Consider the statement
This is the law of identity translated to arithmetic.
More generally we could say
Whatever x is; it must be equal to itself.
Is this always true? Can there be situations when it doesn't hold?
In Jowetts introduction to Platos Theatatus he writes that
of the three laws of thought, the first being A=A, is an identical proposition - that is a mere word or symbol claiming to be a proposition; ...[they] are untrue; because they exclude degrees, mixed modes and double aspects under which truth often presents itself to us.
For, x=x, referring only to itself as symbol, is merely formal and does not refer; once we ask it to refer the situation becomes more complex; when x is a river; what does it mean for a river ton be equal to itself? It changes from moment to moment.
Still, this is not what I'm asking about; I'm asking how far can one loosen the classical meaning of equality; the law of thought that Jowetts mentions and still be rendered meaningful in logic.
One possibility can be shown by examining the following situation:
A blue box on a table in front of which Jameel is sat on a chair. I ask him to close his eyes; and when he has I turn the box around; and when I ask him is it the same box; he being of a suspicious mind says well it looks like the same box; perhaps you've exchanged it for an identical box; or perhaps you've turned it upside down...
This is at least two determinable situations; by permutation and by rotational symmetry; which can be brought together into one idea of symmetry.
In a way one has expanded the notion of equality.
What other possibilities are there?