Effect of the Choice of Connectives on the Relation between the Logic of Constant Domains and Classical Predicate Logic

TL;DR

This paper explores how the choice of propositional connectives affects the relationship between constant domain Kripke semantics and classical predicate logic, extending previous propositional results to first-order logic.

Contribution

It establishes that monotonicity of connectives is necessary and sufficient for the equivalence of valid sequents in constant domain Kripke models and classical logic in first-order logic.

Findings

Monotonic connectives ensure equivalence of valid sequents in all constant domain Kripke models and classical logic.

The paper extends propositional results to first-order logic.

Monotonicity is a key condition for logical equivalence in this context.

Abstract

It is known that not only classical semantics but also intuitionistic Kripke semantics can be generalized so that it can treat arbitrary propositional connectives characterized by truth tables, or truth functions. In our previous work, it has been shown that the set of Kripke-valid propositional sequents and that of classically valid propositional sequents coincide if and only if all available propositional connectives are monotonic. The present paper extend this result to frst-order logic showing that, in the case of predicate logic, the condition that all available propositional connectives are monotonic is a necessary and sufficient condition for the set of sequents valid in all constant domain Kripke models, not the set of Kripke-valid sequents, and the set of classically valid sequents to coincide.