> Well, I suppose it depends on your definition of standard.
Of course :) I believe my distinction between validity and truth is the one generally used in the literature (I have listed four examples above), and the one that would be understood by most working mathematicians and analytic philosophers who care about mathematical logic.
We both agree that there is a clear distinction between formulae that are true in some model (specified, or inferred from context) and formulae that are true in all models; the latter are not particularly interesting to most mathematicians once one has agreed on the logic (e.g. classical, constructive, etc.) in which one operates, hence I think it’s reasonable to use “true” to refer to the former, as indeed many authors do.
> That's called satisfiability.
Many logicians say that a formula is true in a model (sometimes true in a structure) if it’s satisfied in that model under all assignments.
Can you find me a reference in the literature where “true” is used to mean “true in all models” consistently?
Of course :) I believe my distinction between validity and truth is the one generally used in the literature (I have listed four examples above), and the one that would be understood by most working mathematicians and analytic philosophers who care about mathematical logic.
We both agree that there is a clear distinction between formulae that are true in some model (specified, or inferred from context) and formulae that are true in all models; the latter are not particularly interesting to most mathematicians once one has agreed on the logic (e.g. classical, constructive, etc.) in which one operates, hence I think it’s reasonable to use “true” to refer to the former, as indeed many authors do.
> That's called satisfiability.
Many logicians say that a formula is true in a model (sometimes true in a structure) if it’s satisfied in that model under all assignments.
Can you find me a reference in the literature where “true” is used to mean “true in all models” consistently?