Re: definition of identity

On Jun 28, 3:47 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Jun 28, 3:09 pm, Zaljo...@xxxxxxxxx wrote:

On Jun 28, 2:46 pm, Ken Pledger <ken.pled...@xxxxxxxxxxxxx> wrote:
You're quite right in seeing a difficulty. That's the reason for
using "first order logic with identity", in which identity is introduced
as a special predicate symbol with its own axioms.

I thought that identity '=' is a relation symbol.

'relation symbol' and 'predicate symbol' mean the same thing.

we don't wright =(x,y) , we usually write x=y.

Those are two ways of notating the same formula.

What are these axioms. Can you write them, or at least point to me a
site on the web that illustrate them.

You very much need to get a good book on logic, rather than bouncing
around among Internet articles. Anyway:

Axiom:
Ax x=x

Axiom schema:
If P is an atomic formula, then all closures of
x=y -> (P <-> P')
are axioms, where P' is just like P except that zero or more free
occurrences of x are replaced by free occurrences of y.

That's an axiomatization of first order identity theory.

However, in first order, if there are infinitely many predicate
symbols of the language, then there is no axiomatization that will
ensure that in any structure the identity symbol maps to the identity
relation on the universe. We have to make it a semantical rule to
ensure that the identity symbol "really" stands for identity. That is
pretty much what the other posters are telling you too.

MoeBlee

ah I c. Thank you.


.

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