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Wednesday, November 29, 2006

A Doxastic Analogue of Curry's Paradox

Consider for a moment the the following proposition (or propositional schema, to be more precise):

(1) If [Insert your name here] believes (1) is true, [Insert your name here] believes every proposition is true.

To make things easier, I'll consider the paradox in my own case. Thus, I consider the following proposition:

(1') If Jason Zarri believes (1') is true, Jason Zarri believes every proposition is true.

So, I ask myself, do I believe (1')? Well, its truth-conditions are that, if I believe it is true, I believe every proposition is true. So if I believe (1') is true, I believe every proposition is true. But I'm pretty confident that I don't believe every proposition is true. I don't believe that the universe is less than five minutes old, for example. So I conclude (safely, so it seems) that I don't believe (1'). But wait a moment: In asserting what I just did, in the statement beginning "So if I believe (1')..." I asserted my belief in the truth of (1'), because the fact that if I believe (1') then I believe everything is precisely what (1') asserts! So it seems I do believe (1') after all. But, so I continue, if I believe (1'), and if I believe the antecdent of (1') -- both of which I have established-- then I am committed to believing the consequent of (1'), namely, that I believe every proposition is true! Thus it appears that, reasoning from fairly innocuous premises and rules of inference, I've reasoned myself into being a trivialist. And yet I remain certain--perhaps obstinately so--that I am not a trivialist: I don't believe everything is true, and if some demonstration purports to show that I do, then something is seriously wrong with it. The question is, then, what?

1 comment:

LauLuna said...

What?

The following term has no referent: "the proposition expressed by (1*)" in:

(1*) if Curry believes the proposition expressed by (1*), then Curry believes all propositions

it has no referent, because no proposition refers to itself.

Thus (1*) expresses no proposition.

Let Q be a quantified expression, in the sense of generalized quantifiers; then Q does not quantify over Q's intension (Frege's 'Sinn'). If 's(Q)' denotes Q's Sinn and 'x(Q)' denotes the universe of discourse over which Q's quantifiers range, we can state:

(Q) ~s(Q) e x(Q)

where 'e' is the set membership relation.

I'd say that (Q), when properly used, disposes of this and other paradoxes.

Best.