Sunday, 17 April 2016

The Logical Mr Carroll

Lewis Carroll, real name Charles Dodgson, has several direct or indirect claims to fame or, some would say, notoriety. He is best known now as the author of the ‘Alice’ books, written to entertain Alice Liddell, the little daughter of Henry George Liddell, who was the co-compiler of the still standard huge English dictionary of Ancient Greek (Henry, not Alice. (Duh)). Carroll was a very fine photographer, and many of his photographs are of little girls not wearing very much. In this post-Freudian witch-hunting age this excites great suspicion among the prurient, though I don’t think anyone at the time thought there was anything odd about it. Certainly Carroll himself would have been horrified and disgusted at the suggestion there might have been anything sexual (as of course there was) in his interest. We now know of course — and the really suspect people are those who strenuously deny it — that all human relations have a (perhaps unacknowledged) sexual element.

Anyway that’s more than enough about that. Carroll / Dodgson lectured at Oxford University in Mathematics and especially in logic. Some of his more elementary work in logic had much of the wit of his stories and verses; here is an example:


(a)  All babies are illogical.

(b)  Nobody is despised who can manage a crocodile.

(c)  Illogical persons are despised.


As the subjects of this puzzle are people, we take the universe as the set of all people. We will rewrite each statement in the puzzle as an implication. First we define simpler statements,


B : it is a baby
L : it is logical
M : it can manage a crocodile
D : it is despised ,


where “it” in this context refers to a general person. Then the three statements can be rephrased as


(a)   B → ~L : If it is a baby then it is not logical.
(b)   M → ~D : If it can manage a crocodile then it is not despised.
(c)   ~L → D : If it is not logical then it is despised.


Our aim is to use transitive reasoning several times, stringing together a chain of implications using all the given statements. We have an arrow pointing from B to ~L, and likewise an arrow pointing from ~L to D; thus we are able to start with B and arrive at the conclusion D. However, the second statement is still not utilized. But since any implication is equivalent to its contrapositive, we may replace the second statement with its contrapositive D → ~M. Then we get the transitive reasoning chain

B → ~L → D → ~M .

We reason that if B is true, then ~L is true, hence D is true, and therefore ~M is true. Our ultimate conclusion is the statement

B → ~M : If it is a baby then it cannot manage a crocodile .

In ordinary language we would more likely rephrase this answer to the puzzle as

“No baby can manage a crocodile.”

Alternatively, we could write the answer as the contrapositive statement

M → ~B : If it can manage a crocodile then it is not a baby.

The translation into words then would be something like

“Anyone who can manage a crocodile is not a baby.”


This is perhaps his most famous photograph of Alice Liddell.


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