  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.
