Logic Training

[NeoGoomba]

Lately, I've noticed that when it comes to thinking "logically", I just don't hack it. I'm a capable individual, and I feel that I can be insightful at times, but from a pure logic standpoint, I'm greatly lacking. Often in debates I find myself feeling like Winston Smith, where I FEEL like I'm right, but I can get slipped up in my own arguments by someone who's a little more clear-headed than myself. I took Intro to Logic to try and see if I could pull it off, but both my instructor was a jerk, and in all honesty I didn't really give it my best try, so that didn't help me accomplish much. Then I took that Battlefield God test and crashed and burned. Whee!

Anyhoo, I was wondering if theres any sort of exercises or problem sets out there to help to tune up my brain, or any sort of beginners logical proofs to learn from, or shall I just leave debates up to you knowledgeable folk?

[Zero]

I find myself this way too. I often "know" I'm right, but I can't explain things logically. I'm also bad at spotting fallacies.

[Spoonist]

Learning the scientific method should help.

That way you can apply how to 'prove' something scientifically on your argument in a discussion.


But really to be able to debate you don't need logic, you need rhetorics to please the crowds.
(To the infuration of the logical who don't understand why none listens to them =)

[Zero]

Learning the scientific method should help.

That way you can apply how to 'prove' something scientifically on your argument in a discussion.


But really to be able to debate you don't need logic, you need rhetorics to please the crowds.
(To the infuration of the logical who don't understand why none listens to them =)

I think the trouble for most debaters that feel they're right, but can't adequately prove it, the trouble is that the rhetorical strategies, emotional ploys, all the crap that someone will try to pull in a public debate... all that doesn't change the validity of an argument, but does make it seem more convincing.

I know that I waste a decent portion of my time trying to make it sound convincing rather than constructing a rational argument.

[Surlethe]

I took a math class which included an introduction to propositional and predicate calculus; they really helped my debating immensely, I think. I find it helpful to read an opponent's post aloud to myself, and then take a moment and ask, "What is he really saying?" Try to distill his point down into its claim, and then identify the sufficient conditions for the claim which he has presented. You can also apply this to yourself by thinking in terms of propositions when you're constructing an argument: e.g., "I claim P, and because of this piece of evidence, Q→P; since Q is true, P is true as well." Really, though, when constructing an argument, I'd start by making sure you know what you're saying; it's no use writing a book if there's no story.

[sketerpot]

Anyhoo, I was wondering if theres any sort of exercises or problem sets out there to help to tune up my brain, or any sort of beginners logical proofs to learn from, or shall I just leave debates up to you knowledgeable folk?

Whether or not you will be doind any debating, it's very important to think logically. Your goal of improving your logical thinking skills is both noble and practical.

I don't know of any repository of exercises, but I can make you one.


Exercise 1: What does this block of text say?

This is an extremely controversial part of a speech given by former Harvard president Larry Summers. There was a big media outcry after he said this:

So my best guess, to provoke you, of what's behind [this dearth of women in the sciences] is that the largest phenomenon, by far, is the general clash between people's legitimate family desires and employers' current desire for high power and high intensity, that in the special case of science and engineering, there are issues of intrinsic aptitude, and particularly of the variability of aptitude, and that those considerations are reinforced by what are in fact lesser factors involving socialization and continuing discrimination. I would like nothing better than to be proved wrong, because I would like nothing better than for these problems to be addressable simply by everybody understanding what they are, and working very hard to address them.

Now, the exercise: look over that quote and summarize precisely what Summers said. If you can do this without horribly skewing it, you're doing better than ~90% of people quoted in the news about this scandal.

Extra credit: does the existance of highly competent female scientists and engineers contradict Larry Summers' hypothesis? Why or why not?


I numbered this "exercise 1" in hopes that other people would contribute some. It'll be interesting to see what people come up with.

[Surlethe]

Exercise 2: Mathematical Proof

If you want pure logic, you're not going to do much better than studying mathematics. Thus,

Prove that, for any whole numbers a and b, a² can never equal 2b².

This is one of those proofs which does not require technical knowledge, but is just enlightening to know.

HINT: This proof is equivalent to saying "√2 can never be written as a fraction." Look at the implication of the proof, and look up the contrapositive.

[SVPD]

I don't know of any repository of exercises, but I can make you one.


Exercise 1: What does this block of text say?

This is an extremely controversial part of a speech given by former Harvard president Larry Summers. There was a big media outcry after he said this:

So my best guess, to provoke you, of what's behind [this dearth of women in the sciences] is that the largest phenomenon, by far, is the general clash between people's legitimate family desires and employers' current desire for high power and high intensity, that in the special case of science and engineering, there are issues of intrinsic aptitude, and particularly of the variability of aptitude, and that those considerations are reinforced by what are in fact lesser factors involving socialization and continuing discrimination. I would like nothing better than to be proved wrong, because I would like nothing better than for these problems to be addressable simply by everybody understanding what they are, and working very hard to address them.

Now, the exercise: look over that quote and summarize precisely what Summers said. If you can do this without horribly skewing it, you're doing better than ~90% of people quoted in the news about this scandal.

I'll give this the college try, I can use the practice myself.

1. There is a conflict inherent in the amount of time it takes to have a family, and the amount of time it takes to conduct top-level research.
2. Aptitude may vary between the sexes for science and engineering fields to some degree.
3. Discrimination/socialization issues are real, but are secondary in importance (and have the effect of reinforcing) to issues of aptitude, family, and employer needs.
4. Summers would like to be proven wrong, because he thinks that would make the issue of the dearth of females in the sciences easier to resolve.

Extra credit: does the existance of highly competent female scientists and engineers contradict Larry Summers' hypothesis? Why or why not?

No.

If we were to assume for argument's sake, that Summers is correct, and females have a lesser aptitude for science than males on average, that does not preclude individual females from excelling in the sciences any more than it precludes men from being incompetant at science. Averages say nothing about individual data points.

[Zadius]

HINT: This proof is equivalent to saying "√2 can never be written as a fraction." Look at the implication of the proof, and look up the contrapositive.

It's not necessarily equivalent, because I could use the irrationality of √2 in the proof: Let a² = 2b², then a²/b² = 2 and a/b = √2. We know that a/b is rational given that a and b are whole numbers, and we know that √2 is irrational thus we have a contradiction.

[brianeyci]

Suppose a^2 = 2b^2, a and b being whole numbers. Then 2 = (a^2/b^2). Then root(2) = a / b. However one times one is one and two times two is four. Therefore there is a contradiction and root of two cannot be expressed as a fraction. QED.

Brian

[Surlethe]

HINT: This proof is equivalent to saying "√2 can never be written as a fraction." Look at the implication of the proof, and look up the contrapositive.

It's not necessarily equivalent, because I could use the irrationality of √2 in the proof: Let a² = 2b², then a²/b² = 2 and a/b = √2. We know that a/b is rational given that a and b are whole numbers, and we know that √2 is irrational thus we have a contradiction.

That's circular, because in (at least the standard) proof of the irrationality of √2, you simply reduce it to the given problem, and then proceed to demonstrate that the given problem is true. Unless there's another way to prove it I haven't seen?

[Zadius]

HINT: This proof is equivalent to saying "√2 can never be written as a fraction." Look at the implication of the proof, and look up the contrapositive.

It's not necessarily equivalent, because I could use the irrationality of √2 in the proof: Let a² = 2b², then a²/b² = 2 and a/b = √2. We know that a/b is rational given that a and b are whole numbers, and we know that √2 is irrational thus we have a contradiction.

That's circular, because in (at least the standard) proof of the irrationality of √2, you simply reduce it to the given problem, and then proceed to demonstrate that the given problem is true. Unless there's another way to prove it I haven't seen?

Yes, there are other ways to prove the irrationality of √2.

[Zadius]

It's not necessarily equivalent, because I could use the irrationality of √2 in the proof: Let a² = 2b², then a²/b² = 2 and a/b = √2. We know that a/b is rational given that a and b are whole numbers, and we know that √2 is irrational thus we have a contradiction.

That's circular, because in (at least the standard) proof of the irrationality of √2, you simply reduce it to the given problem, and then proceed to demonstrate that the given problem is true. Unless there's another way to prove it I haven't seen?

Yes, there are other ways to prove the irrationality of √2.

Bunch of proofs.

[Kuroneko]

That's just bad form. The question is can you do it. That said, here's the same exercise in different terms.

Def.: A continued fraction [a0;a1;a2;...] is the limit of the sequence A_n = a0+1/(a2+1/(a3+1/(...a_n))).
Given: a number is rational iff its continued fraction is eventually zero--i.e., [a0;a1;...;an;0;0;0,...] (one can view this as finite).
Prove: √2 is irrational.

[TheBlackCat]

Anyhoo, I was wondering if theres any sort of exercises or problem sets out there to help to tune up my brain, or any sort of beginners logical proofs to learn from, or shall I just leave debates up to you knowledgeable folk?

In my opinion, a good place to start would be GRE study guides. The GRE analytical writing section is pretty much exactly what you are looking for: you are supposed to analyze someone else's argument for flaws and to figure out whether it is valid, as well as defend your own position on an issue without making similar mistakes. It might be good practice and should be easy enough to find in any bookstore. The one I have seen also have a decent coverage of some of the most common fallacies. I should point out that ALL the arguments they present in "Analyze an Argument" and all the issues they want you to defend can be found on the ETS website for free (look under the GRE section), so don't buy a book just to get those. GRE books would be more for answers that you can use to compare to your own to quiz yourself, but if you are just looking for the issues and arguments the ETS website is the place to go.

Another source I recommend is Being Logical by D. Q. McInerny. It is a good, concise, and well-written basic overview of rules of logical, logical arguments, as well as many of the more common logical fallacies. It is more in-depth than GRE study guides (although nothing that would be news to anyone with a solid grounding in logic), I would probably use the two hand-in-hand (Being Logical for learning the material and the GRE guides for practicing it and quizzing yourself).

The Skeptic's Dictionary and Skeptic Wiki.org both have very good lists of logical fallacies and how they are applied in real arguments. There is also a Skeptic's Dictionary book, but it is identical to the website (actually, it has less articles) so think twice before paying money for it.

A more advanced book, but one that covers a few of the more damaging and easily overlooked fallacies in much more detail, is How We Know What Isn't So by Thomas Gilovich. This is more a list of real-life case studies regarding a few very specific but very prevasive fallacies, so it should not be relied upon as a book to teach logic. However, it does show you have fallacies are applied in situations where their fallicious nature is not readily apparent.

[Surlethe]

That's just bad form. The question is can you do it. That said, here's the same exercise in different terms.

Def.: A continued fraction [a0;a1;a2;...] is the limit of the sequence A_n = a0+1/(a2+1/(a3+1/(...a_n))).
Given: a number is rational iff its continued fraction is eventually zero--i.e., [a0;a1;...;an;0;0;0,...] (one can view this as finite).
Prove: √2 is irrational.

If there were any doubt of the irrationality of √2, then the problem would have not been posed as a statement; but the problem was posed as a statement, and thus √2 is irrational.

More seriously, can we prove this with induction?

[Kuroneko]

If there were any doubt of the irrationality of √2, then the problem would have not been posed as a statement; but the problem was posed as a statement, and thus √2 is irrational.

But the logical conclusion would only be that there is no doubt about it, not whether or not it is actually the case.

More seriously, can we prove this with induction?

If you want, although the situation is simple enough to not require it.

[mr friendly guy]

Exercise 2: Mathematical Proof

If you want pure logic, you're not going to do much better than studying mathematics. Thus,

Prove that, for any whole numbers a and b, a² can never equal 2b².

This is one of those proofs which does not require technical knowledge, but is just enlightening to know.

HINT: This proof is equivalent to saying "√2 can never be written as a fraction." Look at the implication of the proof, and look up the contrapositive.

If a² = 2b², then b = a/√2
Since a and b are both whole numbers , b must be a whole number multiple of √2.
Since √2 is an irrational number (ie it cannot be expressed as a fraction), no whole number multiplier of it can lead to another whole number.
Thus a contradiction exists.

[mr friendly guy]

Lately, I've noticed that when it comes to thinking "logically", I just don't hack it. I'm a capable individual, and I feel that I can be insightful at times, but from a pure logic standpoint, I'm greatly lacking. Often in debates I find myself feeling like Winston Smith, where I FEEL like I'm right, but I can get slipped up in my own arguments by someone who's a little more clear-headed than myself. I took Intro to Logic to try and see if I could pull it off, but both my instructor was a jerk, and in all honesty I didn't really give it my best try, so that didn't help me accomplish much. Then I took that Battlefield God test and crashed and burned. Whee!

Anyhoo, I was wondering if theres any sort of exercises or problem sets out there to help to tune up my brain, or any sort of beginners logical proofs to learn from, or shall I just leave debates up to you knowledgeable folk?

On another note, I have found Darth Wong's site on logical fallacies is a good starter. Also there are numerous other sites out there.

Also when Darth Wong places some hate mail on the board and asks us to debunk it, it might be good practice. Most of those people as well as making several factual errors also make many logical errors as well.

[B5B7]

I am currently reading a book called 'The Art and Craft of Problem Solving' by Paul Zeitz [JOHN WILEY & SONS, INC].
It has some excellent suggestions on the topic of thinking logically, and provides some strategies and many exercises.
Whilst much of it is concerned with solving mathematical problems it also has a number of logic puzzles.
[Incidentally, it has the proof about square root of 2 being irrational in it also].

Now many of you are thinking - what has this to do with debates that use word based logic? The answer is technique - it provides approaches one can use - a bit like lateral thinking.
Possibly another thing you can do is find examples of famous debates eg some of the evolution-creationist debates.
It is also a matter of self belief, and of practise.

And yes reading skeptics magazines is also agood idea - that reminds me that I have a book called 'Why People Believe Weird Things' by Michael Shermer [MJF BOOKS] that I can also recommend.

[brianeyci]

Suppose root(2) is rational. Then its continued fraction [a0,a1;a2;...;aN] is eventually zero. Therefore one of the terms of the sequence must be zero--take this to be aN. Since the limit converges to zero, by the pinching theorem [a0;a1;a2;...;a(N-1)] <= [a0;a1;a2;...;aN] <= [a0;a1;a2;...;aN;a(N+1)] such that a(N+1) not equal zero. If aN is equal to zero, then there is an obvious contradiction since aN = 0 => [a0;a1;a2;...;aN;a(N+1)] = [a0;a1;a2;...;a(N-1);a(N+1)] => [a0;a1;a2;...;a(N-1);a(N+1)] > [a0;a1;a2;...;a(N-1)]. Therefore root(2) is not rational and root(2) is irrational.

It is probably wrong, but I just woke up and for now that's my best lol. I think the idea is right.

Brian

[NeoGoomba]

Suppose root(2) is rational. Then its continued fraction [a0,a1;a2;...;aN] is eventually zero. Therefore one of the terms of the sequence must be zero--take this to be aN. Since the limit converges to zero, by the pinching theorem [a0;a1;a2;...;a(N-1)] <= [a0;a1;a2;...;aN] <= [a0;a1;a2;...;aN;a(N+1)] such that a(N+1) not equal zero. If aN is equal to zero, then there is an obvious contradiction since aN = 0 => [a0;a1;a2;...;aN;a(N+1)] = [a0;a1;a2;...;a(N-1);a(N+1)] => [a0;a1;a2;...;a(N-1);a(N+1)] > [a0;a1;a2;...;a(N-1)]. Therefore root(2) is not rational and root(2) is irrational.

It is probably wrong, but I just woke up and for now that's my best lol. I think the idea is right.

Brian

*Head explodes*

Thanks for the help Brian

[Surlethe]

If aN is equal to zero, then there is an obvious contradiction since aN = 0 => [a0;a1;a2;...;aN;a(N+1)] = [a0;a1;a2;...;a(N-1);a(N+1)] => [a0;a1;a2;...;a(N-1);a(N+1)] > [a0;a1;a2;...;a(N-1)]. Therefore root(2) is not rational and root(2) is irrational.

I don't see the contradiction; aN = 0 --> [a0;a1;a2;...;aN;a(N+1)] = [a0;a1;a2;...;a(N-1);0;a(N+1)] = [a0;a1;a2;...;a(N-1)+a(N+1)] is still less than [a0;a1;a2;...;a(N-1)], I think.

[Molyneux]

Exercise 2: Mathematical Proof

If you want pure logic, you're not going to do much better than studying mathematics. Thus,

Prove that, for any whole numbers a and b, a² can never equal 2b².

This is one of those proofs which does not require technical knowledge, but is just enlightening to know.

HINT: This proof is equivalent to saying "√2 can never be written as a fraction." Look at the implication of the proof, and look up the contrapositive.

I assume that a!=b?
Because if a=0, and b=0, then a²=2b²

[Kuroneko]

Suppose root(2) is rational. Then its continued fraction [a0,a1;a2;...;aN] is eventually zero. Therefore one of the terms of the sequence must be zero--take this to be aN.

That's true, but the condition is stronger--the sequence becomes zero for all n≥N for some N. As noted above, the whole sequence is eventually zero in the manner [a0;a1;...;an;0;0;0,...]. One can view this as the continued fraction terminating, in much the same manner as 5.34 = 5.34000.... is the limit of the sequence 5, 5+3/10, 5+3/10+4/10², 5+3/10+4/10²+0/10³, 5+3/10+4/10²+0/10³, ..., where all the terms after a certain point are the same. Only now, the terms over which the limit is taken are constructed differently.

Since the limit converges to zero, ...

It doesn't; the limit involves terms that are the same after a certain point. But the limit isn't what's important to the problem itself. It just tells you how to interpret infinite continued fractions.