A Logical Problem

[Eleatic Stranger]

Now, this is one of those older philosophic problems, and I think I've got a solution to it but I'm not entirely sure. First though I'd like to see what you all make of the problem.

The King is currently engaged in a war with another country. His forces are arrayed below him, and the enemy forces over on an adjoining hill. A battle is imminent. He must decided whether to send a large force of men or a small force of men into battle with the enemy. He reasons as follows:

1. I will either win or lose this battle.
2. If I win this battle, then it is better to have won with a small force.
3. If I lose this battle, then it is better to have lost with a small force.
4. Therefore, whether or not I win or lose, it is better to send a small force into battle.

Now, clearly something has gone wrong here. But what?

(Note: I know that the problem is that reasoning like that he's going to lose - that is why the conclusion is false. However, what went wrong in the reasoning?)

 

[chance]

The false reasoning is in the subjective statements of ‘better’. Is 'better to win" overruling, the objective of just winning.

 

[Ashles]

The King is working from the assumption of winning or losing being events that will impartially happen to him.

He is discarding his own involvement in these events thus viewing them as events to be reacted to. When you start off from an incorrect start the reasoning can be technically logical but rendered illogical due to the false primary assumption.

For example you could exaggerate the statements from "a small force" to "One limping dog" to make that false assumption clearer.
The logic would still apply equally well, but it fails to account for the fact that the size of the force sent in will effect the likelihood of the outcome, so you can't consider the outcome first and then think what force what be best to send based on that.
One limping dog will lose (no matter how hardy and battle trained he is), so at what point would a small force become a small force potentially capable of winning?

This involves more complex decision making than the question implies and the concept of small versus large is not a clear cut issue.

This win or lose situation is not a coin toss, it is based on decisions that the King himself will make based on information about the enemy. The question also implies that the battle could go either way, in which case he would always be better off sending in a larger force.

If the opposing forces were massive then the King would probably surrender and throw away no soldiers at all.


But assuming the King has no possible information to go on about the enemy and going from an illogical start:

The trick in the question is comparing 2 different variables (win/lose, large/small) but ignoring the second variable for the first question.
So although the question compares win/lose, and THEN big/small, it should really be comparing win/big, win/small, lose/big, lose/small as these are really the possible outcomes.
This is disguised by the wording: "then it is better to HAVE lost with a small force".
Losing/small were both existent at the same time, you can't decide one after the other, therefore all four potentials are equally possible so we have to compare all four simultanously, not the win/lose first, and THEN the big/small.

I think that's what I mean.

 

[Eleatic Stranger]

Ashles -

On the fact that the King is clearly being stupid because, well, if he commits the smaller force he's probably going to lose then, yes, that's true. However, that's just the reason we recognize something is going wrong in the reasoning (ie, we know perfectly well that the last sentence isn't a good conclusion). The problem is in the argument he offers himself - after all, even though he has a significant bit to do with the outcome, it's still true that it'll either be a win or a loss.

After all, the puzzle deals with Logic and how he's going wrong, not with the fact that he clearly is going to get in some trouble. As it's set out, he seems to be going from three true statements to a false one -- how so?

 

[DaveW]

Sounds to me kind of like a false dichotomy. He is presenting himself ony 2 choices: 1) Send a small force and win, or 2) Send a small force and lose. The reality, as Ashles said, is that the true decision should be more complex than that. He obviously has other choices, such as - send a large force and win (with better probability of a win), - send a large force and lose (with better chance of not losing), -run away, -etc.

 

[Ashles]

After all, the puzzle deals with Logic and how he's going wrong, not with the fact that he clearly is going to get in some trouble. As it's set out, he seems to be going from three true statements to a false one -- how so?

I thought my last paragraph discussed that. It is the wording at fault - it is assumed that the win/lose variable is analysed, THEN the small/large variable.
This is impossible.
It is really FOUR possibilities - Win/Big, Win/Small, Lose/Big, Lose/Small - these are what must be simultaneously compared by the self imposed logic of the puzzle.

Plus the sentences aren't actually linked as well as they seem to be. Sentence 2 merely compares winning with a large force versus winning with a small force.
Sentence 3 compares losing with a large force versus losing with a small force.
Sentence four seems to be comparing all possible outcomes of winning or losing, but in actuality it isn't. It is merely comparing the OPTIMUM outcome of winning with the OPTIMUM outcome of losing, but failing to consider any other possibilities.
Therefore when it says 'whether or not I win or lose' it is not really analysing all the possible scenarios.

 

[epepke]

Now, clearly something has gone wrong here. But what?

I don't know that this is a pure logical problem. It seems a probabilistic problem. In that light, there are two fallacies: 1) Failure to recognize that it is better to win with a large force than lose with a small force, and 2) failure to recognize that the probability of winning depends on the size of the force.

 

[Jyera]

The King is probably very clear about what he is doing.
And his logic is perfect. And his beliefs intact,

But Eleatic Stranger (ES) has two conflicting beliefs that confuses him.

1. On the one hand, ES believe that a Large force is needed to increase the chance to win. So, the possible decision to send in a small force, when he has the choice to send a large force, doesn't make sense to him.

2. On the other hand, ES concur with the King in the belief that it is "better" to avoid casualties. This is seen in ES, "agreement" that the "logic" is okay despite "something is wrong with the conclusion."

The two beliefs , in this situation, conflicts each other.
You cannot want to increase the risk of casualties by sending a larger force and yet want to minimise the risk casualties by sending a small force at the same time.

The King had probably sorted it out in his first statement.
Least casualties is better to him. And he throws away the belief that a large force will necessary wins.

ES, on the other hand wants to hold on to both beliefs.
(that "large force will win", and "good to minimise casulties")

And in fact, both belief has faded to become less of a conviction, and more of mere knowledge.
ie. "A large force MIGHT win and not WILL win"
and "It MIGHT be good to reduce casualties, and not MUST reduce casulties"

The belief provided the strong premise/assumption for action.
Without strong belief/premise, logic becomes correct but meaningless and confusing.

 

[FireGarden]

I don't think the king has made a mistake. It IS better to send a small force. In theory.

But (until his scientists have finished developing that quick-kill plague that's carried by limping dogs and only kills foreigners), he may have to choose something more practicle for today.

Summary,
What's best in theory is not always best in practice.

 

[Ashles]

The king thinks:

1. The enemy either does or doesn't have WMD
2. If the enemy attacks me it is better to have attacked them first
3. If the enemy doesn't attack me I will not have their oil so it is better to have attacked first
4. Whether the enemy attacks me or not I have solved international terrorism


Where are my cynicism-inhibiting tablets?

 

[csense]

The King is currently engaged in a war with another country. His forces are arrayed below him, and the enemy forces over on an adjoining hill. A battle is imminent. He must decided whether to send a large force of men or a small force of men into battle with the enemy. He reasons as follows:

1. I will either win or lose this battle.
2. If I win this battle, then it is better to have won with a small force.
3. If I lose this battle, then it is better to have lost with a small force.
4. Therefore, whether or not I win or lose, it is better to send a small force into battle


Now, clearly something has gone wrong here. But what?

The answer is simple.

To lose implies the soldiers are dead.
To win implies the other soldiers are dead.

Therefore, the statement: If I lose this battle, then it is better to have lost with a small force, is correct.

The statement: If I win this battle, then it is better to have won with a small force, is incorrect

 

[Jeff Corey]

Re: Re: A Logical Problem

The anwser is simple.

To lose implies the soldiers are dead.
To win implies the other soldiers are dead.

Therefore, the statement: If I lose this battle, then it is better to have lost with a small force. is correct.

The statement: If I win this battle, then it is better to have won with a small force. is incorrect

But what about the ones what run away? They are less likely to be dead.
Ergo, according to the rule about the most dead soldiers, they don't lose, specially if they all run away.
/Brave Sir Robin.

 

[csense]

Re: Re: Re: A Logical Problem

But what about the ones what run away? They are less likely to be dead.
Ergo, according to the rule about the most dead soldiers, they don't lose, specially if they all run away.
/Brave Sir Robin.

LOL Hi Jeff...I think I'll leave such complications to the posters here. I can only deal with the terms and categories of the problem itself.


edited to add: I forgot that the smilies were turned off in this particular forum.

 

[Jeff Corey]

Re: Re: Re: Re: A Logical Problem

LOL Hi Jeff...I think I'll leave such complications to the posters here. I can only deal with the terms and categories of the problem itself.


edited to add: I forgot that the smilies were turned off in this particular forum.

;}

 

[Atlas]

I think some others have been saying something like this too. But in my opinion, the King's logic is good if, and only if, the chances of winning with a large force do not exceed the chances of winning with a small force.

 

[csense]

I think some others have been saying something like this too. But in my opinion, the King's logic is good if, and only if, the chances of winning with a large force do not exceed the chances of winning with a small force.

Hi Atlas

As I said to Jeff, I can only deal with the terms and categories of the problem itself.

Briefly:

At the outset, the argument is either intelligible, or it is not.
If it is not, then there must be something in the argument that doesn't agree with itself.

The terms of the argument, win, lose, better...are all based and defined by the negative: casualties.

And where I've already defined win and lose previously, the terms of better implies the least amount of casualties .

If that is not implied or inferred, then the argument is simply nonsense.

That said, the third statement in the argument is categorically correct and agrees with the terms. The second statement in the argument is not categorically correct and does not agree with the terms of the argument. It is either incorrect, or it simply does not apply....but it can not be a correct statement.

 

[Skep]

Calls for too much speculation

Part of this whole problem is that it lacks context and definitions. All of the posted answers have to make speculation about what "better" and "wining" and "loosing" mean, so there is no perfect answer.

1. I will either win or lose this battle.

This apparently arbitrarily omits the possibility of a draw. But since this problem is presented as a syllogism, we can just choose to accept the statement as true for argument's sake. However, the relative importance of this battle is not given, therefore it is impossible to know how critical winning this battle is and how to allocate resources, i.e. troops.

2. If I win this battle, then it is better to have won with a small force.

Why? Does he have another use for his troops? Does it just look more intimidating to the enemy? This statement is flawed if not justified in some way. There is no stated reason why this should be true. Further, statement #2 implies a presumption that the likelihood of winning the battle is independent of the number of troops sent. There is no justification for this, either.

3. If I lose this battle, then it is better to have lost with a small force.

This statement seems to be the only one that seem to be true, yet it is only useful if the king is able to judge the likelihood of wining in advance and that the likely hood of winning is independent of the number of troops. And if the king were to know in advance that the battle was most likely to be lost, then sending in 0 troops would be the best answer.

4. Therefore, whether or not I win or lose, it is better to send a small force into battle

This last statement is dependant on the questionable premise #2 that it is better to win with a small force and that the likelihood of victory was independent of the number of troops. Since statement #2 is not supported, the conclusion fails.

 

[Atlas]

1. I will either win or lose this battle.
2. If I win this battle, then it is better to have won with a small force.
3. If I lose this battle, then it is better to have lost with a small force.
4. Therefore, whether or not I win or lose, it is better to send a small force into battle

Csense, your point has made me rethink my previous post. But I still do not favor your position.
I believe what ES has given us is an ill formed syllogism. If statement 2 is the major premise, statement 3 should be a minor premise using a term from the major premise.

Something like this: If A then B. If B then C. Therefore If A then C.

We are given instead: IF W then BWSF. IF L then BLSF. Therefore If W or L then BSF.
It is so ill-formed it is not syllogistic at all.

This following is not an exact model but I think it is similar enough to expose the flawed setup.

If Republicans want to win then they should not vote for Kerry.
If Democrats want to lose then they should not vote for Kerry.
Therefore, win or lose, Republicans and Democrats should not vote for Kerry.

(edit: OK, my example is not so great. Help me out. I'll see if I can improve it.)

 

[drkitten]

I believe what ES has given us is an ill formed syllogism. If statement 2 is the major premise, statement 3 should be a minor premise using a term from the major premise.

Something like this: If A then B. If B then C. Therefore If A then C.

We are given instead: IF W then BWSF. IF L then BLSF. Therefore If W or L then BSF.
It is so ill-formed it is not syllogistic at all.

Wrong analysis. Or, more accurately, you're right that it's not syllogistic; it needs to be approached using a different formalism if you're going to do it "formally."

But you're also mis-formalizing; W and L are not independent propositions; L is equivalent to not-W (-W).

It's a simple theorem of propositional logic that if W implies P, and if -W implies P, then P.

 

[Atlas]

Wrong analysis. Or, more accurately, you're right that it's not syllogistic; it needs to be approached using a different formalism if you're going to do it "formally."

But you're also mis-formalizing; W and L are not independent propositions; L is equivalent to not-W (-W).

It's a simple theorem of propositional logic that if W implies P, and if -W implies P, then P.

I thought about that but couldn't carry it home so to speak.

It remains as you agreed "unsyllogistic", but what does it become? A major premise with a minor premise that is it's contrapositive?

BWSF = Better to Win with Small Force

If W then BWSF. If -W then -BWSF. Therefore, W or -W, BSF.

I thought leaving L in it's unsubstituted form was clearer to explain, but I agree that with you that because of statement #1, "L" is equivalent to "Not W (-W)"

Somehow there is a form fallacy being presented. I was promoting the idea that formal syllogisms relate to 3 variables...

If A then B. If B then C. Therefore, If A then C. But ES has presented one with 4: W, L, BWSF, BLSF, and perhaps a 5th one in the conclusion BSF.

Perhaps it is still 4 with W, -W, BWSF, and -BWSF but when I originally analyzed it this way it made me think like many others that it was a violation of the law of indirect reasoning: If valid reasoning from a statement S leads to a false conclusion, then S is false. And most of us feel Statement #2 is false. Still, for the king such a subjective statement might be true - so I decided my best argument was to call it ill-formed. And leaving the equivalence (L= -W) unsubstituted seemed the most clear.

I also thought to argue that Statements 2 and 3 are actually both Major premises and must be separated into their own syllogisms.

Given: All battles which are not won are battles which are lost.

If I am to win this battle, then it would be better to have won with only a small force.
It would be better to have won with only a small force because it is the strategy that will bring me the most prestige.
Therefore, to acquire the most possible prestige, I must battle with only a small force.

If I am to lose this battle, then it would be better to have lost with only a small force.
It would be better to have lost with only a small force because it is the only strategy that would still leave me with adequate protection.
Therefore, to insure I retain adequate protection, I must battle with only a small force.

But again, this was only another way to point up that the original presentation is not as syllogistic as it appears. So anyway, I'd like to articulate a coherent argument from the (L = Not W) perspective, but I get kinda tied up in NOTs.