An autobiography of Polyadic Algebras

In the 1950’s I set out to revolutionize logic. I didn’t succeed (not yet!), but it wasn’t for want of trying. Logic has always fascinated me. In the common-sense sense of the word logic was easy; I had no inclination to commit the obvious errors of reasoning, and long chains of deductions were not intimidating. When I was told about syllogisms, I said “sure, obviously”, and didn’t feel that I learned anything. The first genuine stir in my heart toward logic came from Russell’s popular books, and, just a few weeks later, from Russell and Whitehead: the manipulations of symbolic logic were positively attractive. (I liked even the name. “Symbolic logic” still sounds to me like what I want to do, as opposed to “formal logic”. What, I wonder, is the history of those appellations? At a guess, the old-fashioned word wanted to call attention to the use of algebraic symbols in place of the scholastic verbosity of the middle ages, and the more modern term was introduced to emphasize the study of form instead of content. My emotional reaction is that I like symbols, and, in addition, I approve of them as a tool for discussing the content — the “laws of thought” —, whereas “formal” suggests a march-in-step, regimented approach that I don’t like. One more thing: “symbolic” suggests other things, “higher” things that the symbols stand for, but “formal” sounds like the opposite of informal — stiff and stuffy instead of relaxed and pleasant. All this may be a posteriori — but it’s the best I can do by way of explaining my preference.) I liked symbolic logic the same way as I liked algebra: simplifying an alphabetsoup-bowl full of boldface p’s and q’s and or’s and not’s was as much fun, and as profitable, as completing the square to solve a quadratic equation or using Cramer’s rule to solve a pair of linear ones. As I went on, however, and learned more and more of formal logic, I liked it less and less. My reaction was purely subjective: there was nothing wrong with the subject, I just didn’t like it. What didn’t I like’? It’s hard to say. It has to do with the language of the subject and the attitude of its devotees, and I think that in an unexamined and unformulated way most mathematicians feel the same discomfort. Each time I try to explain my feelings to a logician, however, he in turn tries to show me that they are not based on facts. Logicians proceed exactly the same way as other mathematicians, he will say: they formulate precise hypotheses, they make rigorous deductions, and they end up with theorems. The only difference, he will maintain, is the subject: instead of