Changes

The [[root]] [[meaning]] of the word 'postulate' is to 'demand'; for instance, Euclid demands of us that we agree that the some [[things]] can be done, e.g. any two points can be joined by a straight line, etc.

The [[root]] [[meaning]] of the word 'postulate' is to 'demand'; for instance, Euclid demands of us that we agree that the some [[things]] can be done, e.g. any two points can be joined by a straight line, etc.

Ancient geometers maintained some distinction between axioms and postulates. While commenting [http://en.wikipedia.org/wiki/Euclid Euclid]'s books Proclus remarks that "[http://en.wikipedia.org/wiki/Geminus Geminus] held that this [4th] Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property". [http://en.wikipedia.org/wiki/Boethius Boethius] [[translated]] 'postulate' as petitio and called the axioms notiones communes but in later [[manuscript]]s this usage was not always strictly kept.

Ancient geometers maintained some distinction between axioms and postulates. While commenting [https://en.wikipedia.org/wiki/Euclid Euclid]'s books Proclus remarks that "[https://en.wikipedia.org/wiki/Geminus Geminus] held that this [4th] Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property". [https://en.wikipedia.org/wiki/Boethius Boethius] [[translated]] 'postulate' as petitio and called the axioms notiones communes but in later [[manuscript]]s this usage was not always strictly kept.

*Date: [http://www.wikipedia.org/wiki/16th_Century 1593]

*Date: [https://www.wikipedia.org/wiki/16th_Century 1593]

==Definitions==

==Definitions==

*1 : demand, claim

*1 : demand, claim

Logical axioms are usually [[statements]] that are taken to be [[universally]] true (e.g., A and B implies A), while non-logical 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 non-logical axiom is not a self-evident [[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, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.

Logical axioms are usually [[statements]] that are taken to be [[universally]] true (e.g., A and B implies A), while non-logical 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 non-logical axiom is not a self-evident [[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, well-understood 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.[http://en.wikipedia.org/wiki/Postulate]

Outside [[logic]] and [[mathematics]], the term "axiom" is used loosely for any established principle of some field.[https://en.wikipedia.org/wiki/Postulate]

[[Category: Logic]]

[[Category: Logic]]