Possibility and Necessity

Some things are true but might not have been: there are eight planets, and there might have been nine. Some could not have been otherwise: $7+5=12$, and nothing is both round and square. Some things are false but could have been true, and some are false and could not.

Modality is the study of this difference, and it enters ontology at two points. It supplies the machinery in which many ontological theses are stated — essence, material constitution, and the necessity of identity are all modal claims. And it raises an inventory question of its own: if possibilities are real, what sort of things are they?

Flavours of modality

"Could" and "must" are not univocal, and a great many bad arguments trade on the ambiguity. The main kinds, ordered roughly from most to least permissive:

The orderings are mostly but not entirely nested. Everything nomologically possible is metaphysically possible; the converse fails, since the laws could have been different. Epistemic modality cuts across the others entirely: it is a matter of our information, not of the world, and "it might be raining" reports ignorance rather than a feature of reality.

The ontologist cares about the metaphysical kind. It is not settled by logic alone, because "water is not ${\mathrm{H}}_2\mathrm{O}$" is logically consistent yet metaphysically impossible; and it is not settled by the laws of nature, because those are contingent. It is the modality of what things are — which is why, on one influential view, it reduces to essence.

De dicto and de re

A structural distinction that carries most of the weight in the ontological applications.

• De dicto: the modal operator governs a whole proposition. "Necessarily, every bachelor is unmarried" — $\square \forall x(Bx\to Ux)$. The necessity attaches to the claim.
• De re: the modal operator governs a predication of an object. "Socrates is necessarily human" — $\exists x(x=\mathrm{Socrates}\wedge \square Hx)$. The necessity attaches to the thing, independently of how it is picked out.

The difference is a matter of scope, and it is easy to state formally and hard to see in English, where "the number of planets is necessarily greater than seven" admits both readings — true de re (of the number 8, that it is necessarily greater than 7) and false de dicto (it is not necessary that the planets number more than seven).

Only de re modality supports essentialism, which is why Quine's attack on the latter took the form of an attack on the former. His three grades of modal involvement stage the retreat he thought unavoidable: treating necessity as a predicate of sentences is harmless; treating it as an operator on statements is tolerable; but quantifying into modal contexts — grade three — commits one to objects having properties necessarily independently of description, which he regarded as unintelligible. The reply is on the essence page.

Necessary a posteriori, contingent a priori

Before Kripke it was widely assumed that the necessary, the a priori, and the analytic coincided. Naming and Necessity separated them, and the separation is what made modal metaphysics respectable again.

Necessary a posteriori. Hesperus is Phosphorus; water is ${\mathrm{H}}_2\mathrm{O}$; this table is made of this wood. Each is discovered empirically and each holds in every possible world, because names and natural-kind terms are rigid designators — they pick out the same thing in every world where it exists. What we learn empirically is which object or kind we were talking about; the necessity follows from the necessity of identity.

Contingent a priori. The stock example is the metre. If "one metre" is introduced by stipulation as the length of stick $S$ at $t_0$, then the speaker knows a priori that $S$ is one metre long at $t_0$ — but the fact is contingent, since $S$ might have been heated and been longer. The description fixes the reference without giving the meaning.

The moral is that necessity is metaphysical while a priority is epistemic, and they answer to different things. Conflating them is what made de re modality look like a confusion.

The modal logics

The formal treatment adds an operator $\square$ for necessity, with $◊p{=}_{\mathrm{df}}\neg \square \neg p$ for possibility. Systems differ by which iteration principles they accept:

S5 is the standard choice for metaphysical modality, and it has a consequence worth flagging: in S5 the modal status of a proposition is not itself contingent, so the accessibility relation between worlds is an equivalence and every world is possible relative to every other. This makes the modal facts absolute rather than perspectival — which is convenient, and is exactly the step that does the work in Plantinga's modal ontological argument, where the possibility of a maximally great being yields its necessary existence.

Whether S5 is correct for metaphysical modality is not obvious. If some things could have been possible that are not actually possible — if the space of possibilities is itself contingent — S4 or weaker is the right logic, and several arguments that rely on S5 fail.

Where modality sits

Modality is used throughout this section as a tool, and it is worth being clear about the direction of dependence, because the section's later pages disagree about it. On the orthodox view, modality is basic and essence is defined from it. On Fine's view, essence is basic and modality is defined from it. On the dispositionalist view developed under powers, modality flows from the natures of properties. Each is a different answer to where the necessity in the world comes from.

The inventory question it raises is the subject of the next page. Talk of what "could have been" invites paraphrase in terms of possible worlds, and by the criterion of commitment that is a commitment to worlds — unless the talk can be understood some other way. What sort of entity a possible world is, and whether there are any, is the most extravagant ontological question in the section.