Re: definition of identity

On Jun 28, 8:52 pm, Zaljo...@xxxxxxxxx wrote:
Hi all

How identity is defined exactly?

In second order logic one can use the following axiom to define
identity:

AxAy( x=y <-> AP(P(x)<->P(y)) )

How this is wrote in FOL?

Here is a try:

A binary relation R is said to be identity relation if and only if
for every formula P ,all closures of

AxAy(xRy -> (P(x) <-> P(y)) )

are true.

Is there an error with this?


There's the problem you are just thinking of: if you don't quantify
over P you don't have what you wish; there could be an R other than
identity satisfying your formula:

take N as the universe of discourse; define R as follows:

xRy <-> x=y+2

Take P to be 'is even'.

Identity is not definable in first order.

Regards


.

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