X axioms .

  • #26
MathematicalPhysicist
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AKG said:
Well, I asked if you knew the difference between a tautology and a theorem, but it appears you don't. A theorem is something that can be derived from the axioms. Any reasonable logic will allow you to derive P from P, so you can derive any axiom from the axioms, hence every axiom is a theorem.
but it's the most trivial theorem, and thus i havent given it such importance as you two have given it.
the notion of p->p is by iteself an undeniable truth (i wonder if you can give an example of a system when such a statement is not true always?).
 
  • #27
AKG
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loop quantum gravity said:
but it's the most trivial theorem, and thus i havent given it such importance as you two have given it.
No one is giving it importance. If you know the definition of a theorem, then in any reasonable logic, every axiom is a theorem. I'll give you the definitions again, for propositional logic:

Tautology: A sentence is a tautology iff the sentence is true for every possible truth-value assignment of its atomic components.

Theorem: A sentence is a theorem iff it can be derived from an empty set of assumptions.

Now it looks like you think that a theorem is an interesting or useful result, and a tautology is a self-evident or trivial result. That's not what the words mean in logic, however. If you want to count the theorems in a logical system, then you better use the meaning of the word "theorem" correctly, I don't know why you insist on arguing about this. On the other hand, if what you really meant to say is that you wanted a way to count the "interesting" theorems (and tautologies) a logic could produce, then you'd have to give a good definition for "interesting" otherwise this is impossible.
the notion of p->p is by iteself an undeniable truth (i wonder if you can give an example of a system when such a statement is not true always?).
As far as I'm concerned, (p -> p) follows by definition of '->' so any reasonable logical system that has something, '*', that means the same thing as '->' will define it and will provide rules of inference such that (p * p) is both a tautology (or whatever the equivalent to "tautology" would be in this system) and a theorem.
 
  • #28
EnumaElish
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loop quantum gravity said:
the notion of p->p is by iteself an undeniable truth (i wonder if you can give an example of a system when such a statement is not true always?).
Thinking aloud, I could define an ordering relationship whereby p is said to "strict imply" q if "p ---> q and not (q ---> p)," couldn't I? In this relationship, "p does not strict imply p" is true. That is, "p strict implies p" is false.
 
  • #29
AKG
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EnumaElish said:
Thinking aloud, I could define an ordering relationship whereby p is said to "strict imply" q if "p ---> q and not (q ---> p)," couldn't I? In this relationship, "p does not strict imply p" is true. That is, "p strict implies p" is false.
You could also define "-->" such that every sentence that contains it is false, so (p --> p) would be false. However, in both your example and in mine, you're using the same symbol to mean something other than what loop quantum gravity means. That's cheating, there's nothing interesting about saying that you can have a system that uses the same symbols to mean different things, so that sentences in the two systems that appear to be the same actually mean different things, and thus have different truth values. What's interesting is not the symbols, but what they mean, so it would only be interesting if you had a system where, regardless of what symbols you used (which is why I used '*' instead of '-->'), you had a sentence that means the same thing as some sentence in his language, but has a different true value. Of course, this would be "interesting" but impossible (assuming we're dealing with "reasonable" systems, although there is much debate as to what makes a system reasonable).
 
  • #30
EnumaElish
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AKG said:
You could also define "-->" such that every sentence that contains it is false, so (p --> p) would be false. However, in both your example and in mine, you're using the same symbol to mean something other than what loop quantum gravity means.
If you re-read my post carefully, you will see that I was using the ---> symbol in its ordinary meaning, "implies." The relation I made up was "strict implies," which doesn't have a symbolic representation as of this time. "Strict implies" and the ordinary "implies" can be used within the same "system." You are correct that "strict implies" is similar to your * symbol.
 
  • #31
AKG
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My * symbol means the same thing as --> (regular implies, not strict implies), and my post doesn't suggest otherwise. Yes, sorry, I missed that you were using --> in the ordinary way, but then I wonder what the point of your post was, and how it was a response to what lqg was saying. Well you said, "just thinking aloud" so perhaps it wasn't meant to be a response to what he was saying?
 
  • #32
EnumaElish
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AKG said:
I wonder what the point of your post was, and how it was a response to what lqg was saying. Well you said, "just thinking aloud" so perhaps it wasn't meant to be a response to what he was saying?
That's exactly what I had intended, you read my thoughts. :smile:
 

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