Question

Test the validity of the following argument.
All philosophers are logical.
An illogical person is always obstinate.
.: Some obstinate persons are not philosophers.​

Answer 1

Answer:

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:

∀x(P(x) → L(x)) (premise)

∀x(~L(x) → O(x)) (premise)

∃x~L(x) (premise)

~L(c) (EI from 3)

~L(c) → O(c) (UI from 2)

P(c) → L(c) (UI from 1)

~L(c) → ~P(c) (contrapositive of 6)

O(c) (MP 4 and 5)

~P(c) (MP 4 and 7)

O(c) & ~P(c) (conjunction of 8 and 9)

∃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?