Ex Contradictione Non Sequitur Quodlibet

1. Do we need to worry about inconsistency? Classical logic, as we all know, cannot survive contradictions. Among the principles that were gradually incorporated into the “properties of correct reasoning” since Aristotle, the Principle of Pseudo-Scotus (PPS), also known since medieval times as ex contradictione sequitur quodlibet (and also called the Principle of Explosion by some contemporary logicians), states that in any theory exposed to the enzymatic character of a contradiction A and ÏA one can derive any other arbitrary sentence B, so that the theory would turn out to be trivial. Another principle called the Principle of Non-Contradiction (PNC) states that there should be theories from which no such contradictions are derivable. To those principles, one could add the Principle of Non-Triviality (PNT), stating that there should be at least one theory and one sentence B such that B is not derivable from this theory. In order to fully understand what those principles mean, what the relationship is between them and what their importance is for modeling the concept of inconsistency let us introduce some formalism. This formalism will be apt for the syntactical approach to the logics of formal inconsistency we discuss in the first three sections of this paper. However, the reader should be aware that it is possible to start from a purely syntactical account, as we do in Sections 4 and 5. Let For be a collection of formulas of a certain language, and call a theory any subset of For. Let a consequence relation Í over For be a relation between theories and formulas of For, that is, Í ⊆ (℘(For) × For), where ℘(For) denotes the power set of For. If Γ is a subset of For, we write ΓÍA when æΓ , AÆ ∈ Í . We write ΓåA when it is not the case that ΓÍA . We define a logic L to be a structure constituted of For and the relation Í . The consequence relation of a given logic is often defined by its axioms and rules, or else from some semantical interpretation associated with the logic.