Encoding many-valued logic in $\lambda$-calculus

TL;DR

This paper extends Church encoding in lambda calculus to represent many-valued logics, including 3-valued and higher, using an infinitary calculus that unifies solvables and unsolvables, with applications to paradoxes and infinite propositions.

Contribution

It introduces a novel infinitary lambda calculus encoding for multi-valued logic, generalizing Church encoding to 3, 4, and 5-valued logics, and applies it to paradoxes and infinite propositions.

Findings

Encoding of 3-valued logic in infinitary lambda calculus.

Refinement to 4- and 5-valued logics.

Application to Russell's paradox and infinite propositions.

Abstract

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 $\bot$, where $\bot$ is a fresh constant. This encoding refines to $n$-valued logic for $n\in\{4,5\}$. Such encodings also exist for Church's original $\lambda\mathbf{I}$-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.