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A while ago I asked a question about using imaginary numbers as truth-values for a peculiar concept known as "the square root of negation"; I just found out that apparently this concept is being studied in detail in the context of quantum computing. I had emailed the author of the SEP article on sentential connectives in which I had first read about "the square root of NOT" and he sent me the .pdf for an article on the topic that seemed to corroborate my intuition about associating deminegation with the {1, i, -1, -i} cycle, but that only gave me a truth-value diamond, not a truth-value sphere.

However, I also noticed that §3.4 of the SEP entry on quantum logic and probability reads:

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Is this equivalent to there being a sphere of quantum truth-values? I feel like this might be the meaning of what I'm reading, but I'm not much versed in probability theory generally (I tried giving myself a "crash course" in the subject earlier today, on account of the impassioned questions about probability that one of our contributors here often posts and to which I sought a well-grounded reply), so I'm not sure.

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    Not exactly. The Hilbert lattice is formed by all closed subspaces, not just one-dimensional ones, so you'll get the Grassmanian. The one-dimensional ones form a complex projective space (antipodal points on the sphere have to be identified), but it is not closed under connectives. And the notion of truth values is not a good fit there. Typically, we do not take all elements of a Boolean algebra as "truth values", but represent them by 0-1 functions, and 0,1 are the truth values. But Hilbert lattice does not have functional representation.
    – Conifold
    Aug 18, 2023 at 8:02
  • @Conifold OK. Should I forgo my idea of a "truth sphere" and replace it with studying this topic instead, though? I'd rather focus on something that we have understanding of, not always just the obscure stuff that pops into my head :P or is there some (useful?) way to adapt the "sphere" map to this territory? Aug 18, 2023 at 12:30
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    I'll say this. Non-classical logics that come up are not (naturally) truth-functional. But they do have a non-Boolean lattice (Heyting algebra, Hilbert lattice) on which connectives operate. On the other hand, non-classical truth-functional logics (Łukasiewicz, etc.) are rather artificial as logics. If you could adapt your intuitions about square roots to connectives on a lattice maybe they'd apply more easily. On the Hilbert lattice the negation is orthogonal complement. What would be the square root of that, even just for 1D subspaces? Maybe some version of "truth sphere" will come up.
    – Conifold
    Aug 19, 2023 at 8:53

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So, the comparison would be to the Bloch Sphere (which maps to the Rieman sphere). But I'd say you need to follow the analogy more closely. The square root of truth would be a truth wavefunction, the distribution of truth-potential, like we have a distribution say of the chance of finding an electron around a proton, but still take measurements that relate to the square of the wavefunction - this allows there to be places with a negative potential for finding an electron without that being unphysical (in orbitals it generally means another electron is there, linking to the binding of paired electrons in shells). So, you could have a linear superposition of truth value potentials, but still adhere to the law of the excluded middle (ie, by analogy to unitarity).

The square root of NOT comes up under vector logic, and is discussed in relation to qubits and quantum computing. So the utility of this kind of thing might be in allowing the modelling of complex dependencies, that only have specific limited outcomes, like say one of two. It would be interesting if that could relate to sifting through large datasets/wavefunctions for the limited cases something was true, like in quantum cryptography-breaking. It makes me think of evaluating whether P=NP is true...

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Couldn't restrain myself ... my shot at an answer:

a × a = a²

a² = -n

  1. Necessary (?) that a = a (definition of square root)
  2. Necessary (?) that a < 0 AND a > 0 (a product that's negative must have one factor that's positive and the other factor hasta be negative)

Conclusion: Impossible to find an a that satisfies both conditions.

However ... mirabile dictu ...

I've heard highschoolers say +0 = -0 = 0. In a sense, then ...
0 = +0 = -0 = -0 × -0 = +0 × +0 = 0 × 0!!!

That is to say, for the special case that we're negating nothing, the square root of NOT is NOT and also NOT NOT (apply double negation).

😁

Trivially thus, the square root of NOT is both itself, NOT, and its negation, NOT NOT.

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  • That's the intuition that causes many to stumble when it comes to imaginary numbers: it just seems "obvious" that negating a negation should yield an affirmative/positive state. What is needed to overcome this feeling is a relative sense of negation, where something is negative in one context but positive in another, and then two negatives relative to A can make a negative relative to B, or the A-cases can both be positive as such but are negative for B, etc. Aug 18, 2023 at 15:48
  • In a sense, mon cheri (😁), -2 = +2 because, drumroll please, (-2)² = (+2)² = 4. So, we could say, hope there are no mathematicians reading this, -2 = +2 and so the square root of -4, sensu amplissimo, is -2 and +2. So, now the problem has morphed from having no choice at all to one where we have two. Deus Magnus Est.
    – Hudjefa
    Aug 18, 2023 at 16:00
  • Your considerations actually do show up in the theory of nilsquare infinitesimals, which are e such that e to the power of 2 = 0, and which e are not equal to 0 and are both < than 0 and > 0 (for the paraconsistent kind) or neither < 0 nor > 0 (for the paracomplete kind). Aug 18, 2023 at 16:05

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