On Ivlev’s Semantics for Modality

The most popular semantics for modality is the so-called Kripke semantics. However, there were several attempts aiming at a reasonable semantics for modality even before Kripke. Jan Łukasiewicz gave two accounts, and, in this paper, we focus on the second account which turned out to be an approach through many-valued semantics. This attempt is now evaluated as a failed project after an observation by Josep Maria Font and Petr Hájek. Recently, Jean-Yves Béziau suggested a four-valued semantics for modality. Moreover, José M. Méndez and Gemma Robles have explored the many-valued approach to modality intensively. Based on these, the aim of the paper is twofold. First, we discuss the observations offered in the previous attempts and point out some of their problems. Second, we show how to avoid the problems in the previous attempts and clarify what is required to motivate or reject the many-valued approach to modality.