Two-valued Weak Kleene logics

Abstract

In the literature, Weak Kleene logics are usually taken as three-valued logics. However, Suszko has challenged the main idea of many-valued logic claiming thatevery logic can be presented in a two-valued fashion. In this paper, we providetwo-valued semantics for the Weak Kleene logics and for a number of four-valuedsubsystems of them. We do the same for the so-called Logics of Nonsense, which are extensions of theWeak Kleene logics with unary operators that allow looking at them as Logics of Formal Inconsistency (LFIs) and Logics of Formal Underterminedness (LFUs). Our aim with this work, rather than arguing for Suszko´s thesis, is to show that two-valued presentations of these peculiar logics enlighten the non-standard behavior of their logical connectives. More specically, the two-valued presentations of paraconsistent logics illustrate and clarify the disjunctive flavor of the conjunction, and dually, the two-valued presentations of paracomplete subsystems ofWeak Kleene logics reveal the conjunctive flavor of the disjunction.