[DiscourseFan]

As someone who tutored a Formal Reasoning class for a top philosophy department, I don't think we should be teaching anyone logic. It's rigid, its made up, and it doesn't help much for most things. I think the way proofs are written in math is cool, and I think the fact that mathematicians are, in the end, aware that it is impossible to rigidly demonstrate anything is perfectly fine and it makes their proofs a lot easier to read and interpret. Formal logic should never be treated as first order or ever used as such, it will only hamper mathematics. I understand the need for a rigorous language to describe mathematics, I understand why people like it so much, and why it has found so much use. It is still not the groundwork (because there is none).

[cubefox]

Would you agree to at least list some of the common natural deduction inference rules in an introductory class, and do a few exercises with them? Like, for propositional logic, modus ponens, modus tollens, contraposition, transitivity of the conditional, conditional proof, proof by contradiction, de Morgan's laws, distributive laws, etc, and some inference rules for quantified predicate logic?

I agree that full formal logic could be too much irrelevant information, but I think many experienced people underestimate how non-obvious the basic inference rules are to novices, and how confused people are about just being told to produce "convincing arguments". The important part is that the argument has to be truth preserving, unlike a "convincing argument" or "proof" an attorney might give in court. It is very hard to understand this difference if one has only a hazy idea of logic and deduction vs induction.

[DiscourseFan]

I think all proofs are just "convincing arguments." Computation would make that seem meaningless, but computers don't operate based on truth, they are repetition machines, just like you and I. But I've been reading an Introductory Topology book recently (Munkres) and the entire first chapter is explicating at a relatively high level of detail all the things you describe and moreso. But its necessary, in some sense. A lot of the set theoretical rules were employed in topology to create more generalized forms of analysis. You have to learn them because otherwise its very difficult to understand the language that's being used. But as I've argued elsewhere, if we are aware that its all based on interpretation, then I think we need to be a bit anarchistic and let students have the freedom to break those rules without knowing it, if they might produce thereby fascinating proofs and arguments. If you give people a schematic for interpretation, they're just going to follow it, and they might have difficulty deviating from it at all. The spark of human creativity comes when people are forced to, as they say, re-derive certain theorems, and while doing so they might accidentally discover some new implication of the logic that was unseen before, and produce something entirely new.