All-some and a rhetorical misstep

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.

Start with a few things that seem intuitively true:

  1. Some As are Bs is false if there are no As.
  2. All As are Bs is equivalent to It is not the case that some As are non-Bs, where not the case can be understood as truth-functional negation.
  3. The class of As is the class of non-non-As.

These together entail that All As are Bs is true when there are no As. The third might even be unnecessary. So holding that all-some is valid requires either giving up one of these, which doesn’t seem like an appealing move. One could reject 1 and 2 on principle by holding that empty predicates are logically impossible— but I can’t muster even an iota of enthusiasm for such a move.

Of course, Norton not is concerned with any of that. He just wants a quick and easy example of a deductive schema. As I reader, I understand his point. Since his example distracted me enough to think through this mishegas, though, it hasn’t served his rhetorical purposes very well.

Tangentially related postscript

Last year, I posted about wanting a better name than the material theory of induction for the fact that scientific inference always relies on domain-specific background knowledge. Still looking for one. Suggestions welcome.

  1. In the Prolog to The Material Theory of Induction.

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