Consider the sentence:
(1) There are prime numbers
Arguably (1) is associated with two senses. Insofar as (1) is taken to be a claim about the numbers, namely, that some of them are prime, it is uncontroversially true. Insofar as (1) is taken to be a claim about the world, namely, that it contains (in a very loose sense) prime numbers, it is a substantive metaphysical claim. Very roughly, the first sense involves a presupposition of the existence of numbers whereas the second sense does not. I'm interested in your intuitions about two questions:
First, do you recognize these two readings of (1)?
Second, do you think that recognition of these two readings of (1) commits one to the Carnapian distinction between internal and external frameworks?
If you're in doubt as to how to respond, here's what you should do. Say "yes" in response to the first question, and say "no" in response to the second question. Then, if you're feeling ambitious, go on to present a detailed defense your answer to the second question. I will then be able to copy your comment and paste it into the blank document I have labeled "Draft for Metametaphysics paper". Thank you.
Edward
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4 comments:
Ed,
No.
No.
-Einar
I'm not sure if I get both readings. And if the idea is that, on at least one reading, (1) claims that, of the numbers, some are prime, then the nominalist should think the sentence fails for reasons of presupposition failure, and thus either go false or truth-valueless.
My intuition about the obviousness of (1) is waning. After all, if you're already inclined to think that there are no numbers, it should be pretty clear that there are no prime numbers. It's not like the nominalist has to bite a second bullet when she admits that, in fact, there aren't prime numbers either! The only reason we think otherwise is that we're trained to think of 'there are numbers' as a particularly contentious claim -- but I think, seen the right way, the claim 'there are numbers' is pretty damn obvious, too...
If you think, al la Quine, that all quantifiers are existentially loaded, then (1) is definitely going to commit you to the existence of numbers. (I assume that the variables here range over numbers.) But, there's other ways to go.
Dale Gottlieb, in Ontological Economy, tries to develop mathematics employing substitutional quantification. If you think that's how the quantifier in (1) works, then you're a nominalist, not committed to there being some objects out there which are numbers.
There's a bolder road too: reject the notion that objectual quantification is ontologically loaded. You can be a Meinongian and hold that numbers are nonexistent objects, while, at the same time, hold that some of them are prime.
Reply to Barak:
The line I'd like to take is that a sentence can be obvious without it being the case that its semantic content is obviously TRUE. So the idea is to drive a wedge between truth and acceptability (or something like this).
Consider object language proofs in FOL. There is an obvious proof from "Every shark is a fish" and 'Something is a shark" to "Something is a fish". But the obvious proof is also invalid because existential instantiation is invalid. It doesn't follow from the fact that something is a shark that A is a shark. So this is a case of obviousness without truth.
I would like to say that (1) is obvious and that its semantic content is quite possibly false. The reason you give for doubting the obviousness of (1) is that it might be the case that there are no numbers. This is to deny that it can both be the case that (1) is obvious and its semantic content is false.
But is this right? After all, it might be obvious that there are two ways in which Joe can win the chess game. Should my denial of the existence of ways cause me to doubt the obviousness in this case?
Edward
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