# Thread: A ^ ~A 'A' and 'not A'

1. It is a fundamental presumption of logic (and probably all reason) that if a statement turns out to be true and not true at the same time, then one of your supporting arguments was false.

Can you show this?

You can't be in a city and not in a city at the same time?

Possibly a language issue, perhaps by definition this is impossible.

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3. Well...

The first part (as you call it a fundamental [presumption) is true by definition.

Therefore it cannot be demonstrated but must be taken as axiomatic.

In Godel, Escher and Bach in one of the conversations between Achilles and the Tortoise, Hofstader shows why this must be so, by pointing out the infinite regress of any attempt to justify something that is taken axiomatically.

4. Does an infinite regress make it false? not trying to be difficult or anything.

5. Originally Posted by Golkarian
Does an infinite regress make it false?
No, because an infinite regress effectively renders the terms "true" and "false" inapplicable. That's what makes it axiomatic. The infinite regress makes the truth value of an axiom impossible to determine conclusively, and so the axiom can only be presumed, or not.

6. Originally Posted by Lyn
Originally Posted by Golkarian
Does an infinite regress make it false?
No, because an infinite regress effectively renders the terms "true" and "false" inapplicable. That's what makes it axiomatic. The infinite regress makes the truth value of an axiom impossible to determine conclusively, and so the axiom can only be presumed, or not.
Can you use simpler terminology for me so I can get the gist of what you are saying?

7. Sure. An axiom, by definition, is a proposition that cannot conclusively be proven true or false. It is assumed to be true so that it can form the basis of other propositions. Sunshinewarrior mentioned Douglas Hofstadter's claim that any attempt to "prove" an axiom will result in an infinite regress, and Golkarian asked whether this infinite regress makes the axiom false. I'm pointing out that the question is misleading because "falseness" (and "truth") does not apply to axioms the same way it applies to other propositions. Axioms are the assumptions that allow us to create a system of truth and falseness in the first place, so applying that system to the axioms themselves leads to all sorts of paradoxes; but these paradoxes are not generally thought to undermine the validity of the axioms.

8. Originally Posted by Lyn
Sure. An axiom, by definition, is a proposition that cannot conclusively be proven true or false. It is assumed to be true so that it can form the basis of other propositions. Sunshinewarrior mentioned Douglas Hofstadter's claim that any attempt to "prove" an axiom will result in an infinite regress, and Golkarian asked whether this infinite regress makes the axiom false. I'm pointing out that the question is misleading because "falseness" (and "truth") does not apply to axioms the same way it applies to other propositions. Axioms are the assumptions that allow us to create a system of truth and falseness in the first place, so applying that system to the axioms themselves leads to all sorts of paradoxes; but these paradoxes are not generally thought to undermine the validity of the axioms.
Well put.

9. I think I got it, thanks.

10. Originally Posted by Lyn
Sure. An axiom, by definition, is a proposition that cannot conclusively be proven true or false. It is assumed to be true so that it can form the basis of other propositions. Sunshinewarrior mentioned Douglas Hofstadter's claim that any attempt to "prove" an axiom will result in an infinite regress, and Golkarian asked whether this infinite regress makes the axiom false. I'm pointing out that the question is misleading because "falseness" (and "truth") does not apply to axioms the same way it applies to other propositions. Axioms are the assumptions that allow us to create a system of truth and falseness in the first place, so applying that system to the axioms themselves leads to all sorts of paradoxes; but these paradoxes are not generally thought to undermine the validity of the axioms.
So the statement "A and not A" would be neither true nor false (like saying "this statement is a lie")?

11. Yes.

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