An introduction to formal logic

CWL said:


Oh my... does this mean that the existence of God and an afterlife cannot be proven by means of logic alone?!? :eek:
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I fear that you are correct. I also fear that the non-existence of God and an afterlife cannot be proven by logic alone. Kant has much to say about this in the Critique of Pure Reason, the section on the logical antinomies.

I do not believe that logic alone can prove the existence of anything except more logical truth. Logic can demonstrate that something is logically impossible, but that may not be of much help. Formalism is limited that way.
 
Godel's proof

This is just a quick overview, cribbed from Nagel and Newman.

Given an axiomatic system sufficiently powerful to support arithmetic (such as that described in Principia Mathematica by Russell and Whitehead):

1. Godel shows how to construct a meta-mathematical statement G (a formula in arithmetic) that says "Formula G is not demonstrable."

2. Then he shows that G is demonstrable if and only if not-G is demonstrable. (since G say "G is not demonstrable")
If a formula and its negation are both demonstrable, then the axioms of the system are inconsistent.

3. Then he proves that G is true, but not formally demonstrable. (It asserts that every integer has a certain property and that can be shown to be true of every integer we examine, but cannot be shown to be true of all integers)

4. Since G is true and formally undecidable, the axioms of arithmetic are incomplete.

5. Then, he shows how to construct a formula A that represents the statement "Arithmetic is consistent", and proves that "A implies G" is formally demonstrable.
Then he shows that A itself is not formally demonstrable.

This means that there is a true statement in the formal system that cannot be demonstrated by the axioms of the formal system.

If G is added to the axiom set, then the same process can generate another expression that is also true but undecidable.

It says nothing at all about the nature of matter, time, space, and the universe as a whole.

It is, however, one of the shining achievements of twentieth century mathematics.

http://www.faragher.freeserve.co.uk/godeldef2.htm
 
A non-computational insight!

TO WHITE FORK

You wrote on page 5, 11-27-2002 02:55 PM: This means that there is a true statement in the formal system that cannot be demonstrated by the axioms of the formal system.
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Soderqvist1: can a Turing machine (algorithm) give us this answer?

You wrote: It says nothing at all about the nature of matter, time, space, and the universe as a whole.
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Soderqvist1: Our universe is a closed system according to the second law of thermodynamics, if not all inferences can be formalized in this universal system, is its consistency proof immaterial, and thus outside the universe?

Godel's incompleteness theorem as quoted from Godel's biography
Either mathematics is too big for the human mind or the human mind is more than a machine. A consistency proof for [any] system ... can be carried out only by means of modes of inference that are not formalized in the system ... itself. Godel's results were a landmark in 20th-century mathematics, showing that mathematics is not a finished object, as had been believed. It also implies that a computer can never be programmed to answer all mathematical questions.
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Godel.html

Is this universal consistency proof a non-local phenomenon? :)
http://www.twm.co.nz/goswam1.htm
 
whitefork said:


I fear that you are correct. I also fear that the non-existence of God and an afterlife cannot be proven by logic alone. Kant has much to say about this in the Critique of Pure Reason, the section on the logical antinomies.

I do not believe that logic alone can prove the existence of anything except more logical truth. Logic can demonstrate that something is logically impossible, but that may not be of much help. Formalism is limited that way.
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We are in complete agreement. To anyone practicing my profession (law) what you are saying is painfully obvious.
 
Peter: One consequence of Godel's proof is that a Turing machine cannot tell us wether these formally undecidable propositions are true or false.

If physics can be formalized, it will contain formally undecidable propositions.

Beautiful, is it not?

You can create more ever more powerful rules of inference to that will allow you to infer the truth or falsehood of currently undecidable propositions, but the validity of those rules will be as uncertain as the propositions you're trying to evaluate.

You can create new mathematical formalisms (Newton and the calculus, for instance), but the problem of undecidability will always be with you.
 
Franko said:
Where fn = a function of

1) Atoms obey the Laws of Physics. (Atoms = fnTLOP)
2) You are made of atoms. (YOU = fnAtoms)
3) You OBEY the Laws of Physics. (YOU = fnTLOP[fnAtoms])
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THE MATH VERSION! A personal favorite of mine! If nobody minds, I'll repost my response.

Note:
A = "Atoms"
T = "TLOP"
Y = "You"

Originally posted by Franko
A = fnT
Y = fnA
Y = fnA = fnT
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this should really be (and pardon the slight change in notation):

A = f(T)
Y = f(A)
Y = f(A) = f(T)

since f(A) = f(T), then

A = T
A = f(A)

Generalizing,

f(x) = x

So,

Y = f(A) = f(T) = T

And we shown, given Franko's assumptions, that

Y = T

or in English, you are the laws of physics and, as such, are god.

If we are to believe Franko's assumptions and definitions, we have proof of athiesm (welll, really, humanism), because that is what the mathematics seems to indicate.
If, on the other hand, we reject his assumptions and definitions, this particular proof has no basis and indicates nothing either way.
Personally, I tend to reject the premises because I don't believe the assumption that,

Y = f(A) = f(T)

As whitefork pointed out to me in a PM, the proper formation should be

Y = f(A) = f(f(T)) = g(T)

where

g(x) = f(f(x))

which suggests that T relates to Y differently than it relates to A.

What a wonderful stroll down memory lane...

Upchurch

edited to add:
The end conclusion, which I seem to have not saved is that either

Y = T

or

g(x) = f(f(x))

Meaning that either everything that is made of atoms is god or the laws of physics work differently at the atomic level than they do at the macroscopic level. Franko never explained which fell into his belief system....
 
Recycling .... proof of 2+2=4 to follow shortly.

Oh dear, scratch that - that old thread has rolled off into the abyss. That's why I bump these from time to time.
 
whitefork said:
Recycling .... proof of 2+2=4 to follow shortly.

Oh dear, scratch that - that old thread has rolled off into the abyss. That's why I bump these from time to time.
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*sigh* whitefork,

What makes you think Bozo will get it this time?

Cheers,
 
Rumpity bumpity, whitey put some good work into this thread. I'd hate to see it go in the great abyss.

Adam
 
Upchurch said:

Meaning that either everything that is made of atoms is god or the laws of physics work differently at the atomic level than they do at the macroscopic level. Franko never explained which fell into his belief system....
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Wow. Check this out. Franko laid some of the ground work for lifegazer.

Kinda.

Ignoring the fact that Franko's assumptions are in complete contradiction to lifegazer's assumptions. Interestingly, both claim not to have any assumptions and that there conclusions are based entirely on reason. hm...

It's almost like lifegazer is the love child of Franko and Interesting Ian, isn't it?
 
Upchurch said:
…snip…

It's almost like lifegazer is the love child of Franko and Interesting Ian, isn't it?
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I suggest you seek out some form of therapy immediately; anybody with a mind capable of even considering a love child of Franko & II is seriously ill and society needs protecting from you.
 
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