The Roles of Logic and Science in Questions of Theology

[_Gadianton]

Godel's Incompleteness Theorem

Zillion's Philosophy Pages

First let me try to state in clear terms exactly what he proved, since some of us may have sort of a fuzzy idea of his proof, or have heard it from someone with a fuzzy idea of the proof..

The proof begins with Godel defining a simple symbolic system. He has the concept of a variables, the concept of a statement, and the format of a proof as a series of statements, reducing the formula that is being proven back to a postulate by legal manipulations. Godel only need define a system complex enough to do arithmetic for his proof to hold.

Godel then points out that the following statement is a part of the system: a statement P which states "there is no proof of P". If P is true, there is no proof of it. If P is false, there is a proof that P is true, which is a contradiction. Therefore it cannot be determined within the system whether P is true.

As I see it, this is essentially the "Liar's Paradox" generalized for all symbolic systems. For those of you unfamiliar with that phrase, I mean the standard "riddle" of a man walking up to you and saying "I am lying". The same paradox emerges. This is exactly what we should expect, since language itself is a symbolic system.

Godel's proof is designed to emphasize that the statement P is *necessarily* a part of the system, not something arbitrary that someone dreamed up. Godel actually numbers all possible proofs and statements in the system by listing them lexigraphically. After showing the existence of that first "Godel" statement, Godel goes on to prove that there are an infinite number of Godel statements in the system, and that even if these were enumerated very carefully and added to the postulates of the system, more Godel statements would arise. This goes on infinitely, showing that there is no way to get around Godel-format statements: all symbolic systems will contain them.

(and more)

Source

Also recognize that Godel is one of many in Symbolic Logic: Theory and Practices

See These

JAK

True to my prediction, JAK rushes onto Google and cuts and pastes a synopsis of Godel's theorom. He then thinks, given he's studied, or shall we say, Googled formal logic for two days now, he's going to show us something new, that Godel isn't the only logician who's walked the earth. No, he's not, but his contribution is the single most important contribution to mathematical logic. He's an extremely unlikely candidate for offering a proof that is circular.

Funny though in all this, JAK found a missed word from me in one of my quotes as a way around owning up to the fact that he doesn't know what a tautology is.

[_marg]

True to my prediction, JAK rushes onto Google and cuts and pastes a synopsis of Godel's theorom. He then thinks, given he's studied, or shall we say, Googled formal logic for two days now, he's going to show us something new, that Godel isn't the only logician who's walked the earth. No, he's not, but his contribution is the single most important contribution to mathematical logic. He's an extremely unlikely candidate for offering a proof that is circular.

Funny though in all this, JAK found a missed word from me in one of my quotes as a way around owning up to the fact that he doesn't know what a tautology is.

Gad, you've offered nothing to the issue of the argument "is logic essential to theology?. You've taken pot shots, used diversionary tactics but I don't recall one post of yours in which you put forward an argument regarding the issue at hand. Did I miss it?

[_JAK]

Gad wrote Thu Aug 30, 2007 10:11 am:
JAK said (and Marg, do you lick everything JAK says from his palms or don't you?),

Fallacious Argument Ad hominem

Guilt by association and attack of person addressing issues

Fallacious Argument Ad hominem

Gad wrote Thu Aug 30, 2007 10:11 am:
Do you agree with yourself, or do you agree with JAK? Because you don't agree with each other. I'm just trying to hold JAK to his own standards. If the argument had been that CC didn't offer any arguments against Armstrong, I would have just continued on with my lurking without saying anything.

Fallacious Argument By Selective Observation
Fallacious Argument By Inflation Of Conflict

Gad wrote Thu Aug 30, 2007 10:11 am:
Some of the reason why I don't really want to take on yours and JAK's finer points in critique of Godel. First, neither of you, I don't think, have the foggiest idea of who Godel was or what his contributions to logic were.

Fallacious Argument Ad hominem
Fallacy Of Irrelevant Claim

Gad wrote Thu Aug 30, 2007 10:11 am:
(hurry JAK, hurry, do 1000 google searches in the next 10 minutes and try and pretend otherwise..)

Fallacious Argument Ad hominem

Gad wrote Thu Aug 30, 2007 10:11 am:
Second, you both make bold and reckless condemnations of Godel neither of you having studied formal logic at all,


Fallacious Argument Hasty Generalization absent Evidence
Fallacious Argument Ad hominem

Gad wrote Thu Aug 30, 2007 10:11 am:
JAKs 10000 google searches and quotes on modal logic was a pathetic and silly exercise.

Fallacious Argument Exaggeration
Fallacious Argument Hasty Generalization in Ad hominem

Gad wrote Thu Aug 30, 2007 10:11 am:
And instead of picking lets say, just the wiki article, and reading it carefully and noting the standard responses to Godel, he tried to do his own thinking and tackle the most noteworthy logician in history with his own original insights based on half a day's study of modal logic.


Fallacious Argument Changing the Subject
Fallacious Argument Failure to Meet Burden of Rejoinder
Fallacious Argument Ignore Issues Presented

Gad wrote Thu Aug 30, 2007 10:11 am:
Sorry, if I can't help but be condenscending (condescending).

Fallacious Argument Ad hominem
Fallacious Argument Evasion Tactic

Gad wrote Thu Aug 30, 2007 10:11 am:
And finally, NO, it's not my responsibility to argue why Godel's argument ISN"T a tautology.

Fallacious Argument Irrelevant Claim

Gad wrote Thu Aug 30, 2007 10:11 am:
It's up for you and JAK to show why it's a tautology, given that to my knowledge, JAK is the first person to discover this flaw in Godel.


Fallacious Argument Not Invented Here
(Ideas from elsewhere are made unwelcome.)

Fallacious Argument Shift the Burden of Proof
Fallacious Argument Causal Reductionism
Fallacious Argument Ignore Evidence presented
Fallacious Argument False Claim
Fallacious Argument Deny Evidence

Godel's Incompleteness Theorem
“Godel's incompleteness is something to expect. It is even something that can be intuitively understood without a mathematical approach and proof: the incompleteness concept appears in clearly recognizable form in Zen Buddhism.”

Fallacious Argument From Assertion
(JAK is the first person to discover flaw)

“In quick succession, Gödel deprived arithmetic of its hope of completeness and its certainty of consistency. These were devastating blows to the concepts of logic and mathematics that prevailed in 1931 when Gödel published his proofs at age 25. But the conviction that genius and time could conquer every conjecture and hypothesis, not only in arithmetic, but in all of mathematics generally, had prevailed for the two or three millenia of mathematical history before Gödel's theorems. For this reason Gödel truly seemed to overturn the glory of this glorious subject and bring an epoch to a close. With time, however, mathematicians have adapted to the post-Gödelian world, and many now find mathematics to be more beautiful incomplete than it could ever have been when considered completeable.”

50 Years Later, The Questions Remain

Hao Wang, Reflections on Kurt Gödel
“Hao Wang says that Gödel had discovered partial proofs in 1941 and 1943, and never wrote them up. Gödel told Wang in 1976 or 1977 that he was ready to reconstruct his proofs when his health improved; but he died first. Hao Wang, "Some Facts About Kurt Gödel," Journal of Symbolic Logic, 46 (1981) 653-59, at p. 657. The man who catalogued Gödel's papers after his death, John Dawson, believes that nothing of large mathematical importance is likely to be found in Gödel's untranscribed notebooks with the single possible exception of the reputed 1942 independence proof of the axiom of choice.”

(bold and underline for focus refuting claim of Gad)

Gad wrote Thu Aug 30, 2007 10:11 am:
Why woudl the link be "causal"? And aren't all initial assumptions truth by assertion? If they aren't, then they can be broken down into simpler components and hence, they wouldn't be assumptions anymore.


Fallacious Argument Shift of Topic
Fallacious Argument by Assertion
Fallacious Argument of Extension
Fallacious Argument Fallacy of Origins
Fallacious Argument of Extension

Gad wrote Thu Aug 30, 2007 10:11 am:
marg asked:
Does the conclusion conclude that a God exists as an actual entity, by actual I mean a thing which exists in reality.

Yes, but "actual entities" in philosophy are slippery. Does "truth" actually exist? Does the number "5" actually exist?

Fallacious Argument By Question
Fallacious Argument By Evasion
Fallacious Argument By Assertion

Gad wrote Thu Aug 30, 2007 10:11 am:
Why would modal logic, which is largely used to explore possible worlds, need these solid links to the actual world? Usually those who head to modal logic do so because the study of the "actual world" isn't able to resolve their questions. The links you want are there, but not as dirt samples.

Fallacious Argument By Question

Gad wrote Thu Aug 30, 2007 10:11 am:
What is an actual God? Is Aristotle's unmoved mover a God? Why or why not?

Fallacious Argument By Question

Gad wrote Thu Aug 30, 2007 10:11 am:
What is a closed system and an open system? Why are closed systems limited? Can you give an example of a formal proof done in a system that isn't closed?

Fallacious Argument By Question
Fallacious Argument Shift of Topic

(However, in another post, I addressed the "closed system" dilemma.)

JAK

[_Gadianton]

Rather than respond to posts, you engage in Argument By Question, Argument From False Authority, Begging The Question (Assuming The Answer, Tautology), Changing The Subject (Digression, Red Herring), Fallacy Of Division, Ad Hominem (attack of persons as substitute for addressing issues), Introduction of the Irrelevant, and Error Of Fact (misquote of JAK). There are likely others which I have omitted.

The main reason you're mad is because you contradicted yourself, you claimed that Godel's argument was circular, then you claimed you never said that and found a way to sidestep the issue because I forgot to put a word in my quotation. I'll be waiting a long time I suppose, for you to look up his formally stated argument on wiki and show me the circularity.

Also, something to keep in mind. I'm no expert on logic, but of the two of us, I am the one who has worked through Godel's incompleteness theorom line by line, along with a few other famous pieces in mathematical logic by Peano, Cantor, Russell, Zermelo, and Kleene. I've also read a couple texts on formal logic, worked problem sets, and modal logic in particular has relevance to areas of interest I have entirely unrelated to religion. So while there are plenty who could kick my *ss in an argument on logic, I know enough to tell when someone has absolutely no clue whatsoever on the subject. And you would be one of those someones.

Tell me I'm wrong. I'll take your word for it. Was yesterday the first day you had ever looked up a definition for "modal logic?" Was today the first day you've ever read a synopsis of Godel's incompleteness theorom? Do you really think that you need to link to definitions of begging the question and so on, as if it would be some kind of revelation for me? The fact that you're completely 'winging' your discussion here is beyond painfully obvious.

At least Marg has read that book by Copi. I mean, God knows that if logic and reason are her passion, she should at least buy one book on formal logic in her life, say, Quine's, "Methods of Logic" and spend a couple months learning something about it. Then amid her repeated lectures to the rest of us on what logical thinking is, she won't all of a sudden get confused on the definition of a tautology. But, Marg has read that one book, and in her strange devotion to your ideas, is able to come up with interpretations of you that you don't deserve, as well she is able to offer some insights that show she kind of gets what's going on and with a little further study and a lot less dogmatism, would probably be decent at philosophy. But you? Dude, I don't even know what to think. All your posts related to logic on this thread have been complete and utter nonsense. Reading a webpage or two on the definitions of some logical fallacies is nowhere near enough to get you through discussions you're trying to have on logic.

A closed system of argument is an argument having no information flowing into or out of it which is not accounted for in the system.

Closed systems, are self-limiting as they provide for all that is in the system.

Do you have a website to document your definition? (as you tend to have lots of websites to link to)

Gadianton asked:
Can you give an example of a formal proof done in a system that isn't closed?

This begs the question. The issue is the extent to which the process of gathering and assembling information is open to whatever is found in research. The issue is not about “formal proof” exclusively or the construction of a syllogism or syllogistic implication.

JAK

You really don't get what philosophers and logicians do or the reason why they do it. It's going to be hopeless to try and explain these things further.

[_Gadianton]

JAK,

I'm leaving this thread now. Really, all the "fallacy of this.." and "fallacy of that.." is getting a bit much. Wade Englund used to be the master at this line of reasoning, never needing to study anything but logical fallacies. Well, for all his faults, at least Wade did it with personality.

Let's just be clear on one thing, yesterday was the first day in your life you began to at least look up something on logic aside from definitions of fallacies. This after, spending many years on message boards speaking in the tone of the master logician. So you should thank me. Maybe when things cool down, you'll take some time and actually learn something.

But let's further be clear, given the exerpts you've quoted on modal logic, they way you tried to use those, and now your critique of Godel, and how you've bolded certain portions of commentary as if they're significant, you still have no grasp whatsoever on the subject and I have no doubts, if you were put in a room without the internet to quote mine, and administered a simple test on formal logic, its history, and it's importance, you'd get an F. I dare say you might not even get a single question right.

JAK, you know nothing, absolutely nothing about formal logic or its history.

Study hard, hope you learn something.

Marg,

I'm leaving this thread, if you want to post one or two questions you think are very important in another one, I'll try and respond.

[_marg]

Gad:

NO, NO, NO. not all deductive arguments are to some extent "tautologies", why would they be? A deductive argument is either a tautology or it isn't. And Godel is not going to argue a silly tautology. get real.

Really Gad you say "NO, NO, NO" ? Well you better tell that to Professor James Hall. And by the way here are his credentials first.

James Hall is the James Thomas Professor of Philosophy, Emeritus, at the University of Richmond, where he taught for 40 years until his retirement in 2005. He received his B.A. from Johns Hopkins University, his Masters of Theology from Southeastern Theological Seminary, and his Ph.D. from The University of North Carolina at Chapel Hill.

At the University of Richmond, Professor Hall was named Omicron Delta Kappa Faculty Member of the Year (2005), Student Government Association Faculty Member of the Year (2005), and he received the University Distinguished Educator Award (2001). He has written many articles and essays and is the author of three books: Knowledge, Belief and Transcendence; Logic Problems; and Practically Profound: Putting Philosophy to Work in Everyday Life. Professor Hall's first course with the Teaching Company was Philosophy of Religion."

Professor Hall specializes in 20th-century analytic philosophy, epistemology, logical empiricism, and the philosophy of religion. At Richmond, he was noted for developing cross-disciplinary courses combining physics, chemistry, economics, psychology, and literature with his own field of philosophy.


For you Gad: In the Teaching Company course called Tools of Thinking, understanding the world through experience and reason Lecture 13 titled "Inferences avoid Equivocation", he says and this is word for word:

" There is a sense in which every deductive argument is circular. If the conclusion you are trying to establish deductively is not in the premises somewhere then how could you argue your way from the premises to the conclusion. The problem with a question begging argument is that the conclusion simply as such, on its face, is just right there. It's not so much that the argument is circular but that the circle is very very small."

[_JAK]

Rather than respond to posts, you engage in Argument By Question, Argument From False Authority, Begging The Question (Assuming The Answer, Tautology), Changing The Subject (Digression, Red Herring), Fallacy Of Division, Ad Hominem (attack of persons as substitute for addressing issues), Introduction of the Irrelevant, and Error Of Fact (misquote of JAK). There are likely others which I have omitted.

The main reason you're mad is because you contradicted yourself, you claimed that Godel's argument was circular, then you claimed you never said that and found a way to sidestep the issue because I forgot to put a word in my quotation. I'll be waiting a long time I suppose, for you to look up his formally stated argument on wiki and show me the circularity.

Also, something to keep in mind. I'm no expert on logic, but of the two of us, I am the one who has worked through Godel's incompleteness theorom line by line, along with a few other famous pieces in mathematical logic by Peano, Cantor, Russell, Zermelo, and Kleene. I've also read a couple texts on formal logic, worked problem sets, and modal logic in particular has relevance to areas of interest I have entirely unrelated to religion. So while there are plenty who could kick my *ss in an argument on logic, I know enough to tell when someone has absolutely no clue whatsoever on the subject. And you would be one of those someones.

Tell me I'm wrong. I'll take your word for it. Was yesterday the first day you had ever looked up a definition for "modal logic?" Was today the first day you've ever read a synopsis of Godel's incompleteness theorom? Do you really think that you need to link to definitions of begging the question and so on, as if it would be some kind of revelation for me? The fact that you're completely 'winging' your discussion here is beyond painfully obvious.

At least Marg has read that book by Copi. I mean, God knows that if logic and reason are her passion, she should at least buy one book on formal logic in her life, say, Quine's, "Methods of Logic" and spend a couple months learning something about it. Then amid her repeated lectures to the rest of us on what logical thinking is, she won't all of a sudden get confused on the definition of a tautology. But, Marg has read that one book, and in her strange devotion to your ideas, is able to come up with interpretations of you that you don't deserve, as well she is able to offer some insights that show she kind of gets what's going on and with a little further study and a lot less dogmatism, would probably be decent at philosophy. But you? Dude, I don't even know what to think. All your posts related to logic on this thread have been complete and utter nonsense. Reading a webpage or two on the definitions of some logical fallacies is nowhere near enough to get you through discussions you're trying to have on logic.

A closed system of argument is an argument having no information flowing into or out of it which is not accounted for in the system.

Closed systems, are self-limiting as they provide for all that is in the system.

Do you have a website to document your definition? (as you tend to have lots of websites to link to)

Gadianton asked:
Can you give an example of a formal proof done in a system that isn't closed?

This begs the question. The issue is the extent to which the process of gathering and assembling information is open to whatever is found in research. The issue is not about “formal proof” exclusively or the construction of a syllogism or syllogistic implication.

JAK

You really don't get what philosophers and logicians do or the reason why they do it. It's going to be hopeless to try and explain these things further.

Gad,

You appear unable to write without engaging in Fallacious Argument.

Those fallacious constructions have obfuscated the claim by CC which was challenged.

It was CC’s position that Godel proved God exists. Your diversionary tactics have not successfully shifted the topic.

Neither you nor CC have produced one scintilla of evidence that Godel proved that God exists. Nor have you established that reason and evidence have relevancy to theology. Truth by assertion unlike application of scientific method does not rely on reason and evidence as does science.

At least two of us here see through your various flawed claims and evasions.

The fallacies of the straw man attack, ad hominem, and shift of topic/issue appear to be primary for you.

JAK

[_marg]

Marg,

I'm leaving this thread, if you want to post one or two questions you think are very important in another one, I'll try and respond.

Gad, I asked you questions in this thread to get you to add your input to the issue. All you've done is arrogantly harrass so far. You've given me no reason to respect your knowledge on Godel or on the issue at hand re: logic being essential to theology, so I would have no interest in asking you questions on these topics.

[_Calculus Crusader]

Gadianton, you’re either dishonest or stupid.

JAK

This is what we book learnin' folks call a false dichotomy.

You know JAK, the only thing worse than an [expletive] moron is a pretentious [expletive] moron like you.

[_marg]

The main reason you're mad is because you contradicted yourself, you claimed that Godel's argument was circular, then you claimed you never said that and found a way to sidestep the issue because I forgot to put a word in my quotation. I'll be waiting a long time I suppose, for you to look up his formally stated argument on wiki and show me the circularity.

Also, something to keep in mind. I'm no expert on logic, but of the two of us, I am the one who has worked through Godel's incompleteness theorom line by line, along with a few other famous pieces in mathematical logic by Peano, Cantor, Russell, Zermelo, and Kleene.

Well then you are the one who should explain why Godel's ontological argument for God isn't circular, since you claim such understanding. I'm not aware that Godel makes any inductive leaps of reasoning, so if not I assume the argument is deductive. If it is deductive it is circular in reasoning, in that the conclusion contains no new information other than what is already in the premises.

Tell me I'm wrong. I'll take your word for it. Was yesterday the first day you had ever looked up a definition for "modal logic?"

What difference does that make Gad? You have the opportunity to argue a position but all you keep focusing on is attack. Do you think it a requirement that an individual be a walking encyclopedia, in order to argue a point? What is wrong with research? What is important is whether someone understands the information. You may have read tons Gad on Godel, but if so, it's not apparent, you are not coming across as being able to synthesize that knowledge into a coherent argument related to the issue at hand.

Do you really think that you need to link to definitions of begging the question and so on, as if it would be some kind of revelation for me? The fact that you're completely 'winging' your discussion here is beyond painfully obvious.

Apparently it is a revelation for you. Apparently you don't appreciate that all deductive arguments are to some extent are circular. If Godel is creating the definitions and axioms how could it not be circular, unless he makes an inductive leap to conclusion, but then it's not a proof, it's no better than a guess or opinion. It offers no useful practical value of understanding of the world.