Robert Howard Kroepel
Copyright © 2003
Lakeside Studios
20 South Shore Road
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.]
Source: The American Heritage® Dictionary of the English Language, Fourth
Edition Copyright © 2000 by Houghton Mifflin Company. Published by Houghton
Mifflin Company. All rights reserved.
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.
Syn: Axiom, Maxim, Aphorism, Adage.
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.