I highly recommend a course / book in symbolic logic. Sometimes it is called propositional logic.
Essentially it teaches you how to formalize arguments into their abstract symbolic forms, and then evaluate the argument on that form. This is a lot easier than trying to determine an argument’s validity purely from its English written version.
I used Klenk’s Understanding Symbolic Logic, but I’m sure there are more modern courses or books.
I don't think symbolic logic is a great tool for most people and most applications. For that, syllogistic logic is a better fit.
The modern Fregean paradigm was motivated by the need for a formalism to solve mathematical problems. It was advanced from a position of complete indifference to the relationship between logic and language/grammar. However, the Aristotelian tradition that dominated logic for two thousand years before Frege is motivated explicitly by the desire to clarify, draw out, and make conspicuous the logical structures within grammar so that arguments can be better evaluated for soundness.
For a rudimentary introduction to this space, Joseph's "The Trivium: The Liberal Arts of Logic, Grammar, and Rhetoric" is a good resource. For something a bit more thorough and specifically focused on logic, Coffey's two volumes of "The Science of Logic" comes highly recommended[1][2].
[2] https://archive.org/details/thescienceoflogi01coffuoft/page/...
The beginning of the book I linked to has a section on converting English phrases into formal logical symbols. That part is relevant to the link and to anyone looking to clarify their arguments when writing.
The rest of the book may not be as directly related, although I did find it useful to clarifying my thoughts and structuring arguments more clearly.
Propositional logic is about Boolean inferences like "If A then B, A, therefore B". The link you provided expands this to the more complex predicate logic, of which syllogisms (discussed in the article) are special cases. E.g. "Everything that is X is also Y. This is X. Therefore, this is Y."
Though I would also note that most introductions to symbolic logic do ommit the (in this context) most important part: On how to translate natural language arguments to formal logic, and the other way round. This task is very much non-trivial. A few textbooks aimed at philosophers do this, though most don't.
Actually, propositional logic is a lot more than that. The core is really about proof theory, natural deduction, and axiomatic systems that form the precursor of first-order logic and the foundations of model theory.
Well, natural deduction is one specific form of proof theory (it's a kind of proof system), but there are others. Moreover, proof theory and model theory are independent (neither relies on the other) and various logicians would say that we don't really need one of them, though they would disagree on which. Proof theorists tend to be skeptical of the necessity of model theory, model theorists tend to be skeptical of the necessity of proof theory. Mathematicians tend to be skeptical of both.
But yes, to the degree that any such theoretical parts are part of predicate logic, they are already part of propositional logic. The former is an extension of the latter. Perhaps similar to how higher-order logic (simple type theory) extends or generalizes both propositional and first-order predicate logic.
By "natural deduction", I assume both of you refer to the system which is related to "sequent calculus" ? [1].
I bet you'd probably see Klenk and Copi, both very old textbooks kept at least reasonably up to date, used in some of today's classes. "The Logic Book" (Bergmann, Moor, Nelson) is used around sometimes. I think there'd probably also be an appetite among some HN people to tackle the MIT OpenCourseware offerings, which at first glance look pretty good and challenging.
I can recommend E.J. Lemmon's Beginning Logic as a first book. It also contains an appendix with a list of important logic books and brief description of them. I'm curious to know whether a more recent, equally well-done, list exists.