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)

0 Comments

Leave a Reply