Suppose you have the chance to bet on the toss of a fair coin, but you have to put up $1000 to win $900. I think each of these is true:
(t) If the coin will land tails, then you should bet on tails.
(h) If the coin will land heads, then you should bet on heads.
(v) The coin will land heads or the coin will land tails.
Now by v-elimination we can conclude that you should write me a check for $1000 and bet. But that is obviously a false conclusion. Yet all the premises are true, and v-elimination is valid.
I've been meaning to blog this ever since, but somehow never got around to it. But that's easily remedied: let me reproduce our exchange (excavated from the 50+ comments thread)...
* * *
RC: distinguish the ‘ought’ of reasons (externalistically construed, as what would be best in objective fact) vs. the more subjective ‘ought’ of rationality (expected utility, etc.).
In the objective sense, the coin argument goes through just fine: you really should bet, and what you should bet on is the winning option. (Go on, what are you waiting for?)
Taking the subjective sense (which is more appropriate for deliberation), the premises (h) and (t) are false, as per Greg’s [comment] #30. Even if the coin will land heads, you rationally shouldn’t bet on it unless this information is actually accessible to you. Jamie’s conditional proof ("Surely you should [bet tails], on that supposition [that the coin lands tails]!") is only convincing if the supposition is supposed to be actually made by the gambler. Certainly if I (qua gambler) may suppose that the coin will land tails, then that’s what I rationally should bet on. But if we’re supposing, abstractly, that the coin lands heads without my having antecedent knowledge of this, then it wouldn’t be rational for my ignorant self to bet at all.
(This line of thought becomes clearer if we append “without your foreknowledge” to the antecedents of (h) and (t).)
JD: Whoa! I thought conditional proof was supposed to be valid for every conditional there is. In conditional proof we do not add the extra assumption that anybody knows the discharged premise, so I find it puzzling that you (and apparently Claudio) think that it’s necessary for validity here.
Anyway, I would be quite happy with an admission that conditional proof fails for these practical conditionals. My view is that it succeeds but modus ponens fails, but I’m okay with the contrary view.
RC: Hi Jamie, sorry I was unclear: my complaint is not with the logic of conditional proof, but rather your claim that “Surely you should [bet tails], on that supposition [that the coin will land tails]!” Let me clarify.
Suppose it’s true that the coin will land tails (without the gambler’s foreknowledge). Does it follow that the gambler rationally should bet tails? No, of course not. Quite the opposite: given his ignorance of the fact, he shouldn’t bet at all. So the conditional “if the coin will land tails then the gambler should bet on tails” is false. Replacing ‘the gambler’ with ‘I’ doesn’t change this. All it does is allow us to equivocate in a way that gives your argument its misleading plausibility.
The equivocation arises when you conflate the gambler with the logician doing the supposing. Compare:
(t1) If the coin will land tails without my foreknowledge, then I should bet tails.
(t2) If I may suppose that the coin will land tails, then I should bet tails.
I take it that (t1) is obviously false, and (t2) obviously true, when using the ’should’ of rationality. Now, (t) really corresponds to (t1), and so is likewise false. But your argument tempts us to treat (t) like (t2) instead.
“Suppose the coin will land tails. Now, on that supposition, I ask you: shouldn’t you bet on tails? Surely you should, on that supposition!”
This conflates the gambler and the reasoner.
Sure, if I qua gambler get to suppose that the coin will land tails, then that’s what I should bet on. That is, supposing (for sake of CP) that I get to suppose (for sake of deliberation) that the coin will land tails, it follows that I should deliberately bet tails. But applying conditional proof to this will only yield (t2), not (t).
N.B. The gambler should bet tails only if, at the time of decision, they hold that the coin will (probably) land tails. This condition is satisfied if, internal to deliberation, the gambler may suppose that the coin will land tails. So far so good. But if you have an external logical debate with the gambler, and ask them to make this supposition for sake of argument, then the supposition will be discharged by the time they return to their practical deliberation. It will no longer be accessible when they have to decide whether to bet. So the above condition is not satisfied in such a case.
Perhaps a parody-argument would help illustrate. Consider the conditional:
(t*) If the coin will land tails, then you suppose that the coin will land tails.
A groundless claim. But apply Jamie’s reasoning: “Suppose the coin will land tails. Now, on that supposition, I ask you: don’t you suppose that the coin will land tails? Surely you do, on that supposition!” Hmm.
JD: Good, thank you, Richard.
I agree with you entirely about the conditionals you mention, and I think you are absolutely right in bringing out the temptation to conflate the supposition that p with the supposition that the supposer supposes that p.
First, in support of what you’ve said, I think that conflation infects indicative conditionals in many places. Compare this somewhat idiomatic form of conditional:
(if-R) If you believe Richard, then the ought has wide scope.
This conditional is intuitively true. But surely it is grossly implausible that whether the scope of the ought could depend in any way on whether the audience believes Richard. Diagnosis: when we consider (if-R), we imagine that we believe Richard, and then ask ourselves whether the ought has wide scope; but imagining that we believe Richard is very hard to distinguish from imagining that Richard is correct, so our answer is that yes, under that supposition, the ought does have wide scope.
You have replaced ‘should’ with ‘rationally should’, and I want to know how the argument works with just plain ‘should’. And in that case, I don’t buy your analysis. Suppose this ticket is, in fact, the winner; in that case, shouldn’t you spend a dollar on it? Yes (I say)! Only you don’t know it, so it’s understandable that you don’t do what you should.
Or try it with ‘better’. If the coin will land heads, then it is better to make the bet and to make it on heads. Suppose the coin will land heads; now, isn’t it better, on that supposition, that you bet on heads?
Finally, is there an equivocation involved in the argument? Well, I think there must be something like an equivocation at some point (this is what I was saying in  about Mike’s remark in ). But I don’t believe that ‘ought’, ‘should’, ‘better’, and so on, are all literally ambiguous in just the same way.*
RC: Jamie, if we adopt a more objective sense of 'should', isn’t the conclusion of the coin argument then unproblematically true? (Cf. my comment #31.) You should “write… a check for $1000 and bet” on the winning outcome. It’s a pity you don’t know which that is, but that doesn’t falsify the conclusion any more than it does the premises (t) and (h).
Perhaps your point is as follows: granted, it is better to bet on the winning outcome. But it is not better to bet, simpliciter, for that leaves open a 50% chance of losing.
That seems right, but may not connect with the original coin argument as you presented it. The most you can strictly infer from (t), (h), and (v) alone is that “you should bet on tails or you should bet on heads”. To obtain your more general conclusion (that you should bet, simpliciter), you need the further premise(s): if you should bet on tails (heads), then you should bet, simpliciter. But this is simply the denial of the above point. So if we take that point seriously, we should reject this new premise.
JD: Hm, you’re right, Richard, but I feel like that’s a loophole that I just have to be more careful about rather than a serious problem.
Well, here’s a try at repairing with minimal amendment. We can use the principle, “If you ought to do A, and the only way you can A is by doing B, then you ought to do B.”
Now, in each branch of the OR-elimination, I’ll use that principle to conclude, “You ought to bet”, so the overall conclusion resting on the disjunction (but not separately on the disjuncts) will be “You ought to bet”.
How’s that? (I hope that instrumental principle is true! I think it is. I’m less sure that it supports modus ponens itself, of course, but if it doesn’t that works for me.)
RC: the puzzle then seems to come down to this: suppose that it’s best to do B and A, but worst to do B and not-A, compared to the neutral option of not-B. As such, is it better to do B?
I’m not sure if there’s really a determinate answer to this. It seems like the right thing to say is, “It depends whether you’ll go on to do A!”
Similarly for your instrumental principle, and its application to the betting case. You should bet on the winning option. So should you bet? Well, it depends whether you’ll be doing so on the winning option or not…
JD: I agree with all that except for the bit about there being no determinate answer. Aren’t there lots and lots of cases in which whether P or Q is the case depends on whether R is true, but this plainly doesn’t mean that whether P or Q is the case is indeterminate?
Still, when you put it in that nice Boolean way instead of the overtly instrumental way that I put it, I agree that what seemed obviously true to me suddenly seems very questionable. It seems to me that there must be plenty of cases in which you ought to do A&B, but, since you are not in fact going to do B, you ought not to do A, since A&~B is a lot worse than ~A&~B.
* * *
Good stuff -- there's something deeply satisfying about making sense of such logic puzzles. [See also my short essay on the idle argument, which discusses a similarly paradoxical argument for foolhardy battle tactics, which I also owe to Jamie.]