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).
√?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.
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?
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 → ⊤
- i → ⊣
- {0, -1} → ⊥
- -i → ⊢
But so with arbitrary other tacks/turnstiles (rotated in whatever other direction) satisfying other possible "coordinates":
- E.g. {1, 0} → _?
- {1, -1} → _?
- {i, -i} → _?
... and so on?
