Literal and Controllable Paraconsistency

Abstract

The principle of explosion asserts that any formula can be derived from any pair of other contradictory formulas. Paraconsistent logic is typically regarded as a logic in which the universal validity of this principle is questioned. Therefore, a key point is determining when the validity can be considered universal to classify a logic as paraconsistent. A pertinent example to illustrate this point is the calculus CB1 that admits the principle but only for negated formulas, i.e., from any set {α, ∼α} any other formula follows if and only if α is of the form ∼γ. Another example is Sette’s calculus P1, which is paraconsistent at the level of variables but not complex formulas. Both serve as compelling examples of the so-called borderline cases.

In this paper, we examine several calculi expected to be paraconsistent at the level of literals. It means that a pair of formulas, α and ∼α, can yield any β if, and only if α is neither a propositional variable nor is its iterated negation. Furthermore, it is assumed that in some calculi presented here, β must adhere to specific restrictions. Once these conditions are satisfied, we refer to calculus as paraconsistent in a “controllable manner”.