Damnation. David B? Evidence for a miracle?
(Note: The above post(s) had been intended for the Libertarianism thread, quite obviously. Our server burped, the UPS squeeled, and now this.)
“It is lucky for rulers that men do not think.” — Adolf Hitler
I feel stupid. 
It wasn’t your fault, Phaedrus. It was just a cyber-glitch.
“It is lucky for rulers that men do not think.” — Adolf Hitler
Lib - OK, I’ll ignore the glitch (as opposed to *Glitch); I haven’t even lurked on the Libertarian thread, and I’m not going to start now.
Cabbage - that’s actually not half bad; I’ll have to remember it!
Hunsecker - I’ll bite: what do you do with a broken limit?
FWIW, any theorem can be proven false by valid counterexample, regardless of the nature of the ‘proof’. If a valid counterexample exists, then the proof is flawed; the only question is where the logical misstep lies.
Firefly:
Unbelievably dogmatic and gratuitous.
Please explain how that which is an element of a set can be offered as a counter-example to that set. A counter-example is not the discovery of a “misstep”, but the factual contradiction of a conclusion by showing that generalizations drawn about the whole set run counter to observations about one or more elements of the set.
“It is lucky for rulers that men do not think.” — Adolf Hitler
How would you not call that a disproof of a theorem? How exactly would you define a disproof of a theorem? I would agree that it doesn’t necessarily mean that there was a misstep in the proof itself; however, it does mean that there is an error either in the assumptions, proof or conclusions. Disproof by counterexample is a prime means of disproving, the other major way being by contradiction.
“Glitch … Energy Shield.” - Bob the Guardian
How would you not call that a disproof of a theorem? How exactly would you define a disproof of a theorem? I would agree that it doesn’t necessarily mean that there was a misstep in the proof itself; however, it does mean that there is an error either in the assumptions, proof or conclusions. Disproof by counterexample is a prime means of disproving, the other major way being by contradiction.
“Glitch … Energy Shield.” - Bob the Guardian
Lib, please don’t hold back; tell us how you really feel.
What the counterexample demonstrates is that what was ‘proved’ to be a subset, isn’t, since an element of the subset has been produced that is outside the larger set. If the proof is valid, there’s no valid counterexample.
Look, we’ve gotten rather heated in our exchanges. I’m going to try to tone things down on my end. What you do is up to you.
Firefly:
[breathing deeply… slowly…]
Good friend, please lend a heedful ear. Deduction, as you can easily verify without accepting my word, is the process of reasoning from the general to the particular.
Thus, given the aggregate attributes of a set, you may draw conclusions about every element of the set that shares the aggregate qualities. There can’t be any counter-examples because the whole set is simply the aggregate of all its elements.
Induction, on the other hand, lends itself to counter-examples because you are generalizing about the attributes of the whole set based on observations about some particular elements. That’s why the induction hypothesis forces you to examine not just n, but n+1 and 0.
For example, if I generalize that all Americans make more than $10,000 a year based on the facts about 1,000 out of 1,000 whom I’ve examined, you need only drag out one American making $9,000 a year to prove me wrong. (Induction)
But if I particularize that, of a set of 1,000 CEOs making $200,000 or more, that Mr. Stone (one of the set) makes at least $200,000, you cannot offer a counter-example because I am not generalizing about the whole set. I am particularizing from it. (Deduction)
“It is lucky for rulers that men do not think.” — Adolf Hitler
Glitch:
Glitch, it is a disproof — of an inductive argument. I didn’t say you can’t disprove anything with counter-examples. I said they don’t apply to deductions. They are used to disprove slothful induction all the time.
They are simply inappropriate disproofs of deductive arguments for the reasons I have explained.
“It is lucky for rulers that men do not think.” — Adolf Hitler
You take it to l’Hopital.
Which you can do by counter-example. I admit in your CEO example that would not apply because it is a hypothetical axiom which may be true and so it is safe to assume it is true. But if you take as your axiom that all CEOs make over $200,000 and then deduce that Mr. Stone, a CEO, makes over $200,000 then I can prove that wrong, by showing the axiom is false, by showing you Mr. Glitch who, a CEO, who makes a mere $80,000.
Glitch:
I see what you mean in that regard. Yes, the axioms themselves, which are derived inductively, can be counter-exampled. The toppling of the axiom would, as I said, topple the whole argument. (Although, and I won’t go into that here, there is a school of thought that asserts that all axia are true. Period.)
But what I’m talking about is the argument’s conclusion which, drawn toward a particular, is contextually antithetical to a counter-example. Note that in your example, you couldn’t conclude deductively that all CEOs make more than $200,000, but only one or more of those in your set.
To generalize about a broader set would be inductive, and would be shown false as you say, by introducing Mr. Glitch.
“It is lucky for rulers that men do not think.” — Adolf Hitler
Hold it now. Axia are not derived necessarily by any means, although they can be derived by some means if you want (statistical studies, etc). They are simply a basis by which you will apply some logic.
I have never heard of any school of thought that assumes you axia are not true. Why choose an axiom that isn’t true, or I should say that you don’t intend to assume is true? I have this mental image of my modern algerbra professor standing before class, writing down some assumptions, doing some formal logic, then after we had written everything down stating “Now, I will assume this axiom is false, which means you wrote everything down for nothing. Ha ha ha ha!!!” … actually that sounds like something he would have done. 
Now, this isn’t to say when you are done with a particular set of axia, you cannot go back and adjust the axia, including adjusting one to assume it is false or negating it. i.e. if in case one A=B, you can then have a new problem where A<>B. But this is a new problem completely distinct from the previous problem. But I cannot imagine coming up with a problem, in which, we will assume false axia.
I agree that a counter-example to a conclusion is silly. But a counter-example to the conclusion by any logical means is silly.
For the inductive case,
#1)Axiom. I know some CEOs that make over $200,000
#2)Logic. Inductively, all CEOs make over $200,000
#3)Conclusion. Therefore Mr. Stone, a CEO, makes over $200,000.
The counter-example, Mr. Glitch who makes $80,000, refutes #2 not #3. It is not a counter-example to the conclusion but a counter-example to the logic step.
Sure I can based on the axiom which is assumed true. As you already know, you can take faultly axia, apply proper logic, and reach a logically true, but real life false conclusion. The deductive logic in that case is perfect if I assume the axiom that all CEOs make more than $200,000; however, by providing a counter-example to the axiom I show the entire proof to be wrong.
Lib:
I think you are also assuming that all deductive proofs are well-formed. We all know that that is not the case. Sloppy logic abounds in human experience. It is quite possible to structure an invalid deductive argument using true axioms. That a flaw in teh proof exists can then be demonstrated by finding a counterexample.
The best lack all conviction
The worst are full of passionate intensity.
*
Spiritus:
Amen to that! Here’s one example of a deductive spoof:
All Roman emperors are dead.
Adolf Hitler is dead.
Therefore, Adolf Hitler was a Roman emperor.
That was an error called undistributed middle. (Roman emperors aren’t the only things that die.)
Glitch:
Actually, your Mr. Stone counter-example did disprove your conclusion that Mr. Stone makes 200 grand.
“It is lucky for rulers that men do not think.” — Adolf Hitler
By the way, statistical arguments are classic inductive arguments. Keep in mind that strict mathematical induction by Peano’s axiom and De Morgan’s methoc, as Cabbage said, is something quite different from general induction.
Regarding axioms, they are true by definition even if they are wrong in the sense of being for the sake of argument. That’s why if A is a false proposition and B is a true proposition, then A implies B is a true implication.
“It is lucky for rulers that men do not think.” — Adolf Hitler
It did show that the conclusion was logically false (of course, it still could be real life true but who cares about real life in logic right :)), but it did so by showing the entire proof to be logically false and it did so by showing that the second step (the logic step) was false.
Deductive:
#1) Axiom. All CEOs make more than $200,000.
#2) Deductive Logic. Any CEO must make more than $200,000.
#3) Conclusion. Mr Stone, a CEO, makes more than $200,000.
Counter-example to the axiom, Mr. Glitch makes $80,000.
Inductive:
#1) Axiom. Some CEOs make more than $200,000.
#2) Induction Logic. All CEOs make more than $200,000.
#3) Deductive Logic. Any CEO must make more than $200,000.
#4) Conclusion. Mr. Stone, a CEO, makes more than $200,000.
Counter-example to inductive logic: Mr. Glitch makes $80,000.
In both cases the conclusion is logically false, even if it is really true (again, that nasty real life) but the key to my point is what the counter-example is a counter-example to. In the deductive it is a counter to the axiom, in the inductive it is a counter to the inductive logic.
Conclusions are usually false because somebody draws a conclusion that doesn’t follow from the logic either on purpose or by accident.
#1) Axiom. All CEOs make more than $200,000.
#2) Logic. Any CEO must make more than $200,000.
#3) Conclusion, Glitch, a martial arts instructor, makes more than $200,000 (I wish).
P.s. - Really there should be a second axiom in both cases above.
#2) Axiom. Mr. Stone is a CEO.
Breathing deeply myself:
Boy, you make deductive proofs sound so trivial. If only.
As Spiritus mentioned, not all deductive proofs are well-formed. And because not everything that people prove deductively is easy to prove, people can make mistakes that aren’t particularly visible.
I know all deductive proofs can technically be done with Venn diagrams, but if it were a manageable way to do most proofs, then people would do them that way, and then everything would be as obvious as you imply. But there’s a big disconnect between the imagery of proof that you use here, and its practice.
Libertarian
I get the impression that you now have accepted the possibility of counter-examples to deductions, so I won’t belabor the point.
I just want to know, what the hell is axia? Oops. Another spoiled priest.
.