Thursday, August 05, 2004

Werewolf battle strategy

Over at Fake Barn Country, they're trying to figure out what's wrong with this argument. I thought it was cool so I'll post it here too:

Premise 1: Either we will win the battle, or we will lose the battle.
Premise 2: If we will win the battle, then we ought to attack with a small force.
Premise 3: If we will lose the battle, then we ought to attack with a small force.
Conclusion: We ought to attack with a small force.

Here was the comment I left them:

I recommend that you reject premise 3: "If we will lose the battle, then we ought to attack with a small force."
Suppose the antecedent is true: We're going to lose the battle. There are many reasons why this might be the case. One of them is that we aren't going to bring a large enough force. Then it's the case that we ought to bring a large force including many werewolves and some hippogriffs. In this case, the ought-claim that is the consequent of the conditional -- "we ought to attack with a small force" is false. Since the antecedent is true and the consequent is false, the conditional is false.

(I get kind of overexcited when people are talking about battle and I have a chance to adopt a werewolf identity. Dennis might understand why.)

Right now I'm thinking that there's an ambiguity in how to interpret the conditional in 3: Is it a material or a counterfactual conditional? In my comment, I regard it as the material conditional. For the counterfactual conditional, you're supposed to consider all the possible worlds where we lose the battle, and figure out which are the best. The best of those are the ones where we bring a small force and lose. Maybe this explains why 3 seems so appealing. But I don't think the argument is valid if you use the counterfactual conditional all the way through -- the fact that the best winning-world and the best losing-world have us bringing small forces does not tell us what to do. Hmm, I think I'll go back to Fake Barn Country and mention this to them.


Justin said...

Here's my take. "Small force" is a context sensitive term. A "small force" for occupying Russia might be a "gigantic force" for occupying Rhode Island. Imagine that it takes at least 1000 soldiers to win the battle: if you bring 1000 or more you're guaranteed to win, if you bring 999 or less you're guaranteed to lose. In premise 2, context determines that what is meant by "small force" is a force that is small for winning, so what is meant is a force of 1000. In premise 3, context determines that what is meant by "small force" is a force small for losing, so a force of 1 or so. Context fails to uniquely determine what "small force" means in the conclusion. So, first point: there's a kind of fallacy of equivocation going on -- "small force" means something different in premises 2 and 3. But second point: suppose that context determines that "small force" can only be read in 1 of 2 ways in the conclusion: it either means force small for losing or force small for winning (but not, for instance, force small for a group picture). Then, you could try accepting the conclusion of the argument, denying that there's any paradox. Read the conclusion as meaning that you should either bring 1000 or 1, but no other number of troops. Any country that has 1000 or more troops to spare should bring exactly 1000 (not more), anyone who has less than 1000 troops should bring exactly 1 (anything more would be a waste, while I'm assuming if you bring 0 there's no battle).

Still, I think there are further twists possible. Imagine that if you bring 999 or less, you have a 0% chance of winning, if you bring exactly 1000 you have a 1% chance of winning, and as you bring more and more troops above that 1000 threshold, your chances of winning go higher and higher, never quite reaching 100% (this might not matter, but I include it just in case). It still seems true that if you win you ought to attack with a small force. But surely that doesn't mean you should only bring 1000 -- that only gives you a 1% chance of winning. And, it might be that no number of troops that gives you a decent chance of winning still counts as a "small force" . Maybe 2000 counts as the upper threshold for small force (anything more ceases to be a SMALL force) but bringing 2000 only gives you a 5% chance of winning, so you shouldn't only bring 2000. Then, it seems false that you ought to attack with a small force. But, it still seems true that if you win, you ought to attack with a small force. So, the problem returns. Context sensitivity can be invoked again here, but I think now things will get a little trickier.

Dennis said...

Interesting one. I'm going to be kinda non-philosophical, but here's my thought.

Imagine a band of werewolves roaming the countryside, pumping out the hedons as they go. They come upon a similarly-sized band of thieves, clearly needing to be stopped. Now, any one werewolf could take the whole band, unless they've got silver bullets, in which case the attackers will be utterly creamed. Thinking quickly through a strangely-familiar argument, they send in just one ethical werewolf.

What's going on here is the valid version of the (clearly sound) argument, and it hinges on the idea that no matter what you do, the battle's going to go down in a certain way. When we grant premises two and three, what we're granting is really "if x is certain, then y." Reducing it to logic makes us think of the predictions about the future as certainties. The kicker then is that granting premise 1 is really granting "it is certain that (x or not x)," not "certainly x or certainly not x," which is the paradox: certainty doesn't distribute over or.

I'm not sure if this is a version of something someone said on fake barn country, but if it is I'd love to know the words.

Neil Sinhababu said...

Justin -- "Small force" is context-sensitive, but even when I don't allow the context to shift, I get the puzzle. Replace 'a small force' with 'Bob alone' in the premises (I imagine that 'Bob alone' isn't context-sensitive), and I think the paradox still appears.

Dennis -- You've given a nice example of a battle for which the argument is good. However, the argument purports to apply to all battles regardless of what kind of creature you are or how your opponents are armed. The real task here is to explain why the argument, which seems to work for all battles, actually fails most of the time.

Anonymous said...

I think Dennis' later point is the paradox breaker, Neil.

Premise 2 should really be stated as: "If we will certainly win the battle, then we ought to attack with a small force." (and similar with Premise 3). Premise 1 is thus correctly written as "We will certainly either win the battle or lose the battle." Premise 2 (and Premise 3, for that matter) don't follow from Premise 1, because it isn't certain that you'll win. It's only certain that you'll either win or lose.

What would follow is Premise 2b: "If we won't certainly win the battle, then we ought to attack with a large force." Premise 3b would follow as: "If we won't certainly lose the battle, then we ought to attack with a large force."

Given this, you should attack with a large force, since you're not certain to win and you're not certain to lose (although there are a couple of steps that I'm glossing over to get here, I think they're pretty obvious).

Richard said...

Dennis said: When we grant premises two and three, what we're granting is really "if x is certain, then y."You might be on to something there, though you will need to spell out in greater detail exactly why premises 2 & 3 require the assumption of certainty, i.e. how it is that they are false without it.

I like Werewolf's justifiction for rejecting premise 3. You can actually go further and ALSO reject premise (2) by employing a similar counterexample. I have a post at Philosophy, et cetera where I discuss all of this in greater detail.

Dennis said...

Anonymous: Thanks for going into that at greater length for me while my internet was down.

Richard: I tried to write an account of why I didn't agree with you, but it turned into an account of why I'd rather think about the problem differently. Your post essentially says that we shouldn't grant either (2) or (3) because the truth of the antecedent can depend on the truth of the consequent. While we can do this, it suits my taste better to assume they are independent (which, as you say, we must do if we want to accept them), and note that (1) is not presenting the appropriate alternative. This seems more essential to me, since I like my logical statements to be absolutely true or absolutely false, but you may prefer to think of it otherwise.

Justin said...

Werewolf: point taken. The context sensitivity stuff might have just been my hangup -- you see, the conclusion of the original argument actually sounds ok to me on one reading, which is what I was trying to get out. It's certainly not ok if you run the "Bob alone" version of the argument, though, so let me give a new response that drops context sensitivity.

Premise 1: Unobjectionable. I don't buy the false alternatives reply.

Premise 2: I'll pass over it to discuss 3, b/c the two premises are so similar.

Premise 3: The analysis of this premise that sounds most natural to my ear is: among those worlds where we win the battle, the best are the ones where we attack with a small force. On this reading, 3 sounds true. It's irrelevant that there are some worlds where we lose precisely BECAUSE we bring a small force -- this would only be relevant if we were comparing worlds where we bring a small force and lose to worlds where we bring a larger force and win. But we're not. Among the worlds where we do lose, certainly the best are the ones where we brought a small force. Also, the stuff about certainty seems misguided. The conditional just STIPULATES that we lose (or win, in premise 2). Consider 2 worlds, one where we bring 5 troops and win and another where we bring 100 troops and win. Bringing 5 troops only gives you a 5% chance of winning, while bringing 100 troops gives you a 100% chance. So, bringing 5 troops doesn't guarantee certain victory. STILL, the world where we bring 5 troops and win is better than the world where we bring 100 and win. So, premises 2 and 3 look good to me.

So, what you need to do is say that the conclusion is ambiguous. Now, it's GOTTA be the case that on one reading the conclusion follows; you should grant this but say that it's not a very interesting reading, having nothing to do with action guidance. On the other reading, which does have to do with action guidance, it doesn't follow. I think the proper response will need to take this line.

Richard said...

Justin, I don't think the conclusion can be (soundly and validly) obtained according to any interpretation.

See this new post for my full argument to that effect.

Rousseau said...

Um, just Premise 1 is wrong (or incomplete). Whether you lose the battle is dependent on the size of the force.

This either or formulation is weird, and can be manipulated too easily.
1) Either they have X strength, Y strength, or Z strength.
2) If they have X, then any force loses.
3) If they have Y, then only large forces win.
4) If they have Z, any force wins.

Seems math and formulas work better for this F
F(x) being probability of winning given size of force x
C(x) being cost of force x

Maximize x for: F(x)-C(x)

Neil Sinhababu said...

Rousseau -- it seems that you're accepting premise 1. The alternatives you describe all involve us either winning or losing the battle. Could you explain what it means for a premise to be incomplete?