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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?:

Every axiom is an assumption, but not every assumption is an axiom.

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memexor
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    Maybe the attempt to make a somewhat formal distiction between these is as moot as for a distinction between lemma, proposition, theorem, corollary. – Hagen von Eitzen Apr 24 '15 at 16:17
  • @Hagen, By the term 'moot', do you mean 'subject to dispute' or instead 'of little importance/relevance' ? – memexor Apr 24 '15 at 16:28
  • You could probably get a better answer to this question on one of the language websites, like english.stackexchange.com . – DanielV Apr 24 '15 at 17:02

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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.

zoli
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  • I agree with your first half, but in practice the word "assumption" is used as a base of any argument whether that base is true or false. We use the word "assumption" specifically because it could be false: "Assuming the defendant paid his taxes, he still would be charged, so his tax history is irrelevant". Assumptions are not assumed to be scientifically true like axioms are. – DanielV Apr 24 '15 at 17:00
  • @DanielV, You are 100% right. Even so, this is exactly why it is moot to start discussing the meaning of (any) concept. As long as there is no contradiction you can attribute any meaning to anything. I admit that my version for "assumption" is not general enough. It is restricted to statements like: "Assuming that " this "geometry" is "Euclidean" the "Pythagorean" theorem is a true statement. – zoli Apr 24 '15 at 17:09
  • @zoli I like how you explain it from a historic perspective. Can you describe "premise" also? – user1534664 Apr 10 '16 at 20:42
  • @user1534664 Premises are assumptions that are not discharged at the end of a proof. They are the assumptions you take as given; as opposed to those you make and discharge in subproofs. – Graham Kemp May 15 '19 at 23:16
  • Thanks for adding those few sentences about premises. – zoli May 16 '19 at 05:28