- 26 Mar '07 22:56 / 2 editsI recently thought that I might’ve been using the term “axiom” incorrectly, so I looked it up. According to Webster’s New Universal Unabridged, it is:

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.

Could you give a brief and clear exposition on the role of axioms—and other foundational propositions—in logic, and the question of a logical system being incapable of proving all its axioms? Along the lines of your wonderful essay on deductive, inductive and abductive reasoning?

Thank you... - 27 Mar '07 02:11

Yes, I shall. Allow me a few hours to gather my thoughts.*Originally posted by vistesd***I recently thought that I might’ve been using the term “axiom” incorrectly, so I looked it up. According to Webster’s New Universal Unabridged, it is:**

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 consequen ...[text shortened]... e lines of your wonderful essay on deductive, inductive and abductive reasoning?

Thank you... - 27 Mar '07 04:48 / 3 editsThis 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. - 27 Mar '07 04:48Part II -- What is the role of axioms?

I will assume that we are familiar with the rules of deductive logic based on applying logical operations to propositions. That is, for any propositions A and B, we are familiar with how to deduce the truth value of these four new propositions from those of A and B: NOT A, A AND B, A OR B, and A IMPLIES B.

For example, given that A is true, we can deduce immediately that A OR B is true for any proposition B.

In general, the truth value of any proposition in a deductive system derives from other propositions, since the rules of deductive logic are nothing more than rules stipulating how to evaluate the truth values of propositions given the truth values of other propositions. However, it is an interesting and important observation that given only the rules of deductive logic, the values "true" and "false" are simply placeholders. That is, in some alternate system they could be interchanged, or renamed "red" and "blue," without any effect. Put another way, the truth value "true" has no meaning imparted by the rules of deductive logic; if you can deduce from A being true that B is also true, that*relationship*is all that you have learned about B --- you don't however know what it means for B (or A for that matter) to be true.

Unless, of course, you have axioms.

Consider two universes of discourse. Suppose in the first we have:

Axiom A1: "All dogs are black."

Proposition P1: "My pet dog is black."

And suppose in the second we have:

Axiom A2: "No dogs are black."

Proposition P1: "My pet dog is black."

In the first universe, P1 is true. In the second universe, P2 is false. The two universes of discourse differ only in the their axioms. This is a silly example meant only to illustrate one thing: it must be the axioms that impart the*meaning*of truth values onto the propositions. That is, what it means to have deduced that "My dog is black" is true depends wholly on the axioms from which you have deduced it.

To reinforce the idea, consider a third universe with:

Axiom A3: "All of my pets are black."

Proposition P1: "My pet dog is black."

Again P1 is true here as in the first universe, but it should be clear that it means something very different to be true in each universe.

In general, whenever somebody claims to have deduced that something is true, that is an empty claim as it stands (with some exceptions*), without indicating the axioms of the system, since it doesn't mean anything to be deduced as true in the absence of axioms, just as it doesn't mean anything to weigh 3 oogleboogles if the oogleboogle standard of weight has not been specified. You should no sooner accept a deductive conclusion as being meaningfully true without knowing the underlying axioms than you would purchase roast beef at $5/ob.

To summarize, while rules of deductive logic relate truth values among propositions, the axioms, serving as the standard of truth, dictate what it means for the true propositions to be true.

*Tautologies are special sorts of propositions, like (A OR Not-A), that must be true in all universes of discourse regardless of the axioms. They are actually super-standards of truth, for if collectively the axioms of a universe of discourse would render a tautology false, that set of axioms is taken to be flawed, rather than the tautology taken to be false, for once a tautology is taken to be false, then any proposition in any universe of discourse can be validly deduced to be both true and false, rendering truth values completely meaningless. - 27 Mar '07 04:49 / 2 editsPart III - "Comparing": The Devil in the Details

Those not interested in or likely to be confused by theoretical subtleties might do well to skip this section. Those interested would do well to petition bbarr for elaboration. I'm including this section in the interest of disclosure about a shortcoming of my notion of axioms.

We have said an axiom is a standard of truth, and you determine a proposition's truth value by comparing it to the axioms. Ideally, this would be easy and straightforward. If all universes of discourse were toy examples, like

Axiom A4: Roses are red AND Violets are blue

Proposition P2: Violets are blue

there would be an obvious manner for comparing them: P2 can be deduced from A4. We can say that anything deducible from the axioms is true.

Or like this simple example:

Axiom A5: Ass, gas, or grass; nobody rides for free.

Proposition P3: I ride for free.

the negation of P3 can be deduced from A5. We can say that any proposition whose negation can be deduced from the axioms must be false.

But consider this universe of discourse:

Axiom A4: Roses are red AND Violets are blue

Proposition P3: I ride for free.

Can you see that we have a problem? Neither P3 nor its negation can be deduced from A4, but those are the only ways we have of comparing a proposition to our standard of truth. So, in this universe, is P3 true? Is it false? Is it well-defined? Does it have any meaning in this universe? Fortunately, while theoretically interesting these are rather arcane questions that practically have no bearing on typical, real-world applications of critical thinking. - 27 Mar '07 04:49 / 3 editsPart IV: Is the dictionary wrong?

The dictionary says that an axiom is a self-evident truth requiring no proof. Of the three cited definitions, this is the one that I take the most issue with, for it is notionally contrary to axiomatic reasoning. To say that something requres no proof conveys the idea that it is true, in virtue of something else, evidence of which is so obvious that it need not even be mentioned. But to accept this dictionary definition is to reject that distinct universes of discourse with differing axioms are possible. Mathematics is rich with numerous and historical counterexamples, two of which come to mind being Euclidean vs. non-Euclidan geometries, and Boolean arithmetic vs. standard integer arithmetic. To say that a digit can have a value of only 0 or 1 is hardly a self-evident truth, yet it is an axiom enabling the very system I am using to communicate this message.

The dictionary says that an axiom is a universally accepted principle or rule. Well, it is serendipitously correct, in that we refer to the scope of a given set of axioms as a universe of discourse. However, to reiterate, it is not the case that a given axiom is an axiom in all universes of discourse. We have seen numerous counterexamples. Yet in another sense, in particular domains of human discussions, certain axioms do universally recur as common standards of truth. For example, if two Christians are debating, there may be some axioms which they cannot both accept as standards of truth, yet both will likely accept as a standard of truth that God exists. So, this defintion is somewhat satisfactory to me although significantly lacking in precision.

The dictionary says that an axiom is a proposition assumed without proof for the sake of studying its deductive consequences. This is a really great effort, although I have to take serious issue it. What the dictionary has described does describe axioms, but it also describes suppositions, which are something quite different. This definition is akin to defining a dog as an animal with four legs. Taking another example from the domain of mathematics, suppositions are propositions sometimes used in the technique of proof by contradiction: for example, to prove that the integer 2 has no rational square root, one begins by supposing that a/b is the square root of 2, followed by a study of that supposition's consequences resulting in a finding that no such a and b can exist. The key difference between a supposition and an axiom is that suppostions are subservient to axioms; suppositions do no impart meaningful truth values; suppostions operate under the meaning of truth imparted by the axioms governing their universe of discourse. - 27 Mar '07 04:49Part V: Clarifications

Commentary was requested regarding "a logical system being incapable of proving all its axioms." I hope that after reading Parts I through IV, one recalls the famous quote of Charles Babbage, "I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question." In short, an axiom constitutes its own proof; there can be no universe of discourse that contains an axiom which cannot be deduced from that universe's set of axioms. Of course, only a person very notionally confused would set about such a task in the first place.

Perhaps the question intended was, "Can all axioms be deduced from tautologies?" The answer to that is No.

Or perhaps the question intended was "Can all true propositions in a universe of discourse be deduced from its axioms?" Again, No. This is one aspect of the Comparison Devil. See Goedel for more details. - 27 Mar '07 05:14

Whack! Whack! Pimp attack!*Originally posted by DoctorScribbles***And I won't believe it until I see you adapt your thinking accordingly.**

Sounds like the Masta' Docta is back!

Layin' out truths that'll make yo' neck crack,

Showin' what otha people's mindsets lack.

Ridin' yo' ass like he's ridin' yo' momma,

Flippin' out rhymes like a Shakespeare drama.

Bustin' yo' lip or wracking yo' brain,

Tryin' to show how yo' thinking's insane...

But he hasn't learned yet; he's still tryin' to try

To fill up the voids in minds gone awry.

Sorry, Doc, there's more demand than supply;

Give up now, befo' you make a black man cry.

Nemesio - 27 Mar '07 05:18

yo-yo*Originally posted by Nemesio***Whack! Whack! Pimp attack!**

Sounds like the Masta' Docta is back!

Layin' out truths that'll make yo' neck crack,

Showin' what otha people's mindsets lack.

Ridin' yo' ass like he's ridin' yo' momma,

Flippin' out rhymes like a Shakespeare drama.

Bustin' yo' lip or wracking yo' brain,

Tryin' to show how yo' thinking's insane...

But he hasn't learne ...[text shortened]... and than supply;

Give up now, befo' you make a black man cry.

Nemesio - 27 Mar '07 05:52 / 2 editsYo, my logic's like dis: I gots to keep bringin' it

It's like Amazin' Grace: tha choir gots to keep singin' it

It's like yo' mama's phone: I gots to keep ringin' it

I sell it on tha cheap but I gots to keep slingin' it

Dis widom ain't free, I ain't give it away

Dr. S be tha piper, and niggaz gots to pay

I come to collect at the end of the day

My Axiom Numba 1 be "Yo mom's a good lay."

I hit it good, two times per post

Impregnatin' mommas like tha Holy Ghost

You got a lil' knowledge but I still got tha most

I'm tha Logical Lion from tha Ivory Coast