Q:

What is the modal interpretation of the converse to a statement?

I have just begun learning about modal logic and I am trying to get my head around Kripke Semantics. I then began to wonder if there is any connection between convereses of statements and modal logic. Here is what I mean: It is basic knowledge of logic that tells us that the statement $P \rightarrow Q$ says nothing about the converse ($Q \rightarrow P$) and the converse is, by default, accepted as true either way because it doesn't forbid either $Q \rightarrow P$ or $Q \rightarrow \neg P$. So for a basic statement like $P \rightarrow Q$ should we interpret its converse as $\neg P \rightarrow \Diamond Q$? Or perhaps as $\Diamond (\neg P \rightarrow Q)$? Or even just as $\neg P \rightarrow Q \lor \neg Q$, to avoid modals altogether? It seems like the correct interpretation might be related to the problem Dyadic Deontic Logic, centred around problems with conditionals, but after some searching around, I couldn't arrive at anything definite.
Furthermore, how does this work for more complicated (possibly modal) statements? For example, what is the converse of $P \rightarrow \Diamond Q$? Or $\Diamond P \rightarrow Q$? 
Many thanks for the responses!

A:

Modern formal logic for the most part doesn't consider "converse" to be a very useful technical concept at all. One can certainly define it as a relation between certain formulas, but this relation doesn't seem to have any particularly nice properties that would justify giving it a name.
One reason for this is that the classical "converse" only involves one implication, whereas in the modern formalization most mathematical reasoning involves formulas with deeply nested implications. For example, we're trained to view $A\to(B\to C)$ and $(A\land B)\to C$ as simply alternative way of writing down the same underlying idea -- and ideed they are semantically equivalent -- but they have different converses, namely $(B\to C)\to A$ and $C\to(A\land B)$. A concept that is so sensitive to differences that we usually consider negligible does not seem to be very useful.
I can see two different answers to your question:

Since modal logic includes classical propositional logic, the natural thing would be not to change the meaning of "converse". The converse of $\phi\to\psi$ is still $\psi\to\phi$, no matter whether there are modal connectives inside $\phi$ and/or $\psi$. (And things that are not of the form $\phi\to\psi$ do not have converses).
Otherwise, you need to describe a concrete technical property that is satisfied by the "converse" relation in propositional logic, and we can then investigate whether there are relations between formulas in modal logic that satisfy the same property. But you have to start by pinpointing that property because there is no standard choice for a property that characterizes "converse" in propositional logic, except for its syntactical definition.

In ordinary mathematical prose one does throw around "converse" a lot -- in contexts such as "To see the converse, assume such-and-such and apply Lemma 123", or "The converse is not true. For example blah-de-blah".
However, there "converse" is not a technical term, but informal, context-dependent jargon that depends on the reader being able to figure out for himself which of the several possible "converses" make sense. If the original statement was "From A and B we conclude C", the possible "converses" in the usual conversational sense could be either "From A and C we conclude B", or "From C we conclude A and B" -- or even, when the writing is especially confusing, "From C and B we conclude A".

