Tonk, Syntax, Semantics, Pragmatics

In 1960, A. N. Prior proposed a binary sentential functor defined by inferential rule (introduction) A ⊢ A tonk B, and (elimination) A tonk B ⊢ A. Later, this idea was discussed by several authors, including in the context of the question of whether the ab ove rule defines this functor in a sufficient manner. This paper shows that if we assume the standard matrix (truth-tables) characterization of classical sentential functors, no valuation agrees with rules generating the sense of tonk. Moreover, these inferential prescriptions are at odds with the principle that if premises are true, the conclusion has to be such as well. A new solution is proposed. It consists in considering tonk-rules via so-called rejection consequence operation, that is, a dual with respect to the standard Cn. The general moral stemming from the proposed analysis says that inferentialism, the view that inference rules are purely syntactical, is dubious, because logic has its basis also in semantic presuppositions as well as in pragmatic ones.