# √◊ (or generally √M, for whatever modal operator(s) M)

I tried Googling "demi-possibility demi-negation" and got nothing (just "demi-possibility" gave results mostly related to demisexuality). And my analysis of demi-negation didn't seem like it would directly carry over to modal logic as such anyway:

1. I differentiated between additive, multiplicative, and exponential demi-negation/demiproof, with conjunctions carrying demi-negation/demiproof for nonstandard truth values of ±1/2; disjunctions carrying {1, i, -1, -i}; and conditionals carrying solutions to xx = ±1 (including then 0 for "neither true nor false nor demitrue nor demifalse nor..."). The truth-table for conditionals was rather appealing in light of non-factivity issues in conditional semantics, but the truth-table for disjunction was puzzling at best, unacceptable at worst (a disjunction would be false if one disjunct was true and the other was false, for example).
2. Offhand, I was inclined to read possibility as involving conditionals which evaluate to 1 even though neither their antecedents nor consequents are true (or false). For example, if the truth table for AB was (0, 0) then since there is a model(?) of arithmetic where 00 = 1, but there is also a model where the expression is undefined, then in some sense AB would represent a mere/abstract possibility of truth, here.

At any rate, then, are there logic experiments with terms like √◊ or √OB, etc.? Or does the difference between a connective and a propositional attitude report(?) make such experiments prima facie unwieldy? (Note, however, that ¬A performs a syntactic role akin to various modalities MA.)

• It's not clear what the semantics of such an operation would be. How would you interpret the "square root of possibility," for example?
– user66933
Aug 4 at 14:52
• Demi-possibility might suggest something like possibly possible, i.e. ◊◊A. In modal systems without transitivity ◊◊A does not entail ◊A. You could try to define an operator that in possible worlds terms indicates that √◊A holds when A holds in a world that is two steps away. However, this does not deliver the result that √◊A when composed with itself yields ◊A. Aug 6 at 1:17
• @Bumble √◊A for a simple proposition A of a coin flipping result of head cannot be expressed in your above proposed definition, where besides the actual world there's only one other different PW one step away. Also in some cases with uncountably many PWs semantics one cannot even count 2 steps in your intended well ordered countable fashion... Aug 6 at 6:07
• @Bumble If you try to define ‘2 steps’ as nonconstructive existence of some potentially longer distance PWs or rejecting dense frames, it may not be useful for ordinal-like model logics such as linear temporal logic or Stalnaker’s variably strict conditionals. I always consider the intro of radicals to be diagonally transcendental, not more restrictive than without. As for the falsity of your demi A with only one or two PWs instead of undefined I may agree due to bivalence convention and in the similar spirit as vacuously true material conditionals, yet many simple things are trivially false… Aug 7 at 8:10
• @Bumble Now your new √◊A definition at a world trickles down to irreflexive and non-transitive serial frame conditions along with this extra conjunction ◊◊A ∧ ¬◊A at the said world. If you just take the possibly simplest result of a model of 2 symmetric PWs where your √◊A holds at one of them along with the actual truth of A, while A is false at the other mirroring PW, which feels like a cross-world quantum entanglement to fit your new demi-possibility prescription as some conservative definitional extension of modal logic to express this kind of alternating truth value at 2 consecutive PWs... Aug 8 at 6:51