### The Theory of Axioms

Robert Howard Kroepel
Lakeside Studios
New Durham, New Hampshire, USA 03855-2107

What is an axiom?

Axiom n.

1. A self-evident or universally recognized truth; a maxim: It is an economic axiom as old as the hills that goods and services can be paid for only with goods and services (Albert Jay Nock).
2. An established rule, principle, or law.
3. A self-evident principle or one that is accepted as true without proof as the basis for argument; a postulate.
[Middle English, from Old French axiome, from Latin axima, aximat-, from Greek, from axios, worthy.]

Axiom: n. [L. axioma, Gr. ? that which is thought worthy, that which is assumed, a basis of demonstration, a principle, fr. ? to think worthy, fr. ? worthy, weighing as much as; cf. ? to lead, drive, also to weigh so much: cf F. axiome. See Agent, a.]

1. (Logic & Math.) A self-evident and necessary truth, or a proposition whose truth is so evident as first sight that no reasoning or demonstration can make it plainer; a proposition which it is necessary to take for granted; as, "The whole is greater than a part;'' "A object can not, at the same time, be and not be.''
2. An established principle in some art or science, which, though not a necessary truth, is universally received; as, the axioms of political economy.
Usage: An axiom is a self-evident truth which is taken for granted as the basis of reasoning. A maxim is a guiding principle sanctioned by experience, and relating especially to the practical concerns of life. An aphorism is a short sentence pithily expressing some valuable and general truth or sentiment. An adage is a saying of long-established authority and of universal application.
Source: Webster's Revised Unabridged Dictionary, © 1996, 1998 MICRA, Inc.

Axiom: n

1: a saying that widely accepted on its own merits [syn: maxim]
2: in logic: a proposition that is not susceptible of proof or disproof; its truth is assumed to be self-evident
Source: WordNet ® 1.6, © 1997 Princeton University.

Axiom: [In logic] A well-formed formula which is taken to be true without proof in the construction of a theory.
Source: The Free On-line Dictionary of Computing, © 1993-2003 Denis Howe.

Is it true that an axiom is self-evident but cannot be proven by inductive or deductive reasoning?

Axioms are observations/intuitions of real and imagined relationships/causalities/coincidentialities which have never been observed to be other than as observed/intuited and described.

I.e. the proof that an axiom is relevant to the real-world and to intuition is the lack of disconfirmation, the lack of real people/objects/events who/which are exceptions to the people/objects/events claimed in the axiom.

Thus, when we observe/intuit an axiom, we are doing so by actual/intuited observation, and until we find real/intuited people/objects/events who/which disconfirm that axiom, it remains in force.

Thus, axioms are actually developed through the use of the inductive reasoning process.

Over a huge sample of specific cases the inductive generality--the hypothesis--is developed and tested and confirmed/disconfirmed, and when confirmed, then the hypothesis is accepted as a fact, as an axiom, until further observation reveals disconfirmation.

If, in the laws of logic, the axiom is when A = B, and B = C, then A = C, then, no exceptions appearing, A = C when A = B and B = C.

Thus, an axiom shows up as an If P, Then Q form of logical argument.

IF (P) A = B and B = C, Then (Q) A = C.

Premise #1: If (P) A = B and B = C, Then (Q) A = C.
Premise #2: (P) A = B and B = C.
Conclusion: (Q) A = C.

For a real-world example, of the If (P) A = B and B = C, Then A = C logical argument concerning axioms, if identical clocks are built in which each clock is precisely and without exception identical to each and every other clocks, all clocks are alike, and each is the same as any other.

If (P) Clock A = Clock B and Clock B = Clock C, then (Q) Clock A = Clock B.

Premise #1: If (P) Clock A = Clock B and Clock B = Clock C, then (Q) Clock A = Clock C.
Premise #2: (P) Clock A = Clock B and Clock B = Clock C.

[Verified by physical evidence, i.e. the examination of Clocks A, B and C and the comparison of Clock A with Clock B and Clock B with Clock C.]
Conclusion: (Q) Clock A = Clock C.

We see herein that the Conclusion (Q) is (1) valid because it follows the If P, Then Q logical argument form/sequence and (2) true because the Premise (P) is verified by the physical evidence provided by the examination/comparison of the clocks to each other.

NOTE: The complaint that the space coordinates of each clock will be different – otherwise they would all occupy the same space coordinates, and that for all clocks to be identical they would have to be identical in all ways/dimensions/measurements INCLUDING space coordinates overlooks the point that there are other specific characteristics of clocks such as color, materials, design, engineering, and rates of functioning that could be precisely identical and which would serve as the characteristics/standards by which identity can be established, and, thus, space coordinates are not necessarily a standard by which identity is seriously determinable.

Deductive logic often takes the logical form of the P = Q = X logical argument.

The most famous P = Q = X logical argument proves that Socrates, the philosopher, is mortal:

Premise #1: (P) All men are mortal.

[Verified by the observation of a large sample of men who have been proven to be mortal because they died.]
Premise #2: (Q) Socrates is a man.

[Verified by eyewitnesses of the physical evidence of Socrates himself.]
Conclusion: (X) Socrates is mortal.

[Valid because it follows the P = Q = X logical argument form and true because the premises have been verified by physical evidence.]

Returning to the clocks example:

Premise #1: (P) Clock A = Clock B.

[Verifiable by observation/examination/comparison.]
Premise #: ((Q) Clock B = Clock C.

[Verifiable by observation/examination/comparison.]
Conclusion: (X) Clock A = Clock C.

[Valid because it follows the P = Q = X logical argument form and true because the premises have been verified.]

In these examples, we have found that If (P) A = B and B = C, Then A = C has not been observed to have been violated and therefore disconfirmed.

The lack of disconfirmation IS confirmation until disconfirmation is discovered.

Hence, by this Theory of Axioms, axioms describing observable and intuited real-world people/objects/events are valid when verified by sample cases which show no observations/intuitions which are disconfirmations of the content of the axioms.