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Is the law of identity the same for negative expressions?

Does 'if not p then not p' have any specific meaning in philosophy?

I am asking because I am trying to work out whether the vagueness of 'p' would mean 'not p' is also vague, and that's intuitively why I think it might not. If you classify things into being "p" or "not p" in an indeterminate way (it's not because of how things actually are), and "not p" is not identical to itself, then maybe the 'not p' need not be vague, even if 'p' does.

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    This is really more a question about vagueness than about identity. Identity is usually understood to mean that every thing is identical with itself and not identical with anything other than itself. It can also be used to express the fact that a proposition always has the same truth value as itself. If p is vague, then not p is going to be vague as well. Vagueness means that the boundary between p and not p is imprecise.
    – Bumble
    Aug 20, 2023 at 21:31
  • you may be right, which i salute you for @Bumble thanks (though i disagree with your conclusion enough to leave it)
    – user67302
    Aug 20, 2023 at 22:17
  • The law of identity applies to objects (e.g. p), not to rules (e.g. not p); applies to the parts and not to the whole system. Vagueness is hermeneutics of the whole.
    – RodolfoAP
    Aug 21, 2023 at 1:51

1 Answer 1

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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:

  1. x(¬(x) ≠ x)
  2. 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):

  1. ¬(¬x) ≠ ¬x

Or, then, compose ¬(x) and id(x):

  1. id(¬(x)) = ¬(x)
  2. ¬(id(x)) = ¬(x)
  3. 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:

  1. "Axiom": there are different forms of difference, i.e. the concept of an absence is not the same as the concept of an opposite.
  2. 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.
  3. 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 UU, 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)).

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    that's a very interesting, useful and well written post. but i really struggle with symbolic notation (for reasons i cannot explain actually) so i cannot actually incorporate it beyond it being cool, some philosophical agreement cheers. anyway.
    – user67302
    Aug 20, 2023 at 22:15
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    @legoman I could say: it is not 100% obvious that the law of identity applies to negation as it does to other logical operations. On the flip side, without negation in play, it might be hard to explain what identity is anyway. As an aside, it is somewhat "customary" rather than technically necessary, in the theory of antimatter, to count electrons as negatively charged per se: they have a charge opposite that of positrons but there seems to be something perhaps relative about this description. Aug 20, 2023 at 22:21
  • i am specifically thinking about non-local absences (and whether or not they are vague if the thing is indeterminate), but like i told @bumble it's not an argument, just a means to clarify what i should know, if that makes sense ha
    – user67302
    Aug 20, 2023 at 22:23

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