Essence and accident modalities meet Belnapian truth values
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.
Versions
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BibTeX
@misc{petrukhin2026essence,
title = {{Essence and accident modalities meet Belnapian truth values}},
author = {Yaroslav Petrukhin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08323},
eprint = {2610.08323},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08323},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Essence and accident modalities meet Belnapian truth values
AU - Yaroslav Petrukhin
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08323
UR - https://arxiv.org/abs/2610.08323
AB - 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.
ER -