Sunday, July 30, 2006

Lewis Carroll Logic Puzzles

(a) No interesting poems are unpopular among people of real taste.
(b) No modern poetry is free from affectation [i.e., unaffected].
(c) All your poems are on the subject of soap-bubbles.
(d) No affected poetry is popular among people of real taste.
(e) No ancient poem is on the subject of soap-bubbles.

What is the conclusion of this syllogism?

Carroll's own way to handle this is chains of implication. He establishes a universe of discourse (all poems), and then can take each of the terms in the argument and put it in a sentence, "It is _____", where 'It' is something in the domain of discourse. He then assigns letters (I = It is interesting, etc.) and works out what each premise and its contrapositive would be:

(a) I → P ; ~P → ~I

and so forth. Then you try to make chains. When you do this you can see clearly that both of these chains are valid:

I → P → ~A → ~M → ~S → ~Y
Y → S → M → A → ~P → ~I

A much easier way to do it, however, once you get the hang of the notation, is the Sommers-Englebretsen term logic. In this notation, No S is P becomes -(+S)-(+)P, All S is P becomes -(+S)+(+P), and each of the terms can be negated independently of the form of the sentence. For instance, No S is non-P becomes -(+S)-(-P). We can collapse the signs just like we do in algebra. So translations for (a) through (e) are (this time the letters are terms, not propositions):

(a) -I-P
(b) -M+A
(c) -Y+S
(d) -A-P
(e) +M-S

To determine the conclusion that validly follows from these premises, we just let negative terms cancel out their positive counterparts. It's obvious that only two terms don't cancel out, -Y and -I. They can be read either as


which says that no interesting poems are poems that are yours, or


which says that no poems that are yours are interesting poems. It's a bit less work and translation than Carroll's method. (I am, however, skipping a brief but important step here, the step of regularity, which guarantees that you cannot get a universal from a premise set with any particular premises or any conclusion from a premise set with more than one particular premise.)

A simpler puzzle, also one of Carroll's:

(a) No ducks waltz.
(b) No officers ever decline to waltz.
(c) All my poultry are ducks.

The Sommers-Englebretsen translations are:

(a) -D-W
(b) -O+W
(c) -P+D

Which leaves us either -O-P or -P-O ('No officers are my poultry', 'None of my poultry are officers').

Some to try at home:

(a) No shark ever doubts that he is well fitted out.
(b) A fish, that cannot dance a minuet, is contemptible.
(c) No fish is quite certain that it is well fitted out, unless it has three rows of teeth.
(d) All fishes, except sharks, are kind to children.
(e) No heavy fish can dance a minuet.
(f) A fish with three rows of teeth is not to be despised.

(a) The only animals in this house are cats.
(b) Every animal is suitable for a pet, that loves to gaze at the moon.
(c) When I detest an animal, I avoid it.
(d) No animals are carnivorous, unless they prowl at night.
(e) No cat fails to kill mice.
(f) No animals ever take to me, except what are in this house.
(g) Kangaroos are not suitable for pets.
(h) None but carnivora kill mice.
(i) I detest animals that do not take to me.
(j) Animals, that prowl at night, always love to gaze at the moon.