/sci/ - Science & Math

>For all x, x=x

What exactly is the point of the law of identity? What does it mean to say that x=x? Does it mean that for all predicates P, that P(x) iff P(x)? All this seems to state is that
~(P(x)^P(x)), which is just the law of noncontradiction. If this is not the case, then does it state that x has some equality property which goes beyond its predicates? But these things would not be relevant in first-order logic.

If it is not either of these, does it only mean that some symbol "x' can only represent one thing? If so, then this seems to be only a statement of syntax, and not imply anything about x itself.

Having a hard time understanding.