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Saturday, July 17, 2010

The Miners Paradox Revisited

I recently posted about the Miners Paradox, which Janice Dowell has been discussing over at PEA Soup. My initial reaction was to reject two of the premises in the argument, thereby undermining the paradoxical conclusion. However, as Janice pointed out to me, this is insufficient, because common sense tells us that the premises are true. That forced me to elaborate upon--though not reject--my initial response.

The issue has to do with ordinary language and philosophical logic. Specifically, how do we know when and how to apply the rules of logic to ordinary speech? While modus ponens may be one of the simplest rules in logic, its application to ordinary language is not always obvious. The Miners Paradox may be instructive in this regard.

I'll repost the paradox, as presented by Janice:

MINERS: 10 miners are trapped in a flooding mine; they are either all in shaft A or all in shaft B. Given our information, each location is equally likely. We have just enough sandbags to block one shaft, saving all the miners, if they are in the blocked shaft, but killing them all if they are in the other. If we do nothing, the water will distribute between the two shafts, killing only the one miner positioned lowest. On the basis of these considerations, (1) seems true:

(1) We ought to block neither shaft.

While deliberating, though, we accept both

(2) If the miners are in A, we ought to block A

and

(3) If the miners are in B, we ought to block B.

We also accept

(4) Either the miners are in A or they are in B.

And (2)-(4) seems to entail

(5) Either we ought to block A or we ought to block B.

Paradox!


My concern is chiefly with (2) and (3). These are conceivable sentences a person might use while contemplating what to do in the scenario, and they do appear to express true beliefs to that person. The question is, what do these sentences mean? Or, to put it in more analytic terms, what propositions do these sentences express?

The Miners Paradox requires the assumption that these sentences express the following propositions:

(2P): All miners-in-A worlds are shut-shaft-A worlds.

(3P): All miners-in-B worlds are shut-shaft-B worlds.

My initial response to the paradox was to show that (2P) and (3P) are false, and I think my argument for their falsity is strong. Yet, as Janice indicated, I should give some account for why we think (2) and (3) are true. My response is to offer alternative propositional interpretations of those sentences. I don't think (2) and (3) mean (2P) and (3P)--at least, not for the folk deliberating in the scenario. Rather, I think they mean (A) and (B), respectively:

(A) If we know the miners are in shaft A, we should shut shaft A.

(B) If we know the miners are in shaft B, we should shut shaft B.

This should be clear if we imagine how a person in the miners scenario would act. Say we are in the miners scenario. We hear a person deliberating with (2) and (3) and we take them to mean (2P) and (3P). We respond, "yes, you're right. If the miners are in A, we should block A. And if they're in B, we should block B. Since they've got to be in one or the other, we should block one of them."

The deliberator will likely respond, "no, because we don't know which one."

At that point, we can say, "but that doesn't matter. Our knowledge has nothing to do with it. As you said, they're in A or B, and if they're in A or B, we should block A or B."

The deliberator says, "I didn't say that."

Us: "Of course you did. You said, and I quote, 'If the miners are in A, we should block shaft A. If the miners are in B, we should block shaft B.' You didn't say anything about whether or not we knew which shaft they were in."

Deliberator: "But of course I meant that we had to know which shaft they were in!"

Us: "But that's not what the linguists tell me you meant. You didn't mean that your knowledge was required to justify the decision."

At that point, the deliberator might say, "No, I meant that if we knew which shaft they were in, then we should close that shaft." Or, perhaps the deliberator will get confused, saying "I'm not sure what I meant, but I'm sure we need to know which shaft they're in, or else we shouldn't shut either of them." Or, perhaps, "Okay, I was wrong before. I didn't mean what I said." In any case, the deliberator remains sure that there is no justification for blocking either of the shafts. (5) is never a compelling conclusion.

The Miners Paradox requires either baffling the subjects whose beliefs are in question, explicitly contradicting their attempts to clarify their meaning, or complicating their understanding of their own prior statements enough so that they reject both (2) and (3), even though they had previously thought both were true. This cannot be right. I conclude that we should replace (1P) and (2P) with (A) and (B), respectively. While the first pair of interpretations are plausibly false, the second pair are plausibly true and seem to be a better representation of how the sentences in (2) and (3) are being used in the scenario. Thus, there is no paradox, and no need to question the rules of logic.