Polarity Semantics for Negation as a Modal Operator
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Polarity Semantics for Negation as a Modal Operator
Abstract. The minimal weakening N0 of Belnap-Dunn logic under the polarity semantics for negation as a modal operator is formulated as a sequent system which is characterized by the class of all birelational frames. Some extensions of N0 with additional sequents as axioms are introduced. In particular, all three modal negation logics characterized by a frame with a single state are formalized as extensions of N0 . These logics have the finite model property and they are decidable. Keywords: Negation, Polarity semantics, Sequent system.
1.
Introduction
De Morgan algebras (also known as quasi-Boolean algebras in [20]) are algebraic models for the Belnap-Dunn four-valued logic with truth constants and ⊥ (cf. e.g., [1, 7–13]). This logic is denoted by BD, and it has been investigated in the framework of abstract algebraic logic (cf. e.g. [14–16,19]). Dunn introduced the polarity semantics for BD based on informational structures (cf. e.g. [12]). Let W be a set of informational states. Every proposition ϕ is interpreted as a pair V + (ϕ), V − (ϕ), where the set V + (ϕ) consists of all informational states that accept ϕ, and V + (ϕ) consists of all informational states that reject ϕ. The logic BD is characterized by a single reflexive informational state under the polarity semantics (cf. [12,13]). If general Kripke-style informational models are introduced to construe negation as a modal operator that is interpreted by accessibility relations in polarity semantics, we obtain weakenings of Belnap-Dunn logic. The deterministic weakening DW as a nine-valued logic was proposed in [17]. Countably many weakenings of BD which are logics for Berman’s varieties are given in [18]. In the present paper, we introduce the minimal weakening N0 of BD which is characterized by the class of all arbitrary birelational frames under the polarity semantics for negation. Many logics for negation
Presented by Jacek Malinowski; Received February 27, 2019
Studia Logica https://doi.org/10.1007/s11225-019-09879-w
c Springer Nature B.V. 2019
Y. Lin, M. Ma
over distributive lattices can be syntactically or semantically characterized as extensions of the common ground N0 . In the polarity semantics for weakenings of Belnap-Dunn logic (cf. [17, 18]), the negation sign ∼ is indeed treated as a modal operator in the sense that it is a negative necessity operator relative to the modal accessibility relation in a model. Do˘sen [4–6] introduced negation as a modal operator based on the positive fragment of intuitionistic propositional logic. The language of Do˘sen’s logic is built from propositional variables using connectives ∼, ∧, ∨ and →. Do˘sen’s Hilbert-style axiomatic system N is obtained by adding the axiom schema (∼ϕ ∧ ∼ψ) → ∼(ϕ ∨ ψ) and the following inference rule to the system for positive intuitionistic logic: ϕ→ψ ∼ψ → ∼ϕ A Do˘sen’s model for N is a triple (W, RI , RN , V ) where RI is a preorder on W , RN is a binary relation on W with RI ◦ RN ⊆ RN ◦ RI−1 , and V is a persistent
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