Paraconsistent logic is a type of non-classical logic that allows for the coexistence of contradictory statements without leading to a logical explosion where anything can be proven true. Specifically, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" systems of logic, purposefully excluding the principle of explosion. Inconsistency-tolerant logics have been discussed since at least 1910 (and arguably much earlier, for example in the writings of Aristotle); however, the term paraconsistent ("beside the consistent") was coined in 1976, by the Peruvian philosopher Francisco Miró Quesada Cantuarias, under request of Newton da Costa, who is often credited as the creator of the field. The study of paraconsistent logic has been dubbed paraconsistency, which encompasses the school of dialetheism.
Definition In classical logic (as well as intuitionistic logic and most other logics), contradictions entail everything. This feature, known as the principle of explosion or ex contradictione sequitur quodlibet (Latin, "from a contradiction, anything follows") can be expressed formally as
Which means: if P and its negation ¬P are both assumed to be true, then of the two claims P and (some arbitrary) A, at least one is true. Therefore, P or A is true. However, if we know that either P or A is true, and also that P is false (that ¬P is true) we can conclude that A, which could be anything, is true. Thus if a theory contains a single inconsistency, the theory is trivial – that is, it has every sentence as a theorem. The characteristic or defining feature of a paraconsistent logic is that it rejects the principle of explosion. As a result, paraconsistent logics, unlike classical and other logics, can be used to formalize inconsistent but non-trivial theories.
Comparison with classical logic The entailment relations of paraconsistent logics are propositionally weaker than classical logic; that is, they deem fewer propositional inferences valid. The point is that a paraconsistent logic can never be a propositional extension of classical logic, that is, propositionally validate every entailment that classical logic does. In some sense, then, paraconsistent logic is more conservative or cautious than classical logic. It is due to such conservativeness that paraconsistent languages can be more expressive than their classical counterparts including the hierarchy of metalanguages due to Alfred Tarski and others. According to Solomon Feferman: "natural language abounds with directly or indirectly self-referential yet apparently harmless expressions—all of which are excluded from the Tarskian framework." This expressive limitation—where in classical logic, tautologies are considered redundant—can be overcome in paraconsistent logic.
Motivation A primary motivation for paraconsistent logic is the conviction that it ought to be possible to reason with inconsistent information in a controlled and discriminating way. The principle of explosion precludes this, and so must be abandoned. In non-paraconsistent logics, there is only one inconsistent theory: the trivial theory that has every sentence as a theorem. Paraconsistent logic makes it possible to distinguish between inconsistent theories and to reason with them. Research into paraconsistent logic has also led to the establishment of the philosophical school of dialetheism (most notably advocated by Graham Priest), which asserts that true contradictions exist in reality, for example groups of people holding opposing views on various moral issues. Being a dialetheist rationally commits one to some form of paraconsistent logic, on pain of otherwise embracing trivialism, i.e. accepting that all contradictions (and equivalently all statements) are true. However, the study of paraconsistent logics does not necessarily entail a dialetheist viewpoint. For example, one need not commit to either the existence of true theories or true contradictions, but would rather prefer a weaker standard like empirical adequacy, as proposed by Bas van Fraassen.
Philosophy In classical logic, Aristotle's three laws, namely, the excluded middle (p or ¬p), non-contradiction ¬ (p ∧ ¬p) and identity (p iff p), are regarded as the same, due to the inter-definition of the connectives. Moreover, traditionally contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory entails all possible consequences) are assumed inseparable, granted that negation is available. These views may be philosophically challenged, precisely on the grounds that they fail to distinguish between contradictoriness and other forms of inconsistency. On the other hand, it is possible to derive triviality from the 'conflict' between consistency and contradictions, once these notions have been properly distinguished. The very notions of consistency and inconsistency may be furthermore internalized at the object language level.
Tradeoffs Paraconsistency involves tradeoffs. In particular, abandoning the principle of explosion requires one to abandon at least one of the following two principles:
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