The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 19 May 2010 23:30 Jim Burns wrote: > Nam Nguyen wrote: >> William Hughes wrote: >>> On May 19, 10:44 pm, Nam Nguyen >>> <namducngu...(a)shaw.ca> wrote: >>>> William Hughes wrote: >>>> >>>>> Before we go on there are a couple of things to get >>>>> clear. Please begin your answers yes or no >>>>> Do you agree: >>>>> In current FOL. >>>>> F is valid in every model of T >>>>> does not imply that T has a model. >>>> >>>> No I don't agree. >>> >>> I suggest you take a course in FOL. >> >> Anyone, including those who couldn't refute >> logical arguments, could say that: >> it's much easier than making logical arguments. > > [Nam Nguyen, upthread:] > <unsnip> >> Let A = "The box has a $100 bill", >> B = "The box has a $100 bill". >> Consider the inference inf = "A => B". >> Now suppose you open the box and find out it >> _actually_ empty. You could say B is "vacuously true" >> till Kingdom comes but the fact still remains >> B is NOT true! > </unsnip> > > Take the course. Then you will learn what > "vacuously true" means, amongst other useful things. > > The vacuously true statement here is A -> B, > not B. Why don't you read the conversation carefully: note the word "could" in my 'You could say B is "vacuously true"'. You could name it anything you want, so long as you acknowledge B is NOT true, which is what I said above, and which somehow has escaped your attention, understanding. It's William Hughes would believe B be true, in his line of reasoning using "vacuously true". > >> A vacuous truth is a truth that is devoid of content >> because it asserts something about all members of who >> a class that is empty or because it says >> "If A then B" when in fact A is false. > http://en.wikipedia.org/wiki/Vacuously_true But we're talking about model truth that follows non-logical paradigm: Tarski' kind of truth, with different definition for being true, false!
From: Jim Burns on 20 May 2010 00:17 Nam Nguyen wrote: > Jim Burns wrote: >> Nam Nguyen wrote: >>> William Hughes wrote: >>>> On May 19, 10:44 pm, Nam Nguyen >>>> <namducngu...(a)shaw.ca> wrote: >>>>> William Hughes wrote: >>>>> >>>>>> Before we go on there are a couple of things to get >>>>>> clear. Please begin your answers yes or no >>>>>> Do you agree: >>>>>> In current FOL. >>>>>> F is valid in every model of T >>>>>> does not imply that T has a model. >>>>> >>>>> No I don't agree. >>>> >>>> I suggest you take a course in FOL. >>> >>> Anyone, including those who couldn't refute >>> logical arguments, could say that: >>> it's much easier than making logical arguments. >> >> [Nam Nguyen, upthread:] >> <unsnip> >>> Let A = "The box has a $100 bill", >>> B = "The box has a $100 bill". >>> Consider the inference inf = "A => B". >>> Now suppose you open the box and find out it >>> _actually_ empty. You could say B is "vacuously true" >>> till Kingdom comes but the fact still remains >>> B is NOT true! >> </unsnip> >> >> Take the course. Then you will learn what >> "vacuously true" means, amongst other useful things. >> >> The vacuously true statement here is A -> B, >> not B. > > Why don't you read the conversation carefully: note > the word "could" in my 'You could say B is "vacuously true"'. > You could name it anything you want, so long as you acknowledge > B is NOT true, which is what I said above, and which somehow has > escaped your attention, understanding. Just take the damned course, read the damned textbook, or whatever. No, you could NOT say B is vacuously true, not if you knew what "vacuously true" meant. Or don't -- and continue looking like an idiot. Look, Nam, it should be a piece of cake for someone who can obfuscate as well as you do. So, just do your homework, okay? > It's William Hughes would believe B be true, in his > line of reasoning using "vacuously true". > >>> A vacuous truth is a truth that is devoid of content >>> because it asserts something about all members of >>> a class that is empty or because it says >>> "If A then B" when in fact A is false. >> http://en.wikipedia.org/wiki/Vacuously_true > > But we're talking about model truth that follows > non-logical paradigm: Tarski' kind of truth, with > different definition for being true, false!
From: Nam Nguyen on 20 May 2010 00:19 Nam Nguyen wrote: > There's a model in which the universe and all n-ary relations > are empty, and this is the model for all inconsistent formal > systems. So relative to an inconsistent theory, x=x is false > on the account of this particular false model. There's a (minor) degree of glossing here. Technically, per each language L, there's one false model for all consistent theories written in that language. In details the false model per a language L(s1, S2, s3, ...) is: M = {<'A',U>, <=,{}>, <s1,{}>, <s2,{}>, <s3,{}>, ...} where U = {}, s1, s2, s3 are n-ary symbol of L. Having had the above caveat, there's only one kind of false models for all inconsistent theories: the kind in which all the U's and n-ary predicates are the empty set.
From: Nam Nguyen on 20 May 2010 00:37 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Alan Smaill wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> Alan Smaill wrote: > ... >>>>> It looks like you are claiming Shoenfield's formulation >>>>> and yours are *equivalent* (after replacing "valid" with "true"). >>>> On pg. 18 (on a "Structure", say, M) he had: >>>> >>>> "We want to define a formula A to be valid in M if all the meanings >>>> of A are true in M". >>>> >>>> So he defined being "valid" as being "true". >>> let's leave the terminology aside, and look at the logic. >>> >>>>> They are not: Shoenfield's version allows a formula to be valid even >>>>> for an inconsistent T, and yours does not. >>>> Where did he assert or stipulate that? >>> When he said: >>> >>> "A formula is valid in T if it is valid in every model of T" >> How does that invalidate a formula is being false in an inconsistent >> T? > > By normal FOL reasoning; > for example FOL with equality shows > > all x. (( x = 0 & x =/= 0 ) -> P(x)) > > for *any* predicate P. But where is the word "true" in all that syntactical, rules-of-inference-based, proof?
From: William Hughes on 20 May 2010 01:00
On May 20, 12:30 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Jim Burns wrote: > > Nam Nguyen wrote: > >> William Hughes wrote: > >>> On May 19, 10:44 pm, Nam Nguyen > >>> <namducngu...(a)shaw.ca> wrote: > >>>> William Hughes wrote: > > >>>>> Before we go on there are a couple of things to get > >>>>> clear. Please begin your answers yes or no > >>>>> Do you agree: > >>>>> In current FOL. > >>>>> F is valid in every model of T > >>>>> does not imply that T has a model. > > >>>> No I don't agree. > > >>> I suggest you take a course in FOL. > > >> Anyone, including those who couldn't refute > >> logical arguments, could say that: > >> it's much easier than making logical arguments. > > > [Nam Nguyen, upthread:] > > <unsnip> > >> Let A = "The box has a $100 bill", > >> B = "The box has a $100 bill". > >> Consider the inference inf = "A => B". > >> Now suppose you open the box and find out it > >> _actually_ empty. You could say B is "vacuously true" > >> till Kingdom comes but the fact still remains > >> B is NOT true! > > </unsnip> > > > Take the course. Then you will learn what > > "vacuously true" means, amongst other useful things. > > > The vacuously true statement here is A -> B, > > not B. > > Why don't you read the conversation carefully: note > the word "could" in my 'You could say B is "vacuously true"'. > You could name it anything you want, so long as you acknowledge > B is NOT true, which is what I said above, and which somehow has > escaped your attention, understanding. > > It's William Hughes would believe B be true, in his line of reasoning > using "vacuously true". Nope. I would claim that A>B is true. However, I would not claim that B is true. - William Hughes |