Mathematical Proof and the Principles of Mathematics/Logic
As mentioned in the introduction, this is not meant to be an exposition on logic. For one thing that would be a book in itself. For another, our goal is to provide you what you need to know to do mathematics proper and you don't need to know the internal workings for that. If you think of logic as a car, we just need to know how to operate the steering wheel, gas pedal and brakes; we don't need to know the principles of the internal combustion engine or where the gas comes from.
To that end, we will avoid using the symbolism of logic, though we will introduce the notation so you will know what it means if you encounter it. We will accept the rules of logical deduction as valid, and leave the justification that they are valid to logicians. Most are intuitively true anyway. There are actually quite a few rules that have to be learned, but we will introduce them as they are used so they don't come at you all at once.