One of the basic features of a negation ~ within a given logic L is
the so-called 'explosiveness', which states that from a contradiction
'P and ~P' everything is derivable in L. Thus, classical logic (and
many other logics) equate 'consistency' with 'freedom from
contradictions'. Paraconsistency is the study of logic systems with a
negation ~ such that not every contradiction of the form 'P and ~P'
trivializes, that is, such that explosiveness is not always the case.
In this talk we discuss several approaches to the question of
paraconsistency, with emphasis on the Logics of Formal Inconsistency
(LFIs), proposed by W. Carnielli and J. Marcos, which internalize in
the object language the very notions of consistency and inconsistency
by means of specific connectives (primitives or not). This generalizes
the strategy of N.C.A. da Costa, which introduced the well-known
hierarchy of systems Cn, for n > 0. Dialetheism, mainly developed by
G. Priest, is a different approach to paraconsistency based on the
existence of 'real contradictions' (called dialetheias). Some
potential applications of paraconsistency to databases, logic
programming and multi-agent systems will be presented.