Quantification and Necessity

Quantification and Necessity

What is the connection between quantification and necessity? At first sight none: if you make a singular statement of necessity such as “This table is necessarily made of wood” you express no general proposition; you speak simply of a particular table and a property of it. The statement seems no more quantified than “This table is made of wood”—you aren’t saying anything of a general nature here. Yet there is a whiff of generality in the air; you aren’t just talking about a specific table. For you would be perfectly prepared to affirm something along these lines: “Any table is necessarily composed of what it is actually composed of”. Similarly for modal statements concerning identity, origin, and natural kind: you see that something general is true of which a particular thing is a specific instance. It isn’t just that Hesperus is necessarily identical to Phosphorous; any object is necessarily identical to itself (the same is not true for statements of location, brightness, etc.). So we might reasonably say that all singular statements of necessity are implicitly quantificational—they all presuppose general modal truths. Put in the material mode, particular modal facts involve general modal facts; or in the conceptual mode, all singular modal propositions are conceptually linked to general modal propositions. There is a kind of tacit quantification going on in the background. Modal thinking about particular things is always modal thinking about general categories of things. It couldn’t be that only this table is necessarily made of wood (all the other wooden tables being only contingently so). The concept of necessity is inherently a general concept—involving all things of a certain type.

But this claim should be distinguished from another claim that is generally adhered to, viz. that modal operators are quantifiers over possible worlds. This too would render necessity inherently quantificational: necessary truth is simply truth in all worlds. For this table to be necessarily made of wood is for all worlds w to be such that this table is made of wood in w. Clearly this thesis is to be distinguished from the thesis enunciated above; that thesis certainly does not entail that modal adverbs are quantifiers over worlds. And it seems evident that such a thesis meets with firm resistance: if I don’t believe in possible worlds I am not thereby inhibited in my modal pronouncements—because I don’t take my use of modal words to have any such entailments. Nothing in mymodal thoughts adverts to the existence of a class of possible worlds over which I must perforce quantify. That’s not what my words mean; it’s just a theory dreamt up by metaphysicians (right or wrong). The theory is not semantically correct. For one thing, where is universal instantiation? Where are the singular terms (demonstratives or proper names) for worlds that might stand in the place of the variable w? How do these worlds feature in the verification of modal thoughts—do I think about them at all when I arrive at modal conclusions? Is “Necessary truths are truths in all possible worlds” analytic? Maybe modal words are semantically primitive sentence operators or predicate modifiers or copula modifiers.[1] But accepting such semantic views leaves it open that there is nevertheless a connection to possible worlds talk: for we might consistently hold that modal truths have consequences for the condition of possible worlds. If this table is necessarily made of wood, then in all worlds it is made of wood—though that is not what the sentence means. That is, if I accept that possible worlds exist in which objects instantiate various properties, I can allow that a necessary truth implies that things are thus-and-so in those worlds. But the kind of fact stated by a modal proposition is not thereby a fact about these worlds; the two facts are separate and distinct. It is just that one kind of fact determines the other kind: the two cannot vary independently (there is a form of supervenience). This accounts for the feeling we have that modal truths match up with truths about possible worlds, without the former collapsing into the latter (or vice versa). There is this kind of generality in the offing, but it is not of the essence (so to speak). You are not strictly quantifying over worlds when you affirm a modal proposition, though you are at liberty to advert to such worlds in your modal ruminations. It may indeed be useful to do so—and such worlds may exist in whatever way they do—but modal talk itself is not committed to them. Thus it is not quantificational in that way. This seems intuitively correct: we are not generalizing over worlds when we make a simple modal statement. And of course modal words don’t look like quantifiers over worlds—which is why you can happily use them while rejecting worlds on metaphysical grounds. The existence and utility of possible worlds is independent of whether modal words are literally quantifiers over worlds semantically speaking.

So we can say the following three things about quantification and necessity: (a) necessity claims always involve quantification over objects of a certain class (tables, persons, objects in general); (b) the necessity operator is not itself a quantifier over possible worlds, so that modal facts are not general facts involving such worlds; and (c) general facts about possible worlds can coexist with singular modal facts and be dependent on them, without there being any identity between the two. Thus the picture is rather more complex and nuanced than standard views typically recognize. Modality and generality are closely connected, though not in the manner suggested by standard possible worlds semantics.[2]

Colin McGinn

[1] See my Logical Properties (2000), chapter 4.

[2] Modal expressions thus differ from spatial and temporal expressions such as “everywhere” and “always”: here it is very plausible to attribute quantificational content thus assigning an ontology of places and times to these expressions. But we don’t have an analogue of point (a) for them: a specific claim about a place or time does not derive from, or involve, a general truth of that kind concerning other places and times (its raining here and now doesn’t imply that it’s raining at other places and times). Modal expressions are more sui generis semantically than has been recognized. To put it differently, modal logic does not reduce to quantificational logic with a domain of possible worlds (i.e. modal facts are not general facts about worlds).