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Curry's Paradox is just a version of the Liar Paradox that is both conditional and self-referential:
Let C be: If C is true, then all cats are blue.
With just this sentence structure and the truth-in/truth-out inference rules (P -> P is true, P is true -> P) proof by contradiction and modus ponens... you can prove that anything is true.
This is a massive problem for modern logic.
The Liar Paradox: "This sentence is false." is related in that it is a self-referential sentence that results in the paradox of generating falsity when it is true and veracity when it is false.
The IEP tells us:
For these optimists, there are four, main, detailed and coherent ways out.
(1) The Liar Sentence is meaningless, so the Liar argument can't even get started because its main assumption (that the Liar Sentence exists or is meaningful) is faulty. Natural language is incoherent, and its underlying sensible structure is that of an infinite hierarchy of levels. Because the Liar Sentence would have to reside on more than one level simultaneously, it's not really a meaningful sentence. This way out of the paradox is taken by Russell in his ramified theory of types and, following Tarski, by Quine in his hierarchy of meta-languages. For Russell, the referential phrase "This sentence" in (1) is the culprit because the phrase is not allowed to refer to the sentence in which the phrase itself occurs. For Quine, instead, the culprit is the phrase "is false" in (1) because the phrase must be satisfied by sentences in a language lower in the hierarchy and not by the very sentence in which the phrase occurs.
(2) Kripke, on the other hand, retains the intuition that the Liar Sentence is meaningful, but argues that it is neither true nor false. It lacks a classical truth value as does the odd sentence "The present king of France is bald." Kripke trades infinite syntactic complexity for infinite semantic complexity. He rejects the infinite hierarchy of meta-languages underlying English in favor of one formal object language having an infinite hierarchy of partial interpretations. The truth predicate is the formal language's only basic partially-interpreted predicate. Each step in the semantic hierarchy is an interpretation of the language, and in these interpretations all the basic predicates except one must have their interpretations already fixed in the base level from which the first step is taken. This one exceptional predicate is intended to be the truth predicate for the previous level. It becomes a truth predicate for its own level when the inductive interpretation building reaches the so-called 'fixed point'. Each step uses all the sentences which had their truth values fixed at the lower steps in order to help fix the truth values of semantically more complex sentences, for example, to fix the truth value of sentences with even longer chains of nested truth predicates. The Liar sentence, even up at the infinite semantic height of the lowest fixed point, still isn't true or false. But at the fixed point, the language satisfies Tarski's Convention (T).
(3) The third way out says the Liar Sentence is meaningful and is true or else false, but one step of the argument in the Liar Paradox is incorrect (such as the move from the falsehood of the Liar Sentence to its truth). Prior, following the informal suggestions of Buridan and Peirce, takes this way out and concludes that the Liar Sentence is simply false. So do Barwise and Etchemendy. They accuse the Liar argument of equivocating by not paying careful attention to the ambiguity of the Liar sentence, namely that it can be interpreted as being the negation of itself or the denial of itself. If it is the negation of itself, then it is simply false, and the Liar argument cannot successfully show that it is true. If it is the denial of itself, then it is simply true. Neither interpretation allows an argument to conclude that the sentence is both true and false.
(4) A fourth and more radical way out of the paradox is to argue that semantic incoherence is not necessarily caused by letting the Liar Sentence be both true and false. This solution embraces the contradiction, then tries to limit the damage that is ordinarily a consequence of that embrace. This way out of the paradox uses a paraconsistent logic.
Curry's Paradox raises further issues that aren't solved by rejecting Liar Paradox statements as meaningless.
If you reject Curry's Paradox statements as meaningless, you lose classical logic in a big way.
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