Books added: more logic & proof

The following books have just been added to the bibliography.

Copi, Irving M. & Cohen, Carl. Introduction to Logic.

Gensler, Harry J. Introduction to Logic.

Tao, Terence. Analysis I.

I have received, and browsed through, two books titled “Introduction to Logic”. They both appear to be readable and reliable. (I had actually received both books the very day the previous logic books were added to the bibliography, and decided not to delay that post. Come on, I needed to look through the new ones!)

In fact, I had gone looking for a current book by Copi, who is the author of “Symbolic Logic” which I own and like (yes, in the biblio). While I was deciding to buy that current book, I found a second one which was also very favorably reviewed on Amazon.

I bought both. It’s what I do.

I got lucky: they are different from each other. Gensler seems closer to a set of lecture notes; Copi & Cohen, as one might expect from a 13th edition, seems much more like a text.

No, I am not saying that Gensler is disorganized — and I’m not saying that Copi & Cohen is stodgy. One is less formal than the other.

I would say that if either one of these is your textbook for a course, then consider buying or borrowing the other one for collateral reading. In fact, in general, if you have a solid reference book as a textbook for a course, then think about getting a copy of Gensler for collateral reading. Conversely, if all you have is lecture notes, then get yourself a copy of Copi & Cohen.

Another way to illustrate the difference is this. If you look up “material implication” in Gensler, the first entry is page 122, the second entry is pages 369-372. The index is accurate, a discussion does begin on page 122 — but the term “material implication” does not occur until page 369.

In other words, unless you already know what “material implication” is, you either cannot find the first discussion of it — or, while reading the discussion, you will not know the name of it. Still, Gensler does talk about it.

(Incidentally, Copi & Cohen include actual definitions in the index, as well as performing the usual task of providing page numbers.)

In summary, both of these books are at present on my “casual reading” table, and I have referred to both of them while drafting the logic posts. (In particular, Gensler has the medieval mnemonic for the 19 Aristotelian syllogisms, while Copi & Cohen have a handy list of the 15 that are considered valid today.)

The third book is a cat of another color.

I would have said that that the mathematics course typically called “abstract algebra” was the customary introduction to rigorous proof in mathematics. And I might have said that “how to prove things” is generally taught by some combination of “if you see enough proofs, you’ll figure out how to do them” and “if you weren’t born knowing how to do it, there’s no point in trying to teach you”.

From the introduction to Tao, I find that nowadays a first course in linear algebra might function as a math major’s introduction to rigorous proof.

But from his introduction I also learn that he found most students to be woefully unprepared to do rigorous proof when they entered a “real analysis” course. So he decided to introduce them to rigorous proof, at the beginning of the course. And he let them cut their teeth on things they knew, i.e. things they had a feeling for: the natural numbers, set theory, rational numbers, real numbers, and so on.

Sure enough, his students were way behind those in the non-honors class — for a while; he said it took 10 weeks to catch them, and I daresay — he does not say — that they left them in the dust by the end.

The point was not that they covered more elementary material before getting about their real business (no pun intended, I swear) — but that they took the time to prove all the elementary material, thereby getting a lot of practice at proofs.

This is a lot of words to say that: the primary purpose of the Terence Tao text is, indeed, to teach real analysis; but its secondary purpose is to make sure that by the time you are trying to do proofs on new material, you’ve had lots of practice on relatively old material.

It seemed more than appropriate to include this book at this time. It belongs here as much as Exner or Hummel, and it covers more advanced mathematics than they do.

There is a problem, of course: this is a book not a professor, and it only has hints; apart from those hints, it can’t help you get out of trouble with any particular proof. Still, I am delighted to find a real analysis text that starts with the premise that most junior math majors aren’t very good at proving things — and, more importantly, that a teacher could try to do something about that.

And of course, it doesn’t hurt that I trust the author implicitly. Terence Tao is a most-recent Fields medalist (2006).

For the record, Tao volume I ends with the Riemann integral; volume II, which I have not explicitly listed, ends with the Lebesgue integral. (Both volumes have hints for the proofs.)

Here are the three books:

Copi, Irving M. & Cohen, Carl. Introduction to Logic. Pearson / Prentice Hall, 2009 (13th ed).
ISBN 0 13 614139 6.
[logic, 25 Mar 2010]
An interesting-looking solid text, which contains, among other things, an excellent discussion of material implication. Answers.

Gensler, Harry J. Introduction to Logic. Routledge, 2010 (2nd ed).
ISBN 0 41599651 8.
[logic, 25 Mar 2010]
An interesting-looking relatively informal text, which begins with syllogistic reasoning. Further reading. Answers.

Tao, Terence. Analysis I. Hindustan Book Agency, 2006.
ISBN 81 85931 63 1
[real analysis, proof, 25 Mar 2010]
I’m sure this is a fine introduction to real analysis, because the author is a Fields Medalist with an interest in undergraduate education. I recommend it specifically because it aims to help students learn to prove theorems. There is a second volume. Answers (actually hints, both volumes).

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