One example in classical logic is the syllogism where if the premises are true then the conclusion is by necessity true:

Socrates is a man

All men are mortal

therefore it is true that Socrates is mortal

Another example is mathematical proofs. Here is the Wikipedia presentation of Euclids proof from 300 BC that there is an infinite number of prime numbers. Perhaps In your terms this proof provides 0% confidence that we will observe the largest prime number.

Take any finite list of prime numbers p1, p2, ..., pn. It will be shown that at least one additional prime number not in this list exists. Let P be the product of all the prime numbers in the list: P = p1p2...pn. Let q = P + 1. Then, q is either prime or not:

1) If q is prime then there is at least one more prime than is listed.

2) If q is not prime then some prime factor p divides q. If this factor p were on our list, then it would divide P (since P is the product of every number on the list); but as we know, p divides P + 1 = q. If p divides P and q then p would have to divide the difference of the two numbers, which is (P + 1) − P or just 1. But no prime number divides 1 so there would be a contradiction, and therefore p cannot be on the list. This means at least one more prime number exists beyond those in the list.

This proves that for every finite list of prime numbers, there is a prime number not on the list. Therefore there must be infinitely many prime numbers.