The End of an Era - That of The Natural Numbers
in [Math]
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From: Daryl McCullough on 31 May 2010 16:50 Nam Nguyen says... > >Daryl McCullough wrote: >> Anyway, this whole discussion was over the question of whether >> x=x is true. I would say, to be picky, that it is not true, but >> it is valid. > >Whatever the reason you have when saying x=x "is not true" is a >direct contradiction with Marshall's belief (if not with Aatu's >and Jesse's as well), in the original challenge I brought up above >about showing any "absolute (formula) truth" That's a complete exaggeration. The distinction between "true" and "valid" is a picky distinction, which people usually don't care about. There is no question that Aatu or any of the other non-crackpots would understand my distinction between "valid" and "true". >There seems to be 3 sides now: > >- The relative side (my side) which states there's no "absolute > (formula) truth". > >- Marshall's et al belief x=x is true in all contexts. > >- Your side: x=x is NOT true (in all contexts). > >I already defended my position with Tarski's definition via factual >set membership (viz a viz empty and non-empty predicates). First you gave a definition from Shoenfeld, which contradicted your conclusion. Nobody disagrees with Tarski's definition as applied to nonempty domains. For empty domains, it is a trivial, degenerate case, and most people don't care about how we define things in those cases. If you want, we can certainly be more explicit, and instead of saying "valid" we can say "valid for nonempty domains". -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 31 May 2010 17:08 Nam Nguyen says... > >William Hughes wrote: >> On May 31, 3:08 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> >> >>> Iow, a formula being true or false here is being being true or false >>> in the context of tautology or a contradiction. >> >> True or false in a tautology is very different from true or >> false in a contradiction. > >You just didn't carefully read what I say here. (Note my "or" was used >3 times!) > >> And since no model has a contradiction, > >But it could have an empty U and empty predicates and in which >case all formulas are interpreted to be false. That's wrong. In the usual semantics for first-order logic, a formula is interpreted to be false if and only if its negation is interpreted as true. -- Daryl McCullough Ithaca, NY
From: Marshall on 31 May 2010 17:14 On May 31, 1:21 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Daryl McCullough wrote: > > Nam Nguyen says... > > > Because you are very confused. For one thing, you have a paranoid > > belief that you have "opponents". All you have is helpful people > > who are trying to get you unconfused. > > Oh no. You might just don't remember the conversations. The following > conversation between Marshall and I on May 13 actually precipitated > this part of the thread about truth (or falsehood) as perceived in x=x. > > Marshall wrote (May 13): > > On May 13, 7:13 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> ... the fact that nobody could have a single example of a _FOL_ > >> absolute (formula) truth. > > > x=x > > > > > Anyway, this whole discussion was over the question of whether > > x=x is true. I would say, to be picky, that it is not true, but > > it is valid. > > Whatever the reason you have when saying x=x "is not true" is a > direct contradiction with Marshall's belief (if not with Aatu's > and Jesse's as well), in the original challenge I brought up above > about showing any "absolute (formula) truth". I was quite clear from the beginning that truth is a model-theoretic term, and that the reason I suggested x=x was because it is true in every model. So, no. No contradiction. > There seems to be 3 sides now: Nope. > I already defended my position with Tarski's definition via factual > set membership (viz a viz empty and non-empty predicates). Factual set membership?! But you're still claiming that every formula in an empty model is false, even when the formula says that factual set membership shows a predicate is empty. If you are taking factual set membership as your standard, then you are contradicting yourself. Not that I'd expect you'd ever admit it. Marshall
From: Nam Nguyen on 31 May 2010 18:05 Daryl McCullough wrote: > Nam Nguyen says... >> William Hughes wrote: >>> On May 31, 3:08 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>> >>> >>>> Iow, a formula being true or false here is being being true or false >>>> in the context of tautology or a contradiction. >>> True or false in a tautology is very different from true or >>> false in a contradiction. >> You just didn't carefully read what I say here. (Note my "or" was used >> 3 times!) >> >>> And since no model has a contradiction, >> But it could have an empty U and empty predicates and in which >> case all formulas are interpreted to be false. > > That's wrong. In the usual semantics for first-order logic, > a formula is interpreted to be false if and only if its negation > is interpreted as true. Sigh. I don't know why people just don't understand simple things! Why should we care about the word "usual" here, for crying out loud. The challenge I put forward is whether or not x=x is true in all contexts in FOL reasoning. That's a yes or no question and one should just answer yes no and give proper explanation. I gave a clear answer "NO" and did defend it. If your answer is "YES" then there's no need to introduce the word "usual": x=x would be true in ALL contexts in such answer. If your answer is "NO", though Marshall may question you, I'd rest my case. [A technical answer might be difficult to come by and that's OK. But the phrasing of the challenge is way too simple to require a rephrasing!]
From: Nam Nguyen on 31 May 2010 18:20
Marshall wrote: > On May 31, 1:21 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> I already defended my position with Tarski's definition via factual >> set membership (viz a viz empty and non-empty predicates). > > Factual set membership?! But you're still claiming that every > formula in an empty model is false, even when the formula > says that factual set membership shows a predicate is empty. If this is where you got confused and not understand my explanation then that's easy to fix. In the post about "A df= (B and C)", May 29th, I had: > 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. > If you are taking factual set membership as your standard, > then you are contradicting yourself. You see, I'm not contradicting myself: whatever your expression you have as a FOL formula (and I really meant _whatever_ ), it's off limit to describe the set-hood or membership that B is about. You just happened to confuse between 1st order formulas and meta expressions I was referring to. (I did also mention the word "preempted", btw.) > > Not that I'd expect you'd ever admit it. There's nothing for me to admit. There's something you got confused which I've re-explained. |