Paradoxes of Expression
In this note, I show how to construct Liar-like and Curry-like paradoxes in a framework Graham Priest has been considering recently, in which he tries to solve the paradoxes by giving up the rule of modus ponens (detachement) instead of the rules of ex fa
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Abstract In this note, I show how to construct Liar-like and Curry-like paradoxes in a framework Graham Priest has been considering recently, in which he tries to solve the paradoxes by giving up the rule of modus ponens (detachement) instead of the rules of ex falso and contraction. The Curry-like paradox presents a serious challenge to the detachment-free framework because it threatens to trivialize the system, just as Curry’s original paradox does for the more standard paraconsistent approach to the paradoxes.
Graham Priest in some recent talks and unpublished work investigates the possibility, also discussed favorably but in less detail in Goodship [4] and defended in Beall [2], of solving the semantic and set theoretic paradoxes by restricting the truth schema, the naive set abstraction schema, and so on by formulating them with a biconditional that does not detach, i.e., that does not satisfy modus ponens [12]. But a detachable truth schema is needed for the usual account of blind endorsement, i.e., of the ascription of truth to sentences that are identified in a way that gives no clue about their content (e.g., when someone holds that everything the Bible says is true). To solve this problem, Priest deliberates whether to add a further, detachable conditional, propositional quantifiers, and an “expression predicate” (Priest [12], Sect. 5.2). In this note, I will show that given a natural principle about expression, both Liar-like and Curry-like paradoxes can then be constructed without any appeal to the truth schema or to its relatives. I will discuss briefly what these paradoxes of expression mean for such detachment-free approaches to paradox.1 Given Priest’s usual paraconsistent approach to paradox that is based on the logic LP (Priest [11], 53ff.), it is natural for him to work with a non-detachable biconditional, as LP does not in general sanction the move from ‘ p ⊃ q’ and ‘ p’ to ‘q’ if 1I
would like to thank an anonymous referee, Johannes Korbmacher, Tobias Martin, Graham Priest, Stewart Shapiro, and Niko Strobach for helpful discussions and comments.
M. Pleitz (B) Philosophisches Seminar, Westfälische Wilhelms-Universität, Domplatz 6, D-48143 Müenster, Germany e-mail: [email protected] © Springer International Publishing AG 2016 H. Andreas and P. Verdée (eds.), Logical Studies of Paraconsistent Reasoning in Science and Mathematics, Trends in Logic 45, DOI 10.1007/978-3-319-40220-8_9
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‘⊃’ is defined in the usual way from negation and disjunction or conjunction. And he explores some metaphysical options that open up when the theory of identity is based on a non-detachable conditional in his book on unity (Priest [13], 16ff.). In his usual approach to the paradoxes, however, the Liar and its ilk are not solved by restricting the truth schema, but by dropping the ex falso rule, so that it becomes acceptable that there are (some) dialetheias, i.e., true contradictions. Later on, Priest and other dialetheists normally add a detachable conditional (Priest [11], 82ff.), probably b
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