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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 A → B 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 A → B 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.)

Kristian Berry
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  • It's not clear what the semantics of such an operation would be. How would you interpret the "square root of possibility," for example? –  Aug 04 '23 at 14:52
  • @SergZ. IDK... the guy Humberstone who I read about demi-negation from seems to start out with a demi-operation as one where you just "split/factor the operation in twain," and then use each double iteration to re-compose the original operator, which is already hard enough for me to get at with negation (much less conjunction or conditionals!). For a perhaps horrible example, my theological background made me ask, "Could the Trinity be differentiated by demi-negation/demi-possibility alternations?" but I'm wary of that part of my background, nowadays. – Kristian Berry Aug 04 '23 at 18:30
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    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. – Bumble Aug 06 '23 at 01:17
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    @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... – Double Knot Aug 06 '23 at 06:07
  • @DoubleKnot That would be a case where √◊A is false, rather than one where √◊A cannot be expressed. Come to that, there are situations where ◊A is false for any A, but we can still state the truth conditions for ◊A. My suggestion is speculative and I haven't tried to work it out. A definition of √◊A along these lines might well have to avoid treating the space of possible worlds as dense, rejecting ◊A -> ◊◊A as a frame condition. Also, 'two steps' might have to be understood as: there is some path of two steps, without excluding longer ones. – Bumble Aug 07 '23 at 04:23
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    @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… – Double Knot Aug 07 '23 at 08:10
  • @Bumble let △ be demipossibility. What about a system where △A is substituted for epistemic possibility, △△A = ◊A is logical possibility, and then we also have ◊△A and △◊A to consider (metaphysical and nomological possibility, if the two strings are inequivalent)? – Kristian Berry Aug 07 '23 at 12:07
  • @DoubleKnot I'm inclined to agree that trying to define √◊A in terms of a given number of steps is unlikely to be useful. How about a system without ◊◊A -> ◊A and ◊A -> ◊◊A as frame conditions, and where √◊A holds at a world when ◊◊A ∧ ¬◊A holds? – Bumble Aug 07 '23 at 12:37
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    @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... – Double Knot Aug 08 '23 at 06:51
  • @Bumble While OP sounds like to stipulatively introduce some brand new (radical/demi) *transcendental-possibility* in Kant's transcendental logic spirit or some Zalta encoded while not exemplified *non-compossibility*… – Double Knot Aug 08 '23 at 07:23
  • @DoubleKnot I'm not quite sure what that means, but it sounds good. – Bumble Aug 08 '23 at 09:25

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