many-valued logics;twist-product semantics;semiring truth values

Semirings provide a lightweight algebraic language for quantitative reasoning in computer science, supporting tasks such as weighted automata, provenance analysis, and soft constraints. However, a persistent obstacle for semiring-valued logics is that semirings do not generally offer a satisfactory internal notion of negation or implication. To address this problem, the present paper proposes a dual-valuation semantics that interprets each formula by a pair of semiring values representing positive and negative support, with negation implemented by swapping the two channels. We survey several consequence relations that can be defined over this semantics, showing that the approach generates a rich family of semiring-based logics. We also discuss material and first-degree implication connectives and outline possible extensions of the framework.

Semirings provide a lightweight algebraic language for quantitative reasoning in computer science, supporting tasks such as weighted automata, provenance analysis, and soft constraints. However, a persistent obstacle for semiring-valued logics is that semirings do not generally offer a satisfactory internal notion of negation or implication. To address this problem, the present paper proposes a dual-valuation semantics that interprets each formula by a pair of semiring values representing positive and negative support, with negation implemented by swapping the two channels. We survey several consequence relations that can be defined over this semantics, showing that the approach generates a rich family of semiring-based logics. We also discuss material and first-degree implication connectives and outline possible extensions of the framework.