Paraconsistent logic — the reframer of the principle of non-contradiction
“In a village, the barber shaves all men who do not shave themselves. Does he shave himself?” (‘Barber paradox’, Bertrand Russell)
In the 20th century, a logician named Jan Łukasiewicz, known for the Polish notation, focused on the development of mathematical logic and philosophical logic by deconstructing and reformulating the traditional structure of classical logic, particularly on the principle of non-contradiction and the law of the excluded middle. Łukasiewicz expanded his study to a point of conclusion and developed what is now known as the multiple-valued logic, where the first insight of his deconstruction and reframing of logical meaning is seen, as it is based on allowing more than two truth values in propositional calculus, as opposed to Aristotle’s logic, which can only allow two values; true or false. In 1910, a philosopher named Nicolai Vasiliev presented a lecture that involved an idea similar to Łukasiewicz’s, but more more explicit to the point of dismantling and reframing classical logic, it was called “On Partial Judgements, on the Triangle of Opposites, on the Law of Excluded Third”, three years later he finishes his project of thought, called “Logic and Metalogic” and “Imaginary (non-Aristotelian) logic”. It’s worthy mentioning that polish logician Stanisław Jaśkowski also had an impact on the beginning of this theory with his work on logical inconsistencies.
After the development of both the multiple-valued logic and imaginary logic, the advancement of the theory of a logic that was not dependent on Aristotle’s fundamental set of rules had a problem; it wasn’t formalized yet. That was the case until 1963, when a Brazilian philosopher, mathematician and logician, named Newton da Costa, successfully formalized and finished the theory of a complete non-Aristotelian logic, naming his paper “Formal Inconsistent Systems” , in which he redefined the original definitions of the law of non-contradiction and the principle of explosion by allowing contradictions to be made and considered in logical sentences. Contemporary logical orthodoxy has it that, from contradictory premises, anything follows (principle of explosion). A logical consequence relation is explosive if according to it any conclusion B is entailed by any arbitrary contradiction A, ¬A. Propositional logic, and most standard traditional logics such as intuitionist logic, follow the principle of explosion. Paraconsistent logic treats inconsistency (contradiction, in this case) as a logical and possibly important piece of information for the entire sentence, as long as the sentence is not explosive, any logic is paraconsistent.
A simple ‘formalization’ of paraconsistent logic would be something similar to:
S is both P and not P, but S is not Q.
The barber paradox can be true and perceived as a contradiction that is valid in a paraconsistent system, meaning the barber is and is not a barber, he shaves and doesn’t shave himself. The insights that can be drawn from this contradictory conclusion are usually contextual, for example, he does shave himself, but utilizing a tool that is not generally used to shave, so the definition of shaving is not particularly present in the scenario, but the result is.
