Modality, invariance, and logical truth (Q1097254)

I shall discuss the conditions under which a model-theoretic characterization of validity for a language leads to a class of logical truths which exemplify certain necessity properties. Our question is this. Let us think of a notion of possibility for a language L as being characterized by a class \({\mathbb{M}}\) of theories in L; a statement of L is possible (in the sense of \({\mathbb{M}})\) if it occurs in some theory in \({\mathbb{M}}\) and is necessary if its negation is not possible. Under what conditions will the model-theoretically valid statements of L be necessary? This question will receive a general answer in terms of certain properties of the expressions of L which are treated as constants in the characterization of validity. Each notion of possibility \({\mathbb{M}}\) for L will be associated with a property, to be termed invariance over \({\mathbb{M}}\), defined for expressions in L. In Section 4 it is shown that if the logical expressions of L are invariant over \({\mathbb{M}}\), then a sentence is valid in L only if it is necessary in the sense of \({\mathbb{M}}\). The languages which I consider arise by supplementing first-order logic with generalized quantifiers. The results can, however, be extended to other languages.