I was reading Jerome Keisler's Model Theory and I have found the following characterization of tautology. He first defines what makes a formula valid and points out it could be very difficult to find out whether some sentence $\phi$ is valid, because "at first glance," you would have to check uncountably many different infinite models. He then proceeds:
This is because validity is a semantical notion, defined in terms of models. However as the reader surely knows, there is a simples and uniform test by which we can find out in only finitely many steps whether or not a given sentence $\phi$ is valid.
This decision procedure for validity is based on a syntactical notion, the notion of a tautology. […]
1.2.5. Let $\phi$ be a sentence and let $S_{0}, … S_{n}$ be all the sentence symbols occuring in $\phi$. $\phi$ is said to be a tautology, in symbols $\vdash \phi$, iff $\phi$ has the value t for every assignment $a_{0}, …, a_{n}$.
This is the first time I see tautology as a syntactical notion. Although he offers a definition which is syntactical, I thought it was plainly consensual "tautology", "assignment", as "interpretation" and "model", were semantical notions. So, straightforwardly, my question is:
Tautology is a semantical/syntactical notion depending on the definition we are using?
Thanks everyone.
Best Answer
It is not so much a question of which definition, as on what your perspective is. In other words, it depends on what you mean by "syntactical" and "semantical". :-)
From the perspective of model theory, it is convenient to consider "tautology" to be a syntactical concept, because it's a matter of the shape (so to say) of a formula, and not on how the formula's meaning relates to a model at all. So it's a concept that is not particularly interesting from a model theorist's point of view -- he will consider it a background concept that comes from the concept of a formula rather than from the models he's really concerned with, and all that counts as "syntax" for him.
On the other hand propositional logic has its own distinction between syntactic and semantic concepts. Here "semantic" is used about things that are about truth values and evaluation of formulas to a truth value, whereas "syntax" is about picking formulas apart and putting them together in new configurations, as in symbolic proofs. In that world, "tautology" is firmly established as a "semantic" concept. (Or so I thought -- but see also Noah Schweber's answer which shows that he considers the word "tautology" to belong to the syntactic concept even when considering only logic. He and I agree about what is syntactic and what is semantic in that context, but not about the preferred definition of "tautology").
Moral: "syntax" and "semantics" are not crisp technical terms, but are fuzzy categories that you use to structure your theory building within each particular field.