# Are quantifiers metaphysically ineluctable?

I am starting to realize that quantifiers are a very deep topic that I must study on their own.

A simple idea of a quantifier is that it determines a quantity of some thing. If I have a collection of some things, and some way to express assertions about them, a quantifier is like a semantic construct that sort of amounts to “number”.

If you can imagine a formal system where the concept of number is a posteriori constructed, as in set theory, then maybe you could imagine being able to construct a definition of a quantifier. But normally, quantifiers are constructs in first-order logic, which is used to specify the construction of things like natural numbers.

Quantifiers appear more abstract, general, and prior to numbers. The two essential ones, “for all”, and “there exists”, can perhaps be thought of as 2 of the most significant kinds of numbers, in terms of properties: universality, and particularity.

I have started to think of quantifiers as metaphysically required. When we say that “for all X, the following holds true: …”, we are basically hearkening back to the ancient philosophical notion of “ideal forms” and “necessary and sufficient conditions”. To say that “for all things which are apples, it is true that they are also a fruit” is a different way of saying, “the abstract class called “apple” is partially defined by the property, “is a fruit”.”

Abstract forms (it seems) necessarily reduce to their defining properties (because there is nothing else they could be constituted by). I can only define “person” in terms of other concepts. So from this angle, to say that a thing has certain necessary conditions is to say that a thing is identifiable with those necessary conditions. It is not just that any given apple will meet a set of criteria, but also that anything meeting those criteria automatically becomes “an apple”.

The problem is that a language of ideal forms struggles to express the basic notion of an instantiation of that abstract class. According to the above, when I say “apples” or “an apple”, it is not necessarily that I am thinking of “the collection of all apples”, in my mind. Maybe, I am thinking of the bundle of properties that define “apple”. So in a way, it is impossible for me to talk about “a specific apple”, without a seemingly ridiculous circumlocution, like “apples” > “crab apples” > “crab apples eaten by me” > “crab apples eaten by me on Saturday, July 14th, 2023”. I believe this is exactly what Russell tried to do with “definite descriptions”, which Kripke later argued against.

It seems like this is exactly where quantifiers come in. As the most elemental structuring principle for “a world”, it seems like a requirement that we are able to define classes of things (ie, objects attain definition from one another by their differentiated characteristics), and also specific instantiating of those things (ie, one such thing, fulfilling those properties).

I am trying to understand on a much deeper level what this is saying, metaphysically. Why could we not have a world of only universals? Why could we not have a world of only particulars? (Could lambda calculus hold the answer?)

Reference:

https://plato.stanford.edu/entries/quantification/

https://plato.stanford.edu/entries/generalized-quantifiers/

If you can imagine a formal system where the concept of number is a posteriori constructed, as in set theory, then maybe you could imagine being able to construct a definition of a quantifier.

in type theories the usual quantifiers are not 'given', instead one defines them - see for example SEP's entry on Church's type theory - and this has nothing to do with numbers

But normally, quantifiers are constructs in first-order logic, which is used to specify the construction of things like natural numbers.

primitive recursive arithmetic is a quantifier-free theory for parts of arithmetic

Are quantifiers metaphysically ineluctable?

Quantifiers appear more abstract, general, and prior to numbers.

I have started to think of quantifiers as metaphysically required.

check the previous examples

Abstract forms (it seems) necessarily reduce to their defining properties (because there is nothing else they could be constituted by). I can only define “person” in terms of other concepts. So from this angle, [...]

seriously, go actually read some stuff and stop posting non-sense

Logicians often consider that a formalisation of Hilbert's program of finitism is formalized in Primitive Recursive Arithmetic (PRA was already mentioned in the other answer) which is quantifier-free. My 2c is to point out that to the extent that PRA is considered a formalisation of finitism, one can think of quantifiers as going beyond finitism. If you wish to deal with infinitary phenomena, you may typically be forced to resort to quantifiers.