Consider the question.
Given the nature of a sentence $S$, it there any way to tell how many different ways you can prove this sentence?
Proofs are not distinct if we have a situation such as: $P \implies Q$ and we want to prove that $A\implies B$ and say we have $A\implies P$ and $Q\implies B$ then $A\implies P\implies Q\implies B$ is the same proof as $A \implies P \implies B$. So the methods of proving such statements must be as "concise" as possible. However this logically means that given a set of axioms, they can directly imply the result. This is an example of conditions.
So my real question(s) is/are:
What conditions are necessary to distinguish between proof (proofs being a set of logical sentences determining the desired result as true) such that the amount of proofs can be counted?
Is it easily determined which conditions imply the cardinality of the set of proofs?
eg. if I say conditions $X$ for proofs I may be able to get countibly amount infinite proofs. (Which wouldn't be useful). Or as the above condition, which implies proofs are just applications of the axioms. (Also not very useful).