Is there a concise way to notate 'There are exactly 482 x, such that Px...' in logical notation?

Question

My prof has taught us that we can express the proposition $⟦$there are exactly two entities characterized by $P$$⟧$ thus:

That proposition looks verbose, despite the fact that it references just two entities. It seems impractical to use such a formula to express a proposition that references a great number of entities (e.g. 482). Is there a more concise way of conveying the number of entities that a proposition references?

Answer 1

The standard abbreviation for "there are at least $482$ objects $x$ satisfying $\phi(x)$" is

$$ (\exists^{482} x)\phi(x) $$

The extra quantifiers of the form $(\exists^{n} x)$ can either be defined directly in the metatheory, or viewed as abbreviations for longer formulas.

So you could write $$ (\exists^{482} x)\phi(x) \land \lnot(\exists^{483} x)\phi(x). $$

There is also a notation $(\exists ! x)\phi(x)$, which means there is a unique $x$ satisfying $\phi(x)$. I have never seen the notation $(\exists !^{482} x)\phi(x)$, but you could certainly define it in your prose.

Although I have seen this notation less, an SEP article suggests $$ (\exists_{\geq 482} x)\phi(x) $$ and $$ (\exists_{=482} x)\phi(x)$$

Answer 2

Let $S_P$ be the set of individuals characterized by $P$ and $S_Q$ the set of individuals characterized by $Q$, $\mid S_P \cap S_Q\mid=482$.

Where $\mid S \mid$ is the cardinality of the set $S$.

You can possibly define these sets as $S_P = \{x \mid P(x)\}$.

Answer 3

$$\exists x_1, \ldots, x_{482}\left(\bigwedge_{i \neq j \leq 482}x_i \neq x_j \wedge \forall y \left(\phi(y) \leftrightarrow (\bigvee_{i \leq 482} y = x_i)\right)\right)$$

Or, since it's not in doubt that there is such a first-order formula, create some ad-hoc shorthand and write "$\exists^{482} x\, \phi(x)$ where $\exists^{482}x$ is a quantifier denoting the existence of exactly 482 distinct such $x$", which is kinder to your reader.