In normal modal logic S5, the K axiom says $\square (p \rightarrow q) \rightarrow (\square p \rightarrow \square q)$.
First of all, is this an abuse of notation? https://en.m.wikipedia.org/wiki/S5_(modal_logic)
The middle implication arrow is meta-logical, isn’t it? It’s saying, “if statement 1 is true, then statement 2 is true”. This is not the same as the intra-logical implication arrow. Shouldn’t it be $\square (p \rightarrow q) \implies (\square p \rightarrow \square q)$?
Secondly, why is this axiom so critical or definitive to S5? It has been conceptualized in a few ways. One, that necessity distributes over implication. But why should it? Does this imply that possibility also distributes over implication? Or that modal operators distribute over other connectives, like and and or?
It has also been interpreted as saying “necessary consequences of necessary truths are also necessary truths”, which I find more intuitive. Still, I find myself wondering if that claim could be derived from something else, as in that expression it sounds self-evident - even *necessary*.
