A note on a deductive scheme of Dummett in classical and fuzzy logics

Abstract

The main subject of logic is the study of the processes of deductive reasoning and, in particular, the validity of the so-called ‘schemes’ of such type of reasoning. In this paper, a first and partial study of the validity of the following deductive scheme is presented,IfA,thenBIfnot-A,thenBB.This study is done assuming that the statements A and B are representable in Boolean algebras, De Morgan algebras, orthomodular lattices, or in standard algebras of fuzzy sets, and with the conditional statements ‘If/then’ translated into several conditional operators by taking always into account the consequence operator.