Fitch's paradox and OWA
in [Logic]
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From: Nam Nguyen on 31 Dec 2009 14:21 Nam Nguyen wrote: > Marshall wrote: >> On Dec 31, 12:18 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>> My point still stands: if it's _impossible_ (as opposed to just being >>> difficult) >>> to assign truth values to a formula then the formula is neither true >>> nor false, >> >> Your point is still wrong. > > Why? Are you saying all formulas (written in the language of arithmetic) > must > have to be truth-definable? Do you have a reason so? Or are you just saying > that - as usual it seems? > >> >> >>> which means that collectively the naturals isn't a _complete_ model >>> of Q or its >>> extensions. >> >> Your conclusion is also still wrong, unsurprisingly. > > What isn't unsurprising is your "refute" does have any technical details > to back it up. I do hate typo; and here's the correct version: "What is unsurprising is your "refute" doesn't have any technical details to back it up." > > Sigh! Does every technical debate have to be personal fight of sort to you?
From: Daryl McCullough on 31 Dec 2009 14:22 Marshall says... > >On Dec 31, 7:10=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> I don't see a rule saying f |- []f. Where did you see that? > >He didn't say that there was an explicitly stated rule of >that form. He said that in step 8 of the derivation, they >use a rule that was explicitly stated as > If |- f then |- []f >but they use it *as if* the rule was > f |- []f No, I don't think they did that. What they did was to assume K(p & ~K(p)), and show that that leads to a contradiction. That's a proof of ~K(p & ~K(p)). So we have |- ~K(p & ~K(p)). Then we can apply the rule "If |- f, then |- [] f" to conclude []~K(p & ~K(p)) -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 31 Dec 2009 15:29 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. > 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. > > >> (The usual ways >> to define them are not fully syntactic, but rely on "the full semantics" >> of 2nd-order logic, or "a standard model" of set theory, both of which >> are more complicated than just relying on "the Standard Model" of >> arithmetic in the first place.) > > Here's a possible definition: > > nat := 0 | succ nat > > x + 0 = x > x + succ y = succ x+y > > x * 0 = 0 > x * succ y = x + (x * y) > > Is there some way this definition is not fully syntactic? Yes: The part "nat := 0 | succ nat" isn't syntactical. [In the context of FOL, being syntactical is being part of a FOL language/formula which this part doesn't seem to be]. > > 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"? > 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 '+'? > > >>> 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)?" Now if you let (1') be defined as: (1') df= (1) /\ A1 /\ A2 /\ ... A9 where A1 - A9 are Q's axioms (a la Shoenfield). Tell us, Marshall, what models or what kinds of models that you think you could assign 'true' or 'false' to (1')? If you really can't - which I don't think you can - then don't you at least think of the possibility that there are arithmetic statements that can't be true or false? Why is it that a statement has to be true or false while _there's no way_ to assign a truth value to it any way? Other than we might have grown up accustomed to it, what kind of reasoning is that? Ok I might sound a bit rhetorical here. But can you technically answer my question about (1')?
From: Nam Nguyen on 31 Dec 2009 16:07 Daryl McCullough wrote: > Nam Nguyen says... >> Daryl McCullough wrote: >>> Nam Nguyen says... >>>> Daryl McCullough wrote: >>>>> By the way, I haven't thought about it a huge amount, but I >>>>> don't have any problems with the paradox, because I don't >>>>> accept the premise: Every true proposition is potentially knowable. >>>>> It seems to me that sufficiently complex true propositions may never >>>>> be known. >>>> But how can we know it's true in the first place, when its being true >>>> can't be known? >>> I didn't say that we can *know* it is true. That's my point---something >>> can be true without anyone knowing that it is true. It might be true, >>> for example, that there is an even number of grains of sand in the world, >>> but we may never find that out. Is e^pi rational? We may never find out. >> Don't want to beat a dead horse so to speak but not knowing a truth because >> its proof (knowledge) is _finitely_ larger than what one can possibly know >> is *not* the same as not knowing a truth value because the statement is not >> *genuinely* truth-assigned-able. The "sand in the world" being an even number >> example above is of the 1st kind: not the 2nd kind. > > That was my point. So, are you with me that there could be statements that are neither true or false, on the ground that we can't assign a truth value to them; i.e., on the ground what we've _intuitively perceived_ as the "natural numbers" is _not adequate_ for us to say they are true or false? > we will never know that they are true. There can also be statements > that are true, but which we have no way of ever knowing that they are > true. For example, I flip a coin, and before I see whether it lands > heads up or tails up, it is run over by train, smashing it into a > flat, smooth chip of metal. Now, there is no way of ever knowing > whether it was heads-up or tails-up. But it is possible that > "It was heads-up before it was smashed" is true. > > Statements can be true even if there is no way to ever know that they > are true. But that's _not_ my point! The statements I have in mind are the ones that can _not_ be assigned true or false, in the first place! Do you see that they aren't of the same kind of statements you've alluded to? > > -- > Daryl McCullough > Ithaca, NY >
From: Barb Knox on 31 Dec 2009 16:08
In article <a3f061ed-3838-4be9-b73a-836141dc640f(a)u7g2000yqm.googlegroups.com>, Marshall <marshall.spight(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. > >�(The usual ways > > to define them are not fully syntactic, but rely on "the full semantics" > > of 2nd-order logic, or "a standard model" of set theory, both of which > > are more complicated than just relying on "the Standard Model" of > > arithmetic in the first place.) > > Here's a possible definition: > > nat := 0 | succ nat > > x + 0 = x > x + succ y = succ x+y > > x * 0 = 0 > x * succ y = x + (x * y) > > Is there some way this definition is not fully syntactic? > It uses no quantifying over predicates, so it can't be > using second order logic. > It certainly seems to me that the above is fully syntactic, > and is a complete definition of basic arithmetic. 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. This is the usual first-order initial-algebra definition, and with the addition of "succ x = succ y -> x=y" and an induction schema gives first-order Peano Arithmetic. First-order logic is indeed formal (i.e., syntactic) in that all inferencing activities consist of finite operations on finite strings. But, via Goedel and others, the Peano axioms do NOT fully characterise the natural numbers N. N is indeed a model (the Standard Model) which satisfies these axioms, but there are also *non-standard models* which satisfy these axioms -- these models contain infinite elements in addition to the usual naturals. You can get some of the flavour of non-standard models by considering the following non-standard model for just succ, where every element has a unique successor and predecessor: 0, 1, 2, ... ..., w-2, w-1, w, w+1, w+2, ... So, we can readily produce purely formal systems that are satisfied by N, but all of them (as far as I know) are also satisfied by other, non-standard, models. Try as we might, those pesky infinite non-standard integers keep cropping up. That is the sense in which I mean that we apparently can not formally fully characterise N. (Note that we similarly cannot formally define "finite", so the dodge of saying something like "the naturals are defined by the Peano axioms plus the restriction that everything is finite" can not be expressed purely formally.) > > > 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, I am not an expert in that field, but I believe that almost all of real analysis can be reconstructed using just computable numbers, e.g. the work of Bishop. > or suggest that statements > that are undecidable one way or the other are somehow > neither true nor false. What they are is undecidable. 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. Thus it is unreasonable to say that an undecidable statement is simply "true" or "false" -- we need to specify a particular model, almost always the Standard one, which we can not fully characterise formally. This doesn't prevent doing interesting number theory, but it is at least somewhat bothersome from a foundational perspective. |