The End of an Era - That of The Natural Numbers
in [Math]
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From: Marshall on 29 May 2010 13:47 On May 29, 10:32 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > Note my "the only way" in the question. If FOL, by the technicalities > vested in all of its layers, doesn't insist that that's the only way, > then technically other ways are equally possible Yes, but: > and Marshall's counter > stipulation that x=x is true in all contexts of FOL is incorrect in one > of those possible ways doesn't follow. FOL puts *some* restrictions on what the mapping can be. One such restriction is that x=x must be true in all contexts. Marshall
From: Nam Nguyen on 29 May 2010 13:55 Nam Nguyen wrote: > Daryl McCullough wrote: >> Nam Nguyen says... >> >>> You're stipulating _one_ mapping between formulas and a set of 2 >>> binary values. >> >> That's what "truth in a structure" does. It maps formulas to "true" >> or "false". > > Right. But note in the degenerated structure of a language, it's still > true that "It maps formulas to 'true' or 'false'"! > >> >>> Is it the only way that make sense? >> >> It's the natural way to do it. The nice thing about this approach >> is that it gives a convenient way to talk about relativized theories >> and substructures. > > Note my "the only way" in the question. If FOL, by the technicalities > vested in all of its layers, doesn't insist that that's the only way, > then technically other ways are equally possible, and Marshall's counter > stipulation that x=x is true in all contexts of FOL is incorrect in one > of those possible ways > > Whether or not any way is natural or not, is actually debatable and > questionable depending on contexts we're talking about: model-theoretically > context, or otherwise. > >> Invalid does *not* mean "false", though. "x > 2" is not false. >> >> If you want to collapse "true" and "valid", that's possible, by >> interpreting a formula with free variables as meaning the same >> thing as if its free variables are universally quantified over. >> That's a common approach when dealing with axioms that have free >> variables---treat Phi(x) as an axiom as meaning the same thing >> as "Ax Phi(x)". That's really just a convention. > > But are meaning and truth (interpretation) are identical, in general? > "There's a unicorn eating hay in the zoo" is quite meaningful, but > is there anything being true in that statement when the zoo is _factually_ > empty? > > >>> The key question is how would we go from 4 to 2 and still make sense >>> in term of set-membership (to satisfy Tarski's)? The answer is we can't. >> >> What are you talking about? > > Let me make a little detour but hopefully by doing that my explanation > would be more "crisp". > > We know if A df= (B and C) then whether or not A is true would depend > on _both_ B and C right? > > Suppose the following 3 statements are _not_ FOL formulas but meta > statements and as stipulated as: > > "F is model-theoretically true" df= "T is true" and "F is interpreted > as true" > > where F is _any_ FOL formula and T is a meta assertion that the universe > U of a structure is non-empty (per a language L of course). > > Do you see the pattern now? Indeed, let's let: > > A = "F is model-theoretically true" > B = "T is true" > C = "F is interpreted as true" > > Note in FOL the individuals of an U and U itself are off-limit > to FOL expressibility: in the sense that they're of the kind > of unformalized entities that we can only have a priori and > that if we try to formalize them what we've formalized just > aren't they. Iow, B is _not_ FOL expression. > > > Would you see in now? It doesn't matter whether or not F is > _syntactically_ tautologous or contradictory, the meta statement > A, by definition of structure, will also depend on B. And if B is > false by virtual of the factual U's being empty then A is false. In other words, in model theoretically truth interpretations a la Tarski, we've been preempted by Tarski's concept of "factual" truth, as reflected by B. There's some subjective flexibility for non-logical truths in the non-degenerated cases where U isn't empty. But when U = {}, there's no flexibility at all, even for syntactically tautologous or contradictory formulas, . If we don't stay with Tarski's concept of "factual" truth, that would be another meta mapping between formulas to {'T','F'}. That would be a different subject, different debate though. *** So, are we no in agreement that there's no absolute truth, even x=x, in FOL to conclude this part of the thread, before moving on to next parts?
From: Nam Nguyen on 29 May 2010 13:58 Marshall wrote: > On May 29, 10:32 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Note my "the only way" in the question. If FOL, by the technicalities >> vested in all of its layers, doesn't insist that that's the only way, >> then technically other ways are equally possible > > Yes, but: > >> and Marshall's counter >> stipulation that x=x is true in all contexts of FOL is incorrect in one >> of those possible ways > > doesn't follow. FOL puts *some* restrictions on what the > mapping can be. One such restriction is that x=x must > be true in all contexts. And I've refuted this in the same post, via the truth of some meta statements A and B. Do you have any counter argument for this?
From: William Hughes on 29 May 2010 14:44 On May 29, 2:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > But when U = {}, there's no flexibility at all So your claim is that There does not exist an x such that blue(x) must be false? You can refer to as many mappings and definitions of "truth" as you want. At the end of the day if all formula are false in a model with empty universe, then There does not exist an x such that blue(x) must be considered false. - William Hughes even for syntactically > tautologous or contradictory formulas, . > > If we don't stay with Tarski's concept of "factual" truth, that would > be another meta mapping between formulas to {'T','F'}. That would be > a different subject, different debate though. > > *** > > So, are we no in agreement that there's no absolute truth, even x=x, > in FOL to conclude this part of the thread, before moving on to next > parts?
From: Nam Nguyen on 29 May 2010 15:03
William Hughes wrote: > On May 29, 2:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> But when U = {}, there's no flexibility at all > > So your claim is that > > There does not exist an x such that blue(x) > > must be false? You can refer to as many mappings > and definitions of "truth" as you want. At the end > of the day if all formula are false in a model with > empty universe, then > > There does not exist an x such that blue(x) > > must be considered false. It must have been the case you either didn't read or wasn't paying attention or wasn't able to understand what I said about the truth preemptive characteristics of the meta statement B in the post. It doesn't matter what I "want" here: that's Tarski's definition that was used by all as the basis of structure (model) definition in FOL. Neither does it matter what you'd want here by the same token. >> If we don't stay with Tarski's concept of "factual" truth, that would >> be another meta mapping between formulas to {'T','F'}. That would be >> a different subject, different debate though. >> >> *** >> >> So, are we no in agreement that there's no absolute truth, even x=x, >> in FOL to conclude this part of the thread, before moving on to next >> parts? > |