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- W2897400641 abstract "We extend the Church encoding of the Booleans and two-valued Boolean Logic in $lambda$-calculus to encodings of $n$-valued sequential propositional logic (for $3leq nleq 5$) in well-chosen infinitary extensions in $lambda$-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 Bohm tree can be recognised. The construction can be refined for $nin{4,5}$. 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 $lambda$I-calculus, because of the simplifying fact that Bohm trees are always finite in $lambda$I-calculus." @default.
- W2897400641 created "2018-10-26" @default.
- W2897400641 creator A5067994951 @default.
- W2897400641 date "2018-10-17" @default.
- W2897400641 modified "2023-09-27" @default.
- W2897400641 title "Many-valued logics inside {lambda}-calculus: Church's rescue of Russell with B{}ohm trees" @default.
- W2897400641 hasPublicationYear "2018" @default.
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