One of the things I ran into when I started looking into Aristotle was Aristotelian logic, and particularly the Aristotelian syllogism. You’ve seen at least one of these, I’m sure, probably this old chestnut:

Major Premise: All men are mortal.
Minor Premise: Socrates is a man.
Conclusion: Socrates is mortal.

I’d run into this one any number of times…but most of the descriptions I’d read got about as far as, “This is what an Aristotelian syllogism looks like. There’s a major premise and a minor premise, and a conclusion, and if the major and minor premises are true the conclusion is guaranteed to be true.”

OK. I can look at the example above and see quite clearly that the conclusion follows from the major and minor premise; but what is it about this form that guarantees the truth of the conclusion? I had no idea. There was clearly a lot more to be known. I went looking, and found this book, which looked like it covered the topic in sufficient detail, along with a whole lot of other stuff about logic. It went with me on several of the numerous business trips I made this past year, and I had a lot of fun with it.

Bennett’s aim is twofold. First, to tell the story of how the study of logic developed over the last three millenia, and second to discuss the relationship between formal logic and the kind of reasoning most people use when confronted with logical problems. It succeeds very well at both, and I recommend it to anyone who is interested in such things.

But about those syllogisms. Here’s what I discovered. The statements in an Aristotelian syllogism have forms like “All X are Y“, “Some X are Y“, “No X are Y“, and “x is a Y“, where “X” and “Y” are categories and “x” is some particular entity, e.g., “is a man”, “is mortal”, and “Socrates”. What Aristotle did was categorize all possible combinations of these kinds of statements involving three terms “X“, “Y”, and “Z“, and then painstakingly showed, by other means, that a subset of them (well under half, if I recall correctly) will always yield a true conclusion if the premises are true. By remembering and sticking to those valid combinations (to which the Scholastics gave such charming names as “BARBARA”), one could be sure that one’s reasoning was correct.

Now on the one hand, this is cool: Aristotle was dealing with existential qualifiers two-thousand years before they were first defined in mathematical logic. On the other hand, the examples I saw seemed fairly obviously true without any particular analysis of their syllogistic structure. Hmmmm.

And then, in the next chapter or so, I found out why I’d never before had occasion to learn about Aristotelian syllogisms: modern logic, the kind I learned about in geometry class, I kind I’ve been using daily as a working programmer for two decades (and as a hobbyist before that), isn’t based on Aristotelian logic, but rather on the logic of the Stoics. The Stoics had only two syllogisms, the modus ponens and the modus tollens, the form of which is so absurdly simple I’d never thought of them as syllogisms. Given two statements, “p” and “q“, modus ponens says:

Major Premise: If p is true, then q is true.
Minor Premise: But p is true.
Conclusion: Therefore, q is true.

The modus tollens is simply the contrapositive of this, “if not-q then p.”

On these two syllogisms hang, if not the law and the prophets, then all of the programming I’ve done and all of the math I studied as a math major.

At this point I began to realize that I while I might enjoy finishing the book, I wasn’t likely to learn anything both new and useful, and I was right on both counts. For example, did you know that Venn diagrams were invented to help analyze Aristotelian syllogisms? Neither did I. Now I realize that the real descendant of Aristotelian logic is not logic per se, but set theory. Such an amazing epiphany, you can’t imagine.

But it’s still a good book, and if you have any taste for logic or logic puzzles it’s worth your time.