This essay attempts to equip the reader with a familiarity with axioms and their role in the critical thinking framework.
I shall first attempt to convey my notion of an axiom starting from scratch. Then I will illustrate how my notion of an axiom relates to premises, propositions, and rules of deduction within the deductive realm of critical thinking. Finally, I will compare my notion to the cited dictionary definitions:
1. a self-evident truth that requires no proof.
2. a universally accepted principle or rule.
3. Logic, Math a proposition that is assumed without proof for the sake of studying the consequences that follow from it.
Part I -- What is an axiom?
To begin, I think the following definition most clearly and succinctly conveys my notion of an axiom:
An axiom is any proposition serving as a standard of truth within some universe of discourse.
If it is not perfectly clear what is meant by that, consider other standards:
- A meter is a standard governing an entire system of length measurements. How long is something? Compare it to a meter stick.
- 440 Hz is a standard governing an entire system of musical tones. What is the name of the tone that is sounding? Compare its frequency to A 440.
- A second is a standard governing an entire system of characterizing time. Did this runner just break a 50-yard dash record? Compare his time to the previous best time.
But the issue at hand is truth, whose standard is given by axioms. How do we decide whether a proposition is true? Compare it to the axioms. We shall return to the notion of "compare" in this context later; for now, consider it an abstract analog to comparing a length of fabric to a meter stick (although you should be able to intuit some trivial comparisons, such as if "All dogs are black" is an axiom, then "All dogs are not black" would not be true as it does not comply with the standard of truth).
That's really all that I think the notion of axiom ought to encapsulate. That is, any proposition that is serving as a standard of truth in some universe of discourse is an axiom in that universe of discourse.