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.
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
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.