Hello,
Recently, I'm studying modal logic for my master's thesis, and my research background is category theory. So, I naturally have a question that why it is said that necessity (box) and possibility (diamond) are dual? Are they dual categorically? If so, what is an adjunction between them?
Thank you in advance :-)
This is easiest explained using Kripke semantics. You live in one world and can see other worlds. A statement is necessarily true if it holds in every world that you can see. Dual to "necessarily true" is "not necessarily not true", i.e., not false in every world that you can see. This is the same as true in some world you can see. This we call "possible".
$\square$ and $\lozenge$ are dual in the same way $\forall x$ and $\exists x$ are dual. $\neg\square\neg\varphi$ is equivalent to $\lozenge\varphi$ and $\neg\lozenge\neg\varphi$ is equivalent to $\square\varphi$.
From the category-theoretic view, the last part of Stefan's answer is particularly relevant. The universal and existential quantifier, though dual to each other in the logical sense, are not adjoint to each other. Rather, they are the right and left adjoints, respectively, of a pullback functor. Similarly, the dual propositional connectives $\land$ and $\lor$ are the right and left adjoints of a diagonal inclusion. The moral of this story is that logical duality is not simply adjointness, though it's certainly related to adjointness.
In the case of the modal operators, as interpreted in Kripke semantics, there are again adjoints involved, but in a rather different way. The left adjoint of "necessarily" with respect to an accessibility relation $R$ is "possibly" with respect to the converse relation $R^{-1}$.
$\Box$ and $\Diamond$ are dual in the natural extension to modal logic of the De Morgan duality from classical propositional logic, which is itself essentially a logical manifestation of the fact that in the two-element Boolean algebra (or any Boolean algebra for that matter) $\langle B,\le,\land,\lor,\neg,0,1\rangle$, $\neg$ is an isomorphism of $B$ to its dual algebra $\langle B,\ge,\lor,\land,\neg,1,0\rangle$. (That is, in the modal case, a similar dual isomorphism applies to modal algebras, i.e., Boolean algebras with operators.)
This meaning of “dual” as order-reversing isomorphism (or the duality in elementary geometry, for another example) has been here long before anyone invented category theory and dual categories, although I do not doubt that one can make the two-element Boolean algebra into a category to explain in that way.
There is a setting as outlined here, where by composing left and right adjoints to the pullback with the pullback, one can form an adjunction between a comonadic necessity and a monadic possibility operator. This is an S5 modal logic.
