In a Consistent System, Can a True Sentence Imply a False One?
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From: MoeBlee on 2 Aug 2010 12:01 On Aug 2, 2:18 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > To Nam Nguyen: > > Over and over again, we find you in dramatic disagreement with a > > number of people who are well informed (some of them professionals and > > some at a professional level of knowledge even if not actual > > professionals; by the way, I don't include myself) about logic - > > sometimes to the point that some of these people just give up trying > > to reason with you. This is a distinct pattern: one controversy after > > another after another in which you have your own staunch position on > > some matter in logic while a certain number of professional logicians > > and others well informed make post after post with their arguments as > > to what is incorrect in your position. > > Iirc, they pointed minor error on a couple occasions which overall didn't > effect my (key) positions arguments, and I think I did likewise to the. > So unless you have a concrete example, I'd think "incorrectness" here > was just a perception. If you choose to characterize the many arguments I refer to as ones in which there merely was pointing out minor errors, then I don't know what to say, except, even IF we accept that the arguments have only been minor, my question still stands in the sense of why do you think it happens so often that - time after time, and over years - there are any number of professionals in disagreement with you on each of these (minor) points. > To answer your questions above, within my own opinions, I suspect > it's not a fluke: people would tend disagree with me or would try > to find as many reasons as possible to disagree with me, because > the concept of relativity of mathematical reasoning in FOL isn't > readily "well-received". Though I don't think it was so much a > not-understanding as a misunderstanding between the 2 sides. Okay, that's an answer. For me, for whatever reasons (I don't need to belabor them here), it doesn't explain the situation I've mentioned, but at least I accept that that is how you view it. MoeBlee
From: Nam Nguyen on 2 Aug 2010 13:12 Nam Nguyen wrote: > > The following was a conversation between DCU and me, Dec. 2005 > ("About Consistency in 1st Order Theories"): > > DCU: wrote: > > >> Exactly what the objective is is not clear to me. It seems possible > >> that you might want to call it something other than "logic". > >> Because whatever it is, it seems that in the thing you're looking > >> for the "logic" is going to vary from person to person, and I > >> suspect it's going to seem to a lot of people like the whole > >> point to _logic_ is to study _correct_ reasoning, which will > >> _not_ vary from person to person. > > To answer your questions above, within my own opinions, I suspect > it's not a fluke: people would tend disagree with me or would try > to find as many reasons as possible to disagree with me, because > the concept of relativity of mathematical reasoning in FOL isn't > readily "well-received". but it _should be_ well received, I may add. -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Nam Nguyen on 2 Aug 2010 15:10 Nam Nguyen wrote: > Daryl McCullough wrote: >> Nam Nguyen says... >> >>> I've had a couple of related questions: >>> >>> Q1: Is it impossible (in principle, hence in practice) to know the truth >>> value of cGC (which is the FOL sentence "There are infinitely many >>> counter examples of GC")? >> >> cGC might be an example of a statement that nobody knows how to prove, >> or disprove. But there is also no good argument that it is impossible >> to prove or disprove. > > Firstly, there seems to be a subtlety about mathematical knowledge here: > there's something (truth value) possible to know even we don't know yet > or practically couldn't know; but there might also be something that's > impossible even in principle to know it. As long as the 2nd possibility > is still a possibility, then there would be good arguments for the > impossibility. [I wrote this post in a hurry to leave the my keyboard. There was supposed to be the "Secondly, ..." part]. Secondly, not all mathematical truths about the natural numbers are definable and hence in general if a particular formula's truth value hasn't already been known, there's the possibility it couldn't be unknown - in the impossibility sense. >>> Q2: Should the answer of Q1 be a yes, is it reasonable to consider >>> the truths >>> about the natural numbers relative, in general? >> >> I don't see why that would follow. > > Suppose for the time being the answer for Q1 is a yes, then the truths > of the formulas such as cGC or some of the theorems thereof would be > relative: > depending what we'd include or exclude such truths as arithmetics of the > natural numbers. Note though some of the arithmetic truths are still > absolute, common or invariant to the choice of the natural numbers: e.g. > Ax[x=/=0 -> x > 0]. In a sense, Tarski's Undefinability of Truth has already established the concept of the naturals as a relativistic concept, with "undefinable" being an alias, if not a codeword, for "unknown-ability", hence "relativity". What we're doing here with cGC is just to suggest a concrete instance of such undefinability, relativity, though in no way this would be an easy suggestion. We may have to come up with suggested criteria why it be the case and even so the criteria would most likely be just a thesis, "principle", to be accepted. On the other hand, from the practicality point of view, of mathematics being a language of physics and sciences, given that the real and complex numbers are definable from the naturals with basic concepts of set operations (e.g. equivalence classes), it'd seem worthwhile to try to come up with criteria to determine what would be absolute truths and what would be relative ones. Imho. -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: Daryl McCullough on 4 Aug 2010 11:19 MoeBlee says... >If you choose to characterize the many arguments I refer to as ones in >which there merely was pointing out minor errors, then I don't know >what to say, except, even IF we accept that the arguments have only >been minor, my question still stands in the sense of why do you think >it happens so often that - time after time, and over years - there are >any number of professionals in disagreement with you on each of these >(minor) points. The last two arguments had a certain similarity to them. First, there was an argument about what it means for a sentence to be true in a vacuous structure (a structure with an empty domain). Since nobody really cares about vacuous structures, it doesn't really matter how you define truth for one. Or you could just refrain from giving a definition in that case. Nobody much cares. But there was a heated argument over two different proposals: (1) Say that for a vacuous structure, the truths are all sentences of the form "forall x, Phi(x)" (together with all compound statements that logically follow. (2) Say that for a vacuous structure, no sentence is true. I advocated for position (1), and Nam advocated for position (2). But there is a sense in which it doesn't matter one bit. The more recent argument was over what it means to "disprove" a statement. I think everyone agrees that if T is a consistent theory, then to say that sentence S is disproved in theory T means the same thing as saying that the negation of S is provable in theory T. The controversy was over what the definition of "disprove" is for the case when T is inconsistent. Again, if you know that T is inconsistent, then there is no further interest in it, so it hardly matters how you define what it means to disprove something in such a theory. But two proposed definitions for this case were: (1) Even if T is inconsistent, you can still use the same definition of "disprove": Sentence S is disproved by T if the negation of S is proved by T. This has the consequence that *every* sentence is both proved and disproved by an inconsistent theory, but so what? (2) You can, instead, restrict the usual definition of "disprove" to consistent theories, and then say that an inconsistent theory is not capable of disproving anything at all. In both cases, we were arguing about how a definition should be applied in some extreme case where nobody is really interested in applying the definition in the first place. So why bother arguing about it? It *seems* to me that the reason Nam was so insistent on his (non-mainstream) opinion about these special cases is because he has a philosophical position that he wants to advance, and he believes that the "wrong" definition clutters up his argument for his position. The philosophical position seems to be about the relativity of truth---that there is no sentence of first-order logic that should be considered "always true", and that there is no meaningful notion of truth in mathematics beyond what is provable within a particular system. In support of the latter claim, he believes in the existence of absolutely unprovable sentences---sentences (of arithmetic, say) that can't be proved in PA, and can't be proved in any mathematical theory whose axioms are justified (in some sense). Of course, you can always prove any sentence by adding it as an axiom, but Nam would not consider that to be a legitimate proof, since you would have no justification for such an axiom. I don't really understand the connection between picking particular definitions for peculiar cases (vacuous structures, or inconsistent theories) and an argument for Nam's philosophical position, and I don't really understand his philosophical position, either, but it does seem that he considers establishing correct definitions of terms to be an important step in justifying his philosophical position. -- Daryl McCullough Ithaca, NY
From: MoeBlee on 4 Aug 2010 17:04
On Aug 4, 10:19 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > The last two arguments had a certain similarity to them. A couple of the other arguments that happen to stand out in my mind were: (1) He wouldn't agree that there is such a thing as the theory of a model (the set of sentences true in a given model), though it was explained to him over a dozen times and is found in virtually any textbook in mathematical logic. (2) The whole thing about deriving (in a language that has '0' as a constant symbol) 0=0 from the sole axiom Ax x=x (or whatever the exact example was). The more general matter that a formula with symbols that do not occur in a given set of axioms may still be derivable from that set of axioms. MoeBlee |