> 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.
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.