What if P = “The prime numbers are infinite or there are exactly 1337 of them” and P’ = “There are not exactly 1337 prime numbers.”
The shortest way to prove P involves proving (P and P’), which is the statement “The prime numbers are infinite.” It would take a roundabout argument indeed to exclude all finite cardinalities besides 1337, without also excluding 1337 and accidentally proving (P and P’).
What if P = “The prime numbers are infinite or there are exactly 1337 of them” and P’ = “There are not exactly 1337 prime numbers.”
The shortest way to prove P involves proving (P and P’), which is the statement “The prime numbers are infinite.” It would take a roundabout argument indeed to exclude all finite cardinalities besides 1337, without also excluding 1337 and accidentally proving (P and P’).