Modal Logics -- RNmatrices vs. Nmatrices

TL;DR

This paper compares non-deterministic and restricted non-deterministic semantics for modal logics, highlighting their similarities, differences, and extensions using three specific modal logics, and discusses current limitations of both approaches.

Contribution

It provides a comparative analysis of two semantic frameworks for modal logics and demonstrates their application to specific modal systems, clarifying their respective advantages and limitations.

Findings

Both semantics use many-valued matrices with tuples of 0s and 1s.

Restricted non-deterministic semantics can uniformly extend minimal modal logic M.

Differences include how extensions of M are constructed and interpreted.

Abstract

In this short paper we will discuss the similarities and differences between two semantic approaches to modal logics - non-deterministic semantics and restricted non-deterministic semantics. Generally speaking, both kinds of semantics are similar in the sense that they employ non-deterministic matrices as a starting point but differ significantly in the way extensions of the minimal modal logic M are constructed. Both kinds of semantics are many-valued and truth-values are typically expressed in terms of tuples of 0s and 1s, where each dimension of the tuple represents either truth/falsity, possibility/non-possibility, necessity/non-necessity etc. And while non-deterministic semantics for modal logic offers an intuitive interpretation of the truth-values and the concept of modality, with restricted non-deterministic semantics are more general in terms of providing extensions of M, including normal ones, in an uniform way. On the example of three modal logics, MK, MKT and MKT4, we will show the differences and similarities of those two approaches. Additionally, we will briefly discuss (current) restrictions of both approaches.