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I'm trying to understand if a couple of statements would be considered axiomatic:

Example 1: "murder is the unjustified killing of a person; if there was a murder, then a person was killed unjustifiably"

Example 2 comes from a source related to math, which claims that this is axiomatic: "2+2=4"

Are the following examples valid axiomatic statements?

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    2+2=4 is not an axiom of arithmetic: it is a theorem, i.e. it is proved from axioms. Apr 15, 2021 at 19:15
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    The first one is a definition. Apr 15, 2021 at 19:15
  • "As used in modern logic, an axiom is a premise or starting point for reasoning." - en.wikipedia.org/wiki/Axiom I assume this means that the first example is axiomatic in the context of "modern logic", given that it's a premise or a starting point of reasoning.
    – Kiril
    Apr 15, 2021 at 19:40
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    Whether something is an axiom depends entirely on the system you're using it in! It just means that within that system, you assume its truth without justification or proof. So, you could hypothetically construct and work in a system of arithmetic in which 2+2=4 was taken to be an axiom. Within such a system, it would indeed be an axiom. But, generally, this is never done, since we have perfectly good systems of arithmetic to work in which let us prove 2+2=4 from more basic-seeming axioms!
    – thorimur
    Apr 15, 2021 at 21:09
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    It is not a good idea to go on vague descriptions in dictionaries or Wikipedia, their purpose is just to give a general idea. If you want more precise conditions for deciding what is or is not an axiom you need to specify the context. "Is X an axiom?" is not a meaningful question, only "Is X an axiom in such and such system?" is. Are you working on exercises from some textbook, course?
    – Conifold
    Apr 15, 2021 at 23:29

1 Answer 1

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In math axioms are usually belong to the most foundational level as inference rules of a deductive system. So in arithmetic, we need below axioms to derive your correct true proposition "2+2=4". Normally mathematicians won't treat a higher level non-simple relation such as "2+2=4" as an axiom in their theory, it should be more appropriately as a theorem or proposition derived from more basic axioms and definitions.

So normally in arithmetic, we need below axioms for your case which is much more efficient than your case as an axiom:

Axiom 1 (game rule): assume the number 1 to be defined, and also the operation x+1 to be defined as commonly hold, ie, the adding of 1 to a given number x.

Axiom 2: define the numbers 2, 3, 4 by the equalities:

(1) 1+1=2; (2) 2+1=3; (3) 3+1=4;

Axiom 3: define the operation x+2 by the relation: x+2=(x+1)+1.

Given all above axioms and definition, we have:

2+2=(2+1)+1 (axiom 3 & 1)

(2+1)+1=3+1 (axiom 2)

3+1=4 (axiom 2)

so 2+2=4 Q.E.D.

Note some mathematicians like to regard some axioms as mere definitions in disguise such as Poincare, so your example 1 can be regarded as definition or axiom, there's no clear cut borderline we can draw in most cases, it's really equivalent for most practical purposes. You see, your proposed axiom "2+2=4" alone cannot help to make further inference about 2+3 no matter how hard you try, while above more foundational axiom can easily infer about "x+n=(x+n-1)+1=..." for arbitrary n as a natural number...

A more official arithmetic axiomatic system in mathematical logic is called Peano axioms (PA). There you'll see the addition operation is similarly defined as my answer above.

In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.

When the Peano axioms were first proposed, Bertrand Russell and others agreed that these axioms implicitly defined what we mean by a "natural number". Henri Poincaré was more cautious, saying they only defined natural numbers if they were consistent; if there is a proof that starts from just these axioms and derives a contradiction such as 0 = 1, then the axioms are inconsistent, and don't define anything. In 1900, David Hilbert posed the problem of proving their consistency using only finitistic methods as the second of his twenty-three problems. In 1931, Kurt Gödel proved his second incompleteness theorem, which shows that such a consistency proof cannot be formalized within Peano arithmetic itself.

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