David Ripley: Curry’s Paradox and Substructural Logic | Who Shaves the Barber? #51

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David Ripley

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.…

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Jc Beall: Logic of Christ | Who Shaves the Barber? #44

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Jc Beall

Christ is a walking contradiction. He is both fully human and fully divine. Indeed, he is both mutable and immutable. According to classical logic, the existence of a true contradiction would imply that everything is the case, no matter how absurd. And so, theologians and Christian metaphysicians have worked for centuries to conceptually make sense of Christ’s dual nature in a way that avoids contradiction.

Philosopher and logician Jc Beall argues that these efforts have been motivated by a naive understanding of logic. There are “subclassical” logics – that is, logics weaker than classical logic – in which contradictions do not entail every arbitrary conclusion. And these aren’t ad-hoc constructions. Beall argues that one subclassical logic – called First Degree Entailment (FDE) – is, in fact, the correct account of logical consequence, for reasons independent of the Christian problem. Beall covers the basics of how FDE works and why it is the universal or “basement-level” consequence relation. This allows us to have our cake and eat it too: we may take Christ to be, quite literally, both mutable and not mutable, at the same time and in the same respect. This isn’t just appealing for its simplicity. Beall suspects that it is essential to Christ’s role that he be literally contradictory.

If you’re interested in Jc Beall’s work and non-classical logic, check out my interview with Greg Restall (part 1 and part 2) on the book Logical Pluralism, co-authored by Beall and Restall.…

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Stephen Read: Bradwardine Solution to the Liar | Who Shaves the Barber? #42

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Bradwardine Liar
Thomas Bradwardine

For much of the 20th century, the Liar paradox has stood as an elusive and stubborn puzzle. The main solutions to it have significant drawbacks, such as blocking meaningful cases of self-reference or abandoning bivalence (the principle that all propositions are either true or false and not both). In recent decades, Stephen Read has rediscovered and defended a solution by the medieval thinker Thomas Bradwardine. If Bradwardine’s argument is correct, the liar sentence is simply false. When properly examined, its falsity does not imply its truth. Bradwardine shows this with a clever argument that does not require us to abandon classical logic or block self-reference. It does rely on a controversial principle, “closure”: any statement implicitly says (or means) everything that follows from what it says. Arguably, whether the Bradwardine solution succeeds or fails to conclusively solve the Liar depends on whether one accepts closure. In this interview, Stephen Read runs through Bradwardine’s argument in some detail, then defends it against a few objections.

Bradwardine’s argument is rather subtle and abstract and can be hard to follow verbally. Here’s a short written version of Bradwardine’s argument, with minimum symbolism, that shows each step and notes where logical principles are invoked.

Be sure to listen to the first half of this interview, where Stephen explains the Liar and its significance and solutions in the 20th century.

Next week: Jason Lee Byas: Against Criminal Justice

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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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