I was reading vellmans how to prove it and he forms a link between formal logic and proof writing. For instance, he decomposes if p then q to not(p and not q) and similarly for other such proof writing statements. However what I don't follow is What's the motivation for writing out truth tables and evaluating conditions for them.
Usually, we write out Boolean statements like p and not q etc. and evaluate the truth tables when we wish to evaluate different conditions of p and q. Here however, we know that p is either true or not and q is determined accordingly. Why truth tables?
My question might seem a bit vague and under thought but I can't seem to grasp the fundamentals itself so all else is shaky too.