Proof

In the previous chapter, we built up the tools of propositional and quantified logic: propositions, connectives, truth tables, the laws of logic, and quantifiers. Along the way, we occasionally ran into the implication — one proposition claiming that another must follow from it — without stopping to give it the attention it deserves.

That attention is where we begin this chapter. From there, we turn those tools toward their real purpose: building arguments whose conclusions are guaranteed to be true, and proving that mathematical statements — not just isolated propositions, but general claims about numbers, shapes, and structures — are true beyond any doubt.