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 JREF Forum An introduction to formal logic

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 Tags logic , formal , introduction

 26th November 2002, 04:29 AM #161 whitefork None of the above     Join Date: Aug 2001 Location: aka kullervo Posts: 2,339 Quote: Originally posted by CWL Oh my... does this mean that the existence of God and an afterlife cannot be proven by means of logic alone?!? 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. __________________ Convictions are more dangerous enemies of truth than lies - Nietzsche
 27th November 2002, 06:55 AM #162 whitefork None of the above     Join Date: Aug 2001 Location: aka kullervo Posts: 2,339 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 __________________ Convictions are more dangerous enemies of truth than lies - Nietzsche
 27th November 2002, 11:37 PM #163 Peter Soderqvist Critical Thinker   Join Date: Nov 2001 Posts: 305 A non-computational insight! TO WHITE FORK Quote: 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. Soderqvist1: can a Turing machine (algorithm) give us this answer? Quote: You wrote: It says nothing at all about the nature of matter, time, space, and the universe as a whole. 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/~...ans/Godel.html Is this universal consistency proof a non-local phenomenon? http://www.twm.co.nz/goswam1.htm __________________ A simple explanation with few explanation grounds is to prefer, except when you need to hide your flaws!
 28th November 2002, 01:31 AM #164 CWL Funkateer     Join Date: Jul 2002 Location: Stockholm, Sweden Posts: 1,370 Quote: Originally posted by whitefork 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. We are in complete agreement. To anyone practicing my profession (law) what you are saying is painfully obvious. __________________ "All is not true which resembles truth" - Swedish Code for Judges (probably composed around 1540) "The obscurely spoken is the obscurely thought" - Swedish poet Esaias Tegnér (1782 - 1846) "Precisely because of human fallibility, extraordinary claims require extraordinary evidence" - Carl Sagan
 29th November 2002, 04:26 AM #165 whitefork None of the above     Join Date: Aug 2001 Location: aka kullervo Posts: 2,339 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. __________________ Convictions are more dangerous enemies of truth than lies - Nietzsche
 7th January 2003, 06:01 AM #166 Upchurch Papa Funkosophy     Join Date: May 2002 Location: Funky Town (STL, MO) Posts: 23,426 Quote: Originally posted by Franko 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]) THE MATH VERSION! A personal favorite of mine! If nobody minds, I'll repost my response. Note: A = "Atoms" T = "TLOP" Y = "You" Quote: Originally posted by Franko A = fnT Y = fnA Y = fnA = fnT 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....
 10th February 2003, 11:59 AM #167 whitefork None of the above     Join Date: Aug 2001 Location: aka kullervo Posts: 2,339 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. __________________ Convictions are more dangerous enemies of truth than lies - Nietzsche
 10th February 2003, 12:07 PM #168 Tricky Briefly immortal Moderator     Join Date: Nov 2001 Location: The Group W bench Posts: 42,361 Quote: "This is a very worthwhile thread and I recommend that everyone read if from the beginning" ----------Isaac Asmov as told to John Edward
 10th February 2003, 12:13 PM #169 Guest   Posts: n/a Quote: Originally posted by whitefork 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. *sigh* whitefork, What makes you think Bozo will get it this time? Cheers,
 10th February 2003, 12:18 PM #170 whitefork None of the above     Join Date: Aug 2001 Location: aka kullervo Posts: 2,339 It's not for bozo, billzo. It's all about ME.... me me me me me. (me). __________________ Convictions are more dangerous enemies of truth than lies - Nietzsche
 10th February 2003, 12:29 PM #171 Guest   Posts: n/a Quote: Originally posted by whitefork It's not for bozo, billzo. It's all about ME.... me me me me me. (me). ok. you're ALL about you. ok. ok. Cheers,
 11th February 2004, 10:55 AM #172 slimshady2357 jim   Join Date: Oct 2001 Posts: 1,094 Rumpity bumpity, whitey put some good work into this thread. I'd hate to see it go in the great abyss. Adam __________________ Intellectual honesty trumps intellectual satisfaction for me any day -- Soubrette The power of logic collapses at the feet of muddy definitions. -- Paul A. 97%
 11th February 2004, 01:15 PM #173 Upchurch Papa Funkosophy     Join Date: May 2002 Location: Funky Town (STL, MO) Posts: 23,426 Quote: Originally posted by Upchurch 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.... 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?
 11th February 2004, 01:21 PM #174 Darat Lackey Administrator / JREF Forum Liaison     Join Date: Aug 2001 Location: South East, UK Posts: 64,778 Quote: Originally posted by Upchurch …snip… It's almost like lifegazer is the love child of Franko and Interesting Ian, isn't it? 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. __________________ If it were all so simple! If only there were evil people somewhere insidiously committing evil deeds, and it were necessary only to separate them from the rest of us and destroy them. But the line dividing good and evil cuts through the heart of every human being. And who is willing to destroy a piece of his own heart? - Aleksandr Solzhenitsyn 1918-2008
 11th February 2004, 01:22 PM #175 Upchurch Papa Funkosophy     Join Date: May 2002 Location: Funky Town (STL, MO) Posts: 23,426 y'sir

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