Axiomatizability of non-normal and quasi-normal modal predicate logics of first-order definable classes of Kripke frames

M. N. Rybakov

Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 3 (2018), pp. 81-94

Résumé

The possibility of effective description of non-normal and quasi-normal predicate modal logics defined semantically by means of classes of Kripke frames with distinguished worlds is considered. It is proved that any non-normal or quasi-normal (in particular, normal) modal predicate logic, complete with respect to a certain first-order definable class of Kripke frames with distinguished worlds, can be embedded into the classical first-order logic. It is shown how to construct such an embedding based on the so called standard translation of modal predicate formulas into formulas of the first-order classical language. At the end of the work, we present some corollaries of this result and demonstrate the possibility of generalization for the described construction to classes of other systems, in particular, to classes of polymodal logics-temporal logic with a pair of modalities ‘always in past’ and ‘always in future’ and logics of knowledge with the operator of distributed knowledge. Some limitations for applicability of the described method are shown, relevant examples are given. Counterexamples are indicated when the conditions of the method applicability for the Kripke complete modal predicate logic are not met but the construction of an effective description of this logic is nevertheless possible.

Détail

Keywords: first-order logic, modal logic, non-normal logic, recursive enumerability, Kripke semantics.
Mots-clés : quasi-normal logic