The logics of formal inconsistency s) are paraconsistent logics which permit us to internalize the concepts of consistency or inconsistency inside our object language, introducing new operators to talk about them, and allowing us, in principle, to logically separate the notions of contradictoriness and of inconsistency. We present the formal definitions of these logics in the context of General Abstract Logics, argue that they in fact represent the majority of all paraconsistent logics existing up to this point, if not the most exceptional ones, and we single out a subclass of them called -systems, as the s that are built over the positive basis of some given consistent logic. Given precise characterizations of some received logical principles, we point out that the gist of paraconsistent logic lies in the Principle of Explosion, rather than in the Principle of Non-Contradiction, and we also sharply distinguish these two from the Principle of Non-Triviality, considering the next various weaker formulations of explosion, and investigating their interrelations. Subsequently, we present the syntactical formulations of some of the main -systems based on classical logic, showing how several well-known logics in the literature can be recast as such a kind of -systems, and carefully study their properties and shortcomings, showing for instance how they can be used to faithfully reproduce all classical inferences, despite being themselves only fragments of classical logic, and venturing some comments on their algebraic counterparts. We also define a particular subclass of the -systems, the -systems, as the ones in which the new operators of consistency and inconsistency can be dispensed. A survey of some general methods adequate to provide these logics with suitable interpretations, both in terms of valuation semantics and of possible-translations semantics, is to be found in a follow-up, the paper [42]. This study is intended both to fully present and characterize, from scratch, the field into which it inserts, hinting of course to the connections with other studies by several authors, as well as to set some open problems, and to point to a few directions of continuation, establishing on the way a unifying theoretical framework for further investigation for researchers involved with the foundations of paraconsistent logic
