The E x p a n d i n g Being (Paradigms)

If you have no axioms, if you don't presume anything, then your perspective is unlimited. Your mind has total freedom. Truth has to be left undefined for it doesn't fit the old criteria of being either an axiom or something "correctly" derivable from either axioms or other true statements.

This is the progression of an expanding being.

1. in order for a statement to be true, there must be proof;
2. all concepts have an opposite concept;
3. you are your ego;
4. you are your body;
5. sometimes belief/faith is knowledge;
6. knowing about something is knowing something;
7. the universe operates on binary logic, etc.

The more axioms you have, the more limited your perspective is.

An axiom is a condition or governing rule about something. The more conditions you set, the narrower your vision. For example, if you place the condition on speed in the form of the sound "barrier," you are closed to the reality which is that the speed of sound is not a limit on speed.

In the extreme case when the number of axioms becomes large, you run the risk of having two axioms contradict each other. If your set of axioms contains a statement and its opposite, you can then literally derive every statement as true.

On the other hand, the fewer conditions you set, i.e., axioms you have, the more expanded your perspective is.

It's a lot like playing chess with fewer and fewer rules: more moves are then possible, translating by analogy to more freedom to see more.

With the exception of two contradictory axioms, the fewer axioms you have translates into fewer things being true. True in the sense of either being an axiom or being derivable from the axioms or other true statements. Note that in that statement there are axioms behind what is derivable in a so-called correct manner, such as in the Aristotelian sense.

If you have no axioms, if you don't presume anything, then your perspective is unlimited. Your mind has total freedom. Truth has to be left undefined for it doesn't fit the old criteria of being either an axiom or something "correctly" derivable from either axioms or other true statements.

You can't be serious. Fanatics have limited axioms, and their perspectives are as narrow as can be imagined. Scientists, on the other hand, have tons of axioms, and they've greatly expanded the limits of knowledge.

An axiom is a condition or governing rule about something. The more conditions you set, the narrower your vision. For example, if you place the condition on speed in the form of the sound "barrier," you are closed to the reality which is that the speed of sound is not a limit on speed.

That's a bit like saying that if you are a firm believer in gravity, you are closed to the "reality" of falling up. Axioms are typically not just theoretical constructs, they are substantiated by empirical testing. Few scientists would take it on faith that the speed of sound in dry air at normal atmospheric pressure is approximately 750 mph, and that they have accepted the speed of sound as axiomatic does not limit their perspectives, it enhances them. That's why we know more than the savages who think that thunder is the sound of the gods playing duck pins.

In the extreme case when the number of axioms becomes large, you run the risk of having two axioms contradict each other. If your set of axioms contains a statement and its opposite, you can then literally derive every statement as true.

No you can't.

...every formula of an inconsistent P is a consequence of the Axioms and Rules of Inference of P.

On the other hand, the fewer conditions you set, i.e., axioms you have, the more expanded your perspective is.

Just the opposite.

It's a lot like playing chess with fewer and fewer rules: more moves are then possible, translating by analogy to more freedom to see more.

Bad analogy. The fewer rules you play with, the more the game isn't chess.

With the exception of two contradictory axioms, the fewer axioms you have translates into fewer things being true. True in the sense of either being an axiom or being derivable from the axioms or other true statements. Note that in that statement there are axioms behind what is derivable in a so-called correct manner, such as in the Aristotelian sense.

I agree. But then why would you prefer a world in which fewer things were true?

You have to realize that truth, the word, is undefineable. Sure you can write out a dictionaries definition but it is a cirular definition. Of course that doesn't mean it exists. And it's not that I would prefer a world with fewer things being true; it's that I'm realizing that's the world we already live in. It's not the answer I wanted nor am comfortable with.

BrianT wrote:

If you have no axioms, if you don't presume anything, then your perspective is unlimited. Your mind has total freedom. Truth has to be left undefined for it doesn't fit the old criteria of being either an axiom or something "correctly" derivable from either axioms or other true statements.

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If you have no axioms, then there's no point in talking about "truth," including the truth that there are no axioms.

Exactly. It's like talking about the concept of BLALL. We might as well be saying
"1+1=2" is BLALL and "1+1=0" is "not" BLALL.

It may be narrow because one axiom is that their way is the only way.

But a scientist is closed to the possibility of anything other than emperical data as constituting evidence. An axiom scientists have is that the scientific method correctly decides hypotheses.

A scientist is closed to many possibilities simply because they think it's impossible. Yet they rarely, if ever, can prove something is impossible; so those are axioms.

Right. You are closed to that possibility. I never said that that was incorrect. That empirical testing can substantiate an axiom is itself an axiom.

Yes, you can.

http://alixcomsi.com/CTG_02.htm wrote:

...every formula of an inconsistent P is a consequence of the Axioms and Rules of Inference of P.

See also Mathematical Logic by Ebbinghaus, Flum and Thomas.

If I had the opposite number of assumptions, i.e., many, then that reduces my vision. For example, if I assume I will never get a Phd, I am limiting my perspective. If I assume the axiom of foundations, I reduce the number of things that can be sets. If I assume there are 3 dimensions, I limit my ability to understand reality. If I assume that the scientific method corrctly decides hypotheses, I am not open to the possibility that there is a better method, a method in which no scientific thoery is ever seen as false down the road. When I assume X is true, then I am closed to the possibility that X is not true. Then consider increasing the number of X's then I get closed to more and more things, rather than more and more open to things.

I'm not interested in the fact that if you change the rules then it's not chess. That's the point. What's "moving" in my scenario are precisely thoughts. When you have more axioms, your thoughts are constrained to fit with those axioms. When you release the number of axioms you have, the "board pieces" (your thoughts) have total freedom. Hell, they don't even have to stay on the board at all. Imagine the freedom to think whatever you want by assuming nothing.

You have to realize that truth, the word, is undefineable.

Sure you can write out a dictionaries definition but it is a cirular definition. Of course that doesn't mean it exists. And it's not that I would prefer a world with fewer things being true; it's that I'm realizing that's the world we already live in. It's not the answer I wanted nor am comfortable with.

Exactly. It's like talking about the concept of BLALL. We might as well be saying
"1+1=2" is BLALL and "1+1=0" is "not" BLALL.

BrianT wrote:

It may be narrow because one axiom is that their way is the only way.

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Which is what you are suggesting we should all do.

Absoutely not. I just listed three modes of operation. I never said being closed to possibilities is wrong. It just limits your perspective and now if you judge that to be a bad thing, then that's your choice. You're putting words in my mouth.

BrianT wrote:

But a scientist is closed to the possibility of anything other than emperical data as constituting evidence. An axiom scientists have is that the scientific method correctly decides hypotheses.

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A hypothesis doesn't make any sense if it is divorced from the scientific method.[/qoute]
Are you assuming that I think that making an assumption is wrong or incorrect?

BrianT wrote:

A scientist is closed to many possibilities simply because they think it's impossible. Yet they rarely, if ever, can prove something is impossible; so those are axioms.

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Proving empirically that something is "impossible" is impossible -- read Hume. Proving that a theory is "false," however, is the basis of science.

That's what I meant: it is impossible to prove that something is impossible.

BrianT wrote:

Right. You are closed to that possibility. I never said that that was incorrect. That empirical testing can substantiate an axiom is itself an axiom.

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And you're willing to discard that axiom?

I'm not advocating either keeping it or not. Just realizing that it is an axiom without judging it as right or wrong.

BrianT wrote:

Yes, you can.

http://alixcomsi.com/CTG_02.htm wrote:

...every formula of an inconsistent P is a consequence of the Axioms and Rules of Inference of P.

See also Mathematical Logic by Ebbinghaus, Flum and Thomas.

Quote:

No need. You said that "If your set of axioms contains a statement and its opposite, you can then literally derive every statement as true." But that could only be true if you did not adhere to the law of non-contradiction. On the other hand, if you reject the law of non-contradiction, then any statement can be true regardless of how many contradictory axioms you might have.

If you look at a truth table... Let P and ~P be to axioms. What I mean is that P^~P implies any statement Q. Keep in mind that if the antecedent of a conidtional is "false" then the conditional is true.

BrianT wrote:

If I had the opposite number of assumptions, i.e., many, then that reduces my vision. For example, if I assume I will never get a Phd, I am limiting my perspective. If I assume the axiom of foundations, I reduce the number of things that can be sets. If I assume there are 3 dimensions, I limit my ability to understand reality. If I assume that the scientific method corrctly decides hypotheses, I am not open to the possibility that there is a better method, a method in which no scientific thoery is ever seen as false down the road. When I assume X is true, then I am closed to the possibility that X is not true. Then consider increasing the number of X's then I get closed to more and more things, rather than more and more open to things.

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No doubt, and if you no longer assume that gravity exists you might float up to the ceiling. Everything might be an illusion, a dream within a dream. But some illusions, like gravity, are remarkably persistent.

Like I said before, I'm not judging a narrow perspective with many axioms. These are just an editorial about axioms, not a prescription.

BrianT wrote:

I'm not interested in the fact that if you change the rules then it's not chess. That's the point. What's "moving" in my scenario are precisely thoughts. When you have more axioms, your thoughts are constrained to fit with those axioms. When you release the number of axioms you have, the "board pieces" (your thoughts) have total freedom. Hell, they don't even have to stay on the board at all. Imagine the freedom to think whatever you want by assuming nothing.

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Or, to put it another way, the best way to know something is to know nothing. I doubt even you believe that.

That really isn't putting it another way. I never mentioned knowledge on this point; I'm talking about assumptions.

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BrianT wrote:

You have to realize that truth, the word, is undefineable.

Is that a true statement?
What do you mean by "true"?

BrianT wrote:

Sure you can write out a dictionaries definition but it is a cirular definition. Of course that doesn't mean it exists. And it's not that I would prefer a world with fewer things being true; it's that I'm realizing that's the world we already live in. It's not the answer I wanted nor am comfortable with.

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Hunh?

I made a typo. "Of course that doesn't mean that doesn't exist. IE Truth can very well exist even though it is undefinable.

BrianT wrote:

Exactly. It's like talking about the concept of BLALL. We might as well be saying
"1+1=2" is BLALL and "1+1=0" is "not" BLALL.

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If you have no basis for establishing that something is "true," why should I care what you say?

That's a question you should have asked yourself from the beginning.

Absoutely not. I just listed three modes of operation. I never said being closed to possibilities is wrong. It just limits your perspective and now if you judge that to be a bad thing, then that's your choice. You're putting words in my mouth.

Are you assuming that I think that making an assumption is wrong or incorrect?

That's what I meant: it is impossible to prove that something is impossible.

I'm not advocating either keeping it or not. Just realizing that it is an axiom without judging it as right or wrong.

If you look at a truth table... Let P and ~P be to axioms. What I mean is that P^~P implies any statement Q. Keep in mind that if the antecedent of a conidtional is "false" then the conditional is true.

Like I said before, I'm not judging a narrow perspective with many axioms. These are just an editorial about axioms, not a prescription.

That really isn't putting it another way. I never mentioned knowledge on this point; I'm talking about assumptions.

What do you mean by "true"?

I made a typo. "Of course that doesn't mean that doesn't exist. IE Truth can very well exist even though it is undefinable.

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If you have no basis for establishing that something is "true," why should I care what you say?

That's a question you should have asked yourself from the beginning.

If you look at a truth table... Let P and ~P be to axioms. What I mean is that P^~P implies any statement Q. Keep in mind that if the antecedent of a conidtional is "false" then the conditional is true.

If both P and ~P are true, then they are not genuine contradictories.

If you look at a truth table... Let P and ~P be to axioms. What I mean is that P^~P implies any statement Q. Keep in mind that if the antecedent of a conidtional is "false" then the conditional is true.

Your terminology is a bit strange. In the original post, I said that if P and ~P are axioms, then literally every statement is true. This is something that's well known.