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arXiv · 1810.07667

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

Fer-Jan de Vries
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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.

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Fer-Jan de Vries. 2018-10-17. Encoding many-valued logic in $\lambda$-calculus. https://doi.org/10.46298/lmcs-17(2%3A25)2021

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