David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Journal of Philosophical Logic 35 (6):653 - 660 (2006)
Many logical systems are such that the addition of Prior's binary connective tonk to them leads to triviality, see [1, 8]. Since 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 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 tonk in some logics also occurs in contexts in which 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.
|Keywords||proof-theoretic semantics trivializing connectives|
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References found in this work BETA
Arthur Prior (1967). The Runabout Inference Ticket. In Peter Strawson (ed.), Analysis. Oxford University Press 38-9.
N. D. Belnap (1977). A Useful Four-Valued Logic. In J. M. Dunn & G. Epstein (eds.), Modern Uses of Multiple-Valued Logic. D. Reidel
Heinrich Wansing (2000). The Idea of a Proof-Theoretic Semantics and the Meaning of the Logical Operations. Studia Logica 64 (1):3-20.
Stephen Read (1994). Thinking About Logic: An Introduction to the Philosophy of Logic. Oxford University Press.
Roy T. Cook (2005). What's Wrong with Tonk(?). Journal of Philosophical Logic 34 (2):217 - 226.
Citations of this work BETA
David Ripley (2015). Anything Goes. Topoi 34 (1):25-36.
Heinrich Wansing & Yaroslav Shramko (2008). Suszko's Thesis, Inferential Many-Valuedness, and the Notion of a Logical System. Studia Logica 88 (3):405 - 429.
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