I’ve just come across some old exam papers from my time at Manchester University, and I was instantly reminded of the most gratifying example of hubris and its consequences in my personal experience. My main subject was geology, but at the beginning of my second year, I was faced with a decision. Which subsidiary subject should I choose to take? All the usual options were on the list—chemistry, physics, botany, zoology, metallurgy—but at the bottom of the page I spotted something unexpected. I could choose to attend a course in logic.
It turned out to be a lot harder than I had anticipated, but this wasn’t due to any intrinsic difficulty. Most of the others in the class were first-year students of mathematics, and much of the course focused on propositional calculus, which is logic reduced to a form of algebra.
You can imagine what happened. The lecturer started by introducing three axioms, from which everything else could be deduced. And the mathematicians in the front row of the lecture theatre were telling him what the next line should be before I’d finished writing down the previous statement. I resented this at the time, because I was writing down stuff that I simply didn’t understand, and I struggled. Fortunately, it did make sense when I was able to read the material at my own pace, and the payback would come in a three-hour examination at the end of the course.
I couldn’t suppress a little chuckle when I picked up the exam paper, which carried the following instructions:
Answer at least ONE question from each section [there were three sections] and ONE other question. At least TWO starred questions should be attempted by mathematicians.The unstarred questions were easy then, although they make no sense to me now, almost half a century later. Here is an example:
13. Explain the meaning of the constant terms occurring in Aristotle’s syllogistic.And here is a question on the same subject for the mathematicians:
*14. Lukasiewicz has suggested the following axioms for Aristotle’s syllogistic:The second question seemed difficult then, but it would be impossible for me to answer now. And it is possible that the mathematicians at whom the question was aimed in the first place would have found it easy then. Still, I think that I was entitled to a little schadenfreude. It’s what should happen to people who are too cocky.
‘A’ and ‘I’ being primitive terms. Show that the axioms are independent of each other.