Modal Logic: Necessity and Possibility

What Could Have Been Otherwise

Propositional logic distinguishes true from false and stops there. It cannot express the difference between something that merely happens to be true and something that could not have been otherwise, yet that distinction runs through metaphysics, ethics and the philosophy of science.

Modal logic adds two operators. $\Box P$ means "necessarily $P$" and $\Diamond P$ means "possibly $P$". They are interdefinable by a duality worth memorising:

$$\Box P \equiv \neg \Diamond \neg P, \qquad \Diamond P \equiv \neg \Box \neg P$$

Necessarily $P$ is exactly: not possibly not $P$. Possibly $P$ is: not necessarily not $P$. Either operator can be defined from the other with two negations, which is the same pattern as the quantifier duality between "all" and "some".

The standard semantics is possible worlds. A sentence is necessary if true in every accessible world, possible if true in at least one. A model consists of a set of worlds, an accessibility relation saying which worlds are possible relative to which, and an assignment of truth values at each world.

Accessibility is what makes modal logic interesting, because different constraints on it validate different principles:

Reflexivity gives the principle that whatever is necessary is actually true, which sounds trivial and is nonetheless a substantive assumption that some readings of the operators reject.

That is the crucial move: the same symbols admit different readings. $\Box$ may mean logical necessity, physical necessity, obligation (deontic logic), knowledge (epistemic logic), or "always" (temporal logic). The right system depends on the reading. In deontic logic $\Box P \rightarrow P$ must be rejected: obligations are frequently unmet, and a logic that made "ought" imply "is" would be useless.

Common pitfall: reading $\Diamond P$ as "$P$ is probably true" or "$P$ might be, for all I know". Possibility here is not probability and not ignorance: $\Diamond P$ says there is some accessible world where $P$ holds, however unlikely or well known.