As per usual, the reason for learning these various procedures and concepts in logic can be a bit mysterious. Why is it important that we know this? It was already unlikely that we’d build truth tables to figure out whether or not an argument we find “in the wild” is valid or invalid, or to test the formal consistency of our beliefs. Much less will we bust out our inference rules and prove that an argument is valid, given that we often must already know it’s valid if we’re going to commit to spending the time to do a natural deduction proof!
This is, in fact, true for most things you learn in college. You won’t regularly calculate the standard deviation of a distribution by hand like you do in Statistics. You won’t regularly perform integral calculus, recite the stages of the hero’s journey, identify auxiliary verbs, or conjugate the past imperfect passive in a different language. The point of learning all of these things isn’t that they will be immediately useful for your everyday life. You’ll pick up all of those useful skills on your own by simply living your life.
The point is not to give you a tool that you’ll regularly use in your everyday life. Much of what you learn in school is about giving you understanding rather than immediately useful knowledge. This is one more item in this general theme: the point is to understand deductive inferences at an intuitive level rather than to build out a toolkit that one can put to use on a Reddit discussion board or on an assembly line.
If you want to understand how a car works beyond “if I push that pedal down it goes forward,” you have to look under the hood and get your hands dirty in order to see how things fit together. The same goes for Natural Deduction: if you want to know how logical reasoning works, it is best to “look under the hood.” The inference rules are these simple components of reasoning that we understand at an intuitive level, so if we find out that a complex inference is in fact a combination of six of these simple rules, then we’ve found out how that complex inference works and we now understand something quite complex at an intuitive level.
A few reasons for learning Natural Deduction are that we can use it to understand more clearly:
1. How premises are related to their conclusions
2. What a complex inference is built out of and what it means for complex inferences to be built out of simpler inferences.
3. What it means for a premise to be irrelevant to an inference
4. How deductive inference works at a more atomic level
In the process of learning Natural Deduction, moreover, one cannot help but internalize some of the inference rules we’ve learned. The more you internalize these rules, the better sense you have of what implies what. When you’ve been doing logic as long as I have, you can pretty quickly see that “if you don’t go out with me, then I’ll be distraught” deductively implies “either you go out with me or I’ll be distraught”. With a little more thought, I can figure that it also implies “it won’t happen that both you won’t go out with me and I won’t be distraught” or “if I’m not distraught, then it means that you agreed to go out with me.’ The more logic we do, the easier these things are to recognize.