Validity vs. tautology in predicate logic

Asked Dec 12 '17 at 21:07

Active Dec 13 '17 at 07:23

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I have been starting my studies on predicate logic, and in a paper I'm reading it is said not all the valid sentences are tautologies.

An example of such a sentence is given:

$\exists x Q \leftrightarrow \lnot \forall x \lnot Q$.

Now in what sense it might be said that it is not a tautology?

I thank you for your answers beforehand.

edited Dec 13 '17 at 07:23

Mauro ALLEGRANZA

asked Dec 12 '17 at 21:07

user513015

Closely related: https://math.stackexchange.com/questions/1953814/according-to-language-proof-and-logic-a-a-is-not-a-tautology-why-not?rq=1 – Nate Eldredge Dec 12 '17 at 21:21
2

You have to check the definition of tautology: usually, it is defined in the context of propositional logic. For FOL, a formula is a tautology if it is obtainable from a tautology of propositional logic by replacing (uniformly) each sentence symbol by a formula of the first-order language.This is not the case with the formula above, that we can get from the propositional $P \leftrightarrow R$, that is not taut. – Mauro ALLEGRANZA Dec 13 '17 at 07:26
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Does this answer your question? Why isn't $\forall x \forall y ~ P(x,y) \rightarrow \forall x \forall y~P(y,x)$ a tautology? – ryang Aug 13 '22 at 03:37

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