## Tuesday, December 3, 2013

### Quasi-quantifiers

In First Order Logic (FOL), there are three aspects to a quantifier:

• grammar: a quantifier attaches to a formula and generates a new formula binding one variable
• inference: we have the FOL universal and existential introduction and elimination rules
• semantics: the Tarskian definition of truth in a model treats quantifiers in a particular way with respect to a domain.
The first two aspects are normally lumped together under "syntactic aspects", but I think keeping them separate is important.

A quasi-quantifier, then, is something has the grammar and inferential structure of a quantifier, but may have different semantics. Every quantifier is also a quasi-quantifier. A quasi-quantifier that isn't a quantifier—i.e., that has aberrant semantics—will be a quantifier. Quasi-quantifiers can be of types, like "existential" or "universal", that correspond to those of quantifiers. One can have formal languages with existential and universal quasi-quantifiers. In fact, to an approximation English is a language with quasi-quantifiers: "there is" is a mere quasi-quantifier. I will argue for the possibility of mere quasi-quantifiers, connect the issue with fundamentality and then make my suggestion about English.

For any natural number n and quantifier E, let sn(E) be the analogue of the FOL sentence using ∃ that asserts that there are n objects. For instance, s2(E) is the sentence

• ExEy(xy&~Ez(zx&zy)).
A sufficient condition for E to be an existential mere quasi-quantifier is that E has the grammar and inferential rules of ∃ but is interpreted in such a way that sn(E) is false in some model with a domain with n objects.

An uninteresting way to get an existential mere quasi-quantifier is by domain restriction. Restrict interpretations in such a way that names must all be in a subdomain of the model and quantifiers are restricted to the subdomain. A non-trivial quasi-quantifier is a mere quasi-quantifier that isn't just a restricted quantifier.

A sufficient condition for E to be an existential non-trivial quasi-quantifier is that E has the grammar and inferential rules of ∃ but is interpreted in such a way that sn(E) is true in some model with a domain with fewer than n objects.

It isn't hard to generate languages with interpretations that make them have non-trivial quasi-quantifiers, though we will have to reinterpret the identity as well. For instance, it's not hard to generate a pair of existential and universal "doppelganging quantifiers"[note 1], that have the same inferential rules as the existential and universal quantifiers, but a sentence gets interpreted in a model as if each item in the model had a doppelganger, where a doppelganger of x stands in the same relations as x, except for identity (x=x but x's doppelganger isn't identical with x), and yet without adding any objects to the domain.[note 2]

Whether a quasi-quantifier is a quantifier depends on how that quasi-quantifier is treated in a Tarski-style definition of truth. Now, when we quasi-quantify also over non-fundamental objects, like holes and shadows, I think the Tarski-style definition of truth will give the truth conditions in terms of how the fundamental objects are (say, perforate or shadowing). This is going to be controversial, but, hey, this is only a blog post.

It follows immediately that when we quasi-quantify also over non-fundamental objects, we have a mere quasi-quantifier. Moreover, it's not going to be a restricted quantifier, so it's a non-trivial quasi-quantifier.

Now the English "there is" to an approximation is a quasi-quantifier. (It's not quite a quasi-quantifier, as the rules of inference for it will not quite match that of ∃ due to vagueness.) Moreover, it quasi-quantifies also over things like holes and defects and chairs, which are non-fundamental. Therefore, it is a mere quasi-quantifier. Nor is it just a restriction of a quantifier, so it is a non-trivial quasi-quantifier.

Once we see this, temptations to quantifier pluralism should be decreased. Of course, we have quasi-quantifier pluralism: There are quantifiers, there are doppelganging quasi-quantifiers, there are English quasi-quantifiers, there are mereological quasi-quantifiers, and so on. But only the first of these are quantifiers.

Now, in the formal examples, like of my doppelganging quantifiers, one can give a paraphrase of the quasi-quantifiers in terms of quantifiers: one just writes out the Tarski definition of truth for each sentence. But in natural language examples, the Tarski definition of truth is not going to be formally statable (at least not in any way tractable to us). And so there won't be a paraphrase of the quasi-quantifier sentences in quantified sentences. Quine won't like that. And what I said above about the Tarski definition when I characterized quasi-quantifiers won't be easy to say in the natural language case. There is much more work to be done here.

And of course just as there is no entity without identity, there is no quasi-quantifier without quasi-identity.

Heath White said...

I am sympathetic to this suggestion. It promises an irenic solution to some tough problems. A few thoughts:

What really seems to be going on is that quantifiers range over fundamental objects while quasi-quantifiers range over non-fundamental objects. The difference is the semantics of the two categories of (quasi-)quantifier, and to make any hay out of this difference we have to use the concept of “metaphysical fundamentality” in semantics. But as you note, it is hard to actually use this concept in real cases for natural language. So one question would be what is the motivation or philosophical payoff for making the distinction.

(On a similar note: it seems to me that if we have existential quantifiers and existential quasi-quantifiers, the corresponding difference in the objects they pick out is not “[real] existence” and “quasi-existence” but “fundamental existence” and “non-fundamental existence.” )

Another wrinkle is that there is not just one level of ontological dependence. There are particles, blowguns composed of particles, bundles composed of blowguns, etc. Do we need quasi-quasi-quantifiers? What happens if there are no fundamental objects, but an infinite regress of fundamentality?

Another thought: one might treat fundamentality as a context-relative notion. E.g. for most practical purposes a herd is composed of animals. Now it may be that animals are composed of, say, atoms, and thus herds are composed of atoms. But for most purposes where we want the truth-conditions of statements about herds, we are willing to treat animals as the stopping-points of analysis and not interested in proceeding to analysis in terms of atoms.

I think this gets to two possible views about what a semantic theory is. On one view, a semantic theory is in effect a comprehensive metaphysical picture of the world. On another view, a semantic theory is a tool for telling an interested party the meanings of words he doesn’t know in terms of meanings of words he does know. The first view needs a strong, context-independent notion of metaphysical dependence. The second view needs only a context-relative notion of epistemological dependence. (E.g. you could probably define cows as the constituents of herds.)

Now, developing a semantics (and corresponding logic) that reflected a comprehensive metaphysical picture of the world is a project one could pursue. But there are many strands of philosophy today that would question the necessity, or interest, or maybe possibility of doing it. So again I would raise the question of what is the motivation or philosophical payoff for pursuing it.

Alexander R Pruss said...

Heath:

Yeah, there are serious difficulties here.

Here's one line of thought that both complicates what I said and points towards an answer to you.

There clearly are no doppelgangers. (I could say that that's true by stipulation. Tell me what there is, and I can construct doppelgangers, or aleph17-gangers, on top of that!) So the doppelganged quasi-quantifiers aren't quantifiers. That's a clear case.

But we can make the clear case muddy. We can state the semantic theory of the truth conditions for the doppelganged language in a doppelganged metalanguage. And this semantic theory will look just like one for a language with ordinary quantifiers.

At this point, I see three ways forward:

1. We've discovered that the distinction between doppelganged quasi-quantifiers and quantifiers is bogus.

2. Say that the real quantifiers are those quasi-quantifiers that quasi-quantify over the domain of quantification in our natural language.

3. Require that the relevant semantic theory is the most fundamental theory of how the world makes sentences true.

Now, (1) is the end of ontology and, what is worse, the rejection of the very intuitive claim that there really are no doppelgangers.

Option (2) has the problem that the domain of quantification in our natural language is highly contingent. We can imagine our natural language evolving in such a way that we would using doppelganged quasi-quantifiers for some reason. So "the real quantifiers" ends up being relative to a particular metalanguage at a particular time, and that's no better than (1).

That leaves (3). I don't think (3) by itself settles the question of whether quantifiers quantify only over the fundamental things. For it could be that the most fundamental semantic theory actually treats fundamental and non-fundamental things on par. I think it doesn't. But that's a further question.

One might have a minimalist disquotational semantic theory which does, after all, and one might insist that that is the deepest and most fundamental semantic truth of the matter. Why would I reject that? Because that leads back to the same sort of conclusions we get in (1) and (2), namely a denial of the intuition of the mereness of the doppelganged quantifiers.

p.s. Animals are fundamental.