by David Day
Although Dodgson had great respect for the classical tradition of Aristotelian logic, he did recognize its limits, and was much excited by the dramatic discovery of the algebraic formulation of logic by the British mathematician George Boole. This discovery of Boolean logic was made during Dodgson’s student years and had a profound effect on his work throughout his life. He was much taken up with this application of algebraic notations and principles to ancient Aristotelian logical problems. In fact, his Symbolic Logic was a work almost entirely concerned with the application of algebra to logic.
The Game in action: In Carroll’s Symbolic Logic.
“Nothing whatever,” said Alice.
“That’s very important,” the King said, turning to the jury. They were just beginning to write this down on their slates, when the White Rabbit interrupted: “Unimportant, your Majesty means, of course,” he said in a very respectful tone, but frowning and making faces at him as he spoke.
“Unimportant, of course, I meant,” the King hastily said, and went on to himself in an undertone, “important—unimportant—unimportant—important—” as if he were trying which word sounded best.
Some of the jury wrote it down “important,” and some “unimportant.” Alice could see this, as she was near enough to look over their slates; “but it doesn’t matter a bit,” she thought to herself.
At this moment the King, who had been for some time busily writing in his note-book, cackled out “Silence!” and read out from his book, “Rule Forty-two. All persons more than a mile high to leave the court.”
Everybody looked at Alice.
In logic—as in law—there is a profound difference between evidence and proof, and in both, evidence must be rigorously tested. This is particularly true in Boolean logic, in which “soundness” and “completeness” are the two most critical properties in the construction of sentences and the validation of evidence.
When the King of Hearts attempts to decide whether Alice’s evidence is “ ‘important—unimportant—unimportant—important’—as if he were trying which word sounded best,” the regent is not being frivolous. The King is quite properly evaluating each word (or sentence) for soundness, or what mathematicians call a “well-formed formula” or a wff. The King’s judgment is based on the “sound” structure of the sentence, not on its meaning in ordinary speech.
The sentence (wff) must be sound and complete before any conclusion (or verdict) can be reached. This is what the Queen of Hearts loudly insists upon—in her final dispute with Alice—when she says, “Sentence first—verdict afterwards.”
The Queen is not being perverse: she is simply attempting to enforce the strict rules of Boolean logic in a formal system known as sentential calculus (what today is called propositional calculus). This system requires the Queen’s ruthless application of axes, by which Carroll means (and repeatedly puns) axioms. Just as traditional logicians have tried to set the true rules of argument since Aristotle’s time, Boole’s new algebraic system defines a valid argument as sets of logically progressive propositions.
This is why there are so many arguments rather than polite conversations in Wonderland. Alice is unknowingly entering into arguments that employ the formal language of sentential calculus. The Gryphon, for instance, constantly uses double negatives in his speech—this is in fact a sentential axiom of double negation. The Queen’s attempt to behead the body-less Cheshire Cat is a bizarre demonstration of what is known as the axiom of the excluded middle.
In the dispute over the validity of the so-called Knave’s letter, Alice says, “I don’t believe there’s an atom of meaning in it,” and the King replies, “If there is no meaning in it, that saves a world of trouble, you know, as we needn’t try to find any.” In Boolean terms, a zero value would be a valid and significant conclusion.
“I’m not a mile high,” said Alice.
“You are,” said the King.
“Nearly two miles high,” added the Queen.
“Well, I shan’t go, at any rate,” said Alice: “besides, that’s not a regular rule: you invented it just now.”
“It’s the oldest rule in the book,” said the King.
“Then it ought to be Number One,” said Alice.
The King turned pale, and shut his note-book hastily. “Consider your verdict,” he said to the jury, in a low, trembling voice.
“There’s more evidence to come yet, please your Majesty,” said the White Rabbit, jumping up in a great hurry; “this paper has just been picked up.”
“What’s in it?” said the Queen.
“I haven’t opened it yet,” said the White Rabbit, “but it seems to be a letter, written by the prisoner to—to somebody.”
DE MORGAN’S LAWS George Boole’s pioneering work on the calculus of propositions was carried forward after his death by his colleague AUGUSTUS DE MORGAN (1806–1871), author of many mathematical works, including Formal Logic: or, The Calculus of Inference, Necessary and Probable (1847). He formulated De Morgan’s laws and the duality principle. If we accept George Boole as our mathematician’s King of Hearts, De Morgan would certainly be our mathematician’s Knave of Hearts. De Morgan was the intellectual heir of George Boole, just as (presumably) the Knave of Hearts was the heir to the King of Hearts.
A passage from De Morgan’s Trigonometry and Double Algebra (1849) is quoted by Helena M. Pycior in her “At the Intersection of Mathematics and Humour: Lewis Carroll’s ‘Alices’ and Symbolic Algebra” (1984). In his précis to symbolic algebra, De Morgan explains: “No word nor sign of arithmetic or algebra has one atom of meaning throughout this chapter, the object of which is symbols, and their laws of combination, giving a symbolic algebra.”
Ms. Pycior compares De Morgan’s statement with Alice’s declaration on the Knave’s letter: “I don’t believe there is an atom of meaning in it.” She then firmly concludes: “The coincidence of language in De Morgan’s algebraic text book and Carroll’s Alice’s Adventures in Wonderland is not accidental.”
Coincidentally, two decades after the publication of Wonderland, Carroll came to know De Morgan’s son, a noted artist and ceramicist. Carroll wrote in his diary in March 1887, “Called on Mr. William De Morgan and chose a set of red tiles for the large fire-place.” It appears the artist knew Carroll’s work as well, as the chosen tiles were decorated with figures from Wonderland and Looking-Glass: the Dodo, the Lory, the Fawn, the Eaglet and the Gryphon.
Carroll also became familiar with the paintings of William De Morgan’s wife, the Pre-Raphaelite artist, Evelyn De Morgan. The De Morgans, like Carroll, were deeply interested in psychic phenomena and spiritualism. Evelyn De Morgan’s paintings often depicted scenes from classical myths concerning life after death, and include her Demeter Mourning for Persephone.
“And yet I don’t know,” says the King, as he continues to examine the evidence. He then begins to deconstruct the evidence in the Knave’s letter by reducing it to what logicians would call atomic sentences, or in Alice’s terms an “atom of meaning.” In Boolean terms, the King is required to reduce everything to atomic units and arbitrarily attribute true or false values to each, although he must not define them—that is, the King will not ask the identity of “he or she or it.”
In Boole’s own words, the atomic sentences “admit indifferently of the values 0 and 1, and of these values alone.” In this way, all propositions are either true (with value 1) or false (with value 0). Astonishingly, Boole’s absurd-sounding system of binary logical on-off switches became the basis for all modern computer operating systems.
The trial of the Knave of Hearts is full of legal, as well as logical and mathematical, puns. In his “Metaphysics” Aristotle explains: “Those who use the language of proof must be cross-examined.” This was mirrored by the White Rabbit’s reminder that the King must “cross examine this witness,” although Carroll makes a punning joke of this by having the King interpreting this literally: staring at the witness with such a cross expression that he complains to t
he Queen, “It quite makes my forehead ache!”
Augustus De Morgan: Makes a symbolic appearance.
“It must have been that,” said the King, “unless it was written to nobody, which isn’t usual, you know.”
“Who is it directed to?” said one of the jurymen.
“It isn’t directed at all,” said the White Rabbit; “in fact, there’s nothing written on the outside.” He unfolded the paper as he spoke, and added “It isn’t a letter, after all: it’s a set of verses.”
“Are they in the prisoner’s handwriting?” asked another of the jurymen.
“No, they’re not,” said the White Rabbit, “and that’s the queerest thing about it.” (The jury all looked puzzled.)
“He must have imitated somebody else’s hand,” said the King. (The jury all brightened up again.)
“Please your Majesty,” said the Knave, “I didn’t write it, and they can’t prove I did: there’s no name signed at the end.”
It has been suggested that Carroll is making a droll legal joke when the table of tarts are placed before the court as evidence: a delicious example of the corpus delicti, or body of proof. However, more significantly, the table of tarts suggests a pun on a table of torts (or tortious liability). Torts are essentially tables of laws of precedent that litigating lawyers must learn and cite during trials involving restitution.
Then too, it is possible to extend the punning from tarts to torts to tauts, as in tautology. Tautology is a key concept in propositional logic; it is a formula that is always true and can be confirmed, or proved, by use of a “truth table.” So we have typical Carrollian serial, or linked, puns: a table of tarts becomes a table of torts, then transforms into a table of tauts before arriving at a truth table.
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra and propositional calculus—to determine whether a proposition is true, and—as his twentieth-century editor William Warren Bartley discovered—Carroll had employed in his Symbolic Logic II. That is why, at the end of the trial, our Boolean King of Hearts “went on muttering … to himself: ‘ “We know it to be true—’ that’s the jury, of course,’ ” then confirms their judgment upon examining the table of tarts (or truth table): “ ‘Why, there they are!’ said the King triumphantly, pointing to the tarts on the table. ‘Nothing can be clearer than that.’ ”
“If you didn’t sign it,” said the King, “that only makes the matter worse. You must have meant some mischief, or else you’d have signed your name like an honest man.”
There was a general clapping of hands at this: it was the first really clever thing the King had said that day.
“That proves his guilt,” said the Queen.
“It proves nothing of the sort!” said Alice. “Why, you don’t even know what they’re about!”
“Read them,” said the King.
The White Rabbit put on his spectacles. “Where shall I begin, please your Majesty?” he asked.
“Begin at the beginning,” the King said gravely, “and go on till you come to the end: then stop.”
There was dead silence in the court, whilst the White Rabbit read out these verses:—
Much of the humour in Wonderland is generated by the absurd obviousness of tautological statements when delivered in ordinary speech. They are logically circular; for example, all tautologies are necessarily true because they are tautologies. However, there are twenty-one tautologies (or axioms) that are essential rules in sentential, or propositional, logic. Alice is bewildered by many arguments that are essentially demonstrations of axioms. She is certainly unfamiliar with their Latin names: modus ponens or method of affirming, modus tollens or method of denying, modus tollendo ponens or method of affirming and denying, reductio ad absurdum or reducing to the absurd, and so on.
Two of these tautologies or axioms, known as De Morgan’s laws, appear to be employed by the Knave of Hearts in his absurd defence against the equally absurd charges made against him. First: since two things are false, it is also false that either of them is true. Second: since it is false that two things both are true, at least one of them must be false. These laws are by their nature quite obvious, and so simple even a simpleton like the Knave of Hearts would be capable of raising them.
In his diary of 1858, Dodgson notes that he has purchased and placed on his reading list “De Morgan on Chances.” He was certainly familiar with the work of the noted Cambridge mathematician and logician Augustus De Morgan, as probability was a major field of study for Carroll throughout his life.
De Morgan was also credited with what is today known as the duality principle. This is employed in the translation of concepts, theorems and mathematical structures into other concepts, theorems and structures, often by involution. Simply put: if a theorem is true, its dual is true. It is an important general principle that has application in every area of mathematics.
“They told me you had been to her,
And mentioned me to him:
She gave me a good character,
But said I could not swim.
He sent them word I had not gone
(We know it to be true):
If she should push the matter on,
What would become of you?
I gave her one, they gave him two,
You gave us three or more;
They all returned from him to you,
Though they were mine before.
If I or she should chance to be
Involved in this affair,
He trusts to you to set them free,
Exactly as we were.
My notion was that you had been
(Before she had this fit)
An obstacle that came between
Him, and ourselves, and it.
Don’t let him know she liked them best,
For this must ever be
A secret, kept from all the rest,
Between yourself and me.”
Lewis Carroll applied the duality principle to literature. He reasoned that if the underlying structure of his writing was mathematically logical, the linguistic structure would retain its logical integrity—and the results would be “as sensible as a dictionary” (as he wrote in Through the Looking-Glass). However, although grammatically logical, its message is usually absurd and comic. For, as Carroll the author—and Dodgson the logician—knew as well as any comic writer, the great secret of nonsense literature is that it is extremely sensible. That is, nonsense is humorous only if it works within a logical framework. Without logic, nonsense makes no sense.
In this courtroom of the King and Queen of Hearts, Alice has unknowingly entered into what the philosopher, logician and mathematician Bertrand Russell described as “the realm of pure mathematics”: “an ordered cosmos, where pure thought can dwell.” It is a heartless and frightening place—and the monstrous Queen of Hearts is well suited to be its ruler.
Logicians like the Queen of Hearts are interested not in the content of an argument but in the features that make an argument valid or invalid. It is a place governed by rules and procedures and form; there is nothing whatever in the rules about the value of emotions, morals or character, nor anything to do with content or substance.
The Queen of Hearts has to be the ruthless executioner or Wonderland could not exist at all. “Axioms cannot tolerate contradictions,” Carroll wrote in his Symbolic Logic; nor can the Queen of Wonderland. Contradiction in any system of logic or mathematics leads to chaos and collapse of the entire system.
“That’s the most important piece of evidence we’ve heard yet,” said the King, rubbing his hands; “so now let the jury—”
“If any one of them can explain it,” said Alice (she had grown so large in the last few minutes that she wasn’t a bit afraid of interrupting him), “I’ll give him sixpence. I don’t believe there’s an atom of meaning in it.”
The jury all wrote down on their slates, “She doesn’t believe there’s an atom of meaning in it,” but no
ne of them attempted to explain the paper.
“If there’s no meaning in it,” said the King, “that saves a world of trouble, you know, as we needn’t try to find any. And yet I don’t know,” he went on, spreading out the verses on his knee, and looking at them with one eye; “I seem to see some meaning in them, after all. ‘—said I could not swim—’ you can’t swim, can you?” he added, turning to the Knave.
The Knave shook his head sadly. “Do I look like it?” he said. (Which he certainly did not, being made entirely of cardboard.)
“All right, so far,” said the King, and he went on muttering over the verses to himself: “ ‘We know it to be true’—that’s the jury, of course—‘I gave her one, they gave him two’—why, that must be what he did with the tarts, you know—”
“But, it goes on ‘they all returned from him to you,’ ” said Alice.
“Why, there they are!” said the King triumphantly, pointing to the tarts on the table. “Nothing can be clearer than that. Then again—‘before she had this fit’—you never had fits, my dear, I think?” he said to the Queen.
Alice invokes the ultimate contradiction and rejects the Queen’s authority. The King, Queen, Knave and all the cards are simply “made entirely of cardboard.” The laws of this heartless court have no power over Alice. She challenges the Queen with “Who cares for you? You’re nothing but a pack of cards!” For Alice, human values and the concerns of the human heart ultimately must trump this heartless tyranny of abstract mathematics. Once confronted with “Alice’s Evidence,” the house of cards collapses in a heap, and the dreamer awakens in the real world.