A Logical Problem

[Skep]

Good first post kk. Welcome to the forum. I think you spelled out your position most clearly. I wasn't as satisfied with the clarity of Eleatic's explanation as yours. Maybe he'll respond. Anyway, I hope you stick around.

Indeed I would have to agree. I have, however, learned a few things from the thread, especially since it has sent me to the bookshelf a few times, but the clear answer as presented by kk stands out as definitive.

 

[Patricio Elicer]

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?)

I just came across this old thread and decided to post because I think I can add my 2 cents.

I haven't read every post line by line, but seems to me that the controversy about the argument being true or false hasn't settled yet (please correct me if I'm wrong).

My contention is that the entire "argument", that is the premises 1,2,3 plus the conclusion 4, all in conjuction, is true. From a strict logical standpoint it is a true argument or a valid argument. I can prove it by means of propositional logic.

 

[Eleatic Stranger]

What do you mean by 'strictly logical'?

 

[Patricio Elicer]

What do you mean by 'strictly logical'?

By that I mean the use of the "Propositional Logic" or the "Mathematical Logic" or the "Formal Logic" (many names), as taught in college math courses.

A "proposition" is a statement that can be either true or false, unambigoulsy. For example, in the argument you presented, "I will win the battle" is a statement that can be true or false, thus, a proposition. Note that it doesn't matter under what criteria someones considers the battle to be a win, it only matters that it is either a "yes" or a "no".

So my contention is that the entire argument, seen under this prism, is a truth, a valid argument. Of course, I can go further by proving my point, if you are interested.

 

[RandFan]

By that I mean the use of the "Propositional Logic" or the "Mathematical Logic" or the "Formal Logic" (many names), as taught in college math courses.

A "proposition" is a statement that can be either true or false, unambigoulsy. For example, in the argument you presented, "I will win the battle" is a statement that can be true or false, thus, a proposition. Note that it doesn't matter under what criteria someones considers the battle to be a win, it only matters that it is either a "yes" or a "no".

So my contention is that the entire argument, seen under this prism, is a truth, a valid argument. Of course, I can go further by proving my point, if you are interested.

Good post.

Logic & Fallacies

A valid argument is one in which the conclusion must be true — can’t be false — if the premises are true. (Note that the premises don’t have to be actually true for an argument to be valid.)

We must ask what are the goals and what are the priorities? One can have conflicting goals. If winning is absolutely crucial then premise #3 is in conflict with our most important goal and would be incorrect. However the importance of winning is not established. Further premise #1 could be a false dichotomy. A draw could be possible. We don't know because we lack sufficient information and must accept that winning or losing are the only two possible outcomes.

Logic & Fallacies

Secondly, logic is not a set of rules which govern human behavior. Humans may have logically conflicting goals. For example:

John wishes to speak to whoever is in charge.

The person in charge is Steve.

Therefore John wishes to speak to Steve.

Unfortunately, John may have a conflicting goal of avoiding Steve, meaning that the reasoned answer may be inapplicable to real life.

 

[Patricio Elicer]

Every argument includes an inference process. A conclusion is infered by a conjunction of premises. In the problem in question there are 3 premises, propositions (1), (2) and (3). The conclusion is proposition (4), it's clearly the conclusion because it's preceeded by the word therefore.

The argument is not just the conclusion, but the entire set of premises and the conclusion.

If the conjunction of premises is true, the only way for the argument to be valid is that the conclusion is true as well. But the interesting aspect of the inference process is that an argument is always valid when the conjunction of premises is false.

For example the following argument, far-fetched as it is, is valid because the conjunction of premises is false:

Premise (1): 2+2=5
Premise (2): "Paris is the French capital city"
Conclusion : "The Earth is flat"

In the argument in question, the conjunction of premises (1), (2) and (3) can't be determined true or false, but it turns out that it reduces to exactly the same proposition as the conclusion (4) reduces to. So we end up with two alternatives:
(a) Conjunction of premises TRUE; Conclusion TRUE. Or
(b) Conjunction of premises FALSE; Conclusion FALSE.
And in both cases the argument is true.

 

[Eleatic Stranger]

If the conjunction of premises is true, the only way for the argument to be valid is that the conclusion is true as well. But the interesting aspect of the inference process is that an argument is always valid when the conjunction of premises is false.

For example the following argument, far-fetched as it is, is valid because the conjunction of premises is false:

Premise (1): 2+2=5
Premise (2): "Paris is the French capital city"
Conclusion : "The Earth is flat"
[/B]

This isn't exactly accurate -- an argument can certainly be valid when the conjunction of premises is false, but your example is not necessarily a valid argument.

Validity simply means that in all cases where the premises are true the conclusion is similarly true. In cases where the premises cannot be true, the condition is trivially satisfied. So as long as the conjunction of the premises is necessarily false the argument is valid no matter what the conclusion is -- though even this is slightly misleading depending on the sort of conditional used. (Lewis' material conditional is, for example, an attempt to solve this oddity and the problems it causes for counterfactual claims.)

So, for example,
1. P
2. ~P
as a set of premises validly implies any conclusion. This is because there are no situations in with both P and ~P are true and in which some further conclusion is false. (P&~P)->Q is a necessarily true sentence for any values of P and Q.

However, that isn't the case for
1. The sky is green colored.
2. Bananas are poisonous.
--
Therefore:3. I am sitting in the food court of the mall.

Both the premises are false, however there are possible situations where both are true (which is to say the conjunction of the two premises is not formally false) and I'm not in the food court at the mall. The argument is not hence valid.

Validity, this is to say, is a formal description -- it only applies to the form of the argument and not it's content. An argument can be (formally) valid and still have false premises (and either a false or a true conclusion) -- and this is called an unsound argument.

The argument in question, however, equivocates - which is to say that in one reading of the semantic ambiguity it is trivially valid, and in another reading of the semantic ambiguity it is also trivially valid, but when the two are mixed up it turns out to be a nonsequitor.

 

[Patricio Elicer]

This isn't exactly accurate -- an argument can certainly be valid when the conjunction of premises is false, but your example is not necessarily a valid argument.

I think we are trapped in a kinda semantic problem. My definition is that an argument (as a compound proposition) is valid when its "Truth Table" yields TRUE values in all possible combinations of the TRUE/FALSE values of the propositions that make it up. And this is the case of the example I provided.

I think you are refering to a sound proposition. The link RandFan provided above may cast some light on that:

So the fact that an argument is valid doesn't necessarily mean that its conclusion holds -- it may have started from false premises.

If an argument is valid, and in addition it started from true premises, then it is called a sound argument. A sound argument must arrive at a true conclusion

However, that isn't the case for
1. The sky is green colored.
2. Bananas are poisonous.
--
Therefore:3. I am sitting in the food court of the mall.

Both the premises are false, however there are possible situations where both are true (which is to say the conjunction of the two premises is not formally false) and I'm not in the food court at the mall. The argument is not hence valid.

Yes, but in this case the problem cannot be resolved by mathematical logic, because for that we need each statement to be unambigously TRUE or FALSE. Only once you establish a criteria to decide unambigously whether or not the sky is green colored and whether or not bananas are poisonous you can apply the method.

 

[Eleatic Stranger]

I think we are trapped in a kinda semantic problem. My definition is that an argument (as a compound proposition) is valid when its "Truth Table" yields TRUE values in all possible combinations of the TRUE/FALSE values of the propositions that make it up. And this is the case of the example I provided.

Yes, and that's accurate -- what I'm pointing out is that that isn't what you said earlier. The truth table merely shows that with any implication it doesn't matter what the antecedant is if the consequent is true for the truth of the conditional, and that if the consequent is false and the antecedent is false the conditional is also true. However, the fact that some conditional comes out true does not mean it's valid -- that requires that it be true under all interpretations. And that's precisely what your earlier statement did not say, and what I was pointing out about it.

But the interesting aspect of the inference process is that an argument is always valid when the conjunction of premises is false.

Falsity is always falsity under some interpretation, and this only says that an argument can be in the form of a true conditional -- but in order to be valid it has to be a conditional that is true under all interpretations. For example:

1. P V Q
2. R -> P
3. R
--
4. Q

This argument could well have true premises and a true conclusion -- and hence the conditional would be, when phrased in one go (((PVQ)&(R->P)&R)->Q) would be true. However it would not be a valid argument because it wouldn't be a true conditional under all interpretations.

See?

 

[Elind]

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.



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?)

The faulty reasoning, I think, is that he makes the assumption that it is possible to win with a small force, without knowing the nature of the opposition. If he knew the size of the enemy there would be no strategy problem. Not very philosophically elegant I admit, but I suspect that most connudrums are really just an informationally incomplete scenario.

It reminds me of that slightly more subtle one called Pascal's Wager, concerning the consequences of kissing god's ass or not. The problem with that wager is that it assumes a particular type of god, whereas a more modern one would probably reverse the traditional rewards for all the sycophants.

So, when in doubt, look for more info, but don't call the CIA

 

[LW]

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

Haven't noticed this before and didn't read through all posts so I may be repeating something that is mentioned before.

The MOST important thing about logic to remember that it is a tool. You can think it as a black box: you formulate your problem using the language of logic, throw it into the Mysterious Black Box Logic Engine, and get back the logical consequences of your formulation.

The quality of answers you get is directly proportional to the quality of your problem modeling. The fact that in this case you get a nonsensical outcome is a strong indicator that the underlying model is poor.

It is not possible to translate any nontrivial real life problem exactly into logic. You have to abstract the problem somehow. If you abstract too little you get a huge set of sentences and can't do any reasoning at all because of the size, and if you abstract too much, you get a small set of sentences that doesn't capture the relevant behavior of reality.

In this example the model is abstracted way too much. In addition, the problem statement contains the quagmire of "better". Formalizing the notion of preference into logic is really difficult thing to do. I have examined roughly 30 different preference semantics for logic programs and they all contain serious flaws, (those flawed semantics include the ten that I have devised myself).

Also, since real world battles include lots of uncertainties, pure logic is not the right tool to use. Spade is a good tool but you don't try to cut down a tree with it. Formal logic and probabilities just don't combine neatly. A much better bet would be to use decision theory where you have a probabilistic model of the environment and your decisions alter the state in some probabilistic way, and then you try to find the set of decisions that are most likely to put you in the state that you would prefer to be in. (Or alternatively those decisions that are least likely to put you in a state where you really don't want to be, these might not be the same decisions).