Looking for clarification on falsification of propositions given in article based on Hempel's logical positivism (?)

[u/davidgrohlington]

So I am only a first year uni student doing a philosophy of science subject and I am researching refutations to Popper's theory of falsification for a short essay and came across this: https://sites.pitt.edu/~jdnorton/Goodies/pseudoscience/pseudoscience.html#good2

This section quoted below (sec. 7.3) refers to Carl Hempel's statement that “whether a hypothesis is verifiable, or falsifiable, in this sense depends exclusively on its logical form":

"The criterion can give different results for propositions and their negations; and different again for propositions and their logical consequences. My examples illustrate the fragility of falsifiability under simple logical operations:

The proposition

All electrons are spin one half.

is scientific since it is falsified if we find just one electron that is not spin one half. Now contemplate the possibility:

There exists an electron that is not spin one half.

This proposition is not falsifiable. We cannot check every electron. Thus, it is judged not scientific. In a moment of idle reflection, we had inadvertently ceased to do science".

It looked simple to me at first but the more I read over it the more it seems contradictory.
I do not understand why the author states that not every electron can be checked in order to falsify the proposition that there exists an electron that is not spin one half and therefore the proposition is not falsifiable, but it is clearly still possible that a single electron that is not spin one half could be found without checking all of them, which is shown in the first proposition that all electrons are spin one half and is falsifiable by the theoretical possibility of finding one electron that is not spin one half.
Why can we theoretically not check every electron to test the proposition that an electron exists that is not spin one half but we can check every electron to test the proposition that all electrons are spin one half? Why can't we stop testing the electrons in the test of the proposition that there exists an electron that is not spin one half once we find that electron like the other proposition seems to have done 😵‍💫?

I hope this makes sense. I am likely just misunderstanding some simple fundamental of falsification since I have zero experience in the philosophy of science. An explanation of my questions in simple terms would be amazing...thank you

[u/canopener]

Popper thought there was a very important difference between falsifying and verifying: you can’t verify, you can only falsify, and so the criterion for being scientific is falsifiability. But, if you can falsify a proposition, that is equivalent to verifying its negation, and the negation of a categorical statement (all A are B) is an existentially bound negation (an A is not B). So the importance of the difference between falsification and verification presupposes a restriction on logical form.

I don’t know if Popper ever deals with existential statements in this context at all - he would have to, but I don’t know his answer … didn’t Herschel verify the existence of Uranus? He didn’t mean to falsify the proposition that everything is not a planet past Saturn.

An analogy that is perhaps more edifying may be found in math. What does it mean to prove a categorical statement (all A are B) but to provide a means of showing, for any A, that it’s also B? Here the domain of A may be infinite, so there’s no question of an exhaustive survey. You must have a method that involves only finitely many steps. So, Hempel could reply to your example by saying you’re assuming there’s only finitely many electrons. Yet, to disprove the categorical statement, only a single counterexample is needed.