A Logical Problem

[Skep]

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.

You may aduce to your hearts content, but it is the job of the argument to justify why a smaller force would "be better."

 

[drkitten]

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.

Hmm. Good point. Will need to re-think slightly.

 

[csense]

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.)

Well, that certainly clears things up.

First off, the reason it doesn't apply is that the argument doesn't need to be formalized in predicate calculus.

I'm not quite sure what you're trying to say here, but regardless, First Order Predicate Calculus does make a distinction between the quantitative and the qualitative insofar as it is restricted to quantification of individuals.


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.

I could be wrong, but my understanding of Bivalence is directly related to propositions and not predicates per se, even though such predicates are contained within propositions.

I'm not quite sure how anyone can say that better absent context of a proposition must be either true of false.


*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.

Well, we're in agreement that better in context is quantitatively defined, but I'm a little confused that your use of utility in context is quantitatively defined. Explain to me how battle usefulness is a quantity rather than a quality, which to me is what is being implied.


Returning to my original point though and in reference again to your statement: in logical terminology there's no distinction between qualitative and quantitative predicates, which may be true of logical forms per se without reference to subject matter...Then if [logical] symbolic notation makes no distinction between quality and quantity, how can it express that which it does not represent.

 

[csense]

[csense]
"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]
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.

Regardless of definitions, which may be similar as a matter of coincidence, they are categorically different since it literally changes the terms of the argument.

Black or white is not the same as black or not black

Run or walk is not the same as Run or not run

Fight or flight is not the same as Fight or not fight

And

Win or lose is not the same as Win or not win

Although the first propositions may specify a contingent, although not necessarily true, dichotomy...the second propositions do not, and if one is going to interpret and use the second propostions using the contingency of the first propositions, then would it not make sense to just use the first propositions.

 

[csense]

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.

I don't recall ever maintaining this. What I did argue, and to paraphrase, is that the argument fails at 1 if the terms of the argument are not defined. How can you understand the propositions, especially the ones to follow in statements 2 and 3, if you don't understand the terms. This only seems logical and elementary to me.

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.

Well, this is just simply ridiculous.

Unless the King (who is narrating the argument) spontaneously breaks into the third person and issues a dictionary definition for win, then your argument would introduce a virtually endless correlation of additional statements...which would also apply to any other term within the argument

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

Then we're in agreement on this

 

[csense]

Re whether the forms A or B and A or not A are equivalent.


If A or B equals A or not A, then A or B also equals B or not B

Therefore, A must equal either B or not B, and since A can not equal B, then A must equal not B.
We can now infer that B equals not A

Therefore, A or B equals not A or not B, which is a logical contradiction since the terms in the second proposition can not be defined.

 

[drkitten]

Re whether the forms A or B and A or not A are equivalent.


If A or B equals A or not A, then A or B also equals B or not B

Therefore, A must equal either B or not B, and since A can not equal B, then A must equal not B.
We can now infer that B equals not A

Therefore, A or B equals not A or not B, which is a logical contradiction since the terms in the second proposition can not be defined.

This is gibberish. (As Pauli put it : "it's not right. It's not even wrong.")

 

[csense]

This is gibberish. (As Pauli put it : "it's not right. It's not even wrong.")

...Which of course justifies the need for any explanation in a critical thinking forum.

 

[csense]

This is gibberish. (As Pauli put it : "it's not right. It's not even wrong.")

You know what, thanks for posting this since it just confirms that I'm wasting my time here.

I've better things to do.

 

[Atlas]

Therefore, A or B equals not A or not B...

Here is what I don't believe... A or B = NOT(A or B)

Is NOT(A) or NOT(B) = NOT(A or B) ??

I don't know... I'm asking.

 

[drkitten]

Here is what I don't believe... A or B = NOT(A or B)

Is NOT(A) or NOT(B) = NOT(A or B) ??

I don't know... I'm asking.

FIrst, "A or B = NOT (A or B)" is a contraditory statement. Simply let C = "A or B", and we see that this statement is equivalent to C = NOT C, which is the definition of a contradiction.

Second,

"NOT A or NOT B" is not the same as "NOT (A or B)"; a very famous theorem by DeMorgan shows that

NOT A or NOT B = NOT (A and B)

You can see it from a truth table, or just by looking at few special cases. "NOT A or NOT B" implies that we're cool with A alone (just not with B), or with B alone (just not with A) --- hence we don't want A and B together -- hence "NOT (A and B)"

 

[drkitten]

Therefore, A or B equals not A or not B, which is a logical contradiction

First,

A or B = not A or not B

is not a logical contradiction.

Let A be True and B be False.

A or B = True or False = True

not A or not B = not True or not False = False or True = True

Therefore, there is a model of {A,B} such that A or B = not A or not B, and the original statement is not contradictory.

So this much of your statement achieves the lofty heights of being wrong.

since the terms in the second proposition can not be defined.

This part of your statement is simply meaningless, and does not even rise to the dignity of error.

The overall argument is almost as badly flawed : "If A or B equals A or not A, then A or B also equals B or not B. Therefore, A must equal either B or not B" is simply false. Again, I will demonstrate a model. Let A be True and B be any condition of unknown truth value. Obviously, A or B is True, as is A or not A. However, the truth of B is not related to the truth of A.

There are therefore two interpretations of the staetment "A must equal either B or not B." First, in a strictly propositional logic where all variables have a defined truth value, this statement is both true and tautological. Whatever value B has, either B or not B must be True (and the other False); whatever value A has, it must be True or False, and thus equivalent to one of B or not B. But this is a tautology and has nothing to do with the argument.

On the other hand, in a less strict logic where the truth of B is conditional and depends on external circumstances, for example, if B were equivalent to the proposition "I am currently on the phone," then A is not equivalent to either the proposition B, or the proposition not B.

Again: "This isn't right. This isn't even wrong."

 

[Eleatic Stranger]

Well, that certainly clears things up.

First off, the reason it doesn't apply is that the argument doesn't need to be formalized in predicate calculus.

I'm not quite sure what you're trying to say here, but regardless, First Order Predicate Calculus does make a distinction between the quantitative and the qualitative insofar as it is restricted to quantification of individuals.

Nope. You don't seem to understand what 'quantification' means -- it's not the same as quantitative.

Look, if we say something is quantitative as opposed to qualitative we're saying that it is capable of being objectively measured (in terms of numbers--quantitative like quantity), as opposed to not capable of being measured easily (in terms of quality). When we talk of quantification what we're talking about, literally, is "To limit the variables of (a proposition) by prefixing an operator such as all or some."(dictionary.com)

And that has nothing to do with whether the predicates involved are quantitative or qualitative --

If, for instance, the predicate is "nice" (N) -- which we can assume is fairly qualitative -- it's perfectly possible to deal with it in a quantified context. For instance:

(x)(Nx)

Means "Everything is nice"

Just like, if we use a quantitative predicate "Longer than 5 Centimeters" we can quantify over that too --

(x)(Lx)

Meaning: "Everything is over 5 centimeters long."

I think you should probably learn how logic works before you start arguing about it.

Well, we're in agreement that better in context is quantitatively defined, but I'm a little confused that your use of utility in context is quantitatively defined. Explain to me how battle usefulness is a quantity rather than a quality, which to me is what is being implied.

You might also want to look up the difference between 'utility' in terms of usefulness, and 'utility' as a technical term in philosophy, which is rather clearly (I would hope) how I was using it here.

And (A or B), where A and B are taken to be of opposite truth value, does entail that - necessarily - (Not A or Not B).

It's not a contradiction, it's a logical truth.

You know what, thanks for posting this since it just confirms that I'm wasting my time here.

I've better things to do.

I suggest that one of those things might be gaining a working grasp of how logic works.

 

[Selvedge]

Let's suppose that I like to smoke in bed.

If my house burns down tonight, with me in it, I want to have spent my last night enjoying the pleasure of a nice smoke under the covers.

If not, then tomorrow I want to have a pleasant memory of an evening's in-bed smoke to look back on.

So, either way, it is better for me to smoke in bed tonight. What is wrong with this cost/benefit analysis?

Nothing! In terms of my having a pleasant evening, it is better that I smoke in bed tonight, regardless of what happens overnight.

But wait a minute -- what if I have a goal other than experiencing pleasure this evening? As a matter of fact, I do -- I want to stay alive until tomorrow. And the benefit of staying alive trumps the benefit of pleasure for me. It is better to have an unburned house and self -- with or without a cigarette, than it is to die in a house fire -- with or without a cigarette.

So, the cost benefit analysis above is simply inadequate. I need to factor in the benefit of dying vs. not dying. And once I do that, I have to consider that not all outcomes are equally likely. If I smoke, I am more likely to die in a house fire than if I do not.

Back to His Majesty:

H.M.'s analysis is fine -- in terms of minimizing the use of troops. And if we accept that win or lose, it is better (for one reason or another) to have used fewer troops, then there is nothing wrong with H.M's reasoning. His conclusion is "right" -- as far as it goes.

But common sense tells us that H.M. is leaving something out. Most people who fight wars want to win them. And the benefit of winning commonly trumps the benefit of minimizing the use of troops. If H.M. takes winning/not winning into account, then he also has to take into account the probability of winning vs. losing with a smaller force vs. a larger force, introducing a whole new set of premises and changing the equation altogether.

We perceive H.M.'s conclusion as "wrong" because we know intuitively that there are factors missing, but perhaps it would be better to simply say that his cost benefit analysis is incomplete.

 

[FireGarden]

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

Suppose I modify that and instead of C = "better to have sent a small force" we have
C="I sent a small force"
D="I did the best thing I could have done"

1. A or B
2. If A then (C and (D or notD) and A)
3. If B then (C and (D or notD) and B)
4. Therefore: (C and A) or (C and B)
5. Therefore: C and (A or B)
6. Therefore: C = "I sent a small force"

He can also conclude (D or notD).
He doesn't seem to even consider sending a large force.
A more sensible argument seems to be.

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

So he concludes (D or notD).
ie: he doesn't know wether he took the best course of action. Even after he knows the result of the battle, he does not know wether he took the best course of action. That seems to make sense.

Even if he wins with 10 men, he can still say "But maybe I could have won with 9, and that would have been better."

 

[FireGarden]

Supposing the king sends in 10 men

If he wins, then according to his value system:
send 10 and win is better than send 11 and win

But is send 10 and win better than send 11 and lose ?
You'd think so, but we haven't been told this is the case

If he loses, then according to his value system
send 10 and lose is better than send 11 and lose

But is send 10 and lose better than send 11 and win ?
You'd think not, but we haven't been told what the king would prefer. (And sometimes the cost of winning can be too high)

Is it because we can't answer the above questions that we're all tied up trying to figure out if the king is being reasonable?

 

[FireGarden]

My first argument above is wrong. I included (D or notD) instead of just D, for some reason. That means it's no longer equivalent to the king's original argument.

C="I sent a small force"
D="I did the best thing I could have done"

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

Where he doesn't consider sending a large force.

Now the argument is euivalent since
"I sent a small force" and "I did the best thing I could have done" implies "It's better to have sent a small force." (Win or lose)

while
"It's better to have sent a small force." and "I do what's best" implies "I sent a small force" and "I did the best thing I could have done."

"I do what's best" is a necessary assumption to get equality. Not too big a leap.

 

[Hypnotix]

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?)

Well, here goes.

Since I was the Duke of York in a previous life, I would first like to ask: are we sure these assumptions were made by a king? A king in charge of an army in the midst of battle? If so what type of king is this - a fairy king (or maybe queen)?

What king would send his men (whether a small or large group) down a hill and then up a hill again? Once at the top of the hill, where the enemy are lurking, ready to pounce, the poor soldiers (whether in a small or large batallion) would be so worn out they wouldn't even have the energy to see the soldiers never mind attack them and win the battle.

Your king doesn't want to lose his men - that's his problem - he doesn't want to fight, he's not interested in winning, hence the apathy. But then again, maybe you're acutually referring to:

Why - the Grand Old Duke of York,
He had ten thousand men,
He marched up to the top of the hill,
and then marched them down again

(all together now..!!)

The grand old Duke of York he had ten thousand men
He marched them up to the top of the hill
And he marched them down again.
When they were up, they were up
And when they were down, they were down
And when they were only halfway up
They were neither up nor down.

Or maybe, just maybe, you're referring to, in fact you must be referring to the King of France (note the word KING):

The King of France went up the hill
With twenty thousand men;
The King of France came down the hill,
And ne’er went up again.

It's as simple as "A.B.C"

 

[kk2796]

Holy !@#$, people. This is incredibly simple, and was solved in the ver first reply, with one word: EQUIVOCATION.

"Better", in the premises, is the equivalent to "minimize casualties". If no equivocation is used, the King's argument is perfectly valid.


1. I will either win or lose this battle.
2. If I win this battle, then I will have minimized my casualties if I won with a small force.
3. If I lose this battle, then I will have minimized my casualities with a small force.
4. Therefore, whether or not I win or lose, I will have minimized my casualties if I send a small force into battle.


DING DING! Correct - the conclusion does follow from the premise!


Now, let's see the argument WITH equivocation, using "minimize casualties" for "better" in premise 2 and 3, and substitue "improve chances for victory" for "better" in the conclusion.

1. I will either win or lose this battle.
2. If I win this battle, then I will have minimized my casualties if I won with a small force.
3. If I lose this battle, then I will have minimized my casualities with a small force.
4. Therefore, whether or not I win or lose, I will have have a better chance of victory if I send a small force into battle.

BZZZZZT! Incorrect - the conclusion does not follow from the premises.

 

[Atlas]

Holy !@#$, people. This is incredibly simple, and was solved in the ver first reply, with one word: EQUIVOCATION.

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.