I think my kid picked logic up again late in January. In particular, he wanted to look again at two recent books intended to help students make the transition to abstract mathematics — i.e. to having to prove things.

Those two books are Exner and Hummel. They were highly recommended for that, out on the sci.math newsgroup, and, therefore, I immediately bought them.

In addition to those two books, I ended up looking at Aristotle (indirectly), Lewis Carroll — yes, for logic! — Paul Halmos on logic as algebra, and a few books on symbolic logic.

Speaking first of the two textbooks about how to prove things….

Exner is the less formal of the two. For example, his second chapter talks about informal proofs, while his third chapter shows the formalities. In addition to dealing with the mechanics of proving things, this book is also trying to teach good habits for reading a mathematical text.

Hummel, by contrast, is a little more formal from the very start: page 19, for example, has a list — a very nice list! — of tautologies. For example, that

$P \implies Q$

is equivalent to

$!P \lor Q\$.

(That is, “P implies Q” is equivalent to “not P, or Q”.)

Both of these books filled a need for me. Every time I had ever looked at a book on mathematical logic, I had bailed when we got to bound and free variables.

Both of these books showed me why we distinquish bound and free variables. (Yes, I will explain, but not here and now.)

Hummel offers applications starting on page 80: sets, functions and relations, number systems, transfinite cardinal numbers, the axiom of choice & ordinal numbers. His introductory material on logic is pretty much just logic.

Exner, by contrast, uses mathematical applications all the way through. The end of the book is some more concentrated applications to sets and functions.

When I first worked through these books, I decided that I wanted to look at Aristotle’s syllogisms using my modern logic tools. My reference for Aristotle is Joseph. It is not an easy read, and I can’t really say I recommend it. It’s what I have, so far. (I have just ordered two modern introductions to logic, and we’ll see if they have anything about syllogistic reasoning.)

It turns out that Joseph may be an unusual reference for classical logic. Aristotle used what are called figures, and he defined only three of them. Medieval logicians used a fourth figure, which can be converted to the first. (See the subsequent discussion of Lewis Carroll.)

On the other hand, I have tried to read Aristotle in English — and it is hard going, so we want to read somebody else about him. Hence Joseph.

(On the third hand, I have tried to read Aristotle in Greek — and he is the hardest Greek author I have ever encountered. Admittedly, I haven’t tried very many Greek authors — but he is head and shoulders harder than his own teacher, Plato.)

I also happened to see “Symbolic Logic” by Lewis Carroll, about the time I had picked up logic a few years ago. Well, on this pass through Lewis Carroll, I noticed that he had a list of 19 syllogisms at the back of the book.

Not the same list as Joseph — but the difference is understandable: Lewis Carroll is using four figures to Joseph’s three figures. He is using the medieval formulation.

Anyway, to return to the main thread, once that I understood why we cared about free and bound variables, I could make more progress in my modern mathematical logic books.

I have three textbooks, but only two seem worthwhile. One is Copi and the other is Suppes. Both were readable — now! — and even pleasant. In contrast to Exner and Hummel, these are books about the choices of axioms for logic, and working out the consequences of the axioms.

If you will, Copi or Suppes provides the foundations for Exner and Hummel.

There are two other books to be added. One is an ancient Chelsea reprint by Paul Halmos, and the other is a nice introduction to it.

I am pretty sure that when I bought that Chelsea reprint of Halmos’s papers, I could not even read the introductory paper. (If I had any notion then what an ideal was, it was not a very substantial notion.) And I am not sure I will ever care enough about mathematical logic to read the following papers in the book, but maybe you will. Then I discovered that there was a related book, by Halmos & Givant, and it came in about two weeks ago. It gives a very nice introduction to the Chelsea reprint, and to Boolean algebra applied to logic.

Interestingly, it has yet a third list of Aristotle’s syllogisms, and I cannot reconcile it with the other two. That’s okay: I’ll use their methods on one of the other lists! (I was afraid they had completely disposed of Aristotle, leaving me nothing left to do. Not so. I still get to work syllogisms all out for myself.)

Oh, of course there is one more book. Boole’s “Investigation into the Laws of Thought”. I probably bought that even before the Chelsea reprint.

Boole is interesting because I am sure we have found alternatives to some of the calculations he laid out. In particular, I strongly doubt that we do divisions by zero! Actually, I’m pretty sure we don’t use division at all. Someday I’ll figure out how to get Boole’s answers using modern methods.

So. If you are trying to learn how to prove things, I recommend Hummel and probably Exner, too. if you are trying to learn how to read math books, I particularly recommend Exner. And in general, Exner looks to be the more introductory of the two. If you want an introduction to modern logic, I would suggest you look at Copi or Suppes.

Oh, bear in mind that I am speaking of Copi’s “Symbolic Logic” rather than his “Introduction to Logic”.

If you want fascinating logic puzzles, then Lewis Carroll. And if you want to understand just what George Boole accomplished — what he had to work with! — of course, get and read him.

## the books

Boole, George. An Investigation of The laws of Thought…. Dover publication of the 1854 edition.
[logic; 3 Mar 2010]
I think it’s fair to say that he started it all, and Boolean algebra is named in his honor. Of course, he predates set theory, but published almost simultaneously with De Morgan. Reading Boole’s examples made me understand how so many theology students could become mathematicians — talk about convoluted! But that’s part of what he was trying to do — extract the key results out of convoluted theological arguments. Plenty of examples; computational algorithms, of course — but they are not what we would do today.

Carroll, Lewis. (Dodgson, C.L.) Lewis Carroll’s Symbolic Logic. Clarkson N. Potter Inc., 1986 (2nd paperback ed.)
ISBN 0 517 53363 4.
[logic; 3 Mar 2010]
One good reason for publishing this book under the pseudonym rather than under the real name is that Lewis Carroll brought his considerable narrative skills to the puzzles which illustrate his points. The book includes a short discussion of the upheaval in logic as it transitioned from classical, on through the work particularly of Boole and De Morgan, and into modern symbolic logic. He presents graphical techniques for solving puzzles. Lots of answers.

Copi, Irving M. Symbolic Logic. Macmillan, 1966 (2nd ed).
[logic; 3 Mar 2010]
I enjoyed this enough to order the most recent edition of his “Introduction to Logic” — I liked the language, the organization, and the examples in this book. I understand that this book is considered out of date, but I have no idea what should replace it. Selected answers.

Exner, George R. An Accompaniement to Higher Mathematics. Springer, 1996. (corrected 3rd printing, 1999).
ISBN 0 387 94617 9.
[logic, proofs; 3 Mar 2010]
“Undergraduate Texts in Mathematics”. This is a marvelous introduction to how to read a mathematics book, and how to approach the construction of proofs. I think it is an excellent introduction to or companion to Hummel. Selected answers.

Halmos, Paul R. and Givant, Steven. Logic as Algebra. Mathematical Association of America, 1998.
ISBN 0 88385 327 2.
[logic; 3 Mar 2010]
“Dolciani Mathematical Exposition”. A short introduction to the use of abstract algebra applied to logic. I particularly enjoyed their discussion of Aristotle’s syllogisms — even though I can’t reconcile their list of the syllogisms with either of the two lists I trust! This seems a comparatively minor point considering the book as a whole.

Halmos, Paul R. Algebraic Logic. Chelsea, 1962.
[logic, abstract algebra; 3 Mar 2010]
A collection of Halmos’ papers; at this stage, only the first is accessible to me — and it requires being comfortable with modern algebra, in particular with rings. I’ll call it advanced undergraduate, more likely graduate level.

Hummel, Kenneth E. Introductory Concepts for Abstract Mathematics. Chapman and Hall / CRC, 2000.
ISBN 1 58488 134 8.
[logic, proofs; 3 Mar 2010]
This is a marvelous introduction to formal logic and to the construction of proofs. I think it is an excellent companion to or follow-up to Exner. Selected answers.

Joseph, H.W.B. An Introduction to Logic. Oxford, 1916 (2nd ed. reprinted 1966).
[logic; 3 Mar 2010]
This is an old-school logic book. First published in 1906, it may well represent the pinnacle of traditional logic.

Suppes, Patrick. Introduction to Logic. Van Nostrand, 1957. Available as a Dover reprint, but which edition?
[logic; 3 Mar 2010]
“University Series in Undergraduate Mathematics.” Clearly written, and I enjoyed it. Rigorous symbolic logic will almost certainly never be on my list of things to do, but this was fun to read.