The End of an Era - That of The Natural Numbers
in [Math]
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From: Marshall on 30 May 2010 10:51 On May 29, 10:33 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > On May 29, 6:23 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> William Hughes wrote: > >>> On May 29, 5:22 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>> William Hughes wrote: > >>>>> On May 29, 4:03 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>>>> 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 > >>>>> Ok, rephrase. > >>>>> At the end of the day your claim is that, > >>>>> using Tarski's defintion of truth, > >>>>> 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. > >>>> Are you saying that > >>>> "There does not exist an x such that blue(x)" > >>>> is a FOL formula of L(T4)? > >>> Yes. Are you claiming it is not. > >> Oh. My mistake. You're right, it is: ~Ex[blue(x]). But that's a > >> just FOL and therefore is false in the structure in which U = {}, > >> due to B is false, correct? > > > Nope. If there is no x then > > > There is no x such that x is blue > > > is true. > > Then you and I aren't talking about the same thing and I was > talking about model theoretical truth, as in: > > >> 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. > > Don't you see that if B is false then A = (b /\ C) would be false, > irregardless of what C might mean? A serious question: What advantage(s) do you see to your approach in which false statements may be classified as "true" by your logic? Wouldn't it make more sense to have a goal of the logic be that it classifies true statements as true? Marshall
From: Nam Nguyen on 30 May 2010 19:28 William Hughes wrote: > On May 30, 2:33 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> William Hughes wrote: >>> If there is no x then >>> There is no x such that x is blue >>> is true. >> Then you and I aren't talking about the same thing and I was >> talking about model theoretical truth, > > > This does not matter You claim that > there is some meaning you can give to "truth" that > makes both > > There is an x such that x is blue > and > There is no x such that x is blue > > false. Of course it does matter; you just mis-characterized or misinterpreted my position. My claim here is that model theoretically truth _and falsehood_ paradigm (a la Tarski) allows us to give a truth value (false) to all formulas in a partucular _degenerated_ case. That's in no way is the same as giving the truth value "true" a degenerated meaning by such a conjunction. As long as we're talking about a wff, which the conjunction above is, there's no such thing as a meaningless formula. (Truth and falsehood in FOL is context sensitive, which is another way to view my claim). > Whatever, your reasoning, whomever you reference, > the final conclusion is silly. Well, if we make arguments based on subjective notion "Whatever, .... final conclusion is silly" then there's really nothing for me to argue.
From: Nam Nguyen on 30 May 2010 19:53 Marshall wrote: > On May 29, 10:33 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> William Hughes wrote: >>> On May 29, 6:23 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> William Hughes wrote: >>>>> On May 29, 5:22 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>> William Hughes wrote: >>>>>>> On May 29, 4:03 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>>>> 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 >>>>>>> Ok, rephrase. >>>>>>> At the end of the day your claim is that, >>>>>>> using Tarski's defintion of truth, >>>>>>> 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. >>>>>> Are you saying that >>>>>> "There does not exist an x such that blue(x)" >>>>>> is a FOL formula of L(T4)? >>>>> Yes. Are you claiming it is not. >>>> Oh. My mistake. You're right, it is: ~Ex[blue(x]). But that's a >>>> just FOL and therefore is false in the structure in which U = {}, >>>> due to B is false, correct? >>> Nope. If there is no x then >>> There is no x such that x is blue >>> is true. >> Then you and I aren't talking about the same thing and I was >> talking about model theoretical truth, as in: >> >> >> 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. >> >> Don't you see that if B is false then A = (b /\ C) would be false, >> irregardless of what C might mean? > > A serious question: > > What advantage(s) do you see to your approach in which false > statements may be classified as "true" by your logic? But where did I say or very much allude to that? I could classify all formulas as "false" in a particular degenerated case in FOL reasoning, but I have no desire whatsoever to classify "false" statements as "true" once they're already classified as false in a single context! (And it's my logic either: that's allowed in FOL whether or not one cares to find the right context to exercise that option is another matter). > Wouldn't > it make more sense to have a goal of the logic be that it > classifies true statements as true? So it seems like a couple of questions with a few "traps" in it, but I'm not going to fall for it. *** If, on the other hand, you have a different serious question such as: "What advantage(s) do you see to your approach of classifying all formulas as 'false' in the degenerated case of a structure M where it's U is empty?" then that's an objective and, imho, rather constructive question that would help both sides. So far nobody has asked me such question! So I don't venture out to answer that, yet. But if you or others do ask I'll be obliged to answer.
From: William Hughes on 30 May 2010 21:12 On May 30, 8:28 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > My claim here is that model > theoretically truth _and falsehood_ paradigm (a la Tarski) > allows us to give a truth value (false) to all formulas > in a partucular _degenerated_ case. Your claim is that for a model with an empty universe There is no x such that x is blue is false. No matter how you get to it, the fact remains that this claim is absurd. - William Hughes
From: Marshall on 30 May 2010 21:22
On May 30, 4:53 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On May 29, 10:33 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> William Hughes wrote: > >>> On May 29, 6:23 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>> William Hughes wrote: > >>>>> On May 29, 5:22 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>>>> William Hughes wrote: > >>>>>>> On May 29, 4:03 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>>>>>> 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 > >>>>>>> Ok, rephrase. > >>>>>>> At the end of the day your claim is that, > >>>>>>> using Tarski's defintion of truth, > >>>>>>> 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. > >>>>>> Are you saying that > >>>>>> "There does not exist an x such that blue(x)" > >>>>>> is a FOL formula of L(T4)? > >>>>> Yes. Are you claiming it is not. > >>>> Oh. My mistake. You're right, it is: ~Ex[blue(x]). But that's a > >>>> just FOL and therefore is false in the structure in which U = {}, > >>>> due to B is false, correct? > >>> Nope. If there is no x then > >>> There is no x such that x is blue > >>> is true. > >> Then you and I aren't talking about the same thing and I was > >> talking about model theoretical truth, as in: > > >> >> 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. > > >> Don't you see that if B is false then A = (b /\ C) would be false, > >> irregardless of what C might mean? > > > A serious question: > > > What advantage(s) do you see to your approach in which false > > statements may be classified as "true" by your logic? > > But where did I say or very much allude to that? My mistake. What I should have said was What advantage(s) do you see to your approach in which true statements may be classified as "false" by your logic? We already have such an example in the "~Ex[blue(x)]" case. The statement is true in the empty model. Your logic assigns false to all formulas in the empty model. Thus your approach assigns "false" to a true statement in this case. I am wondering what advantage you see in assigning false to a true statement. In fact, it seems to me that if there is a simple frontrunner reason for studying logic at all, it is just exactly to *avoid* doing this sort of thing. But perhaps that's just my imperialist, narrowly utilitarian view. I am sure Jesse or Aatu or someone like that will correct me if I'm wrong here. Marshall |