On Fri, 7 Jul 2017 05:45 pm, Chris Angelico wrote: > On Fri, Jul 7, 2017 at 4:43 PM, Steve D'Aprano > <steve+pyt...@pearwood.info> wrote: >> On Fri, 7 Jul 2017 01:41 am, Marko Rauhamaa wrote: >>> In Second-Order Logic, you can define identity directly: >>> >>> ∀x ∀y x = y ↔ ∀P (P(x) ↔ P(y)) >> >> Translating to English: >> >> For all x, for all y, x equals y if and only if for all P >> (P(x) if and only if P(y)) >> >> >> That might be sufficient for second-order logic, but it won't do for >> programming. Defining if-and-only-if for functions that can return more than >> two values (true and false) requires having a definition of equality, which >> would make the definition circular. > > It sounds to me like this has defined equality, not identity, right?
In mathematics, I believe that equality and identity in this sense are the same, and we could spell the mathematical operator "=" as "is" instead. Mathematicians normally use the word "identity" to refer to things like the identity function f(x) -> x, or the identity matrix [1 0][0 1] say, or the multiplicative identity (usually 1), rather than talking about "the identity of a value" since that would be redundant: the identity of the value is the value of the value which is the value. But programming languages can have values which are equal but not identical, such as 1 and 1.0. -- Steve “Cheer up,” they said, “things could be worse.” So I cheered up, and sure enough, things got worse. -- https://mail.python.org/mailman/listinfo/python-list
