Ever since A. N. Prior (1960) introduced tonk it has been a crucial part of the debate regarding the semantics for logical expression. This expression is assumed to jeopardizes the inferentialist project, being a connective that can be introduced by prima facieacceptable rules, but whose introduction renders unacceptable results: it leads to the trivialization of the system. Different strategies have been sketched to deal with tonk, essentially, posing restrictions on the acceptability of rules of inference based on considerations of how rules should look like, or based on considerations on the context in which they are introduced. This last approach is illustrated in Belnap (1962), where tonk is rejected on the grounds that it conflicts with previous commitments on the notion of deductibility: transitivity and non-triviality. Should the consequence-relation be non-transitive or trivial, our impressions about tonk would be different. Cook (2005) accepts the challenge of developing a non-transitive logic where tonk can be incorporated without leading to triviality. That logic is a fourth-valued logic. In this work I will provide a logic inspired by Cook’s Tonk Logic which is also non-transitive and can be incorporated without the threat of triviality, but which restricts to two values.

References

Belnap, N. (1962) “Tonk, plonk and plink”, Analysis 22, 130-134.

Cook, R. (2005) “What’s wrong with tonk (?)”, Journal of Philosophical Logic 34, 217-226.

Prior, A. N. (1960) “The runabuout inference ticket”, Analysis 21, 38-39.