Is the proposition $\forall x: P(x)$ a tautology?

Question

If $P(x)$ is a predicate and for all x: $P(x)$ is true, does that make the proposition $(\forall x: P(x))$ a tautology?

Or is it not a tautology because P(x) can be defined to be false, in which case $(\forall x: P(x))$ can also be a contradiction since it always evaluates to false?

Edit: Mistook tautology for 'totality'

Answer 1

No. Putting it crudely, a necessary condition for a sentence counting as a tautology is that it is true as a matter of logic alone (some would require more). But for a given $P$ with a given interpretation, if could be the case that $\forall xPx$ because of how things just happen to be.

If $P$ means "is male", and the domain is Presidents of the US, past and present, then $\forall xPx$ is true: but that's not a tautology on any account but a regrettable contingent fact!

Answer 2

I, personally, would restrict "tautology" to classical propositional logic. For classical predicate logic, I would talk about $\forall x.P(x)$ being valid or not. Validity (and tautology) means being true in all interpretations, which, in this case, would mean for any interpretation of $P$.

It doesn't make sense to talk about $P(x)$ being "true" without first fixing an interpretation of $P$. We could restrict our attention only to interpretations where $P(x)$ is true for every $x$ in the domain (which is also given by the interpretation). The notion of validity is not similarly restricted though.

Without first specifying an interpretation, it doesn't make sense to talk about $P(x)$ being true "for all $x$" unless we simply mean $\forall x.P(x)$ is valid/provable. (The soundness and completeness theorems of classical first-order logic show that validity and provability are equivalent.) It is not clear what $x$ should "run over". The only choice that jumps out at me at this level would be something like "for every closed term $t$, $P(t)$ is provable". This is a dramatically weaker statement than "$\forall x.P(x)$ is provable". As an extreme example, typical presentations of ZFC set theory have no closed terms at all, so the former statement would be vacuously true for every predicate.

Alternatively, you could allow $x$ to "run over" arbitrary terms, i.e. including open ones. That would include, in particular, variables. Most presentations of proof systems for classical first-order logic make demonstrating the inter-derivability of $P(y)$ and $\forall y.P(y)$ very easy. (There is a side condition that $y$ not occur free in any of the assumptions allowing you to derive $P(y)$ to make this equivalence.) Note, I'm not saying $P(y)$ holds "for any value of $y$", but that the single open formula $P(y)$ is derivable (with respect to a suitable notion of derivation of open formulas). In practice, if $P$ is an unknown atomic predicate, this will not be derivable unless something has gone horribly wrong.