Fitch's paradox and OWA
in [Logic]
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From: Marshall on 31 Dec 2009 13:08 On Dec 31, 7:10 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Jan Hidders says... > > >So what I wanted to say with the above is the following. You are of > >course right that what (C) really says is: > > >(C) if |- f then |- []f > > >And, assuming that for all f it holds that |- f iff ||- f, this is in > >fact confirmed by the model theory. However, in the inference process > >of the paradox as described on the Stanford page the rule is used as > >if it says f |- []f or |- f->[]f, and that would have the much > >stronger model-theoretic meaning that I described. > > 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 Reading that page, it looks like what he is saying accurately describes the step taken, but I know very little about modal logic. > I don't think that's a sensible modal logic rule. That's his point, as I understand it. Marshall
From: Marshall on 31 Dec 2009 13:10 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. > which means that collectively the naturals isn't a _complete_ model of Q or its > extensions. Your conclusion is also still wrong, unsurprisingly. Marshall
From: Marshall on 31 Dec 2009 13:39 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? > (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. > > 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. Marshall
From: Nam Nguyen on 31 Dec 2009 14:13 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. Sigh! Does every technical debate have to be personal fight of sort to you?
From: Nam Nguyen on 31 Dec 2009 14:16
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 meant "What is unsurprising ..." > > Sigh! Does every technical debate have to be personal fight of sort to you? |