Wednesday, July 25, 2012

Number and Existence

I go away for a week or two and look what happens. Discussing the Maverick's post on number and existence, and Brandon's comment, Michael Sullivan argues against conflating statements about number with existential statements. Consider:

(a) The number of cats in the room right now is two.

(b) Of the four hobbits that set out for Mount Doom, the number that arrived is two.

The two statements are true: his cats are two and Frodo and Sam are two, and in the same sense of 'two'.
But obviously the two hobbits don't have existence in the way that the cats do: my cats have actual existence and the hobbits don't and never did.
Thus we cannot conflate existence with number.

Contra: we clearly can reduce statements about number to existential statements, even when they are in a book. For example

(1) Tolkien said that two hobbits arrived at Mount Doom
(2) Tolkien said that a hobbit arrived with another hobbit at Mount Doom
(3) Tolkien said for some x, y: x was a hobbit, y was a hobbit, not x = y, and x arrived at Mount Doom and y arrived at Mount Doom

But we can't infer from any of these that there are such things as hobbits, or that hobbits exist. What about the claim that 'Hobbits don't have existence in the way that cats do'? Wrong: it's not that hobbits have a different kind of existence. They don't have any existence at all. The book says that, or pretends that hobbits exist. Indeed, it pretends or states that two hobbits – two existing hobbits – arrived at Mount Doom. But what it says is literally false. Nothing of the sort really happened. No hobbits arrived at any mountain. There is an implicit 'says that' or 'pretends that' operator around 'true' fictional statements such as 'two hobbits arrived at Mount Doom', which blocks any inference to existing things.

The problem is that we easily confuse such operators with spatial operators like 'In Europe', 'In London' and so on. We tend to move easily from statements like (4) below to (6), via (5).

(4) According to The Lord of the Rings, two hobbits arrived at Mount Doom
(5) In The Lord of the Rings, two hobbits arrived at Mount Doom
(6) In the universe of The Lord of the Rings, two hobbits arrived at Mount Doom

Now 'In Europe, there are hobbits' implies 'there are hobbits'. But 'In the universe of The Lord of the Rings there are hobbits' doesn't, because there are no hobbits anywhere. Objection: Can we not say that there are hobbits somewhere, namely in the universe of LOTR? Reply: yes, if 'somewhere' means 'it is said somewhere that …'. But then we are equivocating on 'somewhere'. There should be a special name for this fallacy, but I don't think there is one.

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