What is P and -P?

1. We, it depends on which interpretation to use. Usually, there's a theorem 24 in S of PL (Slupezky, Borkowsky):

(A ←→ B) ←→(~A←→~B)

a an individual case of it:

(A ←→ A) ←→(~A←→~A)

So, I guess there must be some other interpretation, maybe unar "–" is different or something.

2. If "P", "R", "S"... ate propositions, and "R" means "it's a rock", then "R = R" might be written as a) "it's a rock equal to it's a rock", or as b) "this/that rock is identical to this/that rock", or as c) "the value of it's a rock equals to the value of it's a rock". Seems we needn't a and c now.

As we can see "this rock" or "that rock" is what where we're pointing at a time. In case if we were speaking about a rock, we could point just formally with no specific or empirical examples. So it matters how this procedure will be going.

Let's suppose we're interested in to check all the cases, can ostensive conditions be dropped for some reasons? I think we have to use higher levels of abstraction then. Honestly, I can name such, or I can try to perform it partially. "R" can be taken as a class of cases relevant to this format, and we need to formalize "identical" as an operation of classes of those cases. How to say one class is identical to another one?

One of the answers is to compare their number, i.e. it cardinal with another; another way to reduce it to a description that the same for both (e.g. {x|x=μ} = {y|y=Ω} where μ=Ω). The same cardinal is not identity as well as some properties that united a class. So, to identify we cannot have to deal with classes, or we need to take "identical" as a name of operation saying that a certain class satisfies it, i.e. {a, b, c...} such as a, b, c... are elements of a set {x|x=π}, where π is a name of "identical".

3. "~P" or "–P" is a denying proposition, e.g. R means "it's a rock", ~R means "it's false that it's a rock" or "it's not a rock". So what is that that is not a rock? – Must be anything that satisfies the conditions. A plate or a stone are as good as candidates for "~R".

I guess it's interesting to analyse such a situation that says that: few of these cases present us identities, or in those cases we watch how some things identify with the others, so we want to ask – what about everything that not identical to those things which identical to each other?

Plainly, if A is identical to B, C to D, and E to F, and everything else isn't like that (that identical to that or cannot perform such an operation), then G, H, I, K... and etc are not identical to each other, however we still don't know are they identical to their oppositions or not: is "~R" identical to "R"? Asking it means to ask: do identical things to not identical things? and is it true for identical things to be replaceble with non-identical?

Considering semantically identification as such operation that is performed (i.e. "true") if and only if certain things can be replaced with some others. And here we must note that semantically identification is not necessary identical to conceptual identification; that's why we can type s-identification and c-identification as different ones. (Btw, the more chances to formalize c-identification, the more pure and clear s-identification will be.)

Ok, if A, B, C, and D belong to a class of identical things (as pairs), then how to be with the others? I guess we're risking to appear in some different categories trying to leap from one form to another. So, A-D are just some special pairs, and the rest is not. To have access to provide it for another types or categories we have to solve a task: how those categories/types relate to each other? If they do it as a scale, then the question is in numbers; etc.

Summary: we must plain the interpretation of identification to get the best results to be able to find clues of inferences about s-identification. We need to detail our narration and discourse with more powerful concepts and formulations to be able to achieve and solve problems with c-identification.