Abstract
Curry’s paradox for “if.. then..” concerns the paradoxical features of sentences of the form “If this very sentence is true, then 2 + 2 = 5”. Standard inference principles lead us to the conclusion that such conditionals have true consequents: so, for example, 2 + 2 = 5 after all. There has been a lot of technical work done on formal options for blocking Curry paradoxes while only compromising a little on the various central principles of logic and meaning that are under threat. Once we have a sense of the technical options, though, a philosophical choice remains. When dealing with puzzles in the logic of conditionals, a natural place to turn is independently motivated semantic theories of the behaviour of “if... then...”. This paper argues that a closest-worlds approach outlined in previous work offers a philosophically satisfying reason to deny conditional proof and so block the paradoxical Curry reasoning, and can give the verdict that standard Curry conditionals are false, along with related “contraction conditionals”.
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Notes
Tarskian theories of truth based on Tarski (1944) will naturally have this consequence. Broadly “contextualist” approaches such as those of Parsons (1974) and Glanzberg (2001), if extended to the Curry paradox, would ensure that Curry sentences did not express propositions, at least in crucial contexts.
Kripke (1975) provides constructions where there will be “ungrounded” counterexamples to the T-scheme, and it is natural to treat Curry paradoxes this way in that system. Classical approaches rejecting bivalence include those of Horwich (1998, pp. 40–42), which denies an unrestricted T-scheme. But there are many, many other theories that restrict the T-scheme: and armed with this restriction, it is very tempting to apply it to Curry paradoxes.
I am tempted to think that some instances of the T-scheme, considered as a biconditional, are false in some contexts. Consider “If nothing were true, then “nothing is true” would be true.” I think in some contexts we should take that conditional to be false. If nothing were true, it may be that dogs would bark and cats would meow, but nothing would be true, including “nothing is true” and the proposition we in fact express with “nothing is true”.
I hedge a bit because my Nolan (1997, p. 554) points out that statement modus ponens fails in my system: and the presence of statement modus ponens is one of the things I think is a classic symptom of susceptibility to Curry’s paradox. But if I suspected in 1997, I forgot again.
I should take this opportunity to note a technical error in my statement of the truth conditions of counterfactual conditionals in Nolan (1997, p. 564), where I stated it in terms of A ⊃ B holding at spheres. While this definition would have been suitable for possible worlds, whether A ⊃ B holds at an impossible world has little to do, in general, with whether A holds or B holds there.
It might be otherwise if one rejected my condition (3) on the similarity relation (Nolan 2003, p. 218), as for example Brian Weatherson does (see page 339 of Weatherson 2009). More reason not to reject my (3), I say, though no doubt there are other ways to treat the indicative closeness relation that will allow one to solve the Curry paradox in the way I will suggest.
Perhaps there is not a closest, because of ties or infinite sequences of ever closer worlds, but let us not worry about that for now.
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Acknowledgments
Thanks to audiences at the “What If?” conference at the University of Connecticut, the University of Melbourne, the Australian National University (twice), the University of Michigan, the University of St Andrews, and to Jc Beall, Carrie Jenkins, David Ripley and Dustin Tucker for discussion. This research was supported under the Australian Research Council’s Discovery Projects funding scheme (Project Number DP130104665).
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Nolan, D. Conditionals and Curry. Philos Stud 173, 2629–2647 (2016). https://doi.org/10.1007/s11098-016-0666-7
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DOI: https://doi.org/10.1007/s11098-016-0666-7