John Norton breezes through an example of a deductive inference so as to characterize induction by contrast. His example of a valid deductive inference form is: “All As are B. Therefore, some As are B.” He even dubs this the all-some schema.1
It is a perplexing example. In old-school Aristotelean logic, the all-some schema is valid. In modern first-order logic, however, A may be an empty predicate. There being no As makes ∀x(Ax→Bx) true and ∃x(Ax&Bx) false, showing that the schema is invalid.
This got me thinking about whether the modern reading of the schema is really better than the classical one. I think it is.
The freewheeling use of the word “induction” is a pet peeve of mine. Sometimes it is used to mean any legitimate, non-deductive inference. Sometimes it is used narrowly be mean the inference from Observed Fs are G to All Fs are G. Sometimes it is carelessly used to mean both and other things besides. While I was sorting through old documents, I found this list of importantly different things that get paraded around under the banner of induction.
I regularly teach a course called Understanding Science, an introduction to some issues in philosophy of science and science studies. One topic is the nature of inference: deduction, the fact that scientific inference is (largely) non-deductive, and the problem of induction.1
Like many other professors, I started making video lectures last Fall. It was hard but unsatisfying work. Even so, 97% of the students in the section of Introduction to Logic that just concluded thought that my video lectures were at least somewhat clear and helpful.1
“I say, Holmes, how did you know that the crucial evidence would be in the galley of the yacht?”
“It was an elementary inference, Watson. As you were so quick to point out, the locked room showed that the murderer could not possibly have committed the crime and escaped. Yet the body of the victim and the absence of murderer showed that they had done so. I was puzzled until I remembered that everything follows from a contradiction, and this allowed me to conclude that the crucial evidence would be wherever I looked.”
“I see,” I said, although I really did not see. “But why the galley of the yacht?”
Holmes looked at me as if I were missing the obvious. “Because I was hungry. If I could find the evidence anywhere, then I might as well find it somewhere I could also make a sandwich.”
“Right then! But what about relevance constraints on logical consequence?”
“Watson, you disappoint me. If there were relevance constraints on consequence, then I could not have solved the crime. I did, so there are not.”
Then I realized that I, too, could derive anything from the contradiction Holmes had exploited. So Holmes conceded that I was clever, poured me a cup of tea, and left me alone for the rest of the afternoon.