Truth tables and propositional calculus can be learned procedurally but they teach conceptual thinking. This was the original intent from Leibniz on. And possibly according to some, Leibniz did not know conceptual thinking in math, since his knowledge of math was mostly what we call high school today.
Leibniz was hoping to settle disputes by calculation.
Indeed, the universal characteristic was intended by Leibniz as an instrument for the effective calculation of truths. Like formal logic systems, it would be a language capable of representing valid reasoning patterns by means of the use of symbols. Unlike formal logic systems, however, the universal language would also express the content of human reasoning in addition to its formal structure. In Leibniz’s mind, “this language will be the greatest instrument of reason,” for “when there are disputes among persons, we can simply say: Let us calculate, without further ado, and see who is right” (The Art of Discovery (1685); C, 176 (W, 51)).
The logic at Stanford group includes the Philosophy and Computer Science Departments.
So in Leibniz’s vision, learning truth tables and propositional calculus is a procedural way to conceptual thinking.
This same thinking is why they are taught in transition to proof classes.
We can use the same algebraic reasoning methods taught in K-12 to do logic in college.
This is why 1950s and 1960s New Math wanted to teach truth tables in K-12. This is a procedural way to teach conceptual thinking.
When symbols are linked to a method, the method can be taught by learning how to manipulate the symbols according to rules.
This is a procedural way to learn the method. The method is acquired as a skill.
If the method is also a way of thinking then the method has been taught procedurally.
Or at least that is the hope. It may be only partially that way in terms of how the human brain and mind operate.
In this way of thinking, there is no mathematical thinking part of the brain. Instead, the brain functions in certain ways and we use those to learn methods.
We still have uncertainty and confusion in the old way about all sorts of things. To some extent, everything we learn in math, we still retain traces of uncertainty about what they really mean.