Starting in this issue, PASS Maths is pleased to present a series of articles introducing some of the basic ideas behind proof and logical reasoning and showing their importance in mathematics. In this article, we shall present a brief introduction to deductive reasoning, and take a look at one of the earliest known examples of mathematical proof. If so, a more efficient version of this page that doesn't present equations as images, and should work with your browser, is available here. Deductive reasoning Given a set of facts that are known or assumed to be true, deductive reasoning is a powerful way of extending that set of facts.
All men are mortal.
Socrates is a man. In this case, the deductive step is based on the logical principle that if A implies B, and A is true, then B is true , a principle that mediaeval logicians called modus ponens . Now, if a conclusion doesn't follow from its premises, the argument is said to be invalid and no reliable judgement can be made about whether the conclusion is true, regardless of whether or not the premises are true. If the argument is valid but the premises are not true, then again the conclusion may or may not be true, but the argument can't help us decide this. Finally, if the argument is valid and the premises are true, then the argument is described as sound , and we deem the conclusion to be true.