Sorry for the late reply... I've been quite busy lately.
As I said, I have difficulty dealing with long written discussions, but I'll reply with some points.
I agree that teaching "logic" when talking about work problems, functions, calculus and geometry is problematic. I still think that speaking about logic outside of mathematics is strange and detached (when would you _really_ encounter something like "All soldiers wear uniforms. All policemen wear uniforms. Are all policemen soldiers?"
We have started a workshop series leading to Logic at the school, and we're using number theory as our basis of discussion. Right now we're trying to formulate and prove that odd + odd = even. We'll see how this goes, and I think it's going in a good path.
And, I don't think that fun is the characteristic measure of proper studies. If that were the case, why not just replace class with recess every lesson?
I think fun is indirectly the key to "proper" studies because it is the key to keeping the audience interested. Of course, I am imagining the students that I have known. I don't know what the atmosphere is like at your workshops - do you crack the whip over the participants? Or are they all grim, self-motivated and determined people? I had visualized it as a diverse collection of people who just happen to wander in. It isn't a mathematical version of "The Suzuki Method", I hope.
Logic is probably useful in legalistic matters. It is useful in the study of philosophy where it is important to understand the difference between "logical" and "true". The average person doesn't encounter humorous word problems in daily life, but you don't encounter "If Bob is seven years older than Sally..." or "If A is divisible by 3 and ..." etc. in daily life either. Speaking about logic outside of mathematics only seems strange and detached to mathematicians! Think about how strange and detached mathematics seems to people who have ordinary interests. The point about "separation of concerns" is to keep it clear to the students that logic is not something that only appears when one studies calculus or set theory or whatever.
(You shouldn't feel obligated to reply according to any schedule - or at all, if you aren't inspired.)
Obviously study needs to be fun. I was saying that it can't be a measurement for the success of a class. The people who come in to our school are indeed a diverse collection of people who happen to come in. They are all interested in some way in mathematics, or they wouldn't spend their time here! The atmosphere is very informal, and people find themselves interested in the questions posed because they are asked in a way that makes them actually interesting. I think people enjoy themselves as well, but our workshops involve deep thought and it's definitely not an ongoing "game". (I guess that's not exactly what I meant, but I'm not sure how I could really explain it...)
In response to tashirosgt: Why study logic? Can any one be truly literate or numerate if they do not understand the principles of logical argument -- principles like proof by contradiction or contrapositive? It is how we use language to derive new knowledge and new questions from what we already know.
As for logic being somehow "outside" of mathematics, nothing could be further from the truth. IMHO, every aspiring mathematician, scientist, engineer or math teacher should have some appreciation, if not a detailed understanding, of the axiomatic method that underlies all of mathematics. They should appreciate that you can start with a small set of self-evident "truths" (axioms) and, by using only the principles of logic, derive all of the rest of mathematics. In principle, anyway.
Does this format require that your discussions are more or less self contained? Or does a a discussion from one week refer to results from a previous week?
We currently have two "forms" of meetings - our workshop series and our cooperative reading. Our workshop series usually last three meetings, which span one long discussion. Each workshop series begins with no assumed background (although we may begin with some workshop series that do assume prior knowledge, if it is something we have discussed in a prior workshop and all participants were at that workshop).
In response to Dan, I hope your logic software doesn't teach that there "self evident" truths.
I think the fundamental contribution that studying logic makes to the liberal arts is that it makes it clearn that the affairs of economics, politics and life in general are beyond the application of logic, simply because there are never enough enough axioms around to allow any spectacular deductions. Studying logic deflates one's expectations that there is some perfectly "logical" way to deal with the local traffic problem or the liberal arts in general. It also helps us deflate the arguments of people who have schemes purporting to do this.
In mathematics, I agree that one cannot get very far without understanding logic and using logic ( Frrequentist statistics may be an exception to this rule..) I would add one caveat. Logic is necessary for skill in math but not sufficient. If you watch skillful people do mathematics such as algebra or analysis at an advanced level, it does not resemble formal proofs involving "if... then"and "for each", etc. .
On self-evident truths: Well, I did put quotes around "truths."
On logic in economics, etc.: A very cynical view. What is the alternative to applying logic to our daily problems? Reading tea leaves? Praying for divine intervention? Heaven help us!
On logic in mathematics: I agree. Experience and intuition are indispensable even in mathematics, but you have to start somewhere. To most people, the principles of logic are not immediately obvious and some conscious effort must be made to master them. I would think that, with experience, they would become second nature, and their application need not be made explicit in most cases.
Ok, I give you credit for handling the word with tweezers. I like to do the same thing.
Economics is a very topical topic! Yes, heaven help us! It is poorly understood subject. My current mathematical preoccupations are daydreaming about mathematics for software to understand pictures and software for "optimal" stock trading. The trading prolem is rather non-topical but theoretically interesting. I'm not talking about market prediction. I 'm talking about: suppose you assume the market is defined by a given probabilistic model. How do you decide how to make trades? It must be a fairly intractable problem because all the literatue that I find on the subject deals with very simplified situations such as two stocks or a stock and fixed interest rate account. ... but I suppose if anyone wants to talk about this it should be in a new thread.