In order to study analysis and related fields you don't need any formal training in logic. At the same time, to learn computer science you don't need any formal training in analysis. Yet most universities require their students to study analysis to some degree. Why?
The reason is not to torture the students, or cull the first year acceptance rate. Or at least ideally these are not the reasons.
When you study abstract mathematics, you learn how to analyze a problem, and how to solve a problem using abstract thinking. The same principle can be applied to anything. Studying mathematics is the study of how to analyze, generalize and solve problems. And the more ways you know, the better you will be.
So while you don't need any formal training in logic and set theory, I would very much recommend at least a very good understanding of the basics of these fields. The basics of propositional calculus, predicate calculus, and naive set theory. These tools are very useful to mathematics, even if you don't apply them directly. They allow you to access better and higher understanding of the problems that you deal with, and how to deal with them.
As for my learned colleague who said that "this is basically common sense", while this is not far from the truth, you'd be surprised how many students I have seen having trouble with understanding the importance of quantifier order, or how to negate a proposition (with or without quantifiers). And Certainly understanding what does it mean for a function to be "not continuous everywhere" is important if you want to do analysis. (And you'd be surprised how many students will not be able to write that statement correctly.)