According to Quine, the question that ontology asks is: “What is there?” And the lazy but obvious answer is: “Everything.”
Now to show my utter ignorance of logic. I’m trying to rephrase Quine’s answer—i.e., “Everything exists”—in first order logic, but find myself stuck. Here’s why.
On the one hand, I shouldn’t write: ∀x Exists(x). For in first order logic, existence is not expressed by means of a predicate, but by means of existential quantification.
On the other hand, this doesn’t seem to work either: ∀x ∃x. That doesn’t look well formed, since quantified statements have the form ∀x …x…, ∃x …x…, ∀x ∃y …x…y…, and so on.
Here’s a third option: ∀x x = x. Or let me try: ∀x ∃y x = y. But these are not what I wanted to say. I want my statement to express the thought that everything exists, not the different thought that everything is identical to itself, or the thought that for anything there’s something identical to it.
So the question is, how do I state “Everything exists” in first order logic? And while we are at it, how about also: “Something exists”?