David Ripley: Curry’s Paradox and Substructural Logic | WSB #51

September 15, 2018

Consider the sentence C: "If this sentence is true, then David Ripley is a purple giraffe". Suppose the sentence is true. Then the antecedent of the sentence ("this sentence is true") is true. According to the inference rule modus ponens, if an if-then sentence (such as C) is true and its antecedent is true, then its consequent ("David Ripley is a purple giraffe") must be true. It follows that if C is true, then David Ripley is a purple giraffe. But this conclusion is C: in other words, by simply supposing how things might turn out if C were true, we have proved that C, in fact, is true. So C is true, and since C's antecedent is the claim that C is true, its antecedent is true as well. Now we can use modus ponens again to show that C's consequent must be true. In other words, David Ripley really is a purple giraffe. QED.

This argument is Curry's paradox. Obviously, the choice of "David Ripley is a purple giraffe" is arbitrary; a sentence of the form of "If this sentence is true, then X" can be used to prove any claim X. Now, in actual fact, David Ripley is not a purple giraffe, but a philosopher of language and logic. According to Ripley, solutions to paradoxes like Curry's (as well as the Liar and the Sorites) fall into two broad categories: those that solve the paradoxes by messing with the meanings of important concepts (such as the meaning of "if-then", truth, "not", etc.) and those that solve them by changing the structural rules of inference by appeal to substructural logics. The latter approach, says Ripley, is preferable because it allows us to keep the intuitive meanings of these important concepts. There are various structural rules that can be modified to avoid the paradoxes, but the one that Ripley prefers is the denial of transitivity. This would mean that even if we prove that a implies b and that b implies c, we have no guarantee that a implies c. Ripley tells a story about assertion and denial conditions to argue that, precisely because of the paradoxes, the denial of transitivity conforms to natural language inferential norms. We conclude with a discussion of "revenge" Curry paradoxes for Ripley's approach, and of frontiers for substructural logic.

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Special thanks to Jackie Blum for the podcast art, and The Tin Box for the theme music.
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Sources:

David Ripley (website)
Curry's paradox (Stanford Encyclopedia of Philosophy)
"Comparing Substructural Theories of Truth" (David Ripley)
"Revising Up: Strengthening Classical Logic in the Face of Paradox" (David Ripley)

Topics discussed:

0:19 - Introduction to David Ripley
3:17 - Logic, reasoning, and ways to approach paradox
11:00 - Curry's paradox
25:19 - The Liar as analogous to Curry
34:34 - Taxonomy of solutions: vocabulary-based vs. structural-based
45:35 - Logical connectives and natural language analogues
49:55 - Optimism vs. pessimism in responses
55:23 - Substructural solutions: non-contractive
59:42 - Substructural solutions: non-transitive
1:06:52 - Assertion and denial conditions
1:20:44 - Gaps and gluts and revenge
1:33:01 - History of and horizons for substructural logic

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