02 Dec 05
1 edit
Originally posted by royalchickenWe could pick axioms at random - but unless we assume the laws of logic also - they would be nonsense. But unlike other axioms - the laws of logic can not be denied unless one denies all knowledge and the ability to communicate knowledge.
The 'self-evident' law of non-contradiction is itself an axiom in the first-order predicate calculus; logic itself is an axiom system, so you still haven't given a 'prior concept' of what 'true' means that is not a consequence of some axiom-system. Truth (in axiomatic reasoning) is always relative to the axioms used (in practice, our axiom-syste ...[text shortened]... t's still an axiom system though, and at least two statements, the axioms themselves, are true.
Then not all axioms will lead to useful knowledge. For instants - to assume Coletti is the final authority on mathematics and logic would get you in a lot of trouble. Also, axioms that are easily proven from prior propositions do not meet the qualities that make them axioms. And finally - while we could course to have P and ~P as our axioms, we would have to abandon logic and intelligibility all together because we have invalidated the law of contradiction with our axioms.
There are test, that do not prove a system is valid, but it can help us compare different systems. First we test for internal integrity. Is the system coherent. For mathematical models, we might lean if there is new knowledge and characteristics that have never been seen before.
For a world-view the axioms play a bigger rule. A world-view should be comprehensive , and coherent, and able to describe the world as we perceive it, out relationships to man (and God) and provide moral objectivity. We should be able to derive from it all the things one needs for a meaningfull life.
It takes a great deal of faith to chose the best axioms for a world-view because a poorly chosen would view can lead to hopeless skepticism. A poor choice of axioms =could lead to leaving a life of hapless despair and misery.
Think of this - lets say you want to incorporate a geometric system into a world-view. Would you adopt hyperbolic geometry, or Euclidean Geometry?
Yes we can pick any axioms, but if the lead to contradictions they are useless. Not all axiomatic systems are the same and alike for providing a useful world-view.
02 Dec 05
Originally posted by ColettiNo, they wouldn't be 'nonsense' unless we also 'assumed' the axioms of a first-order predicate calculus (or perhaps took some other axiom-scheme governing propositions as our set of rules). The concept of nonsense, like that of truth, is contingent on axioms.
We could pick axioms at random - but unless we assume the laws of logic also - they would be nonsense.
There are test, that do not prove a system is valid, but it can help us compare different systems. First we test for internal integrity. Is the system coherent. For mathematical models, we might lean if there is new knowledge and characteristics that ...[text shortened]... icism. A poor choice of axioms =could lead to leaving a life of hapless despair and misery.
But unlike other axioms - the laws of logic can not be denied unless one denies all knowledge and the ability to communicate knowledge.
Please justify this. Be careful not to beg the question.
Then not all axioms will lead to useful knowledge. For instants - to assume Coletti is the final authority on mathematics and logic would get you in a lot of trouble. Also, axioms that are easily proven from prior propositions do not meet the qualities that make them axioms. And finally - while we could course to have P and ~P as our axioms, we would have to abandon logic and intelligibility all together because we have invalidated the law of contradiction with our axioms.
The last part is true. Thus an axiom-system containing an axiom of non-contradiction, the statement P, and the statement ~P is an inconsistent system. The system containing only P and ~P cannot be called inconsistent, because we have no axioms to define a notion of consistency. In general, to be properly rigourous (and there is no point in reasoning axiomatically without trying to be properly rigourous), one must set forth axioms for everything.
The first part is confused, because axiomatic reasoning is not about 'useful knowledge' or any other sort of knowledge. It is only about following typographical rules.
For a world-view the axioms play a bigger rule. A world-view should be comprehensive , and coherent, and able to describe the world as we perceive it, out relationships to man (and God) and provide moral objectivity. We should be able to derive from it all the things one needs for a meaningfull life.
I think you've got this backwards. I'm sort of sympathetic to this point of view, but trying to axiomatise an entire 'worldview' in a way that captures all of the nuance we observe is probably a fool's errand. To start with, a good comprehensive model of the world would probably not have an axiom of non-contradiction. I think (although this is a very undeveloped opinion I hope to explore as I become more acquainted with the technicalities) that the best formal model we can use to think about the fvcked-up mess we live in will come from Bayesian statistics or possibility theory.
Think of this - lets say you want to incorporate a geometric system into a world-view. Would you adopt hyperbolic geometry, or Euclidean Geometry?
It depends on the scale with which I want to view the world, I suppose, and would probably differ from occasion to occasion; if I want to walk somewhere, a Euclidean view of the world is probably best. If I take the tube, I'm better off thinking topologically rather than geometrically. A good world-view is more dynamic than a system of axioms can completely capture.
In fact, can you describe the axiom system from which you deduce your worldview?
02 Dec 05
Originally posted by royalchickenI certainly can not prove the laws of logic without begging the question, since any proof would require I assume them to prove them.
But unlike other axioms - the laws of logic can not be denied unless one denies all knowledge and the ability to communicate knowledge.
However, I challenge anyone to disprove them or even justify eliminating them without contradicting themselves. No communication is possible without logic.
Mathematics presupposes the validity of the law of non-contradiction. Make one statement and you assume the law of non-contradiction. All postulates, equations, words, etc. if they are to have any meaning, one must assume the law of non-contradiction.
One might model the world using probabilistic theories, but these theories themselves require the laws of logic to be intelligible.
02 Dec 05
Originally posted by ColettiYes, mathematics does include the axioms of a first-order predicate calculus. One cannot throw out these axioms in mathematics; on a more intuitive level, this is probably why I like mathematics -- compare it to more complicated, nuanced and hard-to-formalise things and the misery thinking about them can bring.
I certainly can not prove the laws of logic without begging the question, since any proof would require I assume them to prove them.
However, I challenge anyone to disprove them or even justify eliminating them without contradicting themselves. No communication is possible without logic.
Mathematics presupposes the validity of the law of non-contradi ...[text shortened]... babilistic theories, but these theories themselves require the laws of logic to be intelligible.
It is impossible to prove or disprove axioms. Therefore, it is meaningless to talk about proving or disproving axiom systems (if they are consistent), of which logic is one. That was not the belief I wanted you to justify though; I want to see an argument, including a rigourous definition of 'communication', that shows that the axioms of logic are necessary for communication to be possible.
As an aside (if you want to talk about possibility theory, then I think you should, and know I should, read more about it), probability theory does require logic in its formalisations, but the notions of 'possibility' and 'necessity' utilise multi-valued logic, which is a different system from standard logic.
02 Dec 05
Originally posted by royalchicken1. The law of non-contradiction, identity, excluded middle.
Can you list these axioms? Presumably you know them off the top of your head, since they define your worldview.
2. The foundation of knowledge is God's revelation.
In religious language the second axiom is: Scripture alone is the Word of God.
02 Dec 05
Originally posted by ColettiYour worldview does not involve the use of other rules of logic which cannot be deduced from 1.?
1. The law of non-contradiction, identity, excluded middle.
2. The foundation of knowledge is God's revelation.
In religious language the second axiom is: Scripture alone is the Word of God.
By 2., do you mean something like 'Those things are true which can be deduced by applying 1. to God's revelation?'
02 Dec 05
Originally posted by ColettiIt would seem that 2 excludes 1. That is, a single contradiction in
1. The law of non-contradiction, identity, excluded middle.
2. The foundation of knowledge is God's revelation.
the 'Word of God' would result in the implosion of your beliefs.
So, let me ask you: was Jesus crucified before or after the Passover
Seder?
Nemesio
02 Dec 05
1 edit
Originally posted by royalchickenYour worldview does not involve the use of other rules of logic which cannot be deduced from 1.?
Your worldview does not involve the use of other rules of logic which cannot be deduced from 1.?
By 2., do you mean something like 'Those things are true which can be deduced by applying 1. to God's revelation?'
Not sure what you mean by the first question. I suppose until someone can show me how the "rules" of logic can be deduced from the laws of logic - I'll say I agree with Aristotelian rules of inference.
By 2., do you mean something like 'Those things are true which can be deduced by applying 1. to God's revelation?'
Yes, that's seems to say it. It's an epistemological means of testing propositional knowledge to see if it is justifiably true. It makes God's knowledge the foundation for man's knowledge.
03 Dec 05
3 edits
Originally posted by NemesioTrue but it would have to be a real contradiction, and not just a possible one. And you would need to show that the two statements can only be understood as two contradictory propositions.
It would seem that 2 excludes 1. That is, a single contradiction in
the 'Word of God' would result in the implosion of your beliefs.
So, let me ask you: was Jesus crucified before or after the Passover
Seder?
Nemesio
If you take statements from two different books, from two different authors, with two different styles of writing - how do you prove they intended to covey propositions with the exact same term using the exact same sense of the terms and then contradictory?
Are you sure you have the correct meaning? Have you correctly identified the truth that the author wants to convey?
And finally - it is impossible for God's knowledge to contradict itself - and Scripture is axiomatically God's knowledge so it can not contradict itself. Any apparent contradiction are a matter of errors in translation, or misunderstanding the truth that the ultimate author is conveying.
In other words - you can not prove a contradiction if the source is by definition non-contradictory. All your apparent contradiction shows is you have failed to identify the truth - not that the truth is not there.
Yes - it's the God-dun-it defense. But that is the implication of the axioms - God's knowledge is the grounds for truth, and true propositions can not be contradictions.
As for when Jesus was crucified - before or after the Passover Seder, it's one or the other. Maybe we can not prove it either way - which means it is not a significant.