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:
- Some As are Bs is false if there are no As.
- 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.
- 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.
- In the Prolog to The Material Theory of Induction.