Representation Theorems for Cumulative Propositional Dependence Logics

TL;DR

This paper proves representation theorems for cumulative propositional dependence logic and team semantics, linking these logics to specific classes of models and expanding understanding of their semantic foundations.

Contribution

It establishes exact correspondences between cumulative propositional dependence logics, team semantics, and specific model classes, advancing the semantic characterization of these logics.

Findings

System C entailments are captured by cumulative models from Kraus, Lehmann, and Magidor.

Entailment in team semantics is characterized by cumulative and asymmetric models.

Equivalence with classical propositional logic-based cumulative logics is demonstrated.

Abstract

This paper establishes and proves representation theorems for cumulative propositional dependence logic and for cumulative propositional logic with team semantics. Cumulative logics are famously given by System C. For propositional dependence logic, we show that System C entailments are exactly captured by cumulative models from Kraus, Lehmann and Magidor. On the other hand, we show that entailment in cumulative propositional logics with team semantics is exactly captured by cumulative and asymmetric models. For the latter, we also obtain equivalence with cumulative logics based on propositional logic with classical semantics. The proofs will be useful for proving representation theorems for other cumulative logics without negation and material implication.