Counterpossibles and contradictory consequences

In a previous post, I proposed formalizing counterpossible reasoning with respect to a deductive system. To counterpossibly assume statement S, we'd add S to the axioms of the proof system P, and modify the inference rules so that any inference yielding not-S yields S instead, giving a modified system PS. I've found a possible issue with this, which I think has implications for counterpossibles more generally, or maybe just torpedoes this specific proposal.

Let's suppose that S is not "really true" -- that we can actually prove not-S in P (which is the interesting case for counterpossibles anyway). S would have some counterpossible consequences -- statements S' that are provable in PS, where not-S' is provable in P. If the only way to prove not-S' is via not-S, then there's no problem, since PS will just prove S instead of not-S, and proceed to prove S' instead of not-S'.

However, if there's another way to prove not-S' in P, a way that doesn't go through S, we have a conflict. P will prove not-S' either via not-S, or via this second path; PS, on the other hand, will prove S' if it goes via S, and will prove not-S' if it goes via the second proof path. (Maybe this would be easier to understand with an example, but I don't have one right now.) So, PS will be able to prove both S' and not-S'. Whoops!

There seem to be at least two options, if we want to stick with the deductive-system view of counterpossibles:

1. A precedence rule: if PS can prove some statement via S or via some other path, then the S-path takes precedence, and the other path is ignored. This has the unfortunate effect of making PS's inference rules "non-local", since conclusions don't just depend on premises, but also on everything else the system can prove.
2. Paraconsistent logic: we can allow PS to prove some statements both true and false, and somehow limit the explosion to make sure that the result isn't a world where every statement besides S isn't both true and false.

Of these two, the second is somehow more appealing to me; it really does seem like counterpossibly assuming S is not "enough" to flip not-S' to S' if there is a way to prove not-S' independent of S. This makes these kind of contradictions seem more like a desirable feature of the deductive-system formalization of counterpossibles than a bug. I would be unsatisfied with this, however, if every counterpossible consequence S' of S became both true and false. I'm not sure, generally, whether most statements in deductive proof systems have many proof pathways (which would be bad news for this method), or whether some systems have some statements that can only be proved in one way.

There seem to be a bunch of paraconsistent logics that I could use, and I don't know anything about the pros and cons, thought I like the idea of rejecting disjunctive syllogism and reductio ad absurdum. Intuitively, I don't think I want to completely limit the explosion; it seems to me that statements "downstream" of S' and not-S' should also be both true and false, but statements "upstream" shouldn't be affected, but I can't say precisely what that means.

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Incidentally, it feels to me like this kind of problem shouldn't affect decision-making programs that need to use counterpossible reasoning. My feeling is that a decision-making system shouldn't be able to figure out that some decision it could make would cause a contradiction, since it "should be in a position" to make any decision it would "like" to. This smells to me a little like free will -- the consequences of a decision-maker's actions irreducibly depend on the action itself, and there aren't proof pathways that circumvent the decision entirely. Maybe that provides a lead on how decision-making programs should use counterpossible reasoning, though I don't know how to cash it out formally.

However, the thoughts above certainly seem to me to be applicable to mathematical counterpossibles, like what "would be true" if 2 = 3 or if root 2 were rational -- in those cases, I think we need to use some paraconsistent logic.

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: