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.


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 Y Chappell 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.

Richard Y Chappell 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.

Blue 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?