Fitch's paradox and OWA
in [Logic]
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From: Nam Nguyen on 31 Dec 2009 20:10 Marshall wrote: > On Dec 31, 4:18 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Dec 31, 3:40 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> Barb Knox wrote: >>>>> They are true or false in any *particular* model. Since we apparently >>>>> cannot formally pin down arithmetic to have just one particular model >>>>> (the Standard one) then there will always be some arithmetic statements, >>>>> the undecidable ones, which are true in some models and false in others. >>>> Agree. The question - and the heart of my argument - is whether or not there >>>> exists a formula F such that it's impossible to know/assert a truth value >>>> in the collection K of _all_ arithmetic models: K = {the standard one, the >>>> non-standard ones}? I've argued that there exist such statements. >>> Why would the existence of such statements imply that there >>> are truth values other than true or false? >> Because a) FOL truth is no longer absolute: it has to be relativized to some >> models; and yet b) what one constructs and _label_ as a model might indeed >> be impossible to be technically verified as a model. How could a statement be >> true or false if in the first place it can't be true-able or false-able? >> >> I think I've asked/raised this question a few times but have yet to hear >> a response to it! > > There is simply no issue here to respond to. Everything you've > said here is either false or else it's the same as the conclusion > you're trying to establish. Great "refute" you seem to have had here! Among "everything" I've said here are a) and b). Why do you think they're false? Or you just said so out of the habit of saying things with no back-up reasons? Btw, usually "conclusion" is "the same" thing as what one would be "trying to establish". You seemed to be surprise of that. Why?
From: Nam Nguyen on 31 Dec 2009 20:31 Marshall wrote: > On Dec 31, 4:03 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Dec 31, 1:08 pm, Barb Knox <Barb...(a)LivingHistory.co.uk> wrote: >>>> Marshall <marshall.spi...(a)gmail.com> wrote: >>>>> On Dec 30, 8:16 pm, Barb Knox <s...(a)sig.below> wrote: >>>>>> Marshall <marshall.spi...(a)gmail.com> wrote: >>>>>> By the nature of the construction of predicate logic, every arithmetic >>>>>> formula must be either true or false in the standard model of the >>>>>> natural numbers. >>>>>> But, we have no satisfactory way to fully characterise that standard >>>>>> model! We all think we know what the natural numbers are, but Goedel >>>>>> showed that there is no first-order way to define them, and I don't know >>>>>> of *any* purely formal (i.e., syntactic) way to do do. >>>>> I was more under the impression that Goedel showed there >>>>> was no complete finite theory of them, rather than no >>>>> way to define them. Are you saying those are equivalent? >>>> Yes, in this context. Since we are finite beings we need to use finite >>>> systems. >>> I have no disagreement with the point about finiteness, but I >>> don't see how that point leads to saying that a theory is >>> the same thing as a definition. That is rather tantamount to >>> saying that theories are all there are, and that's just not >>> true. There are things such as computational models, >>> for examples. It seems entirely appropriate to me to >>> use a computational model as the definition of something, >>> which is why I gave a computational model of the naturals >>> as a definition. >> You seemed to have confused between the FOL definition of models of formal >> systems in general and constructing a _specific_ model _candidate_. In defining >> the naturals, say, from computational model ... or whatever, you're just >> defining what the naturals be. It's still your onerous to prove/demonstrate >> this definition of the naturals would meet the definition of a model for, >> say Q, PA, .... So far, have you or any human beings successfully demonstrated >> so, without being circular? Of course not. > > Showing that the axioms of PA are true in my definition is > straightforward, using only structural induction, It might be straightforward to you and you might call it "Cheney induction" instead of "structural induction" but it's irrelevant and the question is the same: how could you demonstrate that your definition would meet the FOL standard definition of model of a formal system? Did you already make that presentation in the thread and I simply missed it? > which in the > case of my two-constructor definition is simply case > analysis of the two cases. > > Try it; it's fun! I'm sure there are a lot of fun things in life but here the interesting thing would be demonstrating your definitions meet the FOL definition of models of formal systems. You haven't tried it; so what you've claimed here isn't interesting! > > > Marshall
From: Marshall on 31 Dec 2009 21:08 On Dec 31, 4:57 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > >> I asked you before: > > >> "(1) There are infinite counter examples of GC. > > >> Tell me what you'd even suspect as a road-map to assign true or > >> false to (1)?" > > > You keep assuming that the mere fact that a sentence is > > undecidable means that it has some definite truth value > > that is not one of {true, false}. Apparently you just take > > this as a given. I, however, regard it as a false statement. > > I'm not assuming anything in asking you the question, Marshall. > If a simple question that you, I, or anyone could either know > or don't know the answer. > > If I'm to answer the question I'd say I don't know of any possible > road-map. If you you think (1) is false, as you seem to have so, > present your road-map, reasons based on the _accepted definitions_ > of FOL models etc...to back it up > > Don't just evade the question and hope that people would understand > your argument! I have no opinion on whether (1) is true or false. I don't believe that question to be relevant to the question of whether statements in arithmetic are either definitely true or definitely false. Suppose I tell you I have a natural number in mind, but it's impossible for you to know which natural number it is. However we all know that this natural number can be encoded as a binary string. Let me ask you a question about this number: does its representation as a binary string contain any characters in it besides "0" and "1"? You don't need to know which number it is to answer this question. Likewise, you don't need to know the truth or falsity of (1) to know that its truth-value is limited to being one of those two. Marshall
From: Marshall on 31 Dec 2009 21:12 On Dec 31, 5:31 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Dec 31, 4:03 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Marshall wrote: > >>> On Dec 31, 1:08 pm, Barb Knox <Barb...(a)LivingHistory.co.uk> wrote: > >>>> Marshall <marshall.spi...(a)gmail.com> wrote: > >>>>> On Dec 30, 8:16 pm, Barb Knox <s...(a)sig.below> wrote: > >>>>>> Marshall <marshall.spi...(a)gmail.com> wrote: > >>>>>> By the nature of the construction of predicate logic, every arithmetic > >>>>>> formula must be either true or false in the standard model of the > >>>>>> natural numbers. > >>>>>> But, we have no satisfactory way to fully characterise that standard > >>>>>> model! We all think we know what the natural numbers are, but Goedel > >>>>>> showed that there is no first-order way to define them, and I don't know > >>>>>> of *any* purely formal (i.e., syntactic) way to do do. > >>>>> I was more under the impression that Goedel showed there > >>>>> was no complete finite theory of them, rather than no > >>>>> way to define them. Are you saying those are equivalent? > >>>> Yes, in this context. Since we are finite beings we need to use finite > >>>> systems. > >>> I have no disagreement with the point about finiteness, but I > >>> don't see how that point leads to saying that a theory is > >>> the same thing as a definition. That is rather tantamount to > >>> saying that theories are all there are, and that's just not > >>> true. There are things such as computational models, > >>> for examples. It seems entirely appropriate to me to > >>> use a computational model as the definition of something, > >>> which is why I gave a computational model of the naturals > >>> as a definition. > >> You seemed to have confused between the FOL definition of models of formal > >> systems in general and constructing a _specific_ model _candidate_. In defining > >> the naturals, say, from computational model ... or whatever, you're just > >> defining what the naturals be. It's still your onerous to prove/demonstrate > >> this definition of the naturals would meet the definition of a model for, > >> say Q, PA, .... So far, have you or any human beings successfully demonstrated > >> so, without being circular? Of course not. > > > Showing that the axioms of PA are true in my definition is > > straightforward, using only structural induction, > > It might be straightforward to you and you might call it "Cheney induction" > instead of "structural induction" but it's irrelevant and the question is > the same: how could you demonstrate that your definition would meet the FOL > standard definition of model of a formal system? Did you already make that > presentation in the thread and I simply missed it? What sort of thing would you accept as an answer? What difficulties do you foresee? If you are convinced it is impossible and that nothing will satisfy you, I'd rather not waste my time. On the other hand if you have a specific idea as to what a correct answer would look like, I might be able to satisfy you. Marshall
From: Gene Wirchenko on 31 Dec 2009 21:23
On Thu, 31 Dec 2009 18:10:33 -0700, Nam Nguyen <namducnguyen(a)shaw.ca> wrote: >Marshall wrote: [snip] >> There is simply no issue here to respond to. Everything you've >> said here is either false or else it's the same as the conclusion >> you're trying to establish. > >Great "refute" you seem to have had here! Among "everything" I've said here >are a) and b). Why do you think they're false? Or you just said so out of the >habit of saying things with no back-up reasons? > >Btw, usually "conclusion" is "the same" thing as what one would be "trying to >establish". You seemed to be surprise of that. Why? I am impressed with the speed that you showed yourself a fool worthy of killfiling. Sincerely, Gene Wirchenko |