This is kind of amusing because NP problems, by definition, must have a polynomial-length certification to a given solution that can be run in polynomial time.
The word "certification" can be read as "proof" without loss of meaning.
What you're asking is slightly different: whether the proof-checking problem in logic is NP. I don't know enough math to know for sure, but assuming that any proof can be found in polynomial time by a non-deterministic Turing machine and that such a proof is polynomial in length, then yes, the proof-checking problem is in NP. Alas, I think that all of that is moot because Turing and Church's answer in the negative to the Entscheidungsproblem [1].
It would help your understanding and everyone else's if you separated the search problem for a proof from the verification problem of a proof. Something is NP when you can provide verification that runs in polynomial time. The lesser spoken about second condition is that the search problem must be solved in polynomial time by a non-deterministic Turing machine. Problems which don't meet the second condition are generally only of theoretical value.
The word "certification" can be read as "proof" without loss of meaning.
What you're asking is slightly different: whether the proof-checking problem in logic is NP. I don't know enough math to know for sure, but assuming that any proof can be found in polynomial time by a non-deterministic Turing machine and that such a proof is polynomial in length, then yes, the proof-checking problem is in NP. Alas, I think that all of that is moot because Turing and Church's answer in the negative to the Entscheidungsproblem [1].
It would help your understanding and everyone else's if you separated the search problem for a proof from the verification problem of a proof. Something is NP when you can provide verification that runs in polynomial time. The lesser spoken about second condition is that the search problem must be solved in polynomial time by a non-deterministic Turing machine. Problems which don't meet the second condition are generally only of theoretical value.
1. https://en.m.wikipedia.org/wiki/Entscheidungsproblem