There are 4 broad categories of certainty levels regarding a claim, creating 3 total levels of positive (claim-affirming) certainty:
Weight of evidence is against the claim
Weight of evidence is for the claim (utilized in deciding civil court cases)
The claim is beyond reasonable doubt (utilized in deciding criminal court cases)
The claim is beyond the shadow of all doubt (philosophically, a necessary truth)
Type 0 claims are uninteresting. Type 1 claims are common in the soft (or “infant”) sciences, such as psychology and sociology. Type 2 claims are common in biomedical sciences. Type 3 claims were common in Ancient Greece.
Type 3 claims are special because they do not require empirical (“a posteriori”) investigation beyond common experience. Common experience already turns them into “a priori” statements.
Type 3 claims can be proven absolutely, without taking a sample of objects or persons. But absolute proof is possible in some empirical investigations of the world. With empirics, you look out at the world, obtaining samples. The sample size is designated by “n” and a public opinion survey might have a size of something like n = 900.
Proof from (n = 0) sample
Claims that are true beyond the shadow of all doubt are statements such as:
“There is no round square.”
No attempted sample (of circles and squares) is required in order to prove it. This type of proof is absolute, admitting of no exceptions in the world, or even in one’s thinking about the world. All self-contradictions are claims about which there is absolute proof.
Proof from (n = 1) sample
If someone says that all cookies in the cookie jar are oatmeal, and you randomly pull out one and it is chocolate chip, instead, then you obtain absolute proof regarding an empirical claim (that it was false) by using a sample size of 1.
But you weren’t guaranteed to obtain absolute proof from your single sample.
Proof from (n = 3) sample
If someone claims that most cookies in the jar are oatmeal, and the jar contains 5 cookies, then a sample of 3 can absolutely prove the claim as being true — if all are oatmeal. It also absolutely proves the claim false when all are chocolate chip. Mixed samples are inconclusive, though they may help by being probabilistic.
Proof from (n = 5) sample
If someone claims that a minimum of 96% of cookies in the jar are oatmeal, and there are 100 cookies in the jar, then a sample of 5 cookies, all chocolate chip, is absolute proof regarding the falsity of that empirical claim.