Predicate logic: Why ∀x(Px ⊃ Px) is not a tautology?

Question

I'm learning predicate logic and my textbook says that some logical truths expressible in the language of predicate logic are not tautologies.

For example, according to my textbook, $∀x(Px ⊃ Px)$ is valid even though it doesn't instantiate a tautological schema.

However, I don't understand why $∀x(Px ⊃ Px)$ can't be regarded as a tautology. Can someone help me?

Thanks a lot

Answer 1

A tautology is a formula of propositional calculus that is true for every assignment of truth values to the propositional letters (i.e. for every interpretation).

Example of tautologies : $A \to A, \lnot A \lor A$.

In first-order logic, a formula is valid iff it is true for every interpretation.

In first-order logic, a tautology is a formula that is obtained by a propositional tautology replacing uniformly propositional letters with formulae.

Example, $Px \to Px$ is a FOL tautology, as well as $\forall xPx \to \forall xPx$ and $\lnot \forall xPx \lor \forall xPx$, because they can obtained from the propositional tautologies $A \to A$ and $\lnot A \lor A$ replacing the propositional letter $A$ with the formulae $Px$ and $\forall xPx$ respectively.

Obviously every FOL tautology is valid, but not all FOL valid formulae are tautologies.

Examples:

$\forall x (Px \to Px), \forall x (x=x)$.