Classical logic includes within it computability, and computability includes all the processes that we currently know how to construct.
By "includes within it", I mean that if you have some process for coming to a conclusion about some question, and that process fits into one of the standard models of computation (and therefore all of them), you can describe it within logic and then logically reason your way to an outcome. You can prove: "This process starts with the data [abc], and follows rules [xyz], so at the next step it have the data [def], so ......." and end with "this process terminates with the conclusion [uvw]." Thus, conclusions made by any other system can be embedded into conclusions made by logic.
Since we do not know how to build any non-computable system, logic is sufficient. If you come up with some kind of strange non-logical system, a logician can use logic to describe it and determine its conclusions perfectly.