Информационная система "Конференции"

Abstracts

Non-classical logics, proof theory and universal algebra

MATRIX MODAL LOGIC

We suggest new classification of modal systems on the basis of non-relational (quasi) matrix semantics which includes known Kripkean systems as well as some other systems for which there are no special names in Kripkean logic. Each logic is also presented in a form of a system of semantic (analytic) rules. We present computer implementation of these analytic rule systems.

The (quasi)matrix semantics offers a way to interpret modal formulas in a truth-tabular way, without invoking possible worlds, or anything like them. The idea of non-relational semantics for modal logic goes back to the paper of J.Kearns [2] in which the author presented modal systems T, S4 and S5 on the basis of the combination of four-valued matrices with special construction, so called “T-valuations”.

Yu.Ivlev [1] suggested the idea of quasi-matrix modal semantics. Instead of the alternative worlds one deals with the alternative interpretation quasi-functions formed by the given interpretation function. This allows to give table definitions for modal operators; there sometime can be vagueness in valuations of modal formulas- in that case the value of the given formula A is considered to be ''undetermined'' - for example, the value p/q means ''either p, or q''. Instead of an interpretation function, interpretation quasi-function takes only one single value from the fraction; in case of the value p/q there are two alternative quasi-interpretations, one in which A takes the value p, and the other in which A takes q. The formula A is true in the interpretation if and only if it is true in each alternative interpretation caused by the given interpretation.

Ivlev [1] constructed various quasi-matrix modal systems for which table definitions are given on the basis of intuition. These systems are incomparable to Kripkean ones, since for example Goedel rule is not valid in them. We present systematic research of modal systems including known Kripkean logics, on the basis of (quasi)matrix semantics.

References:

(1) Ivlev, Yu.: Modal logic, Moscow University Publ., Moscow, 1991, pp.220;

(2) Kearns, J.: Modal semantics without possible worlds, Journal of Symbolic Logic, 1981, Vol.46, N1, pp.77-86;

(3) Kouznetsov, A.: Quasi-matrix deontic logic, Proceedings of the Seventh International Workshop on Deontic Logic in Computer Science (DEON'04), Lecture Notes in Artificial Intelligence (LNAI 3065) Springer-Verlag, 2004, pp.191-208.

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