[Source:] Having studied maths gives you a particular way of thinking through problems that Intro to Logic just doesn't.
Will someone please explain and explicate the quote above?
Please pardon me if I ought to have posted this in Philosophy SE; I posted here because I am conjecturing that you have more scholars of maths than Philosophy SE?
In the United States, some universities offer "Introduction to Logic" courses. These courses are often offered to undergraduates who are not majoring in mathematics, as a way that the undergraduates can satisfy requirements that they take a certain number of "math" and/or "philosophy" classes. Some of these courses teach:
Similar length Math and Computer Science courses cover a lot more.
In practice, a good introductory university-level course in relational databases covers about four times as much material, and may cover (in an incidental manner) all of the logic puzzles and basic concepts taught in the "Introduction to Logic" course.
Similarly, many secondary schools teach "Geometry" as a "how to write a mathematical proof" class. The proofs are at a level that non-mathematicians consider thorough; mathematicians often have much more detailed proofs for the same concepts. In the process of doing these proofs, the students learn most of the logic concepts taught in the "Introduction to Logic" course discussed above.
Statistics courses teach approaches that allow drawing inferences with less than 100% confidence.
A good math course teaches "ways of thinking through problems" that are not taught in the "Introduction to Logic" course discussed above. For example:
The process of using the relationships (found above) to solve for what you want to know follows a logical process. This process often includes naming intermediate concepts, and looking at things from multiple points of view.
"Checks by substitution" follow a logical process.
I think the only sense in which the quote is accurate is if you interpret "maths" broadly and "Intro to Logic" narrowly. Intro to Logic would only introduce limited proof techniques tailored to elementary logic. Mathematics is far richer in both content and techniques than would be encountered in Intro to Logic.
But if you replace "Intro to Logic" with "mathematical logic," then the claim is less supportable. Certainly mathematical logic has a content more narrow than all of mathematics, but this is true for any field of mathematics. The proof techniques used throughout mathematical logic are quite broad, although certainly not encompassing all proof techniques encountered throughout mathematics.
To get a sense of the breadth of mathematic logic, look at the answers to the MO question, "What are some important but still unsolved problems in mathematical logic?"
To supplement @JosephO'Rourke's good answer, I would note that the "logicist" philosophy of mathematics is by no means inclusive of mathematical practice... by a long shot... despite various enthusiasts' tendency to implicitly or explicitly claim that first-order logic is the underpinning for mathematics, that formal set theory (axiomatized in first-order logic) is the next layer, ... and, supposedly, everything else is superstructure. An afterthought.
But, srsly, apart from the haute-bourgeois pretenses (however clever) of the (slave-holding, not-working...) classical Greeks, it is easily arguable that the true appeal of mathematics for human civilization is its explanatory power about actual phenomena in the world... even phenomena human-abstracted from chaotic physical events.
I would claim that "mathematical logic", and, to a much lesser degree, "logic" in a more general sense, is a program of backfilling upon realization of weaknesses in prior understanding. Not with the expectation that very much is truly in error, but that the same truths need deeper foundations from some viewpoint...
Thus, in particular, to only look at "(mathematical) logic" is to look at a sort of symptom rather than cause or most substantial effect. Thus, analysis of this symptom, no matter how energetic, can scarcely manage to divine the real events, causes, and causalities.
