Identifying what do we usually do? What definition could we pass here? What does it mean conceptually?
I think firstly we must find some answers on this question to see how it relates to logical formalization, and then how manipulation with forms can give us some new information about the question.
Usually, identifying something is picking something up from bunching of other things. It is the process of finding and picking something, and while we're doing it we remember that there is something to which this picked object matches more or less, but at the same time we suppose to believe that if one matches to the other one the very well, perhaps we're about to say that there is identical item to another one.
As the definition we could formulate something. No knowing English well, I can't start doing it right now. I guess you or anyone else here would complete it better. For a try, I guess we can say that to identify x with y we must have such tools to make such a correspondence to be performed. But as the definition, I guess it's a thing A and a thing B are identical if and only if each time when we're dealing with A, we can use B, and no matter what do we do with B it does not matter. (I guess this definition is rather metaphysical, i.e. philosophical. So as I said I couldn't performed it because of lack of some important knowledge.)
Metaphysically I note it partially, but also it would be good to reveal some problems on it, because dealing with concepts we have to define them as more perfect as it would be possible. Conceptually, we're taking into account that no differences between A and B if both of them are identical. And if there are identical there's no chance to us (or anyone else) to define it. And at the same time: how do we know about any identities at all, if we do not have any tools to differ them?? So, probably this process is going on on some consciousness level, i.e. such "identification" is possible on a consciousness level. But, anyway it can't be said that this is the right one stop. No, even if such definition goes through conceivableness, we must say that it also doesn't seem to be such transparent. Indeed, how that A and B are being divided - with which tools, with which instruments? Only what can we say is that: we're giving names to A and B trying to claim about them as "identical things", but we never know about it anything.
I guess I gave quite enough start stuff to go further. Logically we need to determine the status of those things with what we're operating our logical inference. So, in the topic example:
(p=p)v(p=/=p)
if we use a defined previously language simple or 0-ranged PL, then "p" is the name of a proposition, "v" is a name of operation of disjunction; "=" - means "__ equals to__", and "=/=" means "__ doesn't equal to___". (Brackets are for punctuations.) The formulation is correct, and it can have two different answers as which it bases on two logical valence or binary analogy. (We could use something else, though; any 0.5 or another values are possible to be reflected here.)
Must note: that when we have more, than one name of proposition (here, it is "p") each time we meet it - each time we put just the same one there - however, we're speculating about - identification! - that means our actions of putting the same value confirms (!) that we've already accepted the law of identifiction! And that's why it doesn't seem very well to deal with it with the PL logic in that way...
So, if "p" bears value of 1, then each "p" bears the same; and the opposition of "p" bears the opposite value to "1". Having different values for "__ equals to__" means that "=" we can invert, or we need to say - each time we're dealing with the opposition on the both sides of the equation/equivalence - that this function requires us to infer "0" value. And if A's and B's values correspond, then we write "1" in this function, i.e. inferencing "1" from it.
Interpretation of "1" and "0" as "true" and "false" in logic, including PL, is absolutely freedom. There is no need to be so strictly here. However, trying to walk through it semantically we must need some more axioms or others definitions (language definitions etc). I think there's no need to do it now, till we haven't finished the previous section.
So, the very first barrier is: a) we're talking about identification during the process in which the tool has already done it (or performs it); b) the value of inference (or the result) are what goes beyond, or might be irrelevant to what are we're looking for. (This (b) isn't so good determined, I guess it requires some more understanding, but for now it is ok. Here we should use some "symmetry/non-symmetry" explanations, because having freedom in values we can just see that those values are both symmetrical to each other or kinda.)
