In 1934, Haskell Curry noticed that the types used in typed lambda calculus, and in its combinatory logic counterpart, followed the same pattern as axioms in propositional logic.
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Pe�a's plan to investigate the grounds of his logical system as a nonclassical combinatory logic has thus far remained programmatic, but the combinatory account fits his metaphysical approach.
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Combinatory logic, functionally equivalent to the lambda calculus, is a branch of symbolic logic having the expressive power of set theory, and with deep connections to questions of provability.
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In the study of illative ( deductive ) combinatory logic, Curry in 1941 recognized the implication of the paradox as implying that, without restrictions, the following properties of a combinatory logic are incompatible:
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In the study of illative ( deductive ) combinatory logic, Curry in 1941 recognized the implication of the paradox as implying that, without restrictions, the following properties of a combinatory logic are incompatible:
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For instance, the notion of reduction of terms in combinatory logic can be transferred to Hilbert-style logic and it provides a way to canonically transform proofs into other proofs of the same statement.
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Combinatory logic can be viewed as a variant of the lambda calculus, in which lambda expressions ( representing functional abstraction ) are replaced by a limited set of " combinators ", primitive functions from which bound variables are absent.
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Conversely, the non provability in intuitionistic logic of Peirce's law can be transferred back to combinatory logic : there is no typed term of combinatory logic that is typable with type ( ( ? ?! ? ) ?! ? ) ?! ?.
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Conversely, the non provability in intuitionistic logic of Peirce's law can be transferred back to combinatory logic : there is no typed term of combinatory logic that is typable with type ( ( ? ?! ? ) ?! ? ) ?! ?.
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""'To Mock a Mockingbird and Other Logic Puzzles : Including an Amazing Adventure in Combinatory Logic " "'( 1985, ISBN 0-19-280142-2 ) is a book by the mathematician and logician Raymond Smullyan.
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