Why must a complement class be infinite on universal statements? (Popper - The Logic of Scientific Discovery)

Question

I`d value the generous help here of the enlightened reader:

(Karl Popper) Logic of Scientific Discovery > 33 Degrees of Falsifiability compared by means of the subclass relation

Here we have the definition of subclass:

(1) A statement x is said to be ‘falsifiable in a higher degree’ or
better testable’ than a statement y, or in symbols: Fsb(x) > Fsb(y), if and
only if the class of potential falsifiers of x includes the class of the
potential falsifiers of y as a proper subclass.

Fsb(y) is a subclass of Fsb(x).

The following text puzzles me:

If (1) applies, there will always be a non-empty complement class.
In the case of universal statements, this complement class must be
infinite. It is not possible, therefore, for the two (strictly universal)
theories to differ in that one of them forbids a finite number of single
occurrences permitted by the other.

The first senctece i understood (If (1) applies, there will always be a non-empty complement class.)

My question is:

Why must this complement class be infinite in the case of universal statements?

In the following picture you can see Fsb(x) containing Fsb(y). The non-empty complement class ist limited by the borders of the rectangle Fsb(x).

The following sentence completely eludes my undertanding:

It is not possible, therefore, for the two (strictly universal)
theories to differ in that one of them forbids a finite number of single
occurrences permitted by the other.

Probably Fsb(y) permits a finite number of single occurrences that would be permited by the other here Fsb(x).

Answer 1

I share your puzzlement. In case (1), a theory x is falsifiable in a higher degree than theory y iff the class of potential falsifiers of x is a proper superset of that of y. What Popper is getting at is that a strictly broader theory has strictly more ways that it could in principle be falsified.

For example, the theory "all apes have hair" is falsifiable in a higher degree than the theory "all chimps have hair", since the former can be falsified by exhibiting any hairless ape, while the latter requires a hairless chimp. Popper is asking us to suppose that the class of potential hairless apes is a strict superset of the class of potential hairless chimps. This is a little odd, since there are actually none of either, and Popper does not make explicit what he means by 'potential'. Maybe if he had been writing later, he might have expressed himself in terms of possible worlds.

But let's be generous and allow that potential apes are apes that might exist for all we know, or maybe we can understand them as possible future observations of apes that might reveal one to be hairless. Popper speaks of strictly universal theories, by which he means theories with no limitations of scope. Not, "all the apes in this zoo", or "all the apes currently alive", or "all the apes I have ever seen in David Attenborough's documentaries", but the simple unqualified 'all'. Popper takes the class of such things to be infinite. We might object that there could never be an infinite number of apes, or even an infinite number of observations of apes, but we can allow that Popper is speaking loosely and means merely that there is no definite upper bound.

Popper then proceeds to say that provided both theories x and y are universal, and hence each has infinitely many potential falsifying instances, then there is no finite class of instances that falsifies one and not the other. This is an incorrect inference, since it amounts to claiming that if two classes are infinite, their difference is infinite. In practice, it may be true that in typical cases the difference is infinite. There are perhaps no limits to the number of observations that one could make of apes that are not chimps in order to determine whether any of them are hairless.

As he often does, Popper labours a point that is not all that important, introduces formal logic that is not needed to understand the point, and then proceeds to make errors with the logic. The Logic of Scientific Discovery would be a much better book if a sympathetic editor with a good grasp of logic redacted it down to about a quarter of its size.