# Vacuous quantification

1. Jul 14, 2007

### Manchot

I'm trying to prove a theorem which makes use of the identity $\exists x (P) \rightarrow P$ (where x is not a free variable of P). Intuitively, I want to believe it, but since I'm trying to do things rigorously, I'd like to be able to justify it to myself. Can anyone offer a suggestion as to how I'd derive the identifier from the usual axioms of first-order logic? (I'm sure that I'm missing something totally obvious). Thanks.

2. Jul 14, 2007

### fopc

I don't understand. P is a naked propositional symbol in FOL? If so, then your done
(regardless of whether it has any quantifiers attached to it, or not). It's a tautology.

Last edited: Jul 14, 2007
3. Jul 14, 2007

### Manchot

P is a predicate in which x doesn't appear. Anyway, I know that it is a tautology, but I'm trying to prove it rigorously from FOL's axioms.

4. Jul 16, 2007

### Manchot

Never mind, I've got it. Thanks anyway.