Saturday, October 01, 2011

Jottings on the Modal Logic of Quantifiers

These are mostly me trying to work through some ideas, perhaps badly, but comments are welcome.

I was kept up a few nights ago thinking about modal logic and quantifiers. Quantifiers, put in broad, broad terms indicate whether we are allowing exceptions for a claim. For instance, if I say, "All dogs go to heaven" that (as usually understood) tells us that "Dogs go to heaven" has no exceptions. If I say, however, that "Some dogs go to heaven," this allows exceptions (as long as there is at least one thing that is not an exception), even though it doesn't guarantee them.

It is not quite this simple in practice, since this is a term-logic interpretation of quantifiers; modern logic introduces other complications with existential import, etc. But the key thing quantifiers do is track exceptionlessness or possibility-of-exception.

From this it actually follows that quantifiers can be represented by a modal logic. A modality is basically a qualification; if you are talking about the modality of a proposition or claim, you are talking about some qualification of its truth (or falsehood). For instance, a proposition might be necessarily true, or possibly false, or true in the past, or any number of other things. But there's nothing that requires that we apply modalities only to whole propositions, and, indeed, we regularly apply them in practice to predicates. "All dogs necessarily go to heaven" is subtly but importantly different from saying "It is necessarily true that all dogs go to heaven"; if you don't see this, just take my word for it for a moment.

In general modalities that qualify one thing (whether propositions, or predicates, or whatever) fall into two broad classes: Box and Diamond. Box is the strong one (necessity, obligation, always), Diamond is the weak one (possibility, permissibility, sometimes). Since quantification is a kind of qualification (in term-logical terms, it is a qualification of the subject term), it makes sense to think of it modally. And we do obviously have a strong modality and a weak modality here: universal quantification ("All dogs go to heaven") is Box-like and particular quantification ("Some dogs go to heaven") is Diamond-like. And they do tend to work like Box and Diamond.

This analogy has actually been known for quite some time; I think von Wright was the first one to note it. But it's usually read the other way, so that Box and Diamond can be treated as if they were quantifiers. This is valuable but (frankly) can get one into weird territory, because you always have to ask, "What are we quantifying over?" And so we get possible worlds and the like. But you can go the other way, too: treat universal quantification as Box and particular quantification as Diamond.

Now, the thing of it is, there are many, many, many systems using Box and Diamond, and they are all very different; Box and Diamond, when we aren't specifying a particular system, are really just broad categories of modality, not modalities themselves. So this raises the question of many different kinds of quantifiers, depending on the system in which we are interested. And indeed, we know this in practice: some quantifier-systems allow subalternation (you can get particulars from universals, e.g., you can conclude that some dogs go to heaven from the truth of "all dogs go to heaven"), while some do not.

Thus a subalternating quantifier-system is one in which a version of what is sometimes known as the characteristic D axiom is true: Box implies Diamond for whatever Box might apply to. Thus, 'A is obligatory' (Box) implies 'A is permissible' (Diamond), 'A is necessary' (Box) implies 'A is possible' (Diamond), and, apparently, "All A" (Box) implies "Some A" (Diamond) when we are allowing subalternation.

Further, you can have reverse-subalternating systems. An example would be Sommers's account of 'wild quantity' for singular terms, which is both subalternating and reverse-subalternating. In a reverse-subalternating system Diamond implies Box. (This is, if I recall correctly, the characteristic CD axiom.) So, Sommers argues, there is no significant difference between "Socrates is dead," interpreted as forbidding exception, and "Socrates is dead," interpreted as allowing exception, as long as we are talking about the same Socrates.

Likewise, you can have a quantifier-system allowing universal instantiation. To understand this, we have to note that there are many modal logical systems that allow you to have claims, terms, or whatever, without either Box or Diamond. These have what can be called a Null modality. In alethic systems, Box is necessity, Diamond is possibility, and Null is just plain truth. In universal instantiation we have what is sometimes known as the characteristic M axiom: Box implies Null. Thus, for instance, "It is necessary that all dogs go to heaven" implies "(It is true that) all dogs go to heaven". In a universal-instantiating quantifier-system you can conclude "Dogs go to heaven" from "All dogs go to heaven". The other major rules we typically find all have their modal axiom:

Universal Generalization: Null implies Box
Existential Generalization: Null implies Diamond
Existential Instantiation: Diamond implies Null

Obviously if these were all true without restriction all the distinctions among the modalities would simply collapse; but, in fact, there are different restrictions on when you can use them, thus preserving the difference between universal quantity and particular quantity.

But here's a bit of a puzzle. Modern logic is universal-instantiating but not subalternating. But in modal logic, standard systems with M are stronger than standard systems with D, so it would seem to follow that any universal-instantiating system is also a subalternating system, unless there's something very unusual about the system we are using. In fact, I think the denial of subalternation is often ambiguous when we are talking about modern logic (even setting aside the confusions created by talk about existential import): subalternating is taken as not merely assuming that Box implies Diamond but also that Diamond implies Null. There are restrictions on these, so if subalternation required both it would be subject to a number of different restrictions that would all have to be met. Likewise, when people talk about subalternation they tend to talk about the square of opposition, in which Universal Affirmative propositions imply Particular Affirmative Propositions; and in modern logic many universal affirmative statements are treated as hypotheticals, which is a further complication. But I don't think that these can be a complete account. It's still the case that the combination of universal instantiation and existential generalization should guarantee subalternation in any situations in which they are both allowed by the restriction -- in which case we should at most take modern logic to be a system with restricted subalternation, not as a system without subalternation. The only alternative is that the modal logic is unusual here. Or perhaps I'm missing something obvious here.

The traditional account, though, allows one to have a modal logic of quantifiers of any kind as long as it is D or stronger. Which one is chosen, of course, will affect what one can infer, of course.

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