Classical Negation and Expansions of Belnap–Dunn Logic

Abstract

Abstract.We investigate the notion of classical negation from a non-classical perspective. In particular, one aim is to determine what classical negation amounts to in a paracomplete and paraconsistent four-valued setting. We first give a general semantic characterization of classical negation and then consider an axiomatic expansion BD+ of four-valued Belnap-Dunn logic by classical negation. We show the expansion complete and maximal. Finally, we compare BD+ to some related systems found in the literature, specifically a four-valued modal logic of Béziau and the logic of classical implication and a paraconsistent de Morgan negation of Zaitsev.

Four-Valued Logics BD and DM4: Expansions

“…In [8] De and Omori consider an axiomatic expansion BD+ of fourvalued Belnap-Dunn logic by classical negation ¬. …”

“…In [8] De and Omori present a Hilbert style system BD+ in the propositional language {∼, ∧, ∨, ¬, →} 2 with modus ponens as the single inference rule (16 axiom schemata).…”

“…In [8] the authors proved (Theorem 5) that PM4N and BD+ are (functionally) equivalent, i.e. in M PM4N x → y = ¬x ∨ y, and ∼ x = ¬x ↔ ( x ∨ ¬x);…”

“…Let's pay attention to the logic BDe from [8] with language {∼, ∧, ∨, ¬ e }, where ¬ e (exclusion negation) obeys the following truth table:…”

“…Moreover, it follows from [1] that we have an adequate first-order axiomatization for logic T ′′ with quantifiers. At last, in [8] Corollary 17 asserts that BD+ is not functionally complete. We can give a more precise description of functional properties of BD+.…”