The End of an Era - That of The Natural Numbers
in [Math]
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From: Nam Nguyen on 19 May 2010 21:44 William Hughes wrote: > On May 19, 4:25 am, Nam Nguyen <namducngu...(a)shaw.ca> 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. "Imply" has nothing to due with model and being true definition. 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! > > Do you agree > > In current FOL, you can only show that > > F is valid in every model of {(x=x) /\ ~(x=x)} > > is false, by finding a model for {(x=x) /\ ~(x=x)} > in which F is false. No I don't agree: it's obscured. By what definition did you mean a model for an inconsistency? > >> But for now would you agree there exist >> false models in which all n-ary relations are empty? > > Yes. Out of curiosity, why would you agree to something that's, in your own words, "completely uninteresting"? > > You have introduced the term "false model" > Assuming that a "false model" of T is something that is > not a model of T, a "false model" can be anything > and is completely uninteresting. You've taken the definition of model of a T for granted: "completely uninteresting" is irrelevant here. I already define it in _technical term_ "in which all n-ary relations are empty". (Iow, don't fight with [name] definition and concentrate of the technical essence merit of the definition). And that is a false model of some consistent formal systems. 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.
From: William Hughes on 19 May 2010 22:18 On May 19, 10:44 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > On May 19, 4:25 am, Nam Nguyen <namducngu...(a)shaw.ca> 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. - William Hughes
From: Nam Nguyen on 19 May 2010 22:42 William Hughes wrote: > On May 19, 10:44 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> William Hughes wrote: >>> On May 19, 4:25 am, Nam Nguyen <namducngu...(a)shaw.ca> 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. In his definition of structure (i.e. model where truth and falsehood would be defined), pg. 18, Shoenfield stipulated 3 conditions something to be called a structure M must consist of, the last one of which says: iii) For each n-ary predicate symbol p of L other than =, an n-ary predicate pM in |M|. (|M| means the universe of M). Why don't you take your own advice and read that section and explain why that condition has the phrase "other than =". Do you understand what that phrase mean in the relationship with logical and non-logical predicates, with the definition of being true and being false?
From: Jim Burns on 19 May 2010 23:10 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. > 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
From: Nam Nguyen on 19 May 2010 23:18
Nam Nguyen wrote: > William Hughes wrote: >> On May 19, 10:44 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>> William Hughes wrote: >>>> On May 19, 4:25 am, Nam Nguyen <namducngu...(a)shaw.ca> 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. > > In his definition of structure (i.e. model where truth and falsehood > would be defined), pg. 18, Shoenfield stipulated 3 conditions something > to be called a structure M must consist of, the last one of which says: > > iii) For each n-ary predicate symbol p of L other than =, an n-ary > predicate pM in |M|. > > (|M| means the universe of M). > > Why don't you take your own advice and read that section and explain > why that condition has the phrase "other than =". Do you understand > what that phrase mean in the relationship with logical and non-logical > predicates, with the definition of being true and being false? Here are also some exhibits that would help you to understand why there's no absolute truth to any formula. E1. A formula F's being true isn't a syntactical notion, since the word "true" isn't part of vocabulary, of grammar of formulas and rules of inference. E2. Since F's being true isn't syntactical and since the word "true"/ "false" has to be _associated_ with F, then being true or being false of F is nothing more or less than a _meta mapping_. And that's the definition of truth or falsehood: a mapping from a formula to either one of 2 binary value, and in general the mapping is rather arbitrary (up to Tarski's concept of truth and up to LEM of course). E3. The arithmetic truths of the naturals would be similarly defined: a mapping between formulas of L(PA) and the 2 binary values. Exactly which mapping out of an infinite collection mappings would be rather _subjective_ (though with certain constrained) and hence arithmetic truths are of _intuitive_ nature. E4. I already explain to you the definition of the false model for all the inconsistent formal systems (theories). So in FOL, x=x being true is a relative notion: there are theories in which it's true and there are ones in which it's false. Which, as I said earlier in the thread, there's no absolute formula-truth in the context of FOL reasoning, and which you and others have not been able to refute. |