@musicandfaith94, I understand the argument you make here. I believe your second premise is false. A person can start with a false premise in order to conclude that the false premise necessarily leads to a contradiction. This is common in arguments that use formal logic.

As a mathematics student, I struggled with this concept for a while. Consider the following argument.

Suppose there are finitely many prime numbers. Let k be the number of prime numbers and the set of prime numbers be {p1, p2, ..., pk}. Let x be the product p1 times p2 times ... times pk. Then x + 1 divided by p1 has a remainder of 1, x + 1 divided by p2 has a remainder of 1, ..., and x + 1 divided by pk has a remainder of 1. Thus, x + 1 is not evenly divisible by any prime number. This shows x + 1 is prime. Since p1, p2, ..., and pk are all positive, x + 1 is larger that p1, p2, ..., and pk. Thus, x + 1 is a prime number not in the set {p1, p2, ... pk} and there are at least k + 1 prime numbers. This contradicts that there are k prime numbers. The overall conclusion of this is that the original supposition (there are finitely many prime numbers) leads to a contradiction. Thus, the original supposition must be false. Therefore, there are infinitely many prime numbers.

Concerning an argument like this, I asked my professor, "What allows us to say, 'There are finitely many prime numbers?'" He did not have a good answer on that particular day. I now understand we are not saying there are finitely many prime numbers. Instead, we are saying if there are finitely many prime numbers, then we reach a contradiction. Therefore, the original premise must be false.

In terms of formal logic (and skipping several steps):

If there are finitely many prime numbers, then there are more prime numbers than there are prime numbers.

There are not more prime numbers than there are prime numbers.

Therefore, there are infinitely many prime numbers.

Abstracting this to symbols gives:

P implies Q.

Q is false.

Therefore, P is false.

Notice this does not require us to assert that P is true, only that P implies Q is true. The same holds for the nihilist atheist. Such a person can claim if true evil exists, then there must be a contradiction. Therefore, true evil does not exist. Doing this does not require the claim that (contrary to their nihilist beliefs) true evil exists.