Paraconsistency and Paracompleteness

Abstract

A logic $\langle \mathcal{L},\vdash_{p}\rangle$ is said to be paraconsistent if, and only if $\{\alpha, \neg \alpha\} \nvdash_{p} \beta$, for some formulas $\alpha, \beta$. In other words, the necessary and sufficient (the latter is problematic) condition for a logic to be paraconsistent is that its consequence relation is not $\textit{explosive}$. The definition is very simple but also very broad, and this may create a risk that some logics, which have not too much in common with the $\textit{paraconsistency}$, are considered to be so. Nevertheless, the definition may still serve as a reasonable starting point for more thorough research.

Paracomplete logic can be defined in many different ways among which the following one may be of some interest: A logic $\langle \mathcal{L},\vdash_{q}\rangle$ is said to be paracomplete if, and only if $\{\beta \rightarrow \alpha, \neg \beta \rightarrow \alpha\} \nvdash_{q} \alpha$, for some formulas $\alpha, \beta$. But again, just as in the case of paraconsistent logic, the definition is very general and may be seen to overlap with the logics that have nothing in common with the \textit{paracompleteness}.

In the paper, we define some calculi of paraconsistent and paracomplete logics arranged in the form of hierarchies, determined by several criteria. We put central emphasis on logical axioms admitting only the rule of detachment as the sole rule of inference and on the so-called bi-valuation semantics. The hierarchies (no matter which one) are expected to shed some light on the aforementioned issue.