These two paradoxes look at first sight very different. I won’t go into the difficulties they pose for the early forms of deontic logic. Instead I want to show, first, that actually they are structurally similar, and second, that the remaining difference suggests a pattern for a deontic logic.
Sophie’s Choice, dramatized by the novel and movie of that name, is the choice faced by a mother in a Nazi concentration camp. She if offered a choice: either her son or her daughter can leave the camp for a Nazi boarding school. If she does not choose, neither will go. She realizes that a child remaining in the camp will not survive. The three moral imperatives that she feels very strongly may be represented as: (1) if you can save the boy, do so!, (2) if you can save the girl, do so!, (3) do not save one child at the cost of the other!
The Miners Paradox also poses a trilemma, for a rescue operation. Ten miners are trapped in a shaft, either shaft A or shaft B, and the mine is about to flood. It is possible to block one shaft, which will save the miners if they are in that shaft. But there is no knowledge whatsoever to indicate in which shaft the miners are trapped. If neither shaft is blocked, there will be some flooding in both, drowning only the lowest miner. The three moral imperatives here are: (1) block shaft A if the miners are in A!, (2) block shaft B if the miners are in B!, (3) do not take an action which will, with certainty, lead to the death of a miner!
The great difference between the two puzzles, at first blush, is that in the Miners Paradox it is not known which of imperatives (1), (2) is really in force. With total ignorance, each of the triggering circumstances has a 50% probability.
But there is a straightforward way to remove that difference. Each action, blocking A or blocking B, has an expectation value of saving five miners (multiplying the probability and the value of saving ten). The no-action option has an expectation value of saving nine miners.
Once put like that, the Miners Paradox has exactly the solution that seems obvious (if morally wrenching nevertheless), namely, not to block either shaft.
In the case of Sophie’s Choice there is also a conclusion that is ruled out by having the worst outcome; in that case, it is the no-action option. But in that case, two imperatives remain, which cannot be jointly satisfied.
So the structural similarity is this: each poses a trilemma of imperatives, and a resolution in terms of the values of the foreseen outcomes. The difference that remains is that in Sophie’s Choice the resolution still leaves an irresolvable moral dilemma.
The moral I wish to draw for deontic logic is this. Both imperatives for action (which are default rules) and values of foreseen outcomes must be ingredients in the semantic analysis; but the analysis must not be contrived so that it will necessarily eliminate all irresolvable conflicts.
Obvious retort: Default rules can be given a priority ranking. Rather than introducing the extra ingredient of values of foreseen outcomes, it suffices to give higher priority to some imperatives relative to the others.
Response: that will not do because the priority depends on factual circumstances. In such a situation the agent can only satisfy some maximal set of mutually compatible (non-conflicting) imperatives. But whether imperatives conflict is often due to circumstances. For example promises which were in principle capable of being jointly honored turn out to conflict because of a change in the agent’s assets.
In the case of Sophie’s Choice we actually see three imperatives that could be jointly satisfied, namely if she had been allowed to let both children go, or alternatively if she had not made a choice (on the basis that then the antecedents of (1) and (2) would be made false). But the commander added the two constraints that only one could go, and that if she did not choose neither would go. In these circumstances the three imperatives are in conflict with each other.
When we specify what it means to conflict, or why one maximal set of mutually compatible imperatives takes priority over another one in the given circumstances, the explanation may refer to prior categorical imperatives. But most often — as illustrated above for each example — it will refer to the values of foreseen outcomes.
“Stay inside!” and “Go out!” conflict because there is no outcome of the agent’s action that satisfies both. If in our value judgment it is better to satisfy the former, that must surely be because staying inside realizes some greater value.
But more importantly, a priority ranking of the individual imperatives does not engender a correct ranking for maximal compatible sets of imperatives.
For example, suppose that “Stay inside!” and “Go out!” have the same priority, while “Do something useful!” has a higher priority than the others. This will not prioritize either of the maximal compatible sets (doing something useful inside, doing something useful outside) over the other. But it may actually be the case that there are much more valuable things to be done outside than inside. In that case the value of the proposition [the actor goes outside and does something useful] is higher than that of [the actor stays inside and does something useful], and that should break the tie.
That obligations due to imperatives do not disappear
This (partial) resolution does not imply a denial that Sophie is subject to three mutually incompatible oughts. They are there, they are all in force in her situation, even though there is one that — all things considered — she should not act on.
This looks like we need to maintain the venerable distinction between Ross’ prima facie and all-things-considered oughts, even while denying that all prima facie conflicts can be resolved (as opposed to: ended by voluntary symmetry breaking). I do not like that terminology, it seems to me to be misleading. The moral imperatives that are recognized in a situation or context, and serve as default rules there, establish the primary obligations.
Two things can happen to these so as to block their effect. First, they can be violated, in which case secondary obligations will normally appear, also as default rules. Secondly, they are defeasible, that is, they may give way to other considerations, or simply to bare facts. If I have made a promise to someone who dies, that promise does not then place me any longer under the same obligation (though I may have a derivative obligation to his heirs or reputation). But secondly, as we saw, they may give way to value judgments about the foreseen outcomes.
What remains is that “what I ought to do” is ambiguous. First there are the imperatives that are in force, in that context, under those circumstances, and these are N-ary obligations (either primary, or having come into force due to violations). Secondly there are the residual moral alternatives for how to act or for how things are to be, after all the circumstances, and moral evaluations of the foreseen outcomes, have been taken into account. So we need a logic that covers both: say, with connective O for what is obligatory in view of the imperatives, and Horty’s symbol ⊙ for what ought to be all things considered.
APPENDIX. Notes about the logic, semantics and axiomatization
Let’s say “maxi-set” for “maximal compatible sets of imperatives.” The first point is that each such maxi-set can be replaced by a single imperative that has the same effect. Then the original imperatives can be discarded. This results in a new model in which the imperatives pertaining to a given situation are all mutually incompatible, and what is true in the new model is the same as what was true in the original model.
So soundness and completeness proofs can be directed at a simplified semantics. There is a candidate axiomatization for O, for this sort of semantic set-up, in my “Values and the Heart’s Command” (JPhil 1973).
The second point is that the introduction of the value-ranking of propositions will have as effect only that, typically, some of those maxi-sets are eliminated when it comes to the truth conditions for ⊙ sentences. But logic does not dictate the ranking, so one of the rankings that is admissible ranks all the foreseen outcomes as equal. In that case, there is no reduction — it is as if we just had the model without the value-ranking. All other coherent rankings are similarly admissible.
So the outcome is this: the logic for ⊙ sentences, taken by themselves, is precisely the same as the logic for O sentences, taken by themselves.
Of course if both connectives are in the language we have to add the bridging principle that ⊙A implies OA.
Most of a philosophically interesting discussion, in this case about imperatives, obligations, values, and ought, is not going to be reflected in the logic, which catalogues only the form of valid inferences. The exception here is that the recognition or dismissal of real irresolvable moral conflicts divides deontic logics into two forms, characteristic of apparently irremediably different moral intuitions.
NOTES
There was a lot to do about such paradoxes about forty years ago, but now there is much more sophisticated literature on deontic logic. What I have been reading:
- John Horty (2012) Reasons as Defaults. Oxford: Oxford University Press
- Ali Farjami (2020) “Discursive Input/Output Logic: Deontic models, norms, and semantic unification”. Presented at the 18th International Workshop on non-monotonic reasoning. Available on ResearchGate.
- Catherine Saint-Croix and Richmond H. Thomason (2020) “Chisholm’s Paradox and Conditional Oughts”. http://web.eecs.umich.edu/~rthomaso/documents/deontic-logic/ctd.pdf
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