#### Discover more from Just Emil Kirkegaard Things

# A small thing about enumerative induction

A thing occurred to me while i was reviewing the idea of enumerative induction becus i mentioned it to a friend of mine. The thing is that such inductions are often presented in the a way like this:

A particular thing of the type F is also of the type G Here is another one And another ... Thus, all things of type F are also of type G

Suppose we formalize it in a simple lazy way:

(∃!x1)(F x1∧G x1) (∃!x2)(F x2∧G x2) ... (∃!xn)(F xn∧G xn) ⊢ (∀x)(Fx→Gx)

But wait, one might as well draw the conclusion:

⊢ (∀x)(Gx→Fx)

Surely that follows just as well. However, if we keep this in mind when reviewing the typical example, then we get a result that differs from normal:

This swan is white. So is this one. And this one. ... So, all swans are white.

But following the above, we might as well just draw this conclusion:

So, all white things are swans.

I see no way to block that inference while letting the other thru, for the simple reason that *F* and *G* above are arbitrary. In second-order predicate logic, it looks something like this (with greek capital letters for predicate variables):

(∃Ψ)(∃Ω)(∃!x1)(Ψx1∧Ωx1) (∃Ψ)(∃Ω)(∃!x2)(Ψx2∧Ωx2) ... (∃Ψ)(∃Ω)(∃!xn)(Ψxn∧Ωxn) ⊢ (∃Ψ)(∃Ω) (∀x)(Ψx→Ωx) ⊢ (∃Ψ)(∃Ω) (∀x)(Ωx→Ψx)

Altho, the formalizations above are broken in a slight way. They don't capture the fact that the predicates in each premise have to be the same. So, one wud have to do it something like this:

(∃Ψ)(∃Ω)(∃!x1)(∃!x2)...(∃!xn)(Ψx1∧Ωx1)∧(Ψx2∧Ωx2)∧...∧(∃!xn)(Ψxn∧Ωxn)