Introduction

Reasoning is what separates a mathematical argument from a mere guess. Every proof, every valid conclusion in mathematics rests on the ability to move logically from what is known to what must follow. This chapter builds the vocabulary and the toolkit for that ability.

There are two broad styles of mathematical reasoning:

• Inductive reasoning — spotting a pattern across several specific cases and generalising it into a rule. This is the idea behind the Principle of Mathematical Induction studied earlier: verify a statement for a base case, then show that truth at one stage carries over to the next.
• Deductive reasoning — starting from statements already accepted as true and applying strict logical rules to arrive at new, certain conclusions. No pattern-spotting is involved; each step is forced by logic alone.

This chapter is entirely about deductive reasoning. Before we can reason deductively about anything, we need a precise idea of what counts as a statement in mathematics, how to build new statements out of old ones, and how to check whether a statement is true. Those three questions — what is a statement, how do we combine and negate statements, and how do we validate a statement — are the spine of everything that follows.