postulate


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postulate:

see axiomaxiom,
in mathematics and logic, general statement accepted without proof as the basis for logically deducing other statements (theorems). Examples of axioms used widely in mathematics are those related to equality (e.g.
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Postulate

 

(from the Latin postulatum, “demand”), a proposition, condition, assumption, or rule that is accepted without proof because of certain reasons. Generally, however, a postulate is accepted with some substantiation, and this substantiation usually serves as an argument in favor of accepting the postulate. The nature of the acceptance may vary. A proposition is accepted as true, as in meaningful axiomatic theories, or as provable, as in formal axiomatic systems. Some instructions are adopted “for execution” as rules governing the formation of the formulas of some calculus or as rules of inference of the calculus that make it possible to derive theorems from the axioms. Some principles abstracted from the data of repeated experience are made the foundation for physical and other natural scientific theories; examples include principles of the type “laws of conservation.” Some positions, prescripts, and norms (legal, for example) are accorded, as a result of other positions, the status of laws. Some religious, philosophical and ideological dogmas are made the foundation for certain systems of belief. For all the diversity of these examples, there is something common to them all: without sparing arguments designed to convince us of the rationality, or legitimacy, of the postulates that we propose, we ultimately simply demand—hence the origin of the word “postulate”— acceptance. In such cases we say that the propositions advanced are postulated.

Such a broad concept, so rich in shades of meaning, naturally has many concrete, more specialized, and therefore extremely varied realizations. The following is a list of some of the most common realizations.

(1) Euclid, who gave the first known systematic axiomatic description of geometry, distinguished between postulates (αιτηματα), which assert the feasibility of certain geometrical constructions, and axioms proper, which affirm (postulate) that the results of these constructions have certain properties. Moreover, he defined axioms as propositions of a purely logical (and not geometrical) character that he accepted without proof, such as “the part is less than the whole.” The dual and not clearly drawn line of delimitation between these similar concepts persisted beyond Euclid.

(2) The terms “axiom” and “postulate” were and are often used synonymously. In particular, Euclid’s well-known fifth postulate of parallel lines is called the parallelism axiom in Hilbert’s axiomatics.

(3) At the same time, the term “axiom” is used by many authors to denote “purely logical” propositions accepted in a given theory without proof. (See, for example, A. Church, Vvedenie ν matematicheskuiu logiku, vol. 1, subsecs. 07 and 55, Moscow, 1960 [translated from English].) By contrast, the term “postulate” is used in reference to specific concepts of a given (usually mathematical) theory.

(4) According to another tradition in mathematical logic, postulates of a formal system (calculus) include axioms written in the language proper (“subjective” language) of the system and the rules of inference formulated in the metalanguage of the given theory (and therefore belonging to its metatheory). (See, for example, S. C. Kleene, Vvedenie ν metamatematiku, subsecs. 19 and 77, Moscow, 1957 [translated from English].)

(5) “Postulates” is the name given to assertions of deductive and (especially) semideductive sciences that cannot be proved, if only because the arguments and facts supporting them are exclusively experimental and inductive in character. In many such cases we speak of the assertion of the equivalency of some intuitively clear but not clearly formulated assertion or concept that is an explication (refinement) of the former and therefore formulable at a fundamentally higher level of abstraction. Examples of the first type are the fundamental principles of thermodynamics and the principle of the constancy of the speed of light and its limiting character; an example of the second type is the Church thesis in the theory of algorithms.

postulate

[′päs·chə·lət]
(mathematics)

postulate

Logic Maths an unproved and indemonstrable statement that should be taken for granted: used as an initial premise or underlying hypothesis in a process of reasoning
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Relatedly, Goodlad (1994) highlighted the importance of preparing teachers to navigate and negate the perpetual tension between the rights and interests of individual stakeholders (e.g., parents, other groups) and the role of schools in a democratic society (see also Postulate 12 in Twenty Postulates, n.d.).
In particular, U signifies that the transform of that sentence of which it is the transform is false, so by Bradwardine's second postulate (that signification is closed under consequence), it signifies that U is false.
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Suspicion of the parallel postulate goes back to Euclid, who was the first person to notice (in writing at any rate) that it was needed for some arguments.
The possible results of the contraction (that satisfy at least the success postulate) are the following:
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Consider some vague predicate, P, and a set of axioms concerning P, C, to be thought of as meaning postulates. The notion of definiteness can now be specified in a number of ways.
If quantum mechanics and Born's postulate are right, three-slit interference doesn't happen, and the pattern observed when all three slits are open should be explained entirely by the combined patterns when single and double slits are open.
Economics Professors Michel Aglietta and Antoine Reberioux postulate that a basic tenet of capitalism--that companies should be managed solely for their shareholders--is fundamentally flawed in the modern-day environment filled with liquid markets, greedy investors, and recurring financial instability.
We also postulate the possibility of ethmoid recirculation.
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I have developed the concept of the linguistic postulate to account for the way in which themes/concepts are manifested structurally across all the levels of a grammar within a given language and culture.