The following attempt of mine at defining these terms, reflects my current understanding of them:
Assumption:
$\quad$ A statement accepted as true without proof being required.
Axiom:
$\quad$ A statement deemed by a system of formal logic to be intrinsically true.
Premise:
$\quad$ An assumption present within a logical argument.
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Are these definitions okay?
Can they be improved upon?
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The following statement seems to be correct given my definitions, but putting my definitions aside, is it actually correct?:
$\endgroup$ 3Every axiom is an assumption, but not every assumption is an axiom.
1 Answer
$\begingroup$From the point of view of the history of science there is difference between the concepts described by the OP.
The ancient Greeks considered those statements to be axioms that they saw to be self explanatory, intuitively clear, not requiring proof..., etc. It turned out, however, that certain theorems could replace certain axioms and vice versa. So, the apodictic truth of the "axioms" became questionable soon.
An assumption is still, in modern times, a statement whose truth (if not contradictory) is assumed for the purpose of composing a clear statement to be proved.
In modern times a set of assumptions cannot be distinguished (would be "moot" to) from a set of axioms. Usually we are talking about axioms rather than assumptions when a given set of statements serves as a firm basis of a theory.
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