Would the imaginary unit be the truth-value of sentences formed using √𝐧𝐨𝐭?

Question

Section 4.3 of "Sentence Connectives in Formal Logic" discusses a concept of demi-negation or what is (for the sake of the text) resolved to a concept of "the square root of negation" (note that "#" is being used as the demi-negation operator):

... we are insisting that a single occurrence of “#” is genuinely responsive to the content of what it applies to. It is not hard to see that there is no (two-valued) truth-functional interpretation available for #, not just in the weak sense that it might be said that (for example) ☐ in S4 does not admit of such an interpretation—that this logic is determined by ( = both sound and complete with respect to) no class of valuations over which ☐ is truth-functional—but in the stronger sense that the envisaged logic of # is not even sound with respect to such a class of valuations, at least if that class is non-constant in the sense defined at the end of note 20.

My interpretation of "#" is that it has to do with (or just is) the "part"/"aspect" of negation that is nonetheless positive/affirmative. It is one thing to say nothing at all, another to say something about nothing (about something to which the word "not" is applicable to some extent). The absence of a negative statement is not the same as the presence of a negative statement. Granted, bringing up a concept like absence/presence seems to advert to a deep negation function, and so the composition of "##" into "~" is given as more like the dissolution of X into two samples of √X. For example, √2 = 1.414... does not mean that one can discretely separate two things into irrationally-numbered parts (physically, as it were), but one can indicate an abstract (continuous) separation hereby. Now, since the √-operation is itself an inverse (of the exponential level of the hyperoperator sequence), and since division can be understood in terms of a function on iterated subtraction, which subtraction is the arithmetic cashing-out of set-theoretic complementation—itself taken for the set-theoretic expression of the original negation-concept—it seems as if we can justify the concept of √not as constituting the meaning of the exponential inverse anyway.^√?

Suppose a truth-value function T(S) and a sentence #S, then. Would T(#S) = i? Since i² = -1, and i⁴ = 1, and having, "1 = ⊤." On the other hand, this seems to require, "-1 = ⊥," whereas normally we think, "0 = ⊥," because, "S is not true," is equivalent to, "It is not true that S," i.e. on the sentential level, the difference between absence and antithesis doesn't "make a difference" (as such). However, in a 4-valued logic, the rotation of the turnstile could be had to indicate that ⊢ signifies a nonbinary truth value, per the demi-negation scheme in the limit, the limit being that the rotation of the turnstile is mapped to {i, i², i³, i⁴}. I suppose then that demi-negation sticks with i while ⊢ goes with i³, which is to way that, "A turnstile-proves B," attributes the full negation of demi-negation to B (from A).

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^√?But since the √-operation isn't the only such inverse, but there are logarithms to boot, one wonders if there is a logarithm of negation too as such.

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Practical bearing: suppose one were to try out the idea that fictional sentences are predominantly i-valued, meaning now demi-negated, and in fact kept in a state of "suspended demi-negation." For example, when one says, "Alice spoke to a disappearing cat," one is taken to not actually just be saying that sentence with a sort of "+"-indicator attached, but, "#(Alice spoke...)." A sentence's being said under demi-negation would then stand outside attributing full truth or full falsity to it, and this is one way for fictional discourse to work out?

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EDIT: so maybe you could go on to define a scheme for introducing other nonstandard truth values/circuits. We might say, for example, that the uptack doesn't just map to -1 but also 0 (on account of the absence/presence equation on negative truth conditions). So we'd have a list starting out like:

1. 1 → ⊤
2. i → ⊣
3. {0, -1} → ⊥
4. -i → ⊢

But so with arbitrary other tacks/turnstiles (rotated in whatever other direction) satisfying other possible "coordinates":

5. E.g. {1, 0} → _?
6. {1, -1} → _?
7. {i, -i} → _?

... and so on?

Answer 1

I emailed Humberstone (the author of the SEP article I linked to in the OP) with this question, and he responded with a copy (lucky for me! I don't have a Springer account yet) this article about demi-negation, which includes the following picture:

Setting T = 1, it more or less immediately follows that the rest of the given truth-values are isomorphically juxtaposed with i, -1, and then -i, so depending on how much one trusts category theory and its preoccupation with equivalence "up to isomorphism," I suppose one has an answer to the OP question, then, and in something of the affirmative. The caveat is that demi-negations, and demi-affirmations in turn, both can be styled as having imaginary truth-values, positive or negative as these might be, yet also unfortunately (or not?), even falsity is a demi-truth value (seeing as -1 by -1 = 1, after all).

So how far the isomorphism actually can be carried, as such, is as yet rather opaque to me, and while I found later that one might go on to talk of a "sphere of probability values" in this connection (re: quantum logic), I do wonder about the point of inflating the number of truth values from two to four and then to infinity. There's something to be said for neatness and precision in (re)presentation, of course, so if we wanted, we might define logic, as distinct from mathematics, as exactly that mathematics-like kind of thinking that shouldn't be endlessly inflated but instead reduced to as few parts as possible. Then, though, logic is identified with a normative factor, which can be identified even if not called by the name of logic; and yet then demi-negation as a whole becomes (marginally) suspect as a logical concept, but is to be found in the vagueness that separates logic and mathematics, no matter how hard we wish to fight vagueness in this or that limit.