It’s logical (again) because there’s a method to the madness.
You are confusing logical and logic. Methods are logical because conceived in a logical way. But I couldn't care less about that because logically valid plus false premises give false conclusion.
My point is that methods are not logic.
Logic isn't a method.
People invent methods and they do it usually from experience with a bit of hopefully logical reasoning but mostly it's experience.
Among other things, Aristotle proposed a method of logic. Not logic itself. A method of logic. Irrespective of the merit of it, it was crucial in the fact that for the first time in human history it identified logic. For us. Because, however incredible that is, millions of people can go about their lives for centuries and indeed millennia without ever noticing that they have a logical capability and that they do, routinely, logical inferences without thinking about it because their brain does logical inferences for them.
So, it's not because there is a method, a logical method, that it is logic. Whoever claims to have a method of logic should provide the justification that this method is correct, i.e. that it is a model of logic.
There is no justification that any of the methods used in mathematical logic would be correct. Articulate mathematicians, mostly philosophers in fact, people like Tarski, Russell, Quine, never articulated any good reason to accept mathematical logic as a model of logic. They didn't even try. They expedited the subject in a few cursory lines. We spend a lot of time to argue about this. They didn't. They assumed a few basic hypotheses, for sure very reasonable, but mostly just wrong. And their followers don't have a brain they could use to consider the issue to begin with and so took these people on trust and here we are.
So, yes, they have a method, and it's logical, but who cares? The premises are false. Not just one or two. Most of what mathematicians assume about logic is wrong. They think they understand proof by contradiction: Wrong. They think there's just one possibility for the material implication: Wrong. They think the Scholastic were wrong: No, they were not. The think Aristotle's logic is limited: Wrong. There is so much they don't understand, it's just staggering, and yet, here they are, talking the talk as if the experts on logic, pompous, arrogant, dismissive, incurious.
Also, I didn't always thought that way. I started out only a few years ago with a very favourable opinion of them. I had to realise the hard way that they are just intellectual workers, mindlessly repeating all their lives the little they have learnt at school.
Nobody cares that there should be a method if the method is wrong. Logical, sure, logic, no.
And of course, once mathematicians invented their own flawed method of logic, they apparently started to think likewise. They successfully fucked their own mind. Read Tarski. And intelligent, articulate, well-read true intellectual. Unfortunately, fo one hundred thinkers, only one produces value. He didn't. Instead of looking critically at the foundation of mathematical logic, he spend his time "deriving". Did you notice this word? Mathematicians don't reason logically. They derive the consequences from the axioms using rules. Not logic. They use rules. A computer could do it. Methodical, sure. But who cares?
EB