On a paraconsistentization functor in the category of consequence structures

Abstract

This paper is an attempt to solve the following problem: given a logic, how to turn it into a paraconsistent one? In other words, given a logic in which ex falso quodlibet holds, how to convert it into a logic not satisfying this principle? We use a framework provided by category theory in order to define a category of consequence structures. Then, we propose a functor to transform a logic not able to deal with contradictions into a paraconsistent one. Moreover, we study the case of paraconsistentization of propositional classical logic.

Paraconsistentization and many-valued logics

“…We review some notions in the domain of paraconsistentization and we follow the presentations in [8] and [9].…”

“…The paraconsistentization of L 3 using the standard procedure was first developed in [10]. In [8], we have presented a sufficient condition to the P-transformation of a given logic L to be a paraconsistent logic. We have to consider the following essential concepts (see [8], definition 4.1, p.246).…”

“…Afterwards, in two recent papers (cf. [8] and [9]), it has been showed how to turn a given logic into a paraconsistent one by a very precise methodology which works for a great variety of systems. In [8], the idea of paraconsistentization is presented in the context of category theory and it uses abstract logic as a main source.…”

“…[8] and [9]), it has been showed how to turn a given logic into a paraconsistent one by a very precise methodology which works for a great variety of systems. In [8], the idea of paraconsistentization is presented in the context of category theory and it uses abstract logic as a main source. In this way, paraconsistentization appears as an endofunctor in the category of logics preserving some basic properties of the initial logic.…”

“…The above papers settled the basic theory of paraconsistentization, up to this level. Despite the fact the there are many unknown methods of paraconsistentization, this article shows how to turn some explosive manyvalued logics into paraconsistent ones using the basic idea initially proposed in [8], that is: given a logic L = F or, L , the paraconsistentization of L is a logic given by P(L) = F or, P L such that: Γ P L α if, and only if, there exists Γ ′ ⊆ Γ, L-consistent such that Γ ′ L α. We deal mainly with systems such as L 3 , G 3 and K 3 defined by means of logical matrices, though it is obviously possible to extend the same approach to the whole hierarchies L n and G n .…”