I'm currently an undergraduate student who wants to do research on the pedagogy of formal logic. As a result, I wanted to know what are some challenges that instructors (or even students for that matter) encounter when teaching introductory formal logic (propositional, first-order, modal). Proofs are the obvious answer, since natural deduction may sometimes not come across as intuitive to students, or just the abstract nature of logic in general. Some students, such as computer science, mathematics, and engineering students may find formal logic easier since it relates to some of their topics (boolean logic, for example), whereas others that aren't as logically inclined may struggle. This is what I have so far. Does anyone have any stories or other examples? Thanks!
Every discipline has its own technical terms, and it is often difficult to become accustomed to the ones used in logic. Validity, entailment, interpretation, model, theory, satisfaction, are among the terms used in logic that differ from their ordinary meanings. In particular, 'valid' is one that nearly all newcomers to logic have difficulty with. People are accustomed to using it as an evaluative term meaning that an argument is good, and it takes a while to get used to using it to describe an argument in which if the premises are true the conclusion follows necessarily.
Logic over the last 120 years or so has evolved away from its traditional roots of being concerned with what distinguishes a good argument from a bad argument. Logic is mainly a formal exercise and much of what used to be considered logic is now part of epistemology. The evaluation of grounds, reasons and justification is no longer something that logicians typically concern themselves with. This distinction is something that causes confusion among students. They may have trouble accepting, for example, that an argument is not invalid because it has irrelevant premises, or premises that don't provide a reason to believe the conclusion. Or that an argument with inconsistent premises is always valid.
Some students may have difficulty with the idea that there are many logics. It is usual to teach classical logic to beginners, since it is the most commonly used logic, but there are non-classical logics. If students advance to learning modal logic they will at least have to get used to the idea that there are many distinct systems of modal logic.
The logic of conditionals tends to cause problems for beginners in logic, though this is partly because it is often badly taught. Students are introduced to material implication, because this is the simplest of all the conditionals and the only one that is a truth function, and they are often left with the misleading impression that it is the only way to represent conditionals in logic. This leads to a great deal of confusion as to why ordinary English statements using 'if' turn out incorrect if you translate them directly using material implication. The so-called paradoxes of material implication are not really paradoxes at all; they are just examples of conditionals that are not material implications.
An obvious point is that formal logic looks like mathematics, so inevitably any math-phobic students will struggle.