Logic and Mathematics are deductive systems, where the conclusion of a successful argument follows necessarily from its premises, given the axioms of the system you’re using: number theory, geometry, predicate logic, etc.
Valid argument - An argument is valid when it contains no logical fallacies.
Sound argument - An argument that is valid and whose premises are all true. In other words, the premises are true and the conclusion necessarily follows from them, making the conclusion true as well.
Formal proof - A set of steps from axiom(s) and previous proof(s) which follows the rules of induction of a mathematical system.