Is $\forall I: I(f)=1$ where $f$ stands for a formula the meaning / definition of a tautology?

Question

I have a question about the following formula. Is $\forall I: I(f)=1$ where $f$ stands for a formula and I for an interpretation the meaning / valid definition of tautology?

Answer 1

There are two notions kicking around here, one syntactic and the other semantic.

• Syntactic tautologies: $f$ is a theorem of the empty theory, with respect to whatever proof system is being used.

• Semantic tautologies: $f$ is true in every structure in the language of $f$ (or, every interpretation).

In my experience, "tautology" refers primarily to the former notion, while sentences with the latter property are called "validities." You will need to look at whatever text you're using to tell whether the definition of tautology you've given is appropriate.

However, by the completeness theorem, these two notions are in fact the same (as long as we're using a reasonable proof system). This is one reason that it's not a big deal that texts sometimes define "tautology" in different ways.

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... At least, in the context of first-order logic (or propositional logic, for that matter). The notions above make sense in much more general contexts: "syntactic tautology" makes sense in the context of any proof system, and "semantic tautology" makes sense in the context of any satisfaction notion (even if the structures involved are quite different from what we're used to!). In general these notions may not line up; e.g. second-order logic (with the standard semantics) has no complete proof system.