Consider this question solved. I will not pick a best answer as all of them have contributed to my understanding of the topic.Im unsure what benefit we have by formally defining the semantics of predicate logic. But I do see value in having a formal proof calculus. My point is that we would not need formal semantics to justify the inference rules of proof calculi. We could define a calculus that mimicks the laws of thought, i.e. the rules of inference that have been used by mathematicians for hundreds of years to proof their theorems. Such a calculus already exists: natural deduction. Then we would define this calculus to be sound and complete.  This can be justified by realising that the formal semantics of predicate logic is just a model. The appropriateness of the model can only be justified intuitively. Thus, by showing that natural deduction is sound and complete with reference to the formal semantics does not make natural deduction more true. It would be just as good if we would directly justify the rules of natural deduction intuitively. The detour using formal semantics gives us nothing.Then, having defined natural deduction to be sound and complete, we could show the soundness and completeness of other calculi by showing that the proofs they produce can be translated to natural deduction and vice versa.Are my reflections above correct?Why is it important to prove the soundness and completeness of proof calculi by reference to formal semantics?
Why do we need formal semantics for predicate logic?
lo.logic;semantics