Many valued essence and accident logic extended with Belnapian truth
Essence and accident modalities meet Belnapian truth values
Logic in Computer Science
Summary
Classical logic usually treats statements as simply true or false. This paper studies a richer logical system that allows for four truth values, including cases where information is inconsistent or incomplete. The authors develop new ways to understand when statements are essentially or accidentally true, false, inconsistent, or indeterminate within this four-valued logic framework. They also provide a formal way to interpret these ideas and prove that their system behaves well mathematically.
What this means in practice
- •For formal methods engineers: Use four-valued essence and accident modalities to model systems with conflicting or incomplete information while preserving logical rigor.
- •For automated reasoning developers: Incorporate the extended Belnapian modalities for better handling of indeterminacy and inconsistency in reasoning tools.
A theory result. No direct application yet.
Authors
Yaroslav Petrukhin
Abstract
This paper investigates many-valued generalisations of the classical essence and accident modalities. In two-valued logic, a proposition is essentially true (resp. false) if, whenever it is true (resp. false), it is necessarily true (resp. false); it is accidentally true (resp. false) if it is true (resp. false) but not necessarily so. Many-valued logics provide a natural setting for introducing further modalities of this kind. We focus on Belnap-Dunn's First-Degree Entailment (FDE), a four-valued system that generalises the classical truth values. More precisely, we consider an extension of FDE with Boolean negation and implication. In addition to modalities of essential and accidental truth and falsity, we define modalities of essential and accidental inconsistency and indeterminacy. We present a four-valued S5-based Kripke semantics and cut-free hypersequent calculi for the resulting logics. We then prove semantic and syntactic embedding theorems for these logics into a four-valued version of S5 with necessity and possibility modalities. These embeddings clarify the intended interpretation of the Belnapian essence and accident modalities and yield soundness, completeness, and cut-admissibility results.