The problem with informal logic is its lack of formality. I wish I had a nickel for every time someone has tried to bust out a fallacy on me when arguing over the internet -- "ad hominem" has to be the most over-used string of letters on the entire world wide web. A good example of how informal fallacies aren't always fallacies might be arguments from authority. Sometimes, it is appropriate to use an authority, and sometimes it isn't. That 9 out of 10 medical doctors say a treatment is necessary makes a strong argument that the treatment is, indeed, necessary all things being equal. If, on the other hand, some one comes along and says that the poster citing these statistics is just trying to sell their drugs (used in the treatment) and has cherry picked the doctors or is otherwise not representing the facts in an unbiased manner, then you have to doubt their argument from authority, after all. (This counter, by the way, is ad hominem.) Just because "they are fallacies" doesn't mean they don't figure into the rational consideration of the matter. And that is because, as informal fallacies, they aren't always fallacies. But, such fallacies and informal logic, in general, is really based on the facts of formal logic along with a few other philosophical positions (including the position that the laws of excluded middle and non-contradiction are actually true, for instance). And, math is, too. So, that's probably why, for instance, apparently graduate students in philosophy all over the place have to demonstrate their proficiency with it to be allowed to stay in the graduate school and pursue their PhDs. So, it's not a minor thing. I you go to graduate school in math, though you may not have to pass a specific qualifying exam on that topic, every qualifying exam you take will make heavy use of your ability to use that kind of logic. There is simply no way you can pass an analysis qualifying exam and not know what how to properly negate something like, for instance: "For every epsilon greater than zero, there exists an integer N such that whenever n>N, |a(n)-a|<epsilon." (Of course, when you actually get to writing this down in practice, you will be using the "upside down As" and "backwards Es" and so on.) So, if for no other reason than that, I think it is safe to say that it is absolutely central to both fields. (And these two fields are pretty central to all intellectual life.) My main problem with studying formal logic is that it tends to be just that -- studying it and not doing it. You learn, as one poster mentioned, what "ponendo ponens" means. But, I don't really care if you know that. I want you to be able to negate the definition of a limit in under a nanosecond. You will have to constantly do stuff like that to pass your qual. It doesn't matter if you know that the upside down A is called a "quantifier". It doesn't really matter if you know how to prove Godels Incompleteness theorem. You just have to be super fast and accurate at manipulating that kind of formal statement -- and above all, accurate -- you really have to get it exactly right. And, you have to do it super fast or it will take you days to figure out a problem that you only have a few hours to solve. And, I think that philosophers do it probably for similar reasons. It is not only in their area to know about symbolic logic, but this one topic above all others strikes directly at a person's ability to reason. Your practice with formulating, negating and otherwise manipulating complex logical statements is your experience with reason. There isn't a special kind of reason for liberal arts majors, here. There is just one kind of reason. And formal logic lies at the center of it. (Don't get me wrong -- I'm not saying that informal logic is inferior to formal logic or something like that. And, it isn't knowing terminology or recognizing symbols that really matters, either. But, the practice with and direct instruction of things like that the contrapositive of an implication is equivalent to the implication and the converse is not equivalent, for instance, certainly is very important -- the training to actually do the logic well.) Formal logic isn't about the symbols or the terminology. It's about being able to handle "all", "some", and "none" properly and things like that. Just because a lot of people might implicitly assume the converse or improperly negate statements in their writing all the time, that doesn't mean that it is now some sort of legitimate mode of reasoning -- "logic for the liberal arts". Formal logic, like math, is a tool you will not use only if you don't have it and a tool you probably use, at least intuitively, all the time without realizing it. In reality, it should form the glue with which all informal arguments are strung together as well as just the actually correct entire argument in lots of cases. You should be able to translate regular English into formal statements and formal statements into regular English as well as properly negate complex formal statements at will without having to think too hard about it. If you have to spend a long time just getting the logic of one statement down, then how will you ever be able to string a multitude of them together into an article or a proof or something meaningful like that?

...your reply to her should be, "And, I will not be able to decide if your parents' decision to public school their children was the right choice until their children are adults with good jobs and healthy adult relationships, either."

Based on what I've seen, if you took 100 homeschooling families and they made their kids sit down and do TT and then compared them to 100 randomly selected public school students via a standardized test, my impression is that they would average about the same. So, I, personally, would not really call it "remedial math" (although, I might characterize it that way while engaging in a little hyperbole). What I would be shocked as sh*t to find out is that it really is actually, in some sense, "good". (I have perused their website fairly thoroughly -- it really looks pretty typical of math programs.) For a bright kid, you can go one of two basic ways: theoretical science or math. I call it "theoretical science" because that's what it really is -- not math. What TT seems to lack compared to Singapore or other programs are problems like "Prove that a*b < (a+b)/2," which though it may look like a "proof", is really just an algebraic manipulation (a tough abstract calculation). But, nevertheless, it is a fairly difficult one and an important result. Or, another problem might be "A swimming pool is divided into two equal sections. Each section has its own water supply pipe. To fill one section (using its pipe) you need a hours. To fill the other section you need b hours. How many hours would you need if you turn on both pipes and remove the wall dividing the pool into sections?" From what I can tell, TT is really pretty light on such problems and/or really coaches a student through it too much. (That's another thing about the TT debates -- "really teaches" my ds math frequently translates into "coaches them through everything". It means something completely different when a student is truly able to figure it out on their own than it does if they are just doing a specific class of problems they have been systematically groomed for.) I believe both of these problems are in Gelfand and probably can be found in other "challenging" programs. Singapore, in particular, is renowned for its hard word problems. (On the other hand, it coaches its students through them a lot, too, with specific tricks for doing classes of problems.) The other direction to go in is more of a pure theoretical math direction that almost no one -- not even most mathematicians -- do. In fact, the first problem above, could be an example, here, if done in a certain way. For instance, on a recent road trip, I had the following discussion with my 11 yo son. I said to suppose that I just had a collection of numbers, P (whatever we decide "the numbers" are). Suppose all I know about P are the following two things: 1) For any number, a, one and only one of the following is true: a is in P, a is the additive identity, or the additive inverse of a is in P. and 2) P is closed under addition and multiplication. (That is, for all a and b in P, a+b is in P and ab is in P.) Then, can -1 be in P? It was totally Socratic, and believe it or not, he came up with the reasons why -1 cannot be in P. The discussion when something like Me: "Can P={-1}?" Him: "No, then -1*-1=1 would have to be in P." Me: "Then, let P={-1,1}." Him: "Then, 1+1=2 would have to be in P." ... (And it kept going, actually, until he finally had to resort to (1) above, but he did get there, pretty much on his own.) At any rate, I give this example to prove the viability of what I am talking about. Even in programs like Singapore or Foerster's, say, they will approach "greater than" and "less than" heuristically. They will do something like show the student a number line and how the larger numbers are to the right and the smaller numbers are to the left. That begs the question -- you essentially have to already know what "greater than" and "less than" are to "get it". (And, fortunately, most kids do have some intuition built up about it, so it's not that big of a problem.) On the other hand, there is a book by this guy Beckenbach that handles inequality axiomatically with the very set P that I mention above. A number is called "greater than" another number if their difference is in P, for instance. So, this is another way to go -- more rigor. (Clearly, I favor the more rigor. The other way might be great training for science or an even better IQ test, or something, but it actually often even interferes with a student's ability to do rigorous math if pursued too much.) TT doesn't seem like it is really very good at either which doesn't necessarily make it such a bad program, but it does probably make it a poor choice for challenging an advanced student.

What set of axioms does Teaching Textbooks use, anyway? What would you do besides proofs in geometry? Would you just do regular coordinate geometry? Some sort of vector spaces? Does Teaching Textbooks talk about methods of proof such as that an implication is equivalent to its contrapositive or proof by contradiction -- that sort of thing? Does it ever get past the T-diagram? Do they at least state all the theorems of standard plane geometry? I'm pretty sure it's not purely synthetic (but it doesn't sound like you want anything like that).