In the theory of functions generally, we distinguish critical points from fixed points. A critical point of f(x) ≠ x, a fixed point of f(x) = x. The negation and identity functions have the following distinctions:
- ∀x(¬(x) ≠ x)
- ∀x(id(x) = x)
So we might say that negation is the function all of whose points are critical points whereas identity is the function all of whose points are fixed points. Now take double-negation (abstract over the usual introduction vs. elimination schemes):
- ¬(¬x) ≠ ¬x
Or, then, compose ¬(x) and id(x):
- id(¬(x)) = ¬(x)
- ¬(id(x)) = ¬(x)
- Therefore: id(¬(x)) = ¬(id(x))
Is (6) true, however? There seems to be room in paracomplete logical space to think that (5) is already misguided: the absence of an identity for x need not be taken for the presence of a negatively "charged" x, perhaps.
We should also distinguish between kinds of negation:
- "Axiom": there are different forms of difference, i.e. the concept of an absence is not the same as the concept of an opposite.
- Let T be the value True and F be the value False. In zeroth-order logic := L0 (AKA propositional/sentential logic per se), ¬T = T¬. Likewise ¬F = F¬. So in L0, negation is commutative. However, in modal logic, ¬◊ := impossibility is not the same as ◊¬ := contingency. So beyond L0, we can differentiate absences from opposites by the positioning of the negation mark.
- But there are, furthermore, different kinds of opposition. For example, two things can be opposites by being at either end of a given sequence, but they can also be opposites by being on sequences that are themselves opposites of each other.
The upshot is that there is some intuitive reason to think that negation, when considered as a "process" in and of itself, either has no identity, or uniquely has an identity whereby it is not identical to itself. This is contradictory, ultimately, so our intuition seems arguably untrustworthy in this case, or is at any rate defeated by our intuitions about contradictions not being true (though these intuitions might themselves depend on intuitions we have about negation-by-cancellation vs. other senses of negation-by that we have).
Another "puzzle" occurs with respect to unrestricted quantification, though. Let ≹ mean "is neither greater nor lesser than" with the qualification "is not equal to, either." In other words, let that symbol be for "is not commensurate with" (we emphasize this distinction insofar as otherwise, when x is neither greater nor lesser than y, then x = y). Now take an absolutely universal set U. Trying to do so is fraught with logical peril; the classic problem is that, using the Separation Axiom in normal set theory, we can carve out a Russell set (of paradoxical infamy) from U. And at any rate, an absolutely open, all-encompassing set seems hard to associate with complete determinacy: if something is determined to be A, is it not closed under A? But then how can we say, "U is closed under the predicate of not being closed under any predicate"? Does U = U or do we instead have a peculiar (or even unique) case such that U ≹ U, i.e. the universal set is not commensurable with itself (or anything, for that matter)?
One of my endeavors in working on set theory has been to see if there is some well-grounded way around the problem of the quintessential indeterminacy of the universe of sets. My conclusion so far is that this task is difficult to achieve bordering on pointless or impossible; so though I am yet optimistic, I would recommend Storer[10] for extended discussion on the question of universal determinacy (or the lack thereof). For a quick quote from that dissertation, along with my musings on how to relate the ideas to the question of vagueness, see this PhilosophySE question.
ADDENDUM: per your comments, you might be interested in the SEP article on boundaries as well as the one on relative identity. You don't have to worry about being on the right-vs.-the-wrong track, luckily, since all of us are perplexed, ultimately, by these questions, too. One might say: philosophy exists to some extent as an indeterminate object itself, whose boundaries are vague (vaguely adjacent to science and religion, and even love (it is not without reason that we call it philosophy)).
p), not to rules (e.g.not p); applies to the parts and not to the whole system. Vagueness is hermeneutics of the whole.