Gödel, Tarski, & Logic. (for grace2u)

grace2u,I thought it would be a good idea to begin a new thread to focus specifically on your assertions that the universe is ruled by some fundamental "laws of logic." In this post I hope to educate you a bit in the work that has been done in metalogic and the implication it has upon logic in general. (Admins, if you think this thread is inappropriately placed, or should simply be included in the original thread once it is re-opened, feel free to move or delete it since I have saved a copy of its text and can easily re-insert it where/when appopriate)First, grace2u, I will state your position as I understand it.You asserted that there exist some "laws of logic" that are universal, absolute and fixed. You also asserted that the fact that these are is evidence of your God, but I will ignore the theistic arguement for a moment since it is really only tangential to this topic. I only intend to show that the things you assert exist in fact do not and provably so.I take your assertion that the "laws of logic" are universal, absolute and fixed to mean that there exists some single set of fundamental axioms from which all logical systems proceed. You seemed to contest this in your last response to me in the former thread however, when you claimed that your "laws of thought" were different than the axioms of elementary logic. The fact is that your laws of thought state A = A, (A & ~A) = 0, and (A or ~A) = 1. The axioms of elementary logic state A = A, (A & ~A) = 0, and (A or ~A) = 1 (among additional axioms). I hope that if you wish to continue to assert that your "laws of thought" are different than the axioms of elementary logic you will kindly show me where the properties you enumerated as your "laws of thought" do not appear within elementary logic's axioms.Now, moving along to the focus of this post, I wish to introduce you to some prominent logicians and their proofs -- the first of which is Kurt Gdel.Gdel is famous for his Completeness and Incompleteness Theorems (theorems of course being statements that are consistently derivable from the axioms of the logical system in which they are formulated). What I intend to focus on here is the Incompleteness Theorem. What it states essentially is that for any set of axioms at least sufficiently complex as to model elementary arithmetic there exist within the system well-formed formulae which are true yet unprovable in the system lest the system suffer inconsistency.Gdel's statement basically said this:G(x) = "The well-formed formula G(x) is unprovable in the system."Once he was able to formulate that statement arithmatically, he was able to show that if the system could prove G(x) true, then it resulted in a contradiction, and in order to prove it false it would have to prove it true resulting also in a contradiction. Therefore the system is incomplete since there exist statements which could be formulated according to its syntactical rules yet were undecideable.Now, you asserted that the axioms of every logical system proceed from these supposedly universal, absolute, and fixed "laws of logic" implying that they must be able to model elementary arithmetic. If this is the case, then those laws must be either inconsistent or incomplete, and therefore they cannot sufficiently model the entire universe. That is to say, truth is not entirely bound to be logical.That brings me to my second famous logician: Alfred Tarski. Tarski also has a theorem which basically says that truth cannot be entirely represented in a logical system. In fact, as the page I linked you to states, Tarski's Theorem can be regarded as a corrolary of sorts to Gdel's Theorem, and it states in no uncertain terms that there must be true statements which are not logical theorems. in other words, truth is not only that which is logically provable, and therefore we can conclude that reality is capable of exhibiting behaviors which are not in strict accordance with ANY logical "laws."Now, I leave you to explain how this can be if we are to suppose that the "laws of logic" are universal, absolute and fixed. If there were such "laws", wouldn't we expect that all true statements must be therefore derivable from them? Wouldn't we expect that all truths would be logically provable?

What I think you're referring to is the self-inclusion or self-reference which characterizes Gdel's statement. Where Russel's paradox involves the "Set of all sets which do not include themselves," I think there is only a similarity to the Gdel sentence, but not necessarily relevance.Or perhaps you're referring to the well-know antinomy "The set of all sets" which -- according to its definition and Cantor's Power Set Theorem -- must include its power set which results in contradiction. When regading reality, however, it (the real universe) must correspond to the set of all sets despite the contradiction if reality is to be regarded as a whole. That is to say, that which is not included in "the set of all and only that which is real" is irrelevant since whatever it may be, it is NOT REAL. Still, Cantor's theorem supplies the power set (which is obviously real) and which must therefore be included in the set. I think what we can conclude from this is that it is always illogical to apply definitions and theorems writ within the universe about relationships observed among segments of the universe to the universe as a whole.I'll certainly be interested in what you can offer, though.

You stated that the "laws of logic" were universal. Are you now retracting that statement? If the universe is not ruled by them, then they're not universal, wouldn't you say? In what sense can they be universal yet not hold everywhere in the universe? Where did I say that I was an atheist? Well I'm not so sure what your claiming is the truth. Are the laws of logic universal or aren't they, according to you? Whatever one I want. I just so happens though, that my species has developed a widely accepted abstract structure to govern the constructions of our language. It's not so surprising really since we all share highly similar pathways for experiencing reality. If I wish to be understood in my communications, it is pragmatic that I adhere to a common structure. That doesn't make it anything more than a construction of the mind. I never said that they don't exist. I'm saying that they are not universal or absolute. I'm saying that they do not exist as objective features of reality. I'm saying they only exist in our minds. That's right. But if I want someone to agree with me, I should likely construct my argument using a system of thought that is common to other individuals. That we all agree on certain rules does not make them universal or absolute. We could all agree on different rules if we wanted. Heck, we do use different rules when describing different aspects of reality, i.e. elementary logic, fuzzy logic, quantum logic, etc... Yet apparently I can't expect the same from you. You know what prompted my opening this thread? The fact that I couldn't get you to answer the following question.And I quote myself thusly, and like so...

quote:

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If logic were indeed universal, then we would expect that there would exist no statement X such that X is a well-formed statement and yet we can not prove X true or false logically, do you agree? In other words, assuming that logic is universal, absolute and fixed, we would expect that every conceivable statement is theoretically decideably true or false, right? A simple yes or no will suffice. If your answer is no, please explain how logic can be universal, absolute and fixed and yet there exist well-formed statements which it cannot prove true or false.

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So what is it? Yes or no.

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This debate has literally nothing to do with solipsism. In many instances impossible. If you were thinking along the lines of classical logic, a statement in quantum logic would appear absurd. So what? That we do assume certain rules at certain times when communicating does not make those rules universal or absolute. We assume different rules all the time when we speak about different aspects of reality. Non-sequitur.

Of course I don't understand your statements. In one thread you asserted that the "laws of logic" were universal. In this, you've said that they're not. Which is it? You'll have to be more specific with regard to the relevance of an "eternal data set" and how it supposedly "can jump systems." What do you mean by these statements? So you're asserting that they exist, yet you don't know that they do, and in fact they are unknowable? Why should I believe you when you assert their existence then? Would you believe me if I told you that there is an invisible pink unicorn in my living room, but I just don't know it, and in fact neither you nor I can ever know it? So what? While it may seem unfortunate to you, if it is indeed the case that we can never reach absolute certainty, it will simply be a fact we'll have to live with. That is no reason to go around postulating the existence of unknowable beings in order to cope with aspects of life that make us uncomfortable. (And honestly, it doesn't make me uncomfortable at all) Right, and from your assertion that the "laws of logic" were universal and absolute, it necessarily follows that the set of these "laws of logic" be able to model every logical system. Otherwise they would not be universal and absolute as you've claimed (and then retracted, it seems). Why? Because you say so? If the set of axioms is capable of modeling elementary arithmetic, then it must be either inconsistent or incomplete. That's what the theorem states, and you've agreed that it does. If this set of axioms is universal and absolute as you've asserted, then it follows that all logical systems must proceed from it. Elementary arithmetic is a logical system, and it is incomplete. Therefore this set of axioms can not be absolute and universal since at least one logical system which proceeds from it is incomplete.[This message has been edited by ::, 11-20-2003]