Society semantics for four-valued Łukasiewicz logic

Abstract

We argue that many-valued logics (MVLs) can be useful in analysing informational conflicts by using society semantics (SSs). This work concentrates on four-valued Łukasiewicz logic. SSs were proposed by Carnielli and Lima-Marques (1999, Advances in Contemporary Logic and Computer Science, 235, 33–52) to deal with conflicts of information involving rational agents that make judgements about propositions according to a given logic within a society, where a society is understood as a collection $\mathcal{A}$ of agents. The interesting point of such semantics is that a new logic can be obtained by combining the logic of the agents under some appropriate rules. Carnielli and Lima-Marques (1999, Advances in Contemporary Logic and Computer Science, 235, 33–52) defined SSs for the three-valued logics $I^{1}$ and $P^{1}$. In this kind of semantics, all the agents reason according to classical logic (CL) and the molecular formulas behave in the same way as in CL (the non-classical character of these logics only appears at the propositional level). Marcos (unpublished data) provided SSs with classical agents for the three-valued Łukasiewicz logic Ł$_{3}$, but in this case, the molecular formulas do not behave classically. We prove here that one can characterize Ł$_{4}^{\prime}$, a conservative extension of Ł$_{4}$ obtained by adding a connective $\blacktriangledown$, by means of a closed society where the agents reason according to Ł$_{3}$. We shall emphasize the importance of recovery operators in the construction of this class of societies. Moreover, we shall relate this semantics to Suszko’s view on the ‘two-valuedness’ of logic.

Many-Valued Logics and Bivalent Modalities

“…The last few years have witnessed a growth of interest in many-valued logics (MVLs). Examples can be found in their application to the analysis of semantic paradoxes [Da Ré et al, 2018;Priest, 1979] or in the study of rationality [Belnap, 1977;Bezerra, 2020;Kubyshkina, 2016]. These fruitful exercises have indirectly responded to the criticism that MVLs have received in the literature, due to the conceptual difficulties in characterizing the meaning of their intermediate logical values [see Pogorzelski, 1994].…”