Many-valued logics inside λ-calculus: Church's rescue of Russell with Bohm trees
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We extend the Church encoding of the Booleans and two-valued Boolean Logic in λ-calculus to encodings of n-valued sequential propositional logic (for 3≤ n≤ 5) in well-chosen infinitary extensions in λ-calculus. In case of three-valued logic we use the infinitary extension of the finite lambda calculus in which all terms have a unique normal form in which their Böhm tree can be recognised. The construction can be refined for n∈{4,5}. The three n-valued logics so obtained are variants of McCarthy's left-sequential three-valued proposition calculus. The four-valued logic has been described by Bergstra. The five-valued logic seems new, but closely related to a five-valued logic proposed by Bergstra and Ponse in the context of Process Algebras. The encodings of these n-valued logics are of interest because they can be used to calculate the truth values of infinite closed propositions. With a novel application of McCarthy's three-valued logic we can now resolve Russell's paradox. We make the speculation that Church could have found a similar encoding of three-valued logic in his own λI-calculus, because of the simplifying fact that Böhm trees are always finite in λI-calculus.
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