Sunday, August 08, 2004

Better Battle Tactics

Some further thoughts on my previous battle tactics post... I want to look at the problem in a slightly different way from before.

In the FBC comments, Jamie offers the following consequentialist version of the argument:
C1: Either we will win the battle or we will lose the battle.
C2: If we will win the battle, then it is better to attack with a small force.
C3: If we will lose the battle, then it is better to attack with a small force.
CC: It is better to attack with a small force.

It is the same general form as before: a 'proof by cases', we could call it. The first premise lists all possible cases, and the remaining premises assert that in each of those cases, it is better to attack with a small force. Thus, we conclude absurdly, it is always better to attack with a small force.

So here's the problem: the argument doesn't actually cover all of the cases. I don't mean that there is any fault with the first premise. (Some at FBC objected that it could be a draw, but that is rather besides the point.) Rather, the 2nd and 3rd premises fail to capture the full range of cases that they purport to cover.

There is more than one way each that we could win or lose the battle. And it simply isn't true that in every case where we will win the battle, it is (or would have been) better to attack with a small force. An obvious counterexample is that discussed previously, where we will win the battle with a large force, but would have lost it with a smaller one. In such a case, although we will win the battle (because we will take a large force), if - contrary to fact - we had taken a small force, we would instead have lost. And that, of course, would not be "better" at all. So the conditional expressed in premise 2 fails to hold in this particular case.

Now, there is a reading of premises 2 & 3 which makes them look more plausible. That is if we read C3 (for example) as saying something like "of all those close possible worlds where we will lose the battle, those where we attack with a small force are better than the others". That could well be true. But the problem is that it's only making comparisons to other losing worlds. It never considers whether in that world itself it would have been better to attack with a large force (and perhaps win instead of lose).

So, by that limited reading of the conditionals, we can no longer conclude that it is in all cases better to attack with a small force. Why not? Because, simply enough, our premises no longer cover all the cases. They are ignoring cases like the counterexample discussed above. And that means that the 'proof by cases' logical form is not being used validly. The premises, by this understanding of them, though all true, are insufficient to ensure the conclusion.

Again, the argument looks valid because it looks as though it covers all cases (given that we either win the battle or lose it). But premises 2 & 3 are only true if we interpret their conditionals in a weak sense whereby they no longer cover the entire set of cases that we might expect from them. The logic requires that they be true in the strong sense, e.g. that C2 asserts something like "In every case [possible world] where we will win the battle, it is better to attack with a small force in that case". But this strong version is quite definitely false - as the counterexamples demonstrate.

So what can we conclude from the weaker premises? We're given that (1) we either win the battle or lose it; (2) The best winning worlds are those where we attack with a small force; and (3) the best losing worlds are those where we attack with a small force. So from here we can validly conclude that the best worlds are those where we attack with a small force.

Is that the same thing as saying that "it is better to attack with a small force"? Well, no. It would only be "better" if we ended up in one of those 'best' worlds. But there's no guarantee that fulfilling that one criteria would achieve that end (in fact it's extremely unlikely that it would). After all, there are also a lot of really crappy worlds where we attack with a small force (e.g. the counterexample worlds). Chances are we'd end up in one of them, which would not be a "better" result for us at all!

(As an analogy: So far as my finances are concerned, the best possible world might be one where I win the lottery. That is, a world where I buy a lottery ticket. But in most worlds where I buy a lottery ticket, I lose, and so have wasted my money. So it would be a mistake to say categorically that it is 'better' for me to buy a ticket. The mere fact that I do so in the 'best' world is insufficient to reach that conclusion.)

Conclusion: the apparent 'paradox' arises from conflating two possible interpretations of the conditional premises. According to the strong interpretation, the argument is a logically valid "proof by cases", but the conditional premises are demonstrably false. Alternatively, the weak interpretation allows us to accept the premises as true, but at the cost of rendering the logic invalid. Instead we should be concluding that "the best worlds are those where we attack with a small force". But this does not mean that it is better to attack with a small force. So, either way, we avoid the absurd conclusion.

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