ENCODING MANY-VALUED LOGIC IN λ-CALCULUS

We will extend the well-known Church encoding of Boolean logic into lambda-calculus to an encoding of McCarthy's 3-valued logic into a suitable infinitary extension of lambda-calculus that identifies all unsolvables by perpendicular to, where perpendicular to is a fresh constant. This encoding refines to n-valued logic, for n is an element of {4, 5}. Such encodings also exist for Church's original lambda I-calculus. By way of motivation we consider Russell's paradox, exploiting the fact that the same encoding allows us also to calculate truth values of infinite closed propositions in this infinitary setting.