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Topic: "defining identity" and AC
Replies: 16   Last Post: Dec 2, 2013 2:18 PM

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Rock Brentwood

Posts: 116
Registered: 6/18/10
Re: "defining identity" and AC
Posted: Nov 20, 2013 8:07 PM
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On Tuesday, November 19, 2013 11:55:55 PM UTC-6, fom wrote:
> A short time ago I started a thread which
> Peter Percival described as trying to
> "define identity".


For the general issue of identity, the standard formulation is fairly well-known and is to be found in second order logic. It is given by the two axioms:
(1) for all x: x = x
(2) for all properties P, for all x: x = y & P(x) -> P(y).

Following Carnap, these two axioms can be combined equivalently into an actual *definition*
for all x, y: x = y <-> (for all properties P: P(x) -> P(y))

Notice, by the way, that the arrow goes only one way. Symmetry, nonetheless, is a theorem.

I almost certain that it is also the case that there is no formulation of any theory of identity in first order logic that is equivalent to the combination of (1) and (2).



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