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In a lot of places equality is defined that for two expressions A=B, A=B means that A and B have the same value (A=B). This relationship seems strange as we are slightly abusing use/mention, if '=' denotes a relationship between expressions then in '1+1=2' '1+1' and '2' are denoting themselves.

Doing this seems a little bit strange considering we generally use '2' and '1+1' to denote values themselves.

If we treat equality as a relationship a mathematical object has with itself, it is not that useful, however the variety of the names and what they inform us of is, for example x+1=2 is equivalent to 2=2 but the former is more interesting.

In this way, is the elementary mathematical use of the equals sign essentially just an identity statement? (Ignoring the intricacies of the idea that different objects that we equate aren't always mathematically the same due to their construction, as I usually get linked this article and it's not really what I'm asking).

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There are actually lots of different equality/equivalence relations one can define (in the following simply R to avoid any associations you might have with the symbol =). They all have in common:

  • reflexivity (xRx for every element x)
  • symmetry (xRy => yRx for every element x,y)
  • transitivity (xRy, yRz => xRz)

= is overloaded in mathematics, sometimes it denotes identity, but not always. When two expressions are compared, a+b = c+d actually means equality between the numbers the expression evaluates to.

Equality of a program or an expression might range from: evaluates to the same thing, computes the same function, computes the same function AND does so with the same complexity (for various measures of complexity).

An equality between logical sentences means, or might mean, the worlds that make them true are identical, not the sentence itself.

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