UPDATE: since first writing this comment, I’ve checked four quasi-randomly selected books from my shelves (Leary and Kristiansen
Introduction to Mathematical Logic, Hodges
Model Theory, Manzano
Model Theory, Avigad
Mathematical Logic and Computation), and they all use
valid/
validity rather than
true/
truth to describe formulae that are true in all models, as I originally pointed out below. To be clear, I’m not at all trying to score points by appealing to the literature, but I think it’s really important to clarify that your definition isn’t standard because it
will confuse people; I myself was confused by this
exact point when I studied logic having previously read the “true but unprovable” description of Gödel 1.
> A statement is true by definition if and only if it is satisfied in every model.
I don’t think this is a standard definition.
Every treatment I’ve seen refers to truth with respect to a model; if no model is specified, it is assumed to be obvious from context. Outside of formal treatments (i.e. in the setting where the 99% of mathematicians who aren’t logicians do their work), the model is the standard model.
First-order formulae that are true in every model are validities.
> A statement is true by definition if and only if it is satisfied in every model.
I don’t think this is a standard definition.
Every treatment I’ve seen refers to truth with respect to a model; if no model is specified, it is assumed to be obvious from context. Outside of formal treatments (i.e. in the setting where the 99% of mathematicians who aren’t logicians do their work), the model is the standard model.
First-order formulae that are true in every model are validities.