Logic

In most mathematics courses a student has taken so far, they have usually been presented with a collection of results — facts they can use whenever warranted. Polynomials, for instance, come with a wide variety of such facts: how to divide one polynomial by another, how to find the roots of a quadratic equation, and so on.

Geometry offers just as many. The Angle Bisector Theorem tells us that the bisector of an angle in a triangle divides the opposite side into two segments proportional to the triangle’s other two sides. A wide variety of theorems describe the chords of a circle. And the famous Pythagorean Theorem relates the three sides of a right triangle.

At some point, a student may wonder why these results are true — how do we know for certain they’re correct, rather than simply guesses that happen to look accurate? The answer is that we use a system of logic to rigorously prove, beyond any doubt, that such results are true rather than mere guesses. With a system of logic in hand, we no longer have to wonder whether the results we’ve come to rely on are mystical in nature, handed down on stone tablets — they are results we can work out ourselves, with infinite precision.

Over the next two chapters, we build up a system of logic that lets us derive the truth of many of the results we’ve already seen, along with results yet to come.