Aristotle’s Categories, presenting the ten categories in the order Aristotle discussed them but in reverse: passion, action, state, position, time, place, relatives, quality, quantity, and substance. First, the White Rabbit is passion, who acts on Alice. Second, the mouse is action, acted-upon by Alice. Third, the bird’s caucus race is state. Fourth, Alice takes the position of the White Rabbit’s servant and fills his entire house. Fifth, the Caterpillar is time, who accepts change and uncertainty. Sixth, the Cheshire cat is space, who shows Alice exclusive and opposed positions. Seventh, the Duchess and baby are relatives or relations.
Eighth, the Mad Tea Party is quality, with the unsound Hatter and Hare who used the best butter. Ninth, the Queen of Heart’s garden is quantity, with the two, five and seven cards forming an addition problem and the Queen threatening everyone with subtraction. Tenth and finally, the King of Heart’s trial of who stole the tarts is substance, as the tarts are still there substantially but the trial and evidence are insubstantial.
Alice’s second adventure Through the Looking Glass illustrates the syllogistic forms found in Aristotle’s Prior Analytics in an order that shows subalternation twice. The four royal pieces, the Red Queen, Red King, White Queen and White King, are the four corners of Aristotle’s Square of Opposition, a visual presentation of logic popular in Europe for centuries. The White Queen, inclusively open like a child, is the universal positive (All, All, All), the Red Queen is the universal negative (All, None, None), the White King is the particular positive (Some, All, Some) and the Red King is the particular negative (Some, None, Some-Not). In the end, Alice sits as an inclusive-exclusive OR between All and None, as the one who must decide for herself, with her powers of logic and reason, some and some not like an adult between the extremes, as Aristotle advises us in ethics. There are countless examples of syllogistic reasoning in both texts, but here are central examples that show each royal chess piece as an Aristotelian corner.
BARBARA, the Positive Universal Syllogism: If All A is B, and All B is C, then All A is C. If all things are possible to think if you Shut your eyes and try very hard, as the White Queen suggests to Alice, and if all impossible things are things indeed, even if they, unicorns and we are all quite mental, then Alice can think six or more impossible things before breakfast if she shuts her eyes, imagines, and tries very hard, as the White Queen implies but doesn’t say directly, meaning what she doesn’t say syllogistically. In Venn diagram form, if A is entirely B, and B is similarly C, then A must also be C.
CELARENT, the Negative Universal Syllogism: If All A is B, and No B is C, then No A is C. If All ways are mine, as the Red Queen says, and None of what’s mine is yours, as the Duchess moralizes, then none of these ways are yours, is what the Red Queen means but doesn’t say, which we understand and infer quite syllogistically from what is given in her words. As a Venn diagram, if A is entirely B, and no B is C, then no A can be C.
DARII, the Positive Particular Syllogism: If Some A is B, and All B is C, then Some A is C. If the White King says he sent almost all his horses along with his men, but not two of them who are needed in the game later, and if Alice has met all the thousands that were sent, 4,207 precisely who pass Alice on her way, then Alice has met some but not all of the horses, namely the Red and White Knights who stand between Alice and the final square where she becomes a queen. As a Venn diagram, if some A is B and all B is C then some A must be C.
FERIO, the Negative Particular Syllogism: If Some A is B, and No B is C, then Some A is not C. If all things are dreams, as Tweedle Dum and Tweedle Dee tell Alice, and some dreams are untrue or not ours alone, then all things are somewhat untrue, and somewhat aren’t ours alone, which is what Tweedle Dum, Dee and the Red King dreaming silently imply, but don’t say. As a Venn diagram, if some A is B and no B is C then some of A is C. As Aristotle says, if we have only some and no all or none, we can’t draw syllogistic judgements completely, leaving us with only a relative, somewhat satisfying conclusion, just as the Red King silently dreams and says nothing to Alice after she happily dances around hand in hand with both twin brothers.
Let us imagine a game played with pieces on a board, a logistical space for moving the pieces. What is on the board is true and certainly the present case. The pieces on the board and in the box beside the board contain all the pieces that can possibly be used, all the possibles and possibilities, such that if we see the pieces on the board and beside the board, we don’t know what pieces will be played but know all the pieces that could be played. A piece beside the board is false and dead, unless it gets onto the board, and then it is true and live in the game, and a piece on the board is false and dead if it is taken off the board. There are also things that are not in the game, the logical universe of discourse, the subject under discussion or consideration, and these are neither true nor false, as the are not in the cards, are not possibilities or pieces that can possibly enter the game or be negated to the side.
With this structure alone, we can exhaust all the connectives we use in formal Sentential Logic the way Wittgenstein envisioned it in his Tractatus and set it with truth tables. First, AND puts pieces on the board, including them together, but it could also, in combination with NOT, take sets of pieces off the board, adding them to the possibilities currently false rather than those currently true. NOT takes pieces off the board, or adds them if it is tied up with other NOTs. OR considers pieces, involved with whether or not pieces are moved on or off the board, completely indeterminate by itself, but determinate in combination with pieces added to or dropped from the board with ANDs and NOTs.
IF-THEN connects pieces possibly on the board together, such that one should lead to the other. Sluga argues that there are connections between things in the moment, such as speciation, with all men being mortal all at once or always, and connections between things over time, such as causation, with one thing leading to another. If the game changes with the possibilities over time, the first would be statements about the relationships between pieces at the same time, and the second different times. Bivalence, IF-AND-ONLY-IF, would simply state the relationship is reciprocal, working both ways, which would work with speciation and causation in a chicken-and-egg situation, where two things lead to each other circularly over time, again and again.
Ludwig Wittgenstein (1889 – 1951) is my favorite of the latest, greatest philosophers, and I learned his work from Hans Sluga and Barry Stroud at Berkeley, who taught me that Wittgenstein’s later thought experiments can lead to much more than he or we have worked out about truth and meaning. Wittgenstein’s thinking can answer many questions about thinking, not completely but more fruitfully, as Wittgenstein says, than other thinkers can.
The turn between Wittgenstein’s earlier and later thought is much like the Indian metaphor of turtles supporting the world and the question that arises from such an arrangement. Locke, Hume, Russell and other European philosophers have brought up the Indian debate about what the world sits on such that it is stable and continues. Some say that it sits on a turtle, an animal that symbolizes the cosmos in India and China, as it is flat on the bottom like the earth, and round on the top like the sky. Others ask what the turtle sits on if the world sits on it, and someone once said it’s turtles all the way down. Some have called this an infinite regress, an endless series that vanishes over the horizon, Buddha called it an unsolvable problem, Plato called it the greatest difficulty for philosophy, and today some call it the foundationalism debate, arguing whether or not knowledge or certainty sit on anything known or certain.
Philosophy is the love and study of wisdom, truth, meaning and thought. Thought interweaves several elements in our world. We sense, see, hear, touch, smell and taste things in our world. We also feel, feeling good, bad, tense and calm about the things we sense. We also remember, sense and feel things that are not in our world, but were. We also reason, building what we remember from sense and feeling into thoughts. In the middle of all this are words, things we hear and see from others that are interwoven with what we sense, feel, remember, and think.
Is sensing a thing without words, feelings or memories a thought? Is looking at an apple thinking? Is looking at it and feeling a feeling thinking? If I look at an apple and feel happy, is that a thought without words or images in mind? Some say yes, and others say no. Once we have several things interwoven, including the words we use to mean things, many call that thought. Some say thought is logical and rational, such that it follows rules, or follows rules when it is right and correct in judgement. Others say that this is the turtle problem yet again.
If things need thoughts to make sense of them, and if thoughts need thoughts, such as rules, or plans to make sense of them, is there thought that makes logical, self-aware, rational sense of thought itself? Are there words that make sense out of how we use words to mean things and know things? Some say yes, and it terminates in the rules and forms of logic, and others say no, and we simply continue to gather and divide things without an underlying logic that brings all of our wants and plans into common, coherent systems, visions or words. As Zhuangzi the Daoist asks, What do our ways or words rely on such that our words mean things?
What do turtles sit on? Some say other turtles. The Buddha in India and Wittgenstein in Britain answered the question with similar, simple metaphors that show us more than any system or logic in images or words can completely in itself. Thought and our world are interwoven, such that it isn’t turtles all the way down, but turtles all the way around. Much as Nicholas of Cusa and Hegel said about a circle, it is an infinite regress, but it is also complete in itself, and continues right in front of us. It isn’t that truth or rules rely or rest on any specific thing, but rather situations of sense, feeling, memory, reason and words mean things all together.
Situations shift, and these shifts show us how things mean things to us better than any specific words can. As Wittgenstein said, there is what can be said, but what can be said is only a part of what can be shown, which is best done not with complete, enclosed systems of words or images but by leading people through many open-ended situations of mind, stagings of thought, what Wittgenstein called thought experiments that involve many and any elements.
Much as Alice is frustrated with her sister’s text without pictures in the opening of Wonderland, words and rules without many interrelated examples of rich situations and the infinite variety found in them confuse us and lead us into considering words outside of actual, useful meaning. Carroll wrestled with Boole’s algebra much as Wittgenstein wrestled with Frege’s logic, and both came to the conclusion that words and systems can trap us like a fly in a bottle.
As Zhuangzi said, once we have the rabbit, we can forget the trap, and then we can involve the trap or not as we like, such that we can have words with others who have forgotten words, remembering and forgetting words and understandings freely as we please rather than sitting on particular words or systems as final, fixed foundations. Wittgenstein enjoyed reading Alice’s adventures to two sisters in Wales where he worked on his final thoughts, and he likely heard and felt Carroll’s deeper meaning, that it is good to use thought, rules and logic to show others how open-ended thought can be, beyond anyone’s particular logic, words, thought or feelings.
Buddha called the interweaving of everything codependent-arising, life as a tangle of many forms of life, as we see in Klimt’s painting Death and Life, which he began in Vienna 1908 and finished in 1915, the time Wittgenstein left Vienna to study logic, mathematics and philosophy with Russell at Cambridge. Klimt was not only one of the most influential painters of Wittgenstein’s Vienna, he painted a portrait of Wittgenstein’s sister, who was also psycho-analyzed by Sigmund Freud. As we might suspect, Wittgenstein’s family had some pull in Vienna, which in Klimt’s day was the city with the latest, greatest culture, replaced in the 1920s by Paris, the 40s by New York and the 60s by San Francisco. Wittgenstein said that life and thought are like an old city, with many forms of life inter-tangled for centuries.
Much as Buddha taught there is no essence or nature that completely defines or causes a thing because it arises out of the relationships it shares with other things outside of itself, Wittgenstein argued that life is like a thread without a single strand running through the entire length, and so we should always beware of the lure of the secret cellar, the proud idea that we have hit bedrock and completely revealed the truth rather than revealed yet another strong connection between different interwoven things. The cure for this proud ignorance, what Heraclitus called the human disease, is a rich variety of interwoven examples and elements that continue to show us more and more about the greater whole, endlessly.
I have been working on Lewis Carroll’s Alice in Wonderland and Through the Looking Glass for many years now to find the logical and philosophical forms hiding inside it, and as I have been teaching logic this semester I have used the class as an excuse to go over Aristotle, Boole and Carroll’s work carefully. In the process, I have found many Aristotelian and Boolean forms that are structural to both works that I have never seen before.
Aristotle’s four “perfect” syllogisms and Boole’s inclusive and exclusive operations of AND, OR and NOT, gathering and dividing as John Stuart Mill would say, form the positions and plot of both Alice books. Carroll was studying the logic of Aristotle, Mill and Boole as he wrote both of Alice’s adventures, visually presenting logic as characters, but also as emotions, as inclusive and exclusive feelings that operate in our thoughts and our world together. Carroll was trying to show us that syllogisms and logical operations are series of emotions, of feelings that gather and divide things in sequences as the underlying structure of thought with the underlying structure of his stories about Alice.
In the first book of Wonderland, Alice works her way from an inclusive AND, the White Rabbit, past the inclusive OR of the caterpillar, the exclusive OR of the Cheshire Cat, to the NOT of the Queen of Hearts, who chops off heads. The various symbols for NOT Boole and other logicians use look a bit like an ax next to a capital letter, a symbol for a group much like a regal head who stands for the common people. Alice says it is all a pack of cards, meaningless manipulation of symbols and pieces regardless of meaning, and disrupts her imaginary dream.
The White Rabbit is like an addition problem, an AND, Alice and her older sister, inclusive of different elements, the two sisters, and exclusive, specialized and late to a specific event at a precise time. This makes the White Rabbit an absurdly rational animal, as Aristotle would say, both man and beast. Alice, bored with her sister reading to herself, charges after the White Rabbit down the rabbit hole, with no thought as to how she would get out again, like a wildly inclusive child, mirroring the absurdly inclusive combination of a rabbit with a waistcoat, and unlike her sister, who is carefully considering a specialized text. Alice dreams she follows the absurdly complex White Rabbit as she can’t follow her sister in reading a boring specialized text that gathers a very narrow sort of element. A child needs emotions, pictures, words and many things to stay interested in a story.
In the second book of the Looking Glass, Alice works her way from the Red Queen, another NOT like the red Queen of Hearts, past the White Queen, a childlike inclusive AND, timid like the White Rabbit, to the end of the board where Alice is the OR, who must inclusively and exclusively choose between inclusive AND, the White Queen on her right, and exclusive NOT, the Red Queen on her left. The Queens test Alice and find she can’t inclusively add or exclusively subtract things the ways they ask her to, they take her to a banquet where food turns into people and people into food, and Alice hates it and turns the table over, upsetting her second dream. Wonderland works from childlike AND past OR to adult NOT, from inclusion to exclusion, and the Looking Glass works from adult NOT past childlike AND to bring the childlike-adult balance of OR, both inclusive and exclusive.
The four royal pieces of the Looking Glass world, the Red Queen, Red King, White Queen and White King, are the four corners of Aristotle’s Square of Opposition, a visual presentation of logic popular in Europe for centuries. The White Queen, inclusively open like a child, is the universal positive (All, All, All), the Red Queen is the universal negative (All, None, None), the White King is the particular positive (Some, All, Some) and the Red King is the particular negative (Some, None, Some-Not). In the end, Alice sits as an inclusive-exclusive OR between All and None, as the one who must decide for herself, with her powers of logic and reason, some and some not like an adult between the extremes, as Aristotle advises us in ethics. There are countless examples of syllogistic reasoning in both texts, but here are central examples that show each royal chess piece as an Aristotelean corner.
BARBARA, the Positive Universal Syllogism: If All A is B, and All B is C, then All A is C. If all things are possible to think if you Shut your eyes and try very hard, as the White Queen suggests to Alice, and if all impossible things are things indeed, even if they, unicorns and we are all quite mental, then Alice can think six or more impossible things before breakfast if she shuts her eyes, imagines, and tries very hard, as the White Queen implies but doesn’t say directly, meaning what she doesn’t say syllogistically. In Venn diagram form, if A is entirely B, and B is similarly C, then A must also be C.
CELARENT, the Negative Universal Syllogism: If All A is B, and No B is C, then No A is C. If All ways are mine, as the Red Queen says, and None of what’s mine is yours, as the Duchess moralizes, then none of these ways are yours, is what the Red Queen means but doesn’t say, which we understand and infer quite syllogistically from what is given in her words. As a Venn diagram, if A is entirely B, and no B is C, then no A can be C.
DARII, the Positive Particular Syllogism: If Some A is B, and All B is C, then Some A is C. If the White King says he sent almost all his horses along with his men, but not two of them who are needed in the game later, and if Alice has met all the thousands that were sent, 4,207 precisely who pass Alice on her way, then Alice has met some but not all of the horses, namely the Red and White Knights who stand between Alice and the final square where she becomes a queen. As a Venn diagram, if some A is B and all B is C then some A must be C.
FERIO, the Negative Particular Syllogism: If Some A is B, and No B is C, then Some A is not C. If all things are dreams, as Tweedle Dum and Tweedle Dee tell Alice, and some dreams are untrue or not ours alone, then all things are somewhat untrue, and somewhat aren’t ours alone, which is what Tweedle Dum, Dee and the Red King dreaming silently imply, but don’t say. As a Venn diagram, if some A is B and no B is C then some of A is C. As Aristotle says, if we have only some and no all or none, we can’t draw syllogistic judgements completely, leaving us with only a relative, somewhat satisfying conclusion, just as the Red King silently dreams and says nothing to Alice after she happily dances around hand in hand with both twin brothers.
If you are interested in more, please read my lecture on Logic, Lewis Carroll and Alice’s Adventures, which is very much under development and in progress at the moment, as can hopefully be understood. It may turn out that all negativity is merely a playful, innocent kitten after all.
Let’s say that something is merely metaphorical if it is simply similar to something but not identical, as the Nyaya logicians of ancient India would say. That means that if I act like a cat, but I am not a cat, it can be said, metaphorically, “Eric is a cat,” as I act like one, but it can’t be said literally AND truthfully that I am a cat, as said. As Saussure the Swiss linguist could say, in French or German, the word “cat” doesn’t look like a cat or sound like a cat, nor does the word nap sound like a nap or look like one, but a catnap, a nap taken by a cat or me later, does look like a napping cat, whether or not I am a cat. This means that when I, a human, take a catnap, I am literally taking a nap, but I am metaphorically taking a catnap. Does this mean when I nap, “Eric is taking a catnap,” is both literally AND metaphorically true in different ways? Can the two be complimentary, or are they exclusive?
If I am a cat, a catnap looks entirely like a cat taking a nap, and if I am not a cat, then it looks like a nap, which I am literally taking, but I am only like a cat, not actually or literally one, as said. So: If I take a catnap with my cat, and you say, “They are taking a catnap together,” did you say something that was metaphorically true for me, but literally true for the cat, or is it both literal and metaphorical for both of us? Does it feel metaphorical to say it about me, and feel literal to say it about the cat? Does it feel or apply to me and the cat differently? Does it depend how it feels to say it, or does it depend on how it is said, and to whom? Nothing seems clear here, no matter how literally or carefully we speak.
My math teacher was trying to hide a graph. She must be plotting something…
This joke works because plot means to draw a graph to visualize information, but it also means to scheme, to plan evil, to hatch a sinister plot. The joke works because a math teacher can plot a graph, which could be involved in plotting a crime, but not usually, which makes the speaker seem suspicious, and in a silly way, as if the plan of the graph could be the plan of a crime simply because the word plot is used to say both. It is possible our math teacher is planning a bank heist, with the suspicious graph. The math teacher is certainly plotting something, the graph, and what is normal isn’t suspicious. This shows us the word plot is used by us in two ways, and the difference is fear, suspicion that a plan is more than a plan, it is a plan for evil, and we brace for evil with fear. A plan is someone being calm and resolute in a way, and a sinister plot is a plan, a resolution, that others fear. The turn from a calm, normal, plotted mathematical situation to unreasonable paranoia and aggression is the jerk of the joke. If we look at language use in particular situations from a pragmatic perspective, and keep an eye on the situation of emotions, and how emotions can change, we can understand what jerks us around and makes us laugh at some jokes and not others.
Thought Itself 