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 $\Box$?


1 Answer 1


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.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .