A logical interpretation of the function subtyping rule

We discussed the subtyping rule for function type in this post. In short, a function type is contravariant in the argument types and covariant in the result type. Several days ago, I happened to notice that the Monotonicity Theorem for implication (see p.70 in [1]) looks very similar to the aforementioned subtyping rule in form. Given propositions $p,p',q,q'$ such that $p'\Rightarrow p$ and $q'\Rightarrow q$, the theorem asserts $$p'\vee q' \Rightarrow p \vee q.$$ Besides, it is immediate to see that $p \vee q \equiv \neg q \Rightarrow p$ and $q' \Rightarrow q \equiv \neg q \Rightarrow \neg q'$, assuming the law of double negation. Now, by writing $p$, $p'$, $\neg q$ and $\neg q'$ as $P$, $P'$, $Q$ and $Q'$, respectively, we obtain \begin{equation} (Q' \Rightarrow P') \Rightarrow (Q \Rightarrow P) \label{subtyping} \end{equation} given that $P'\Rightarrow P$ and $Q\Rightarrow Q'$.

The connection between the above fact and the subtyping rule of function types can be established by treating the implication connective "$\Rightarrow$" as both a function-type constructor and a subtype-of relation. Indeed, if Apple is a subtype of Fruit then being an Apple implies being a Fruit. It is thus reasonable to use "Apple $\Rightarrow$ Fruit" to express the subtyping relation between Apple and Fruit. In view of this, the logical entailment in \eqref{subtyping} leads us to the following statement for function types: If $P'$ is a subtype of $P$ (i.e. $P' \Rightarrow P$) and $Q$ is a subtype of $Q'$ (i.e. $Q \Rightarrow Q'$), then function type $Q' \Rightarrow P'$ is a subtype of function type $Q \Rightarrow P$ (i.e. $(Q' \Rightarrow P') \Rightarrow (Q \Rightarrow P)$). In other words, a function type is contravariant in the argument types and covariant in the result type.

Reference

1. Gries, David, and Fred B. Schneider. "A logical approach to discrete math." Springer, 1993.