Fitch's paradox and OWA
in [Logic]
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From: Nam Nguyen on 31 Dec 2009 19:03 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.
From: Marshall on 31 Dec 2009 19:15 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, which in the case of my two-constructor definition is simply case analysis of the two cases. Try it; it's fun! Marshall
From: Nam Nguyen on 31 Dec 2009 19:18 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!
From: Marshall on 31 Dec 2009 19:39 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. Marshall
From: Nam Nguyen on 31 Dec 2009 19:57
Marshall wrote: > On Dec 31, 12:29 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall 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. >> Godel didn't show any of the 2 you've mentioned. > > "Any effectively generated theory capable of expressing > elementary arithmetic cannot be both consistent and complete. > In particular, for any consistent, effectively generated formal > theory that proves certain basic arithmetic truths, there is an > arithmetical statement that is true, but not provable in the theory." > > So there cannot be a complete finite theory of basic arithmetic. > > >>> Are you saying those are equivalent? >> If I'm the one answering this question then "No": defining a model of a formal >> system is not the same as demonstrating anything about a formal system that's >> supposed to be about the model. Naturally. > > Well we agree on one thing. That's unusual. > >>> It certainly seems to me that the above is fully syntactic, >>> and is a complete definition of basic arithmetic. >> That's *not* the canonical knowledge of arithmetic: what happens to the usual >> syntactical symbol '<', in your "complete definition"? > > It's easy to extend this with <. > > >>> Are >>> there statements that are true of this definition that >>> can't be captured by any finite theory? Sure there >>> are, but that has nothing to do with whether it's >>> a proper syntactic definition. To say it's not a syntactic >>> definition, you have to point out something about >>> it that's not syntactic, or not correct as a model >>> of the naturals. >> Setting aside the missing "<", what you've defined up there is >> *in no way* conforming with the _FOL definition of a model_ which >> the naturals is supposed to be collectively. For example, what's >> the set of 2-tuples that would correspond to your '+'? > > The goal was to provide a syntactic definition of the > naturals, which I did. The goal was not to provide > a FOL model. Nonetheless it's pretty easy to > get there from here. For example: > > {((x, y), z) | x+y=z} > > >>>>> If it's actually the case (that every statement of basic arithmetic >>>>> is either true or false) then it's not a shortcoming to say so. >>>>> On the contrary, that would be a virtue. >>>> Speaking philosophically (since I'm posting from sci.philoisophy.tech), >>>> entities which in some sense exist but are thoroughly inaccessible seem >>>> to be of little value. This applies to the truth values of any >>>> statements which can never be known to be true or false. >>> While I have sympathy for that position, I don't think it's >>> tenable in the long run. Or anyway, it's not tenable to go >>> from "of little value" to suggesting that we should, say, >>> not attend to the real numbers because of the existence >>> of uncomputable numbers, or suggest that statements >>> that are undecidable one way or the other are somehow >>> neither true nor false. What they are is undecidable. >> First order undecidable formulas are in a different class than those >> that aren't model-able, aren't truth assigned-able. >> >> 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! |