Abstract
In his ‘Proof and Paradox’ (Dialectica 36, 1982), Tennant draws a distinction between paradox and inconsistency. In this paper, two examples are discussed which show Tennant’s formulation of the distinction to be problematic. Tennant’s distinction is reformulated so to account for these cases as well. This is done by using ‘A natural extension of natural deduction’ (J Symbolic Logic 39:4, 1984) devised by Schroeder-Heister to allow not only formulas but also rules to be assumed in the course of a derivation.
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Notes
- 1.
The two alternative definitions of ‘normal’ are discussed in detail in Tranchini (n.d.).
- 2.
In Tranchini (2014), I suggest a way in which Tennant’s ideas can be used to extend Dummett-Prawitz-style proof-theoretic semantics so that it can be applied to paradoxical phenomena as well.
- 3.
‘I shall always assume that eliminating detours such as the one mentioned, that lead from a sentence occurrence to another occurrence of the same sentence, is an acceptable part of any procedure of normalising proofs’. (Tennant 1982, p. 270)
- 4.
For the problems that Tennant’s analysis of paradox has to face when confronted with the phenomenon observed by Ekman, see also Schroeder-Heister and Tranchini (n.d.).
- 5.
References
Belnap, N.D. 1962. Tonk, plonk and plink. Analysis 22(6): 130–134.
Chihara, C. 1979. The semantic paradoxes: a diagnostic investigation. The Philosophical Review 88(4): 590–618.
Ekman, J. 1998. Propositions in propositional logic provable only by indirect proofs. Mathematical Logic Quarterly 44: 69–91.
Kripke, S. 1975. Outline of a theory of truth. Journal of Philosophy 72(19): 690–716.
Prawitz, D. 1965. Natural deduction. A proof-theoretical study. Stockholm: Almqvist & Wiksell.
Read, S. 2010. General-elimination harmony and the meaning of the logical constants. Journal of Philosophical Logic 39: 557–576.
Schroeder-Heister, P. 1981. Untersuchungen zur regellogischen Deutung von Aussagenverknüpfungen, PhD thesis, Bonn University.
Schroeder-Heister, P. 1984. A natural extension of natural deduction. The Journal of Symbolic Logic 49(4): 1284–1300.
Schroeder-Heister, P.: 2011. Implications-as-rules vs. implications-as-links: an alternative implication-left schema for the sequent calculus. Journal of Philosophical Logic 40(1): 95–101.
Schroeder-Heister, P. 2012. Proof-theoretic semantics, self-contradiction, and the format of deductive reasoning. Topoi 31(1): 77–85.
Schroeder-Heister, P. and L. Tranchini. n.d. Ekman’s paradox. Under review. Available at http://ls.inf.uni-tuebingen.de/psh
Tennant, N. 1982. Proof and paradox. Dialectica 36: 265–296.
Tranchini, L. 2014. Proof-theoretic semantics, paradoxes, and the distinction between sense and denotation. Journal of Logic and Computation. doi: 10.1093/logcom/exu028.
Tranchini, L. n.d. Harmonizing harmony. To appear in the Review of Symbolic Logic.
Tranchini, L. and P. Schroeder-Heister. n.d. Ekman’s paradox and general natural deduction rules. Under review. Available at https://sites.google.com/site/lucatranchini/
von Plato, J. 2000. A problem of normal form in natural deduction. Mathematical Logic Quarterly 46(1): 121–124.
Acknowledgements
I thank Alberto Naibo and Peter Schroeder-Heister for having pointed to Ekman’s paper. His ‘paradox’ has been also independently observed by Matteo Plebani, whom I thank for his brilliant work as discussant of my talk ‘Proof-theoretic semantics, paradoxes and the distinction between sense and denotation’ at the V Latin Meeting of Analytic Philosophy, Lisbon, November 2011. Thanks also to an anonymous referee for several suggestions which helped improving the readability of the paper.
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Tranchini, L. (2015). Paradox and Inconsistency: Revising Tennant’s Distinction Through Schroeder-Heister’s Assumption Rules. In: Lolli, G., Panza, M., Venturi, G. (eds) From Logic to Practice. Boston Studies in the Philosophy and History of Science, vol 308. Springer, Cham. https://doi.org/10.1007/978-3-319-10434-8_7
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