Syllogism from Lewis Carroll

Question

I was reading Code by Charles Petzold, and on page 86 he references a syllogism from Lewis Carroll:

• All philosophers are logical;
• An illogical man is always obstinate;
• Therefore, some obstinate persons are not philosophers.

I formalized this argument in predicate logic as follows:

• Let P(x) be the propositional function "x is a philosopher".
• Let L(x) be the propositional function "x is logical".
• Let O(x) be the propositional function "x is obstinate".

Then the premises become:

• ∀x(P(x) → L(x)), and
• ∀x(~L(x) → O(x)), respectively.

The conclusion is:

• ∃x(O(x) & ~P(x)).

I attempted deriving this conclusion, but was unable to. I then wondered if a derivation was at all possible, and used prover9 to find a proof, but the result was negative (which could mean that my input was invalid).

I then added an additional premise, ∃x~L(x) (there is at least one illogical person), and was easily able to derive the conclusion:

1. ∀x(P(x) → L(x)) (premise)
2. ∀x(~L(x) → O(x)) (premise)
3. ∃x~L(x) (premise)
4. ~L(c) (EI from 3)
5. ~L(c) → O(c) (UI from 2)
6. P(c) → L(c) (UI from 1)
7. ~L(c) → ~P(c) (contrapositive of 6)
8. O(c) (MP 4 and 5)
9. ~P(c) (MP 4 and 7)
10. O(c) & ~P(c) (conjunction of 8 and 9)
11. ∃x(O(x) & ~P(x)) (EG of 10)

My question then is: are the original premises sufficient for deriving the conclusion, or is the additional premise necessary?

Answer 1

Your second premise could be vacuously satisfied- i.e. there might be no illogical men in the world: without a witness (guaranteed only when you add the aditional premise ∃x~L(x)), the existential quantifier in your conclusion would not be true (in the sense of Tarski). As this remains a possibility, Caroll's argument is indeed not formally logical- this is, I suppose, what one might expect from someone writing 40 years before sentential calculus was formally axiomised.

Answer 2

Someone suggested that I answer the question directly. The problem you're having is a consequence of one of the changes that were made to "mainstream logic" between the time of Lewis Carol and the time when modern predicate logic was defined. Before Frege, Russel, et al, universal propositions were thought to have existential import if they were true. "All unicorns have one horn" is false simply because there are no unicorns, according to the older, Aristotlean rule. The statement is true according to the more modern view.

The matter really boils down to whether you allow empty terms or not. That is, does it make sense to make assertions about things that don't exist. Do unicorns have hearts? Do they sometimes suffer strokes? Are they raced the same way horses are? There is a legitimate intuition that it is nonsense to talk about empty terms and non-existent animals. But it is also legitimate to speak about empty terms hypothetically for the sake of argument, such as if you want to engage in reductio ad absurdum reasoning.

But the point is crucial if you want to translate statements to predicate logic. If someone says that "All philosophers are logical," do you really believe that he suspects that "philosophers" could be an empty term? Even if this is a cynical dig at modern philosophers these days, the term isn't empty if there ever were or ever will be philosophers. So to translate this assumption into modern logic, you would make the existence of at least one philosopher explicit: ∃x(P(x)) & ∀x (P(x) → L(x)).

You should do the same thing with the second premise. Sure, you could wonder if the author really meant that there are no illogical men. But I would suggest that with most common "practical" reasoning, we don't really deal with empty terms. What you should never do is just translate a sentence naively into predicate logic and then think that the argument isn't valid. Predicate logic itself makes several assumptions that aren't always the case. You need to at least know what those assumptions are.

Answer 3

I'm afraid I don't have the patience to work through your proof in detail, but I will mention that problems of this nature are more easily solved by Lewis Carroll's "Game of Logic" schema than by other formalizations-- in fact, I suspect that the example was drawn from that book.

It is out of copyright, and a scan can be found here; I'd highly recommend it if you are interested in logic, as Carroll has a knack for drawing out the unexpected.

Answer 4

Here's one way of deriving the conclusion the Aristotelian way, as Carroll himself would have done :

p : philosophers, l : logical, o : obstinate

 1. All p is l            Premise
 2. All non-l is o        Premise
 3. All non-l is non-p    1, Contraposition
 4. Some non-l is non-p   3, Subaltern
 5. Some o is non-p       2, 4, Syllogism
 6. Some o is not p       5, Obversion (Conclusion)