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Assumptions. The overall claim is that, "X is possible," when X is some proposition, can be translated into, "If X is the object of a true conditional or disjunction, then the description possible applies to X." So for example, suppose that, "If the moon is made of green cheese, then space mice would be tempted to eat it," is a true conditional. But so then we would not (in this immediate context) refer to the antecedent as possible, but instead the consequent is an example of possibilia relative to the antecedent.

Unrestricted conditionality, then, would be necessity: "If [insert any stable proposition], then X," means that X obtains under all conditions whatsoever, i.e. is necessary.

As far as disjunction goes, we would say e.g., "If the moon is made of minerals or green cheese, then there is a possibility of the moon being made of green cheese (abstract though this possibility is)." I imagine that we would need to compromise on the principles that give us the standard explosion argument for the LNC, perhaps by adopting a relevance logic, to keep all this tidy enough; otherwise, we would end up with, "If a logical explosion occurred, then the LNC would be false," but we don't want to say that there is some sort of actual, though abstract, possibility of the LNC being false (at least not on account of the abstract possibility of a logical explosion!).

Questions. Supposing we can keep all that tidy enough, what happens if we iterate the modal operators (or predicates) so construed? I have a hazy idea in my head that we can correlate the "levels" of modality, with those iterates. So maybe, "If a logical explosion occurred..." goes with epistemic possibility, as the most ethereal, and one-off, iteration of the possibility descriptor. For present purposes I will conflate epistemic and logical modality; so let's say that something's being possibly possible (possible under a conditional/disjunction representing an anterior epistemic possibility?) is metaphysical possibility. Then possibly possible possibilities are nomological, or if we consider a force like free will as "the ability to do otherwise modulo a principle of alternative possibilities," a modal triplex can also map to the concept of free-will possibilia.

Now, I don't know that there are any substantive descriptions out there of levels of modality besides the epistemic/logical, the metaphysical, and the nomological/volitional. So I wouldn't know what to say of a fourfold iteration, and so on down the line, to say nothing of infinite iterations. But then I'm not sure any of these constructions, even as per the opening assumptions, actually do go through.

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    Already there seems an issue in your first section defining your relative possibilia, assume the universally accepted law of identity and replace your consequent with x!=x, in such case your possibility modal reduction to the so-called "function of conditional operator" seems impossible... Jun 29, 2022 at 2:05
  • I keep getting the nagging feeling that my definition is viciously circular, too. Intuitively, it seems to me as if the concept of "iffiness" is similar enough to the concept of possibility, for some form of the definition to go through, but I've never figured out how to clearly spell the "equation" out :( Jun 29, 2022 at 4:28
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    1. Are you familiar with the work on counterfactuals? 2. I'm not sure what to make of the fantastical nature of your examples, or whether the physical impossibility is relevant. Would you have the same issues with an example sentence like "If Trump had won the election, then the Keystone Pipeline project would not have been shut down"? 3. I have my doubts as to whether the laws of predicate logic apply to these sentences with internal modality. That is, after all, why modal operators were introduced in the first place. Jun 29, 2022 at 8:24
  • Unfortunately, I only have weak knowledge of counterfactuals. And yes, I would say that the possibility of the pipeline project continuing, is relativized to materially/objectively true conditionals, in terms of the antecedents being meaningful. This is where one of my expression problems arises: I have to say that incoherent antecedents are always false, but then I seem to implicitly invoke logical necessity over and above conditional relativity. Jun 29, 2022 at 8:58

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Some observations about your assumptions...

If we are asked to suppose that the counterfactual conditional, "If the moon is made of green cheese, then space mice would be tempted to eat it," holds, then I would say this does entail that we are also being asked to suppose that there is a counterfactual possibility that the moon is made of green cheese. Our moon is not actually made of green cheese, but we might allow that there is a possible world with a moon that is a counterpart of ours that is made of green cheese in that world. Under Stalnaker's account of conditionals, the counterfactual conditional would hold if the PW in which the moon is green cheese and mice eat it is closer to the actual world than the PW in which the moon is green cheese and mice don't eat it.

The consequent part of the conditional is possible relative to the antecedent since if there is no moon made of green cheese, then the mice are unable to eat it. This tells us nothing interesting about the nature of possibility; it is just a simple dependency. One might equally say that if I had a bicycle somebody might steal it, but since I don't, it is not possible.

A conditional with a necessary antecedent does not entail a necessary consequent. It would be somewhat odd, but not untrue, to say, "if 2+2=4 then Abraham Lincoln was the 16th president of the United States". The antecedent is necessary, in some sense, but the consequent part is not.

It is a matter of some disagreement between logicians as to whether we can countenance counterpossible conditionals, which is to say, conditionals with an impossible antecedent. On some accounts, these are trivially true, or fail to have a truth value, but others consider them to be acceptable. The issue is related to that of whether it makes sense to speak of impossible worlds.

It does not seem unreasonable to me to conditionalise on the features of logic itself. For example, it is correct to say, "if we use intuitionistic logic, the rule of double negation elimination does not apply", or "if we use the logic of paradox, this proposition is not excluded from being true and false". Conditionals allow us to make any supposition we choose and then state what holds within the context of that supposition. The supposition may transcend any kind of metaphysical, epistemological or even logical constraints.

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