In terms of helping students to understand propositional and predicate logic, with quantifiers, is there any research regarding when it is most advantageous for students studying mathematics, to first be explicitly taught the basics in logic?
In part, my question is motivated now, in relation to this question revealing a student's lack of understanding of basic propositional logic, and specifically, their lack of understanding of implication.
I understand that in undergraduate mathematics education, propositional and predicate logic with quantifiers may be covered over four weeks of a discrete math class, or first formally introduced in the early portion of an "introduction to proof" class offered in a math department, or first introduced after Calc I- III, DE, perhaps in a proof-based linear algebra course. But there are many undergraduate math programs that don't explicitly require majors to complete a full semester (or two) in logic, or math logic.
I am concerned that perhaps students are not taught early enough about logical operators like implication, or that $\leq$ means $\lt$ OR $=$, so that any expression $a\leq b$ means $a \lt b$, or $a=b$, and if either is true, then $a\leq b$ is true. Not to mention, there are students struggling with $0\lt x \lt 12$, which relies on, implicitly, an understanding of the logical operation of "AND": If $0\lt x \lt 12$, then $(x> 0$ AND $x\lt 12)$, that is, $0\lt x \lt 12$ is only true if BOTH $x>0$ AND $x\lt 12$ are true.
(But notice in all these scenarios, it is assumed that students already understand the definition of material implication: "if p, then q". And that isn't always the case for many students.)
I've just listed only a few important concepts that students seem to struggle with in understanding, even in high school.
If anyone has significant experience with respect to teaching logic, or is aware of programs that introduce basic logic explicitly, to students prior to college, or is aware of research in math education relating to when it is most advantageous to introduce logic to students studying math, I'd love to hear your input.