This question already has an answer here:
What is it that mathematicians, and more likely perhaps philosophers, give as an explicit justification that any method of formal logic, which is actually used by mathematicians, or even by automatic theorem provers, to prove anything, be it some theorem or some argument, would be effectively the best method to make valid deductions.
As far as I have been able to ascertain, there is no such justification.
Apparently, all logical proofs, formal and informal, seem to rely ultimately on the intuition of at least some human being as to what formulas are logical truths, and more likely on the consensus of the specialists since Aristotle as to what formulas are logical truths, consensus which itself seems to rely ultimately on the intuition each specialist may have as to what formulas are logical truths, such as for example p and q implies p, the Modus Tollens, Aristotle's syllogisms etc.
While closely related to my previous question on the justification of systems of "logical calculus", the focus here is not on the usefulness of the methods used but on the foundation of formal logic as articulated or even theorised from the perspective of each of the various methods used.