It is important to be aware of the reasons that we study logic. There are three very significant reasons. First, the truth tables we studied tell us the exact meanings of the words such as "and," "or," "not" and so on. For instance, whenever we use or read the "If..., then" construction in a mathematical context, logic tells us exactly what is meant. Second, the rules of inference provide a system in which we can produce new information (statements) from known information. Finally, logical rules such as DeMorgan’s laws help us correctly change certain statements into (potentially more useful) statements with the same meaning. Thus, logic helps us understand the meanings of statements, and it also produces new meaningful statements.
Logic is the glue that holds strings of statements together and pins down the exact meaning of certain key phrases such as the "If..., then" or "For all" constructions. Logic is the common language that all mathematicians use, so we must have a firm grip on it in order to write and understand mathematics.
But despite its fundamental role, logic’s place is in the background of what we do, not the forefront. From here on, the beautiful symbols \(\wedge\), \(\vee\), \(\Rightarrow\), \(\Leftrightarrow\), \(\sim\), \(\forall\) and \(\exists\) are rarely written. But we are aware of their meanings constantly. When reading or writing a sentence involving mathematics we parse it with these symbols, either mentally or on scratch paper, so as to understand the true and unambiguous meaning.