Wednesday, March 16, 2011

Simmons' paradox

The following, which is akin to the Liar paradox, is a paradox of self-reference by Keith Simmons, described on p. 231 of his ‘Reference and Paradox’ in JC Beall (ed.) Liars and Heaps (Oxford 2003):
Suppose I’ve just passed by a colleague’s office, and I see denoting phrases on the board there. That puts me in the mood to write denoting phrases of my own, and so I enter an adjacent room, and write on the board the following expressions:

......pi
......six
......the sum of the numbers denoted by expressions on the board in room 213.

Now I am in fact in room 213, though I believe that room 213 is my colleague's office. I set you the task of providing the denotations of these expressions.
Simmons goes through your (fictional) reasoning; here is what I think:
......The first expression, ‘pi’, referred to 3.14159..., and the second to 6. So if the third expression does denote a number, say N, then N = N + 9.14159... Given that by ‘number’ we mean finite number, it seems that the third expression can’t denote a number. So those three expressions denote only pi and six. But then the sum of the numbers denoted by those three expressions is 9.14159..., and so the third sentence does seem to denote a number after all. Or rather, it does because it doesn’t; and furthermore, it seems to denote 9.14159..., but therefore it seems to denote 18.283..., or rather, 27 and a bit, etc.
......I have, however, been implicitly assuming that reference is an all-or-nothing affair. Usually we can – and indeed, should – take it to be so, but is it so in general? Imagine, for example, a man staggering through a desert. He sees a mirage, which he takes to be a pool, and as it happens there is a pool, just where he takes one to be, but it’s obscured from his view by the mirage. As he staggers towards it, he’s constantly thinking ‘that pool looks cool’. As he nears the pool, its image gradually replaces the illusory one, without him noticing, so that the referent of ‘that pool’ gradually changes to the pool. And so at some point he may have referred only vaguely to it.
......Such a case would of course be exceptional, but so are scenarios designed to be paradoxical. And it does at least seem possible that the third expression of Simmons’ paradox referred only vaguely to 9.14159... It would also have referred, even more vaguely, to 18.283 (and so on), but while that’s even odder, it too seems possible. And if the alternative is paradoxical, vague reference may not be too odd. And such a resolution would cohere with ‘this is not true’ being vaguely true (about as true as not), and ‘is heterological’ being as heterological as not (see my previous posts this year).

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