A Logical Problem

[Skep]

Re: Re: A False Tautology

Did you mean..

1) A or Not A
2) If A Then C
3) If Not A Then C
4) Therefore: C

I think that's what you were getting at.

Yes, I merely changed variable C to B because there are only two, now.

 

[Eleatic Stranger]

In addition to the other reasons concerning the inherent problems with comparative analysis of qualitative/ quantitative....

...it is the only way to be sure logically speaking.

Seriously -- in logical terminology there's no distinction between qualitative and quantitative predicates, especially not when you assume an extensional semantics.

I'm really not sure why you think it's a problem, but I will note that it is distinctly possible to define "better" quantitatively -- in terms of net utility, as I believe I gestured at above -- while avoiding serious oddities in defining what counts as a win.

I'll post what I think the problem with the argument is in a little bit -- but at the moment I have a time conflict. You'll probably have to wait a couple hours.

 

[drkitten]

I'm really not sure why you think it's a problem, but I will note that it is distinctly possible to define "better" quantitatively -- in terms of net utility, as I believe I gestured at above -- while avoiding serious oddities in defining what counts as a win.

You might be surprised by this. I think that it's much more difficult than you think to formalize this properly.

The reason, of course (and part of the flaw in the King's original argument above) is that "better" is inherently comparative. Better than what? If the statement is "if I win, it is better to have sent a large force," then a proper analysis of "better to have sent a small force" will show that the two then-clauses are not identical.

So what we really have is a formal argument of the form

A or not-A
if A then B
if not-A then B'
hence B or B'

which by the fallacy of equivocation is made to appear that B and B' are identical (B), and hence that B is proven.

I doubt that you can correctly formalize "better" without resolving this equivocation.

 

[Eleatic Stranger]

You might be surprised by this. I think that it's much more difficult than you think to formalize this properly.

The reason, of course (and part of the flaw in the King's original argument above) is that "better" is inherently comparative. Better than what? If the statement is "if I win, it is better to have sent a large force," then a proper analysis of "better to have sent a small force" will show that the two then-clauses are not identical.

So what we really have is a formal argument of the form

A or not-A
if A then B
if not-A then B'
hence B or B'

which by the fallacy of equivocation is made to appear that B and B' are identical (B), and hence that B is proven.

I doubt that you can correctly formalize "better" without resolving this equivocation.

Now, I don't think this is exactly right, but I do think it's hitting the right point: the problem as I see it is a fallacy of equivocation, and also one on the word "better".

The problem with how you've phrased it is that, as I see it, "it is better to have sent a small force" is just leaving the last term of the comparative sentence implicit -- namely, "It is better to have sent a small force than a large one.", and when phrased that way the sentential form is identical in the second and third premises.

The form, in other words, is identical to the one I've posted -- the equivocation comes in in a sneakier manner, between the second and third premises, and the conclusion. It is an equivocation between "better" in the sense of actual utility, and "better" in the sense of "rationally expected utility".

To see how this works out, take the following example:

1. I will either win or lose the battle.
2. If I win the battle, I would rationally have expected the better utility from sending a small force (than a larger one).
3. If I lose the battle, I would rationally have expected the better utility from sending a small force (than a larger one).
4. Therefore, the rational expected utility from sending a small force is greater than the rationally expected utility from sending a large force.(Or in other words, it is better to send a small force than a large one.)

Now, in this phrasing the second and third premises are both blatantly wrong -- the rationally expected utility from sending the large force is greater than the rationally expected utility from sending the small force, since the small force is more likely to lose. Furthermore, the actual outcome has no bearing on this fact.

Alternatively, look at what happens if we phrase it in terms of actual utility:

1. I will either win or lose the battle.
2. If I win the battle, the actual utility from sending a small force is greater than the actual utility gained from sending a larger one.
3. If I lose the battle, the actual utility from sending a small force is greater than the actual utility gained from sending a larger one.
4. Therefore, the actual utility from sending a small force is greater than the actual utility gained from a larger one.

In this example the second and third premises are both true -- and the conclusion is true as well. However, the truth involved is a trivial one that has nothing to do with, prior to the event,what act one should rationally expect to yield the greater utility (barring, as we are, actual knowledge of the outcome itself).

The problem comes in when we note that the king is taking the first three lines of the 'actual utility' reading of 'better' form of the argument, and the conclusion of the 'rationally expected utility' reading of better form of the argument.

That, as I see it, is the problem involved in his argument.

 

[csense]

Seriously -- in logical terminology there's no distinction between qualitative and quantitative predicates, especially not when you assume an extensional semantics.

I'm really not sure why you think it's a problem, but I will note that it is distinctly possible to define "better" quantitatively -- in terms of net utility, as I believe I gestured at above -- while avoiding serious oddities in defining what counts as a win.

I'll post what I think the problem with the argument is in a little bit -- but at the moment I have a time conflict. You'll probably have to wait a couple hours.

Now that is one of the more interesting responses I've read yet.

I've posted my opinion, and my only interest now is the opinion of others...not necessarily of my opinion mind you. I'm not a stranger to people disagreeing with me, or of scratching their heads over things I've said. I do my best, try to explain, and move on from there.

That said, I think your critique of the argument will be an interesting read and I await it.

What would be even more interesting to me however, is if you or anyone else might rewrite the following using qualitative terms, instead of quantitative terms and still retain it's meaning:

I'll post what I think the problem with the argument is in a little bit -- but at the moment I have a time conflict. You'll probably have to wait a couple hours.

 

[drkitten]

Now, I don't think this is exactly right, but I do think it's hitting the right point: the problem as I see it is a fallacy of equivocation, and also one on the word "better".

The problem with how you've phrased it is that, as I see it, "it is better to have sent a small force" is just leaving the last term of the comparative sentence implicit -- namely, "It is better to have sent a small force than a large one.", and when phrased that way the sentential form is identical in the second and third premises.

... and that's where the equivocation comes in; you have two sententially identical forms, but with different referents. Specifically, you still have left crucial aspects of the situation implicit.

A better (more complete) formulation is :

"It is better to have sent a small force and won than (it is to have sent) a large force and won." (or "and lost," as the case may be.) The difference that clarifies the equivocation is exactly this won/lost distinction, or the difference between the full form of B and B', in my earlier notation.

Conflating these two leads directly into Simpson's paradox, otherwise sometimes called the "Berkeley admissions paradox," where two statements that are true of subgroups are not true of the groups as a whole, because the statements "P is true for subgroup 1" and "P is true for subgroup 1" and "P is true for subgroup 1 and subgroup2 combined" are not necessarily linked.

The equivocation you propose isn't necessary, nor is it in fact explanatory in context. The equivocation is much more simple; you didn't unpack and formalize the semantics enough.

Would you like me to dig up some numbers for you to play with to see how this works? (Alternatively, this site here has a nice explanation and some sample data.)

 

[csense]

1. I will either win or lose the battle.
2. If I win the battle, I would rationally have expected the better utility from sending a small force (than a larger one).
3. If I lose the battle, I would rationally have expected the better utility from sending a small force (than a larger one).
4. Therefore, the rational expected utility from sending a small force is greater than the rationally expected utility from sending a large force.(Or in other words, it is better to send a small force than a large one.)

Now, in this phrasing the second and third premises are both blatantly wrong -- the rationally expected utility from sending the large force is greater than the rationally expected utility from sending the small force, since the small force is more likely to lose. .

Why would you assume this.
Maybe the King's soldiers are better quality fighters.

 

[Eleatic Stranger]

... and that's where the equivocation comes in; you have two sententially identical forms, but with different referents. Specifically, you still have left crucial aspects of the situation implicit.

A better (more complete) formulation is :

"It is better to have sent a small force and won than (it is to have sent) a large force and won." (or "and lost," as the case may be.) The difference that clarifies the equivocation is exactly this won/lost distinction, or the difference between the full form of B and B', in my earlier notation.

But that then (1) fails to be a formulation of the actual argument at hand, and (2) ends up with you posing premises like "If I win, it is better to have sent a small force and won than it is to have sent a large force and won." That isn't any more descriptive, it's just redundant. I feel that in general it's best to leave redundancies implicit.

Conflating these two leads directly into Simpson's paradox, otherwise sometimes called the "Berkeley admissions paradox," where two statements that are true of subgroups are not true of the groups as a whole, because the statements "P is true for subgroup 1" and "P is true for subgroup 1" and "P is true for subgroup 1 and subgroup2 combined" are not necessarily linked.

The equivocation you propose isn't necessary, nor is it in fact explanatory in context. The equivocation is much more simple; you didn't unpack and formalize the semantics enough.

Would you like me to dig up some numbers for you to play with to see how this works? (Alternatively, this site here has a nice explanation and some sample data.)

I'm not sure what in the world you're talking about when you say 'formalize the semantics' (one doesn't formalize the semantics -- especially not when it comes to a sentential calculus), but unpacking in general isn't really required to a significant degree for translating arguments into sentential form. You basically just look for the operators and give the sentences letters.

And as far as applying fallacies of statistical reasoning (wrong levels of specifity in groups, etc), I don't see that that's relevantly applicable to a strict logical argument -- which is what this one is.

I think you're trying to approach this in a way which will let you set the entire thing out on a grid and rank the various combinations in terms of utility (win/lose, small force/large force -- which would probably go in something like this order: Small force/win; Large force/win'; small force/lose; large force/lose). The thing is, that's just not what the king is doing -- he's just letting the win/lose option remain steady and seeing how the small force/large force one plays out. And like I said, as long as you're thinking in terms of strict utility that's fine, if useless (but when you try to do that in terms of rationally expected utility you're suddenly reasoning ass-backwards, as the sensible way to do that would be to hold steady the option you have some direct control over.)

You can say perfectly well that the King should have reasoned in a different way, but that isn't a fallacy. To say that he's reasoning poorly you have to take the argument as it's presented -- and as such I don't think Simpson's paradox really holds here.

 

[csense]

Well, even then they'd still be more likely to win if there were a larger number of better quality fighters, right?

Depends

Are you equalizing the [battle] utility of both King's soldiers?
(And if so, what is left but a general comparison of quantification in the argument...as I suggested.)

But, that still doesn't answer the question of why it is better to send a small force if he loses.

Anybody care to take a bet that the answer to that lies in quantifying [to] lose?

 

[csense]

By the way, care to comment on my challenge and support your statement:
"in logical terminology there's no distinction between qualitative and quantitative predicates"

If you believe this is true, which you seriously do, then it should be a simple matter for you.

I'll be awaiting your response

 

[Skep]

.. and that's where the equivocation comes in; you have two sententially identical forms, but with different referents. Specifically, you still have left crucial aspects of the situation implicit.

I can't help but think that people are over analyzing this problem on way too high a level. As a logic problem:

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

The proposition "If A then B" in #2 must be entailed, that is “better to have won with a small force” must be entailed by winning the battle. Alas, without some further statement to justify why it should be better to win with a small force, since none is given, the argument fails at #2. Analysis of #3 and #4 is not necessary, nor is any other analogy or reference to a special logical conundrum.

 

[csense]

The proposition "If A then B" in #2 must be entailed, that is “better to have won with a small force” must be entailed by winning the battle. Alas, without some further statement to justify why it should be better to win with a small force, since none is given, the argument fails at #2. Analysis of #3 and #4 is not necessary, nor is any other analogy or reference to a special logical conundrum.

The order of statements 2 and 3 is irrelevent, and since most people here agree that statement 3 is correct, then reversing the order would pose somewhat of a problem for you [or at least for them]

Generally speaking though, and in keeping with the theme of your post, the argument fails at 1 if win and lose are not defined.

 

[csense]

I can't help but think that people are over analyzing this problem on way too high a level.

By the way, this I most definitely agree with.

 

[Skep]

The order of statements 2 and 3 is irrelevent, and since most people here agree that statement 3 is correct, then reversing the order would pose somewhat of a problem for you [or at least for them]

Well, yes, but I had to start somewhere. In this case #1 was a good place. Since #1 is a tautology (a statement that is inherently true, like saying “I will or I won’t” or “I’m correct unless I’m not.”), the lack of definitions for "win" or "loose" doesn't become a problem until people try and dissect #2 and argue about what "winning" or "loosing" means—which, as it turns out, isn’t even necessary since #2 breaks down on its own.

So, I'd still say that, if read in order, the problem doesn't break down until #2. Of course, you can reorder all of the arguments (though, not the conclusion) and look at it in different ways, but that wouldn't affect my opinion that #2 is the problem (not because it is second in order, but because of the wording in the specific phrase we have been calling #2).

But, it sounds like we are generally of like mind on this problem.

 

[csense]

Since #1 is a tautology (a statement that is inherently true, like saying “I will or I won’t” or “I’m correct unless I’m not.”), the lack of definitions for "win" or "loose" doesn't become a problem...

Either A or B

Tell me why this constitutes a true dichotomy without defining A or B

 

[Skep]

Either A or B

Tell me why this constitutes a true dichotomy without defining A or B

It doesn't have to be a true dichotomy. I never argued that it was. I said it was a tautology. That is "A or Not A," hence the meaning of A isn't relevant.

You could counter by saying that "I'll win or I'll loose" is not equivalent to "I'll win or I won't win" since losing could be defined differently than not winning. Not winning could incorporate not fighting at all, a draw, and loosing. However, from what I've read of the posts so far, I haven't seen anyone try to argue that.

Further, I'd say that #1 is stipulative, i.e. that we must accept the premise in order to examine the argument. In that sense I think it is fair to characterize the stipulation as "A or Not A," in which case it is a tautology where the definition of A doesn't matter since statement #1 will always be true for purposes of the argument.

 

[csense]

You could counter by saying that "I'll win or I'll loose" is not equivalent to "I'll win or I won't win" since losing could be defined differently than not winning.

Yes, you're correct, therefore A or not A is not the proper logical form for this argument since it changes that which it represents.

 

[Skep]

Yes, you're correct, therefore A or not A is not the proper logical form for this argument since it changes that which it represents.

Well, I would respond by pointing out your contention that "winning" and "loosing" are not defined, therefore you can't categorically say that "I'll win or I'll loose" is not equivalent to "I'll win or I won't" (i.e. A or Not A). So, statement “A or Not A” may, in fact, not misrepresent what proposition #1 represents.

However, more to the point, I haven't seen a proof that #1, in it's original form is invalid on its face, as you contended earlier.

You can argue that #1 needs definitions for winning and loosing, but it is not required that proposition #1 be whole and complete and contain those definitions. It is only one statement in the argument. Another statement could have been introduced to the argument define winning and loosing. Because this statement does not have to be part of #1, #1 cannot be considered invalid by itself.

Therefore, I'll still hold that the flawed proposition is the one we call #2.

 

[drkitten]

You can say perfectly well that the King should have reasoned in a different way, but that isn't a fallacy. To say that he's reasoning poorly you have to take the argument as it's presented.

Well, the King's argument is demonstrably fallacious, in that he gets from true premises to a false conclusion, which is the very definition of a fallacious argument.

In order to determine where the fallacy lies, we need to understand exactly what the meaning of the premises and conclusions are, and hence the need to study the semantics. Despite your claim that my unpacking is simply listing "redundancies," my reformulation of the argument actually shows that two strings that are textually identical ("it is better to have sent a small force than a large") are actually different in meaning. This, in turn, is the definition of the fallacy of equivocation.

I don't see how you can claim that I'm reformulating "the actual argument at hand" -- in what way does the *meaning* of the King's words vary from my more explicit version?

 

[Eleatic Stranger]

By the way, care to comment on my challenge and support your statement:
"in logical terminology there's no distinction between qualitative and quantitative predicates"

If you believe this is true, which you seriously do, then it should be a simple matter for you.

I'll be awaiting your response

First off, the reason it doesn't apply is that the argument doesn't need to be formalized in predicate calculus. Secondly, the reason there's no distinction is because predicates are predicates, and unless you want to deny bivalence or assert that qualitative statements like 'better'* are nonsensical then they darn well count as predicates to be treated in by the calculus in the exact way that all other predicates are.

(*And, as I've noted, 'better' can be simply defined in terms of 'utility', which is at the very least ideally a quantitative term. So like I said, it doesn't matter to the logic, and even if it did, you're doing it wrong.)

I can't help but think that people are over analyzing this problem on way too high a level. As a logic problem:



The proposition "If A then B" in #2 must be entailed, that is “better to have won with a small force” must be entailed by winning the battle. Alas, without some further statement to justify why it should be better to win with a small force, since none is given, the argument fails at #2. Analysis of #3 and #4 is not necessary, nor is any other analogy or reference to a special logical conundrum.

This only holds, though, if you are attempting to use a material conditional (and have some sort of good logical definition of it), which standard first order logic doesn't contain. Conditionals in logic only state that the a state of affairs in which the antecedant is true and the consequent is false are impossible. Also I've adduced several reasons why winning with a small force creates greater net utility than winning with a large force -- and no one has provided any reason to believe that this is unreasonable.

Well, the King's argument is demonstrably fallacious, in that he gets from true premises to a false conclusion, which is the very definition of a fallacious argument.

No, that's the criteria for determining if the argument is fallacious -- that the King reaches a false conclusion from true premises would mean that the King's argument was demonstrably fallacious (and showing that he does so would count as the demonstration required), but reaching a false conclusion simply shows that the argument is in some way either unsound or invalid.

In order to determine where the fallacy lies, we need to understand exactly what the meaning of the premises and conclusions are, and hence the need to study the semantics. Despite your claim that my unpacking is simply listing "redundancies," my reformulation of the argument actually shows that two strings that are textually identical ("it is better to have sent a small force than a large") are actually different in meaning. This, in turn, is the definition of the fallacy of equivocation.

It certainly is true that we need to get to the informal level to see where the fallacy sneaks in -- which is obvious from the fact that the formal structure is a distinctively valid one.

You're reformulation, however, simply moves the antecedant into the consequent of the conditional claim in premises 2 and 3. That is why it is redundant. Actually the argument proceeds onwards in a strikingly similar fashion --
Where: A:"I win the battle"; B: "I lose the battle"; C: "It is better to have sent a small force than a large one."

1. A or B
2. If A then (C and A)
3. If B then (C and B)
4. Therefore: (C and A) or (C and B)
5. Therefore: C and (A or B)
6. Therefore: C

And we end up at precisely the same problematic point -- 'it is better to have sent a small force than a large one' -- after only a few more intermediate steps.