Representation Theorems for Cumulative Propositional Dependence Logics

TL;DR

This paper establishes and proves representation theorems for cumulative propositional dependence logic and for cumulative propositional logic with team semantics and shows that entailment in cumulative propositional logics with team semantics is exactly captured by cumulative and asymmetric models.

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.

Figures (1)

• Figure 1: Landscape of classes of entailment relations. For not defined classes, consult SMK ( ?) or the supplemental material.

Theorems & Definitions (25)

• Definition 1: Team semantics of $\mathrm{PL}$
• Proposition 2
• Example 3
• Proposition 4
• Definition 5: Shoman ?, Dix and Makinson ?; ?
• Definition 6
• Definition 7
• Proposition 8: KLM, ?
• Theorem 9
• Proposition 10