> 3. The definition I suggested, where we say P is true iff it holds in some “standard model”;
By the way, I wish you would answer my previous objection about that definition in the context of set theory. What is the standard model of ZFC? (or ZF?) As far as I know, you can't prove that a model for ZF exists (unless you assume some powerful axioms, in which case you won't be able to prove that a model for the extended theory exists).
Edit: Another situation where that definition is problematic is the case of an inconsistent theory. Obviously, an inconsistent theory cannot have a standard model since it does not have a model at all. Whereas with my definition, we get the usual "Ex falso" as expected.
The standard model that most set theorists have in mind is something like the Von Neumann Universe, V. Note that this is a proper class, so it's not a structure as usually considered in model theory.
We (hopefully) can't prove V is a model of ZF in ZF, because that would amount to proving consistency and fall foul of Gödel 2, but the axioms of ZF come from an attempt to axiomatise our understanding of set theory in the sense of it being the study of the objects that make up the Von Neumann Universe.
> Obviously, an inconsistent theory cannot have a standard model
Indeed. Paraconsistent logics are an attempt to deal with inconsistency from a proof-theoretic stance, but I'm far from an expert and I don't know what models of paraconistent theories look like.
By the way, I wish you would answer my previous objection about that definition in the context of set theory. What is the standard model of ZFC? (or ZF?) As far as I know, you can't prove that a model for ZF exists (unless you assume some powerful axioms, in which case you won't be able to prove that a model for the extended theory exists).
Edit: Another situation where that definition is problematic is the case of an inconsistent theory. Obviously, an inconsistent theory cannot have a standard model since it does not have a model at all. Whereas with my definition, we get the usual "Ex falso" as expected.