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Quick recap: the paradox in question can be formulated like so (I think; look in the SEP article on deontic logic for a better recap if this doesn't work):

  1. If Jill kills Jane, she ought to kill Jane gently.
  2. Jill kills Jane gently if and only if Jill kills Jane at all.
  3. Jill kills Jane.
  4. Therefore, Jill ought to kill Jane gently.

My intuition is that this and some other paradoxes (or "problems"/"puzzles") in deontic logic, such as that of epistemic obligation (you end up obligated to make an evil fact real in order to fulfill your duty to know which evil facts to fight against), might be resolved by appealing to the distinction between sentence-types and sentence-tokens. In the Jill-kills-Jane case, for instance, the idea is that the hypothetical (1) holds only as a sentence-type, and that a token of its discharged consequent doesn't hold. An (I'll admit unclear) analogy would be with the following take on mathematical facts/truth, in the formalist umbrella/camp: Weir, by contrast, explicitly embraces formalism (1991; 1993; 2010; 2016), moreover formalism in the game formalism tradition. His position, if situated with respect to fictionalism, can be seen as one in which ‘consequence’ is read, in the formalist tradition, syntactically, in terms of formal derivability. As a first approximation, the position is that a mathematical sentence is true if there exists a concrete derivation of a token of it, false if there exists a concrete derivation of a token of its negation. Since truth and falsity conditions make no appeal to abstract proofs, this type of formalism is firmly anti-platonist [SEP, "Formalism in the Philosophy of Mathematics," sec. 8].

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  • Could you spell out what "holds only as a sentence-type, and a token of its discharged consequent doesn't hold" means exactly, and how it disposes of the paradox. – Conifold Oct 29 '20 at 20:08
  • A link to a clear introduction to the paradox would be welcome. As is, I simply don't see where there is a paradox. If you kill someone, you better do it without making a mess (cruelty in murder is an aggravating factor and leads to more severe punitions, at least in my country's judiciary system). That does not mean you ought to kill them in the first place... – armand Oct 30 '20 at 1:27
  • @armand The OP version is confusing. In the standard version the premises are:"Jill ought not kill Jane", "If Jill does kill Jane she ought to do it gently", "Jill did kill Jane". It then follows from the standard laws of deontic logic that "Jill ought to kill Jane" period, which is odd in itself, and a contradiction, see SEP. – Conifold Oct 30 '20 at 8:07
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    @rs.29 Obligations are not time bound, they do not cease being obligations just because they are violated, and obligation to kill is a problem regardless of when things are debated. "Jill will kill Jane" leads to the same contradiction anyway. The issue is that rules of deontic logic lead to a contradiction on a seemingly consistent set of premises. The general sense is that standard rules simply do not faithfully represent reasoning about contrary to duty secondary obligations upon violation of primary obligations, see SEP and Goble, Murder Most Gentle. – Conifold Oct 30 '20 at 10:29
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    @rs.29 It makes no difference what people do or do not understand, the question is how to change the formal rules so that the contradiction does not result, but what ought to be derivable is still derivable. Whatever "conditionality" and "hidden premises" are there has to be written into them, and it has to work across the board, not just for this example, as armand pointed out. – Conifold Oct 30 '20 at 22:04
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I don't see how appeal to a type/token distinction helps to resolve the paradox. Why would sentence 1 fail to hold in a particular instance for a particular Jill and Jane?

The problem you describe is one of many paradoxes that occur when attempts are made to formulate a logic of obligation. Obligation is a kind of modality. Saying "it is obligatory that..." invites comparison with "it is necessary that..." and other modalities. However, while the logic of necessity is comparatively easy to formalise using conventional modal propositional logic, the logic of obligation is not.

One of the important differences is that obligations can conflict with each other. Another is that obligation is a matter of degree: some obligations are stronger and take priority over others. Another is that obligations are not monotonic: in ordinary propositional logic, "if A then C" entails "if A and B then C", but with obligations this does not hold. Another difference is that with necessity we do not have to be concerned with things that are necessarily false, i.e. impossible, because impossible things don't happen. With obligations, on the other hand, things that are obliged to be false, i.e. forbidden, do happen and we need to be able to express conditional obligations in such cases.

One way of circumventing many of these problems is to represent conditional obligation as a primitive dyadic operator O(B|A), rather than as a strict implication. This is similar to how we represent conditional probabilities, which also express matters of degree and are non-monotonic. Another approach is to abandon the aim of expressing absolute obligations in favour of comparative preferences between competing obligations.

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Jane not "actable" - limits of deontic logic

Let's write some sentences using traditional normative statuses :

  • You must kill Julius Caesar

  • You are permitted to kill Julius Caesar

  • You must not kill Julius Caesar

  • You are permitted not to kill Julius Caesar

  • You may kill or not kill Julius Caesar as you wish

Does any of these sentences have any value ? Well, according to Shannon's information theory none at all. Caesar is already dead, without time machine you cannot kill him or not kill him. Digression - with a time machine no killing is permanent, therefore "evilness" of such deed is no longer absolutely certain.

Back to our world - Caesar is no longer killable. Any decision about his murder is therefore beyond the reach of deontic logic. Again, deontic logic in real life is mostly used in law and similar fields, and clearly case of killing dead Caesar is beyond their interests. But even if approach this problem from purely mathematical point, we must understand that every formal logical system has set of formulae that could or could not be evaluated by it . Example of the latter is well-known "This sentence is false". Self referring sentences are not evaluable by traditional logic.

In the similar manner, deontic logic implicitly assumes that proposition we are evaluating is possible, i.e. that decision to act or not to act on it could be made. But in example of dead Caesar or dead Jane, this is no longer the case. In sentence No.3 of OP example, Jill kills Jane. Jane is no longer "actable" considering murder. Therefore, sentence No.4 ("Therefore, Jill ought to kill Jane gently") is no longer evaluable by deontic logic. Like "this sentence is false" , whole system of 4 sentences from OP's question is self-referring, changing original hidden proposition from sentence No.1 that Jane is killable, i.e. Jane is still alive.

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