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.

@article{paperbot1303,
  title = {Encoding many-valued logic in $\lambda$-calculus},
  author = {Fer-Jan de Vries},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 17, Issue 2},
  year = {2021},
  doi = {10.46298/lmcs-17(2:25)2021}
}