Another paradox which is intensively studied in deontic logic and recent metaethics is the Miners’ Paradox,9 an example that runs as follows:
Ten miners are trapped either in shaft Aor in shaft B, but we do not know which one. Water threatens to flood the shafts. We only have enough sandbags to block one shaft but not both. If one shaft is blocked, all of the water will go into the other shaft, killing every miner inside. If we block neither shaft, both will be partially flooded, killing one miner.10
9See for instance Kolodny and MacFarlane (2010), Willer (2012). 10See Kolodny and MacFarlane (2010), page 1.
The reason this paradox is so prominent is that it incorporates other familiar paradoxes generated by SDL: paradoxes involving conditional obligations, dis- junction of obligations, and conflict of obligations. Why is the Miners’ Paradox a paradox? Lacking any information about the miners’ position, it seems right to that the outcome should be:
[5] We ought to block neither shaft.
However, in deliberating about what to do, we accept: [6] If the miners are in shaftA, we ought to block shaftA. [7] If the miners are in shaftB, we ought to block shaftB. We also accept:
[8] Either the miners are in shaftA or they are in shaftB. But [6]-[8] seem to entail:
[9] Either we ought to block shaftAor we ought to block shaftB. And this is incompatible with [5]. Thus we have a paradox.
In order to block the Miners’ Paradox, several escape routes have been proposed.11 In particular, it has been suggested that two differentoughts are
involved in the paradox: asubjective ought and anobjective ought.
Intuitively, subjectivism and objectivism differ with respect to the body of information in light of which the deontic modal is evaluated. Under the objectivist reading, a sentence likeX ought to doαindicates thatαis the best option available to the agentX in light of all facts, known or unknown; while, under the subjectivist reading, X ought to do α indicates that αis the best option available to the agentX in light of whatX knows. If one adopts a purely objectivist reading of the deonticought, the paradox does not arise, since the very premise [5] is rejected. Under the objectivist perspective, [5] does not hold: in light of all facts, the best option is to block one shaft, the one the ten miners are in. On the other hand, if one adopts a purely subjectivist reading ofought, premises [6] and [7] do not seem to be acceptable. For instance, take [6]: If the miners are in shaftA, we ought to block shaft A. While the antecedent refers to a fact that might be unknown to the agentX, the consequent establishes a moral obligation in light of the knowledge of the agentX. Thus the conditional is not acceptable, since it may be the case that the antecedent is true and the consequent is false.
Both objectivism and subjectivism have, however, some difficulties. As Kolodny and MacFarlane (2010) also point out, objectivism seems too strong, since it does not allow deontic reasoning for partially-informed agents. If the objectivist view had to be adopted, only an omniscient agent with a complete knowledge of all facts would be able to determine whetherX ought to doαis
true. On the other hand, subjectivism seems to be too weak, since it does not validate conditional obligations likeIf α, then X ought to do β.12 That appears to be a remarkable loss, since those conditional constructions are very common in ordinary deontic reasoning, in particular in deliberation. Subjectivism is, therefore, too weak if it cannot deal with those common constructions.
However, that does not imply that some of the claims made by objectivism and subjectivism are not worthy to be considered. In what follows we show that, in the case of the Miners’ Paradox, it is possible to provide a solution which takes into account some aspects of the objectivism’s and subjectivism’s views, but uses one single deontic operator that remains neutral between the two positions.
The Miners’ Paradox is about agents who consider what they ought to do in a context of partial factual information. Again, we can understand the paradox’s scenario in terms of a conversational exchange between a moral authority (the speaker) and a listener. Depending on what the moral authority commands, the listener may decide to accept or reject certain obligation sentences. In particular, we can see that even if the listener has accepted all the premises of the Miners’ Paradox, the paradoxical conclusion [9] is not supported in the updated information state.
We formalize sentences [6]-[8] as follows: [10] Ifa, Obligeblocka
[11] Ifb, Obligeblockb
[12] a∨b
whereaisThe miners are in shaftA,bisThe miners are in shaftB,blocka is
We block shaftAandblockb isWe block shaftB.
Assume that the moral authority has already utteredIf the miners are in shaft A, we ought to block shaft AandIf the miners are in shaft B, we ought to block shaft B, and the listener has already accepted those obligations. Consider the information stateSn=hn, H, H∗n, v, Betterisuch that:
• H =H∗n={hi, hj, hk, hl} • v is such that: – (i, n)∈Vv(a),(i, n)∈Vv(blocka),(i, n)∈Vv(¬b),(i, n)∈Vv(¬blockb) – (j, n)∈Vv(b),(j, n)∈Vv(blockb),(j, n)∈Vv(¬a),(j, n)∈Vv(¬blocka) – (k, n)∈Vv(a),(k, n)∈Vv(¬b),(k, n)∈Vv(¬blocka),(k, n)∈Vv(¬blockb) – (l, n)∈Vv(b),(l, n)∈Vv(¬a),(l, n)∈Vv(¬blocka),(l, n)∈Vv(¬blockb)
• Better(hi, n) =Better(hk, n) ={(hi, hk),(hi, hj),(hi, hl),(hk, hl),(hj, hk),
(hj, hl),(hl, hk)}
12Subjectivism may validate only weaker conditionals such asIf X knows thatα, then X ought to doβ. See Kolodny and MacFarlane (2010) , page 118.
• Better(hj, n) =Better(hl, n) ={(hi, hk),(hi, hl),(hk, hl),(hj, hi),(hj, hk),
(hj, hl),(hl, hk)}
• for every m 6= n, Better(hi, m)= Better(hj, m) = Better(hk, m) =
Better(hl, m) =H×H.
The common ground consists of four histories: hi in which the miners are in
shaftAand we block shaftA (hence no miner dies),hj in which the miners are
in shaftB and we block shaftB (hence no miner dies), andhk andhl where
one miner dies. From the point of view ofhi andhk atn,hi represents the best
course of events; while from the point of view ofhj andhl atn,hj is the best
one. Therefore, we get that:
• Sn|=a∨b
• Sn|=Ifa, Obligeblocka
• Sn|=Ifb, Obligeblockb
In the stateSn, all the premises of the paradox are accepted. However, we get that:
• Sn6|=Obligeblock a
• Sn6|=Obligeblock b
To see that, it is sufficient to consider the result of updating Sn withOblige blockaandObligeblockb. Consider the case ofSn[Obligeblocka].13 Sn[Oblige blocka] =Sn+1=hn+ 1, H, H∗n+1, v, Better0isuch that:
• Better0(hi, n+ 1)= {(hi, hk),(hi, hj), (hi, hl), (hk, hj), (hk, hl),(hj, hk),
(hj, hl),(hl, hk),(hl, hj)}
• Better0(hj, n+ 1)= {(hi, hk),(hi, hj), (hi, hl), (hk, hj), (hk, hl),(hj, hk),
(hj, hl),(hl, hk),(hl, hj)}
• Better0(hk, n+ 1)= {(hi, hk),(hi, hj), (hi, hl), (hk, hj), (hk, hl),(hj, hk),
(hj, hl),(hl, hk),(hl, hj)} • Better0(h
l, n+ 1)= {(hi, hk),(hi, hj), (hi, hl), (hk, hj), (hk, hl),(hj, hk),
(hj, hl),(hl, hk),(hl, hj)}
SinceBetter0(hj, n+ 1)6⊇Better(hj, n) andBetter0(hl, n+ 1)6⊇Better(hl, n),
it follows thatSn6|=Obligeblock a.
SoSndoes not supportObligeblockanorObligeblockb. That is, neither we
ought to block shaftAnor we ought to block shaftB. Even if all the premises of the Miners’ Paradox hold in the information stateSn, the paradoxical conclusion
[9]Oblige blocka∨Obligeblockb does not hold. Therefore no paradox arises. Notice that, however, we get that:
13The case ofSn[Obligeblock
[13] Sn|=Oblige(blocka∨blockb)
This is an intuitive result. It means that,in absolute terms, the listener knows that we should block one shaft (as no miner killed is better than one miner killed), but given herpartial informationabout the world, she cannot conclude which shaft has to be blocked. She rejects, indeed, the obligation of blocking shaftA and the obligation of blocking shaftB. In fact, in our semantics the contrast between what we ought to do in absolute terms and what we ought to do relative to our partial information can be represented in terms of a scope distinction between one single deontic operator and the disjunction. We can, therefore, account for the distinction to which subjectivism and objectivism appeal, without accepting a proliferation of different deontic operators.