Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-28T15:21:10.987Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  03 February 2015

University Of Aberdeen


This paper explores how a semantics for Prior’s infamous connective tonk should be, a connective defined by inference rules that trivialize the logic of a deductive system if that logic is supposed to be transitive. To avoid triviality, one must reject transitivity and in a relatively recent paper, Roy Cook develops a semantics for tonk with non-transitive entailment. However, I show in this paper that a cut-free sequent calculus for tonk - the arguably most natural and simplest deductive system for a non-transitive logic - can neither be complete with respect to Cook’s semantics nor with respect to a semantics with non-transitive entailment based on a semantics for vagueness and transparent truth developed by Cobreros et al. It is argued that the failure to adequately represent tonk is connected with the fact that tonk is not uniquely defined in a cut-free sequent calculus system unless the logic is in addition non-reflexive. To remedy this, the paper develops a semantics with non-transitive and non-reflexive entailment based on the idea that complex formulae are true or false relative to them being assessed as premise or as conclusion.

Research Article
Copyright © Association for Symbolic Logic 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



Belnap, N. (1962). Tonk, plonk, plink. Analysis, 22, 130134.Google Scholar
Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41, 347385.Google Scholar
Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2013). Reaching transparent truth. Mind, 122, 841866.Google Scholar
Cook, R. (2005). What’s wrong with tonk(?). Journal of Philosophical Logic, 34, 217226.CrossRefGoogle Scholar
Dummett, M. (1991). The Logical Basis of Metaphysics. Cambridge, MA: Harvard University Press.Google Scholar
Gupta, A. (2009). Definitions. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Spring 2009 edition).Google Scholar
Hjortland, O. T. (2009). The structure of logical consequence: Proof-theoretic conceptions. PhD thesis, University of St. Andrews.Google Scholar
Humberstone, I., L. (1988). Heterogeneous logic. Erkenntnis, 29, 395435.Google Scholar
MacFarlane, J. (2012). Relativism. In Fara, D. G. and Russell, G., editors. The Routledge Companion to the Philosophy of Language. New York: Routledge.Google Scholar
Negri, S., & von Plato, J. (2001). Structural Proof Theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Priest, G. (2008). An Introduction to Non-Classical Logic: From If to Is (second edition). New York: Cambridge University Press.Google Scholar
Prior, A. N. (1960). The run-about inference-ticket. Analysis, 21, 3839.Google Scholar
Restall, G. (2005). Multiple conclusions. In Hajek, P., Valdes-Villanueva, L., and Westerstahl, D., editors., Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress. London: Kings College Publications.Google Scholar
Ripley, D. (2012). Conservatively extending classical logic with transparent truth. The Review of Symbolic Logic, 5, 354378.Google Scholar
Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91, 139164.Google Scholar
Ripley, D. (2014). Anything goes. Topoi. doi: 10.1007/s11245-014-9261-8.Google Scholar
Schroeder-Heister, P. (2004). On the notion of assumption in logical systems. In Bluhm, R. and Nimtz, C., editors. Selected Papers Contributed to the Sections of GAP.5, Fifth International Congress of the Society for Analytic Philosophy, Bielefeld, September 2003. Paderborn: Mentis, pp. 2748.Google Scholar
Steinberger, F. (2011). Why conclusions should stay single. Journal of Philosophical Logic, 40, 333355.Google Scholar
Stevenson, J. T. (1961). Roundabout the runabout inference-ticket. Analysis, 21, 124128.Google Scholar
Tarski, A. (1944). The semantic conception of truth. Philosophy and Phenomenological Research, 4, 341375.Google Scholar
Troelstra, A. S., & Schwichtenberg, H. (2000). Basic Proof Theory. Cambridge: Cambridge University Press.Google Scholar