Mathematical proof

Other theories

In traditional Aristotelian logic, deductive reasoning is inference in which the conclusion is of no greater generality than the premises, as opposed to adductive and inductive reasoning, where the conclusion is of greater generality than the premises. Other theories of logic define deductive reasoning as inference in which the conclusion is just as certain as the premises, as opposed to inductive reasoning, where the conclusion can have less certainty than the premises. In both approaches, the conclusion of a deductive inference is necessitated by the premises: the premises can't be true while the conclusion is false. (In Aristotelian logic, the premises in inductive reasoning can also be related in this way to the conclusion.)

An axiom is a sentence or proposition that is taken for granted as true, and serves as a starting point for deducing other truths.
In certain epistemological theories, an axiom is a self-evident truth upon which other knowledge must rest, and from which other knowledge is built up. An axiom in this sense can be known before one knows any of these other propositions. Not all epistemologists agree that any axioms, understood in that sense, exist.

In logic and mathematics, an axiom is not necessarily a self-evident truth, but rather a formal logical expression used in a deduction to yield further results. To axiomatize a system of knowledge is to show that all of its claims can be derived from a small set of sentences that are independent of one another. This does not imply that they could have been known independently; and there are typically multiple ways to axiomatize a given system of knowledge (such as arithmetic). Mathematics distinguishes two types of axioms: logical axioms and non-logical axioms.