Four-valued logics of indicative conditionals
Miguel Muñoz Pérez
TL;DR
This work extends three-valued logics of indicative conditionals to a four-valued setting by enriching twist-structure semantics with additional truth-values. It first reviews the canonical three-valued twist representations for OL, DF, CN/F and then develops four-valued families (OLg, DFg, CNf, Fg, and their variants OLf, DFf, CNf, Ff) arising from semantic-gap and falsity extensions, respectively. The paper provides twist-representation theorems showing how each four-valued logic can be realized as a twist algebra over combinations of factor algebras (e.g., Diamond and related constructions) and discusses how negation interacts with the expanded value set. It also analyzes algebraizability issues, Boethius-type properties, and potential future research directions, including axiomatization and alternative approaches like non-deterministic matrices. Overall, it clarifies how four-valued indicative-conditionals can be constructed from established three-valued twist frameworks, linking semantic interpretation with precise algebraic semantics.
Abstract
We detail some ways in which the study of three-valued logics of indicative conditionals can be extended by further adding a new truth-value. Our approach heavily relies on twist constructions, which have been already used in the literature in order to provide algebraic semantics in the three-valued case. Here we follow the inverse path: we first specify how these twist constructions, expanded with the new truth-value, induce new logics and then we prove the corresponding twist representation results.