Idempotent Full Paraconsistent Negations are not Algebraizable

Using methods of abstract logic and the theory of valuation, we prove that there is no paraconsistent negation obeying the law of double negation and such that ¬(a ∧ ¬a) is a theorem which can be algebraized by a technique similar to the Tarski-Lindenbaum technique. 1 What are the features of a paraconsistent negation? Since paraconsistent logic was launched by da Costa in his seminal paper [4], one of the fundamental problems has been to determine what exactly are the theoretical or metatheoretical properties of classical negation that can have a unary operator not obeying the principle of noncontradiction, that is, a paraconsistent operator. What the result presented here shows is that some of these properties are not compatible with each other, so that in constructing a paraconsistent negation as close as possible to classical negation, we have to make a choice among classical properties compatible with the idea of paraconsistency. In particular, there is no paraconsistent negat...

• Type: Journal
• Year: 1998
• Where: NDJFL
• Authors: Jean-Yves Béziau
• Keywords: Double Negation, NDJFL 1998, Negation, Paraconsistent Negation