Forrester's Paradox

My colleague, Jim Forrester, has his own paradox (Forrester, J. 1984.Gentle murder, or the adverbial samaritan, Journal of Philosophy, 81: 193-197. If you have subscription, you can view it at JStor.). Lucky guy.

Each of the following seem to be quite reasonable deontic principles:

(i) [O(p --> q) --> .p --> Oq] (where O is a deontic operator meaning "it is obligatory that")

(ii) (p --> q) --> (Op --> Oq) (inference rule)

(iii) Op --> ~O~p

Unfortunately, they seem to lead to paradox. For suppose that we have a legal system which, while prohibiting all forms of murder, nevertheless regards brutal murders to be less wrong than gentle murders. On the supposition that there is such a legal system, we get the following argument:

(1) O~(Smith murder Jones) [assumption]

(2) O(Smith murders Jones --> Smith murder Jones gently) [assumption]

(3) Smith murders Jones --> O(Smith murder Jones gently) [(i) & (2)]

(4) Smith murders Jones [assumption]

(5) O(Smith murder Jones gently) [(3) & (4)]

(6) Smith murders Jones gently --> Smith murders Jones [logic]

(7) O(Smith murder Jones gently) --> O(Smith murders Jones) [(ii) & (6)]

(8) O(Smith murders Jones) [(5) & (7)]

(9) ~O~(Smith murders Jones) [(iii) & (8)]

And now we have our contradiction. As Jim notes, (i) seems to be the questionable deontic principle--though he thinks it holds up well enough in the case as described.

I'm not so sure. Although I do think that something like (i) is pretty widely accepted, I'm not sure it has the right formal structure to push the paradox through. Specifically, I'm not sure that the wide scope reading of O in the antecedent of (i) is ever acceptable.

Let me change the example (because I find it clearer). In the animal welfare debate, one sometimes runs across people who have become convinced that meat-eating is wrong but who still persist in their carnivorous ways. In an attempt to make themselves less morally culpable, they often say things such as "Well, at least they kill them quickly." The idea being that, even if it is wrong to kill animals just because they are yummy, if we are going to do it anyway we ought to do it quickly.

Now in this case, I find it extremely difficult to find a plausible representation of this reasoning which would involve the conditional O(kill animals --> kill animals quickly). To my ear, this is categorially wrong. What does it mean to say that a conditional is obligatory? Acts are the sorts of thing that are obligatory or not, and conditionals don't offer up any acts in the right sorts of "positions". It seems, rather, that what is obligatory is that conditions be made such that if we kill animals, then we kill them quickly. But this can't be reasonably represented as an instance of O(p --> q); it is just an instance of Op.

[Addition: the oddness I am gesturing at is also reflected in the oddness of formalizing "It is obligatory that Smith not murder Jones" as (1). (1) translates as "it is obligatory that it not be the case that Smith murder Jones," which I find horrendous. It seems like there must be some Davidsonion analysis that makes all of this perspicuous.]

We can represent the latter proposition a bit more explicitly, if you like, by making notationally clear what the conditional circumstances are. Thus: O(condition[kill animals --> kill animals quickly]). But I don't think it follows from this plus the fact that we kill animals, that we are obligated to kill them quickly. That is, I deny (i*):

(i*) [O(condition[p --> q]) --> .p --> Oq]

Rather, it seems to me that the principle is acceptable only if Op; that is, only if:

(i**) [O(condition[p --> q]) --> .Op --> Oq]

But then, clearly the argument won't go through.

Comments

[Mike]

Might be worth noting that standard deontic logic does not validate (i) even if we assume that the actual world is among the deontically perfect worlds.
(i) [O(p --> q) --> .p --> Oq]
Since our world is obviously sub-optimal, we should not expect a detachment principle like (i) to hold. And (iii) is not valid unless there is a unique moral standard for each world. That is effectively to hold that there is a single correct moral principle for each world. SDL does validate (iii).
What you seem to be after in your representation of the deontic conditional is the dyadic operator in (i').
(i') O(Q / P), (read) in all of the morally best worlds at which P is true, Q is true. And it might be the case that in all of the best worlds in which you murder someone, you do so gently. But then we seem to get an analogue of Forrester's Paradox. Since Q entails P and since (ii') is true,

(ii') |-(Q ->P) -> [O(Q / P)-> O(P/P)]

So in the best worlds at which Smith murders Jones gently, Smith ought to murder Jones gently. Yikes!

[Mike]

Here's the analogue I mentioned.

1. O(P/P) [it is obligatory that Smith murder Jones given that he has done so]

But intuitively (for whatever that's worth),

2. ~O(P/P) [it is not obligatory that Smith murder Jones, given that he has done so].

[Kenny Easwaran]

It seems to me that the questionable premise is the second one. It certainly seems false that just because p-->q happens to be true that O(p)-->O(q) would be true. It would be a more plausible condition if N(p-->q) were true, where N is a necessity operator, possibly even logical necessity. This seems to be the case in the deduction given, because necessarily, if Smith murders Jones gently then Smith murders Jones. But even here, it seems a bit odd to me to think that obligation can "see through" the logical connection here. After all, even if it is obligatory that one save the lives of the five people tied to the trolley track, and the only way to save their lives is to push the fat man in the way of the trolley, it doesn't seem obviously obligatory that one push the ft man in the way of the trolley.

[Mike]

Of course that inference rule is either (i) if
|-p->q then |-Op->Oq or (ii) N(p-->q)->(Op->Oq). Otherwise it is clearly invalid. Judging by the example (in which we have an entailment from mudering gently to murdering) I think he has in mind (ii).

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