Encoding many-valued logic in {\lambda}-calculus

Abstract

We extend the well-known Church encoding of two-valued Boolean Logic in $\lambda$-calculus to encodings of $n$-valued propositional logic (for $3\leq n\leq 5$) 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 their B\"ohm tree as their unique normal form. We refine this construction for $n\in\{4,5\}$. These $n$-valued logics are all variants of McCarthy's left-sequential, three-valued propositional calculus. The four- and five-valued logic have been given complete axiomatisations by Bergstra and Van de Pol. The encodings of these $n$-valued logics are of interest because they can be used to calculate the truth values of infinitary propositions. With a novel application of McCarthy's three-valued logic we can now resolve Russell's paradox. Since B\"ohm trees are always finite in Church's original $\lambda{\mathbf I}$-calculus, we believe their construction to be within the technical means of Church. Arguably he could have found this encoding of three-valued logic and used it to resolve Russell's paradox.Comment: 15 page