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In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either selfevident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.
In mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "nonlogical axioms". In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Unlike theorems, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow (otherwise they would be classified as theorems).
Logical axioms are usually statements that are taken to be universally true (e.g., A and B implies A), while nonlogical axioms (e.g., a + b = b + a) are actually defining properties for the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom," "postulate", and "assumption" may be used interchangeably. In general, a nonlogical axiom is not a selfevident truth, but rather a formal logical expression used in deduction to build a mathematical theory. To axiomatize a system of knowledge is to show that its claims can be derived from a small, wellunderstood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.
Outside logic and mathematics, the term "axiom" is used loosely for any established principle of some field.
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