On logics of formal inconsistency and fuzzy logics

Paraconsistency is the study of logics (as deductive systems) having a negation operator ¬ such that not every contradiction {φ,¬φ} trivializes or explodes. In other words, a paraconsistent logic is a logic having at least a contradictory, non-trivial theory. Among the plethora of paraconsistent logics proposed in the literature, the so-called Logics of Formal Inconsistency (LFIs), proposed in [3] (see also [2]), play an important role, since they internalize in the object language the very notions of consistency and inconsistency by means of specific connectives (either primitive or not). This generalizes the strategy of da Costa, which introduced in [5] the well-known hierarchy of systems Cn, for n > 0. Besides being able to distinguish between contradiction and inconsistency, on the one hand, and non-contradiction and consistency, on the other, LFIs are nonexplosive logics, that is, paraconsistent: in general, a contradiction does not entail arbitrary statements, and so the Principle of Explosion φ,¬φ ! ψ does not hold. However, LFIs are gently explosive, in the sense that, adjoining the additional requirement of consistency, then contradictoriness does cause explosion: ©(φ),φ,¬φ ! ψ for every φ and ψ. Here, ©(φ) denotes the consistency of φ. The general definition of LFIs we will adopt here, slightly modified from the original one proposed in [3] and [2], is the following: