I have been trying to familiarize myself with the foundations of mathematics, which led me to discussions about propositional, first-order, and second-order logic. I understand that semantics is related to model theory and the satisfiability of models; but I feel that I'm not taking away what I am supposed to.
I understand that in language, semantics defines the "meaning" of words and phrases. Is this analogous to the use of the term in mathematical logic? If so, how does one rigorously talk about the meaning of a statement in logic or math?
Additional insight, sources, and reading recommendations would be greatly appreciated.
Best Answer
Syntax has to do with the formal structure of sentences, proofs, etc., as mere strings of symbols. For example, a proof is a finite sequence $p$ of strings, each one of which either is an axiom or can be derived from earlier strings in $p$ using the rules of inference (these are syntactical conditions, depending only on the strings of characters involved, without reference to any meaning one might give them).
Semantics has to do with the meaning of these sentences—for example, as true or false in some particular model under some interpretation.
The fundamental theorem of first-order logic is the Completeness Theorem, which relates these two completely different ways of looking at languages:
Completeness Theorem: If $T$ is a first-order theory, then $T$ has a model iff $T$ is consistent.
Here, "$T$ has a model" is a statement of semantics, since it has to do with the truth or falsity of the members of $T$ in some model.
In contrast, "$T$ is consistent" is a syntactic statement, since it means merely that there is no proof of a contradiction using, as axioms, sentences of $T$ and the standard axioms of first-order logic. (A "proof" here is as defined in the first paragraph; a proof is a finite sequence of strings, with each string either an axiom or derivable from earlier strings by rules of inference—this is entirely syntactical, as it is a mere question of formal string manipulation.)