 should both be provable, to which it is natural to add immediately that # should be congruential according to the generalized consequence relation associated with this proof system. Without the latter addition, we could just think of “##” as an idiosyncratically long-winded way of writing “¬”. With it, 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.[43]  In the paper cited, this congruential logic of what is there and in an earlier paper there referred to as demi-negation is discussed and equipped with a Kripke-style semantics (essentially re-working an earlier 4-valued matrix semantics) the upshot of which is that classical propositional logic or any fragment thereof is conservatively extended by the two zero-premiss rules inset above—or alternatively, assuming ¬ is in the fragment, by the equivalence of ¬φ and ##φ—and a congruentiality rule for #. Nevertheless it is hard to imagine, “from the inside”, speaking a language with a connective for which this provided a suitable formal treatment. The description suggested at p. 257 of Dalla Chiara, Giuntini, and Greechie (2004), a study in quantum logic, of such a connective as expressing a kind of tentative negation, seems wrong, since iterating tentative negation would not produce (untentative) negation: If you tentatively negate the tentative negation of φ you would appear to be headed (however tentatively) back in the direction of φ rather than towards ¬ φ. On the same page of Dalla Chiara et al. (2004), Deutsch et al. (2000) are approvingly quoted as saying the following—in which, since composition is more like multiplication than addition, demi-negation is referred to as the ‘square root’ of negation: