Counterpossibles

I'm casually interested in counterpossible reasoning (or "logical counterfactuals"), asking what would be true if something logically impossible were true. Here's an example due to Carrie Jenkins:

1. If the square root of 2 were rational, it could be expressed as n/m, with n, m integers.
2. If the square root of 2 were rational, it couldn't be so expressed.

It sure seems to me, and to her, that 1 is true and 2 is false. However, most logics don't deal with this sort of thing naturally, i.e. with normal conditionals; if we assume that root 2 is rational, then prove that it is irrational, then we can prove any other statement from this contradiction -- if you build counterpossible worlds this way, they all turn out to be worlds where everything is both true and false.

I'd especially like to know about counterpossible reasoning because it seems like it could be part of a theory of reasonable decision-making. Suppose that a computer program is making a decision; it should evaluate what would happen if it chose option 1, 2, or 3. However, since it is a program, it actually only chooses one of these options, so all but one of these choices are logically false (or maybe they are all false, if the program fails to halt). The program should be using counterpossible reasoning to figure out what would happen if it made each choice.

It seems suggestive to me that the program "doesn't yet know" which choice it will make while it's reasoning, and so it will reason the same way for the choices it will choose and the choices it won't. Maybe this gives us some clue, or limits the ways that it can do counterpossible reasoning? On the other hand, this kind of counterpossible reasoning might be specific to decision-makers, and be unsuitable for e.g. imagining counterpossible mathematical systems like Carrie's root-2-rational world I gave above.

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I have a few desiderata for counterpossible reasoning (which I really haven't done serious reading on, so for all I know these are well-known proposals):

1. There should be exactly one world W defined by counterpossibly assuming S. (A world is a set of assignments of truth values to all statements.)
2. W should assign each statement either "true" or "false", not both or neither.
3. S should be true in W.
4. If S is actually true, then W should just be the normal assignment of truth values to statements.
5. If S is not actually true, then in most cases W should assign other statements S', S'', etc. the opposite of their normal truth values. These are the "counterpossible consequences" of S. All other statements are "counterpossibly independent" of S.

5 is interesting, because it means that we can't go for a W that is minimally different from the actual world in its truth assignments; this would be the world where only S is different, which is not very satisfying. These desiderata clearly aren't enough to pin down a particular way of reasoning; defining 5 better is an obvious next step, but I'm not sure how to do it.

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One possibility that is tempting to me is to work in a proof system P instead of a complete world W (which are different because of incompleteness).  To counterpossibly assume S, add S as an axiom of P, and add a rule to P that says something like "whenever a rule in P says you could conclude not-S, conclude S instead". I think this satisfies 2, since we can never conclude not-S and the proof system is otherwise unchanged. I'm somewhat worried about 4; the mangled proof system seems like it could have trouble with proofs by contradiction, since it can't assume Q and conclude not-S to get a contradiction. This construction would also make statements not provable in P counterpossibly independent of S relative to P, which seems OK to me. Is this construction satisfactory overall? I'm not sure, and I'm out of blogging time for today! Maybe I'll come back to it -- writing this post makes me more excited about the possibility.

I'll leave you with something that seems to me like a nice application of counterpossible mathematical reasoning: