Axiomatic

Axiomatic (pronounced ak-see-uh-mat-ik)

(1) Of or pertaining to the nature of an axiom.

(2) That which is self-evident or unquestionable; the obvious.

(3) Containing maxims; aphoristic.

(4) In formal logic (or any logical system), as axiomatic system, a set of axioms from which theorems can be derived by the application of by transformation rules.

(5) In mathematics, relating to or containing axioms (now less common).

1797: From the Ancient Greek ἀξιωματικός (axiōmatikós), from ἀξίωμα ((axíōma), genitive axiomatos) (a self-evident principle), the construct being axiōmat (stem of axíōma) + the suffix ikos (and the related ic).  The now less common form axiomatical was known as early as the 1580s.  The ikos suffix was from κός (kós) with an added i, from i-stems such as φυσι-κός (phusi-kós) (natural), through the same process by which ῑ́της (ī́tēs) developed from της (tēs), occurring in some original case and later used freely.  It was cognate with the Latin icus and the Proto-Germanic igaz, from which came Old English iġ (which in Modern English ultimately was resolved as y), the Old High German ig and the Gothic eigs.  The ic suffix forms adjectives from other parts of speech.  It occurred originally in Greek and Latin loanwords (metallic; poetic; archaic; public et al) and, on this model, was used as an adjective-forming suffix with the particular sense of “having some characteristics of”, as opposed to the simple attributive use of the base noun (balletic; sophomoric et al), “in the style of” (Byronic; Miltonic et al), or “pertaining to a family of peoples or languages” (Finnic; Semitic; Turkic).  The -ic suffix was from the Middle English -ik, from the Old French -ique, from the Latin -icus, from the primitive Indo-European -kos & -ḱos, formed with the i-stem suffix -i- and the adjectival suffix -kos & -ḱos.  The form existed also in the Ancient Greek as -ικός (-ikós), in Sanskrit as -इक (-ika) and the Old Church Slavonic as -ъкъ (-ŭkŭ); A doublet of -y.  In European languages, adding -kos to noun stems carried the meaning "characteristic of, like, typical, pertaining to" while on adjectival stems it acted emphatically; in English it's always been used to form adjectives from nouns with the meaning “of or pertaining to”.  A precise technical use exists in physical chemistry where it's used to denote certain chemical compounds in which a specified chemical element has a higher oxidation number than in the equivalent compound whose name ends in the suffix -ous; (eg sulphuric acid (H₂SO₄) has more oxygen atoms per molecule than sulphurous acid (H₂SO₃).  Axiomatic & axiomatical are adjectives, axiomatize & axiomatize are verbs and axiomatically is an adverb.  Clumsy forms (sometimes hyphenated) like nonaxiomatic & unaxiomatic are created as required.

In mathematics (notably in including geometry, algebra, and set theory), an axiomatic system (“deductive system” or “formal system” seem to be the more fashionable terms) is a set of axioms or postulates, which, coupled with rules of inference, can be used to derive theorems or statements from those axioms.  In mathematics, there are collections of equations which can be used to document the processes but in any form of applied logic these systems provide a rigorous foundation for reasoning and proof, using what can be reduced to a mathematical process.  In axiomatic systems, axioms are assumed to be true without proof and the rules of inference are used to derive new statements from the axioms; theorems derived from the axioms are then considered to be true, based on the validity of the axioms and the rules of inference.

Axiomatic: If crooked Hillary Clinton is using a cell phone, she will be deleting something.

In the discipline of philosophy, even those parts which are not inherently mathematical (such as formal logic), the axiomatic system works in a similar way in that a statement, proposition, or principle that is considered self-evident or universally accepted without needing to be proven.  Axioms are thus often used as the starting point for logical reasoning or the foundation upon which a system of thought or theory is built, assumed to be true and are not subject to further analysis or questioning within the context of the system they are part of.  There are a number of highly technical rules which define whether a axiomatic system can be described as “consistent” but that means that within its own terms it contains nothing contradictory.  In other words, from the elements of any axiomatic system, it’s not possible to be either proven or disproven. This differs from one labeled “independent” in that that status is defined by them not being proven or disproven from other axioms in the system.  An axiomatic system is labeled “complete” if for every statement, either itself or its negation is derivable from the system's axioms (implicit in which is that every statement is capable of being proven true or false).

Not axiomatic: The real Lindsay Lohan and Take Two Interactive’s alleged doppelganger in Grand Theft Auto V (GTA5).

Lawyers too like the word “axiomatic”, possibly because concepts like “foreseeability” and “causation” are such an essential part of their training.  The use though exists within different parameters to that of mathematics.  In Lindsay Lohan vs Take Two Interactive Software Inc et al (APL-2017—00027 and APL-2017-00028 (November 2017)), the New York Court of Appeals held that a certain section of an act “categorically excludes works of fiction, a protected category of expression beyond the narrow scope of the statutory phrases advertising and trade”, noting the US Supreme Court (USSC) had “reversed course to recognize First Amendment protections for fiction”.  The Court of Appeal explained that after the USSC “limited Section 51 claims for fictionalization” to factual works that place persons in a false light, subsequent case law both isolated the commercial interest protected by the right of publicity and recognized “the right of publicity does not attach” where “it is evident to the public that the events so depicted are fictitious.”  The judgment noted with approval the decision of the California Supreme Court which “famously” recognized fiction writers may “more persuasively be able to more accurately express themselves by weaving into the tale persons or events familiar to their readers”, adding “correctly”, that “the choice is theirs”.  “This categorical protection is now axiomatic”.  Once can see what the judges meant and of course they were correct but what can be held to be “axiomatic” in law can differ from the same thing in mathematics because in the world of numbers, there is no superior court able to rule 2+2=5.  Their position is more akin to the philosophers who for centuries until 1697 could regard as inviolate the axiom to “all swans are white and all non-white birds are not swans”.