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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 $\Box$?

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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.

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