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The phrase "axiomatic argument" is correct and usable in written English.
It can be used in contexts where you are discussing a statement or proposition that is accepted as true without controversy or question, often in philosophical or logical discussions.
Example: "In his essay, he presented an axiomatic argument that the laws of nature are consistent and unchanging."
Alternatives: "self-evident claim" or "fundamental assertion".
Exact(1)
Latour's axiomatic argument that '[n]o place dominates enough to be global and no place is self-contained enough to be local' (p.204) is - in contrast to Burawoy's and Scholte's argumentation - invoked as a plea for abstaining 'from ever using the local/global repertoire' (p.206).
Similar(58)
Stoic syllogistic is an argumental deductive system consisting of five types of indemonstrables or axiomatic arguments and four inference rules, called themata.
The axiomatic tone of this statement belies its quietly subversive implications.
This was "a reminder," presumably, because there was no need to debate something so axiomatic.
Aristotle somewhat similarly had argued that the logical principle of noncontradiction, which he took to express a highly general truth about the world, must be accepted as axiomatic on the ground that its correctness is presupposed in any argument directed against it.
The hope for metamathematics was that, by using only intuitively convincing, weak number-theoretic arguments (called finitary methods), unimpeachable proofs of the consistency of such theories as axiomatic set theory could be given.
They begin with the axiomatic definition of Entropy – still falsely portrayed as the "measure of information" – then dive into involved technical arguments on "typicality" and "asymptotics".
Deploying a range of sophisticated logical arguments developed over the last 25 years, Christian philosophers have revived faith as a subject of rigorous academic debate, steadily chipping away at the assumption -- all but axiomatic in philosophy since the Enlightenment -- that belief in God is logically indefensible.
Axiomatic, even.
It sounds axiomatic.
Their innocence is axiomatic.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com