The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 19 May 2010 03:25 William Hughes wrote: > On May 19, 3:27 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> William Hughes wrote: > >>> In the current FOL >>> F is valid in every model of {(x=x) /\ ~(x=x)} (*) >>> does not imply {(x=x) /\ ~(x=x)} has a model. Therefore, >>> you cannot use the fact that {(x=x) /\ ~(x=x)} does not >>> have a model to show (*) is false. Indeed, in the current >>> FOL, you can only show that (*) is false >>> by finding a model for {(x=x) /\ ~(x=x)} >>> in which F is not valid. >> What's your definition of "valid" in this context is? > > "valid" = "true", So That's what I suspected but wasn't quite sure. (Shoenfield's syntax and terminology conventions sometimes more complicated that necessary, imho). > > Indeed, in the current FOL you can only show that (*) > is false, by finding a model for {(x=x) /\ ~(x=x)} > in which F is false. It's kind of late and I have to go but I'll be back with more discussions on the issues. But for now would you agree there exist false models in which all n-ary relations are empty?
From: Alan Smaill on 19 May 2010 06:54 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. >> How would you express this in FOL? > > What do you mean by "this" here? The statement: "A formula is valid in T if it is valid in every model of T" -- Alan Smaill
From: William Hughes on 19 May 2010 08:42 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. 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. > But for now would you agree there exist > false models in which all n-ary relations are empty? Yes. 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. - William Hughes
From: Daryl McCullough on 19 May 2010 10:44 William Hughes says... >> What's your definition of "valid" in this context is? > >"valid" = "true", So I think that there is a subtle distinction between "valid" and "true". A sentence is *true* relative to a particular interpretation. A sentence is *valid* if it is true for *every* interpretation. So true is basically a two-place relationship, relating interpretations to formulas, while valid is a one-place relationship: valid(F) == forall I, true(I,F) -- Daryl McCullough Ithaca, NY
From: William Hughes on 19 May 2010 11:20
On May 19, 11:44 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > William Hughes says... > > >> What's your definition of "valid" in this context is? > > >"valid" = "true", So > > I think that there is a subtle distinction between "valid" > and "true". A sentence is *true* relative to a particular > interpretation. A sentence is *valid* if it is true for > *every* interpretation. > > So true is basically a two-place relationship, relating > interpretations to formulas, while valid is a one-place > relationship: > > valid(F) == forall I, true(I,F) > A subtle and useful distinction. However, it is too subtle for this bear of little brain. In this discussion, I use valid and true interchangably. - William Hughes |