CEFISES Seminar: Pawel Pawlowski, “One framework to rule them all”
Channel: CEFISES at UCLouvain
Date: 05/03/2023

Series: Logic and Philosophy

Speaker: Pawel Pawlowski (Ghent University)

Title: “One framework to rule them all. How to overcome Dugundji’s theorem by using non-deterministic semantics for modal logics.”

Abstract:

Kripke’s or possible world semantics are now commonly associated with modal logics. Its impact on the development of modal logics is undisputed. However, this was not always the case historically. During the so-called syntactical period of modal logics, two fundamental approaches were formulated for our discussion. The first approach aimed to find natural many-valued semantics for existing modal systems, such as Lewis’ systems S1-S5. The second approach directly applied a many-valued framework to define new modal logics. The first approach was deemed impossible in 1940 when James Dugundji modified Godel’s proof, showing that modal systems between S1 and S5 did not have many-valued semantics.

The second approach was proposed by Lukasiewicz in 1953, who defined four-valued modal systems. This framework had some philosophical and technical problems. Both lines of research were forgotten until 1981 when Kearns published his work that resurrected and combined both lines of research. Kearns’ idea was similar to Lukasiewicz’s approach, with one main difference. Instead of the usual many-valued approach, Kearns constructed a non-deterministic many-valued approach to modal logic.

Non-deterministic semantics are generalizations of the matrix approach to many-valued logics. The main difference between the usual many-valued semantics and the non-deterministic one is the interpretation of connectives. In the deterministic case, connectives are interpreted as functions in the set of values, resulting in a truth-functional and extensional framework. In non-deterministic semantics, connectives are interpreted as functions into the power set of values minus the empty set, allowing connectives to assign a non-empty set of values to a formula rather than a unique value. The resulting framework is quasi-extensional. Such a framework allowed Kearns’ to characterize modal logics T,S4, and S5. His results were recently rediscovered in 2016 independently by Omori and Coniglio groups.

Our talk presents and extends recent results regarding non-deterministic semantics for modal logics. We show what modal logics can be characterized within the eight valued frameworks proposed by Omori and Skurt. This framework is an exciting alternative to possible-world semantics and needs further development.