Connectives Stranger than Tonk

Abstract

Many logical systems are such that the addition of Prior's binary connective \({\tt tonk}\) to them leads to triviality, see [1, 8]. Since \({\tt tonk}\) is given by some introduction and elimination rules in natural deduction or sequent rules in Gentzen's sequent calculus, the unwanted effects of adding \({\tt tonk}\) show that some kind of restriction has to be imposed on the acceptable operational inferences rules, in particular if these rules are regarded as definitions of the operations concerned. In this paper, a number of simple observations is made showing that the unwanted phenomenon exemplified by \({\tt tonk}\) in some logics also occurs in contexts in which \({\tt tonk}\)is acceptable. In fact, in any non-trivial context, the acceptance of arbitrary introduction rules for logical operations permits operations leading to triviality. Connectives that in all non-trivial contexts lead to triviality will be called non-trivially trivializing connectives.