# Order of Operations Modal Logic

In modal logic we have: $$\Diamond P \equiv \neg \Box \neg P \\ \Box P \equiv \neg \Diamond \neg P \tag{1}$$

Does that also imply the following? $$\Diamond P \equiv \neg \left( \Box \neg P \right) \Leftrightarrow \neg \Diamond P \equiv \Box \neg P \\ \Box P \equiv \neg \left( \Diamond \neg P \right) \Leftrightarrow \neg \Box P \equiv \Diamond \neg P \tag{2}$$

How should I think regarding "order of operations" in modal logic in general? Or in particular, if I want to convert expressions with $$\Diamond$$ to equivalent ones with $$\Box$$. My hypothesis is that the above conversion is valid, but I can't find any reference that supports it.

What I want to do is to express $$\Diamond P \land \Diamond \neg P$$ with only $$\Box$$, if my hypothesis $$\left( 2 \right)$$ holds true, then I think I can do this:

$$\Diamond P \land \Diamond \neg P \Leftrightarrow \neg \Box \neg P \land \neg \Box P \tag{3}$$

Also with the substitution $$Q = \neg P$$, I can derive:

$$\Diamond P \land \Diamond Q \Leftrightarrow \neg \Box \neg P \land \neg \Box \neg Q \Leftrightarrow \neg \Box \neg P \land \neg \Box \neg \left( \neg P \right) \tag{4}$$

Is both $$\left( 3 \right)$$ and $$\left( 4 \right)$$ valid? And again, what do I need to consider when converting expressions from $$\Diamond$$ to $$\Box$$?

Your conversions are indeed valid. Note that there isn't any order of operations issue here, since there is only one order of operations that makes sense: "$$\neg(\Box(\Diamond p))$$" makes sense, but what about "$$(\neg\Box)(\Diamond p)$$" - what even is "$$\neg \Box$$" as a single entity? The operators $$\neg,\Box,\Diamond$$ act on sentences, not each other, so they all have to associate to the right.
• OK fine, you can whip up an approach by which unary syntactic operators can act on each other, and this is sometimes a good thing to do. But even as this introduces ambiguity, it simultaneously removes it: the way (for example) "$$\neg\Box$$" as a single operator would be defined is as the operation sending $$p$$ to $$\neg(\Box p)$$, and associativity would fall out trivially.
If you want to be a bit more explicit about things - which is probably a good idea at first - the classical "dualities" would be better written as $$\Diamond p\equiv \neg(\Box(\neg p)),\quad \Box p\equiv \neg(\Diamond(\neg p)).$$ Because of the unique readability point mentioned above, though, these parentheses are suppressed almost always.
An order of operations issue arises when you consider multi-ary operations like $$\wedge$$ and $$\vee$$: does $$\Diamond p\wedge q$$ mean $$(\Diamond p)\wedge q$$ or $$\Diamond(p\wedge q)?$$ But this ambiguity isn't relevant to the more restrictive context of this question.