One formulation of the law of Identity states that a thing is equal to itself (e.g., "A=A").
The "thing" must always be represented (with a letter, a word, a number, a picture, etc.) in order to be communicated. These representations will have physical, measurable properties, and no two of them -- for instance, two spoken or written "A"s -- will have exactly identical physical properties. If you attempt to circumvent this mirror image comparison with, for example, an "A" with an arrow doing a U-turn back upon itself, you still must make a mental comparison, and that comparison takes time, and as Heraclitus famously puts it, you can't step in exactly the same river twice (in other words, the first thought "A" is gone by the time you think of its twin). So, without sprawling this out further with more examples, why doesn't it make more sense to assume that "a thing is NOT equal to itself"?
I am probably just talking around some hackneyed epistemological issue. Can anyone sort out my fuzzy thinking a little (with a minimum of truth tables)?