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.