In modal logic, we write $A \Box \phi$ for "$A$ knows that $\phi$ is true"

But we can also want to write the statement "$A$ knows whether $\phi$ is true": $(A \Box \phi) \lor (A \Box\neg\phi)$.

Is there a generally accepted way to abbreviate this?

EDIT: in fact it would be helpful for me if there was notation (perhaps the same one), that could signify "knows which", for when the formula is quantified over by some variable (e.g. a natural number): "A knows for which $x$ $\phi(x)$ is true": $\forall x\in X, (A \Box \phi(x)) \lor (A\Box \neg \phi (x))$

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    $\begingroup$ Interesting question. I don't know of such a notation. The fact that here, here, here and here no such notation is used or introduced though the context would have permitted it seems to indicate that if there is such a notation, it's not in widespread use. $\endgroup$ – joriki Aug 21 '18 at 5:16
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    $\begingroup$ $\lnot(\text{Me}\Box\exists\text{such notation})$, and in case there truly is no such thing, you are allowed to invent your own. Just write something like "Let $A\triangle \phi:=(A\Box\phi)\lor(A\Box\lnot\phi)$ signify that $A$ knows whether $\phi$ is true." and then you can use it freely from there. $\endgroup$ – Arthur Aug 21 '18 at 5:17

In the area of modal logic, I think it is more common to use the notation $K_a \varphi$ to formalize "$a$ knows that $\varphi$". For "knowing whether", I've seen people using $J_i$ ("Knowing Whether," "Knowing That," and The Cardinality of State Spaces), $\Delta_i$ (Contingency and Knowing Whether), and $Kw_i$ (Knowing Whether).

Prof. Wang (webpage) has more to say about the "knowing which" case.


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