Can an open formula be a tautology?

Question

A tautology is a statement which is true by dint only of the logical connectives contained therein. My question is about an open formula which contains an unquantified variable. For example:

• P: ($x$ is a cat) or not ($x$ is a cat).

This appears to be a tautology, and is certainly true in any universe when we precede the statement by a "$\forall x$".

QUESION: Is P considered to be 1) a tautology and/or 2) an open formula?

Answer 1

Of course this depends on the definition of "tautology" in predicate logic. The definition I learned years ago is the same as what Wikipedia says: "In the context of predicate logic, many authors define a tautology to be a sentence that can be obtained by taking a tautology of propositional logic and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable)." By that definition yes, $P$ is a tautology, being a substitution instance of the propositional-logic tautology "A or not A".

(Regarding your doubt about open formulas, note it says "formula" above, not "sentence".)

We should probably note that this definition of "tautology" is purely syntactic, has nothing to do with assigning truth values in structures for first-order logic. Which it seems to me is as it "should" be; it "should" be purely syntactic. (Although all the other definitions we've seen in this thread involve the semantics of first-order logic I've never seen a definition of "tautology" anywhere else that was not syntactic.)

Answer 2

An open formula is not possible to evaluate at all. However one often define that $M\models \varphi(x)$ if $M\models \forall x\varphi(x)$. In this case if we translate $\forall x\varphi(x)$ to propositional logic, we only get a single propositional variable $P$ which certainly is not a tautology. So in general I would say No.

However a formula such as $\forall x(P(x)\rightarrow P(x))$ is equivalent to $\forall xP(x)\rightarrow \forall xP(x)$. If we translate $\forall xP(x)\rightarrow \forall xP(x)$ to propositional logic we get $R\rightarrow R$, which is a tautology. Thus we may possibly be able to say that $P(x)\to P(x)$ is a tautology, since if we add $\forall x$ it is equivalent to a formula which may be translated to a tautology. But if this is allowed, depends on how you define a tautology in predicate logic.