I've been having thoughts for a while about what constitutes a proof. Formal logic usually consists of incredibly detailed steps and, as such, is usually not utilized that often in everyday life.
However, we often ask people to do things such as "prove that you were not at place X at time T". While some people may use purely objective logic to do this (such that it's undeniable to everyone that the proof is presented to), there may be places in which a subjective proof is more practical. When I say subjective, I mean that the proof is a conversation between the prover and the person who the proof is being presented to. At the end of such a proof, the validity need only be undeniable to the subject of the proof. I would say that the Turing Test is an example of a subjective proof, since an observer could just be watching two random sentence generators and mistake them to be humans. The determination of the validity of the proof is left to the person who the proof is being presented to. The proof is also mainly intended for that person/subject.
I haven't found anything on this, so are there more proofs of this nature? Does the field of logic allow subjective proofs or are all proofs strictly objective (if you prove something for someone, it must be provable to everyone)? Any help on the subject is greatly appreciated.