The End of an Era - That of The Natural Numbers
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From: Marshall on 29 May 2010 17:01 On May 29, 1:22 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > > 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)? (I wouldn't think so). Lookit him dodge, folks! He's a blur I tell you. Marshall
From: William Hughes on 29 May 2010 17:04 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. - William Hughes
From: Nam Nguyen on 29 May 2010 17:13 Marshall wrote: > On May 29, 1:23 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On May 29, 10:58 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> Marshall wrote: >>>>> On May 29, 10:32 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>> Note my "the only way" in the question. If FOL, by the technicalities >>>>>> vested in all of its layers, doesn't insist that that's the only way, >>>>>> then technically other ways are equally possible >>>>> Yes, but: >>>>>> and Marshall's counter >>>>>> stipulation that x=x is true in all contexts of FOL is incorrect in one >>>>>> of those possible ways >>>>> doesn't follow. FOL puts *some* restrictions on what the >>>>> mapping can be. One such restriction is that x=x must >>>>> be true in all contexts. >>>> And I've refuted this in the same post, via the truth of some >>>> meta statements A and B. >>> No you didn't. >> And your _technical reasons_ for saying that is ...? > > My technical reason for saying that you didn't refute > that x=x is true in all FOL contexts is that you didn't. If that's what you call "technical reason" then let me borrow what AK said before and say it _to you_: > thank you for the helpful reminder that it's useless to attempt to > discuss logic with some people. > > Have you figured out yet that you are wrong about > vacuous truth yet? You dropped the ball on that > subthread. Don't bet on it. Just reopen subthread and I'm sure you'd be shown again there's nothing there that I was wrong that's worth mentioning. Then again, the way you understand what technical reason would mean, perhaps don't bother. I'm a bit tired of your "technical reasons", to be frank.
From: Nam Nguyen on 29 May 2010 17:23 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? > > - William Hughes
From: Alan Smaill on 29 May 2010 17:27
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Alan Smaill wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Alan Smaill wrote: >>>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>>> >>>>> Alan Smaill wrote: >>>>>> Fine, take U = {0,1,2}, and take everything else as above. >>>>> Well, so far you've only spelled out U (and in effect <'A',U>). >>>>> You've not spelled out the mapping (ordered pair) <'blue',p_blue> >>>>> where p_blue is an actual _set_. Iow, if R is p_blue, can you >>>>> spell out the predicate-set R? >>>> The set is the extension of the relation R: >>>> >>>> { x in U | R(x) } = {1} >>> Now then, let's extend the language L(t4) to L(T4b) so it >>> has another 1-ary symbol 'non-blue', and extend T4 into >>> T4b so it has another axiom: non-blue(c1) <-> ~blue(c1). >>> >>> Can we keep the model M4 for T4b? If not what can we keep, >>> and what should we add? >> >> You need to say what the meaning of the new predicate is. > > What would you think "non-blue" usually mean? It's your predicate, I wouldn't introduce it myself. If you want to express the property of "not being blue" when you already have a predicate blue, then why not use the negation of FOL? >> This can be done by extending the old structure, eg by takinh >> extension of not-blue as { x in U | not R(x) } = {0,2}, >> which then provides a different structure which is a model for T4b. > > Right. But to be precise M4b now would have <'non-blue',C({1})> > where C({1}) is the complemantary set of {1} (in U), which is > {0,2}. The point is, Alan, the 1-ary predicate {0,2} is a common > part of _both_ M4 and M4b, serving the same purpose: to interpret > ~blue(c1) - or any equivalent formula - as true. Iow, difference > on this between M4 and M4b is just the _name_ of first component > of <x,{0,2}>: which name x should have? You could choose a "foreign" > name by extending the language, or a "domestic" using FOL symbol > redefinition, or in my case a variance of redefinition: an "intrinsic" > name such as "the-complementary-set-in-U-of-the-predicate-symbolized- > by-'blue'", which in this case I use an short alias '~blue'. > > And everything would still conform with the definition of structure > (model). I have no idea where this is taking us, but, yes, you can extend the initial structure in this way to get a model in the extended syntax with your extra axiom. >>>> Are you claiming that your notion of model is equivalent to >>>> Shoenfield's? >>> Of course I do. >> >> It was not clear to me if you thing you are correcting Shoenfield, >> or following him. Several of your claims are just not consistent with >> what he wrote, as others have pointed out. > > Who are they, and _exactly_ what did they point out, in relation to > my following structure definition as described by Shoenfield? Did > they _exactly refute_ what I've said here? Have I countered those > refutes? It's hard to know what you're referring to if you're precise. Let's see where your present line is leading you. >>> And I'm still in the process of doing the explanation >>> so I hope you don't mind answering the new question above. >>> >>>>>> Do you agree that it follows from his definition that a constant >>>>>> is interpreted as an element of the domain, >>>>> Suppose you have a theory T5 = {Ax[~(x=e)]}, which element of your >>>>> "domain" U (whatever U might be) would get interpreted as e? >>>> Could be any object in U. >>>> But whatever it is, the structure is not a model for >>>> T5, i.e. it will not satisfy the statement Ax[~(x=e)]. >>>> Just follow Tarski's definition. >>> Not a true model of course. But there's a false model for it. >> >> "false model" is confusing terminology. > > Is "a model in which the universe U is empty" very confusing to > you? Is that what you mean by false model? If so, why not use that terminology? Since you claim to be following Shoenfield, then here is a place where you clearly diverge: he says: "Let L be a first-order language. A _structure_ *A* for L consists of the following things: (i) A nonempty set |*A*|, called the _universe_ of *A*. The elements of |*A*| are called the _individuals_ of *A*." So that his structure is *non-empty*. Since for him, a model is a structure with extra properties, it follows that for him there are *no* models with empty universe. >> If you mean a structure where every predicate has empty extension >> (is false everywhere), then something like "everywhere-false structure" >> makes more sense. > > I only meant every predicate is an empty set, which is a simple fact, > given U = {}. And if a predicate of an empty U is empty, the so is > its _complementary predicate_ which means all formulas would be > interpreted as false in such (degenerated) structure! As others have pointed out, Shoenfield's and Tarski's definition for the truth of a negated sentence flat out disagrees with you here. p 19: " If A is ~ B [negation of B, my comment], theh *A*(A) is H_~(*A*(B))" where H_~ is the truth function for negation. Since you follow Shoenfield, then consider what happens when you take a structure S with empty universe, and a statement like "some x blue(x)": that will be false in S, right? So what will the truth value of "~ some x blue(x)" be in S, using the definition above? >> There is then a separate question as to whether >> such a structure is a model of a given T or not, in Shoenfield's sense. >> >> p 22: >> >> "By a *model* of a theory T, we mean a structure for L(T) in which >> all the non-logical axioms of T are *valid*." > > Sure. that's what he said. But would there be any reasonable cause > to believe it's not the case that by "mode" there he only meant T be > assumed consistent hence "model" would refer to a non-degenerated > structure? No, what reasonable case would that be?? After all, he proves the completeness theorem for 1st-order logic: (2nd form, p 43): "A theory T is consistent iff it has a model." So, he is saying explicitly that if T is not consistent, then it does *not* have a model. (The "iff" is "if and only if" here, so that the implication goes both ways.) >>> _Everything is n-ary_ in his treatment: including 0-ary (constant) >>> symbol, 0-tuple, 0-ary function value for a constant itself. >> >> Fine. >> >>> So if an >>> _n-ary predicate is a set of n-tuples_, then the predicate is always >>> a set (even if it's empty)!. So the 2nd component _is a set_ in all >>> cases. >> >> I was not asking about predicates, but about constants (0-ary function >> symbols). > > Right. > >> >> From p 18 of my copy: >> >> "Let L be a first-order language. A _structure_ *A* for L consists of >> the following things: >> >> (i) A nonempty set |*A*|, called the _universe_ of *A*. The >> elements of |*A*| are called the _individuals_ of *A*. >> >> (ii) For each n-ary function symbol f of L, an n-ary function f_a >> from |*A*| to |*A*|. (In particular, for each constant e of L, e_a >> is an individual of *A*.) >> >> (iii) For each n-ary predicate symbol p of L ..... " >> >> So, the meaning of a constant is an _individual_ of *A*, ie a _member_ of >> the universe. In particular, you cannot have the meaning of a constant >> as the empty set (unless the empty set itself is a member of the universe). > > If you reflect on pg. 10 where he said about "predicate", "0-tuple", > "0-ary function" and the like you'd see that in the degenerated case of > U = {}, and _only_ in that case, all predicates and individuals would be > defined as the empty set. Why on earth would Shoenfield say: "In particular, for each constant e of L, e_a is an individual of *A*." if he meant that individuals could denote the empty set??? Someone *can* come along and propose a different notion of model from Tarski/Shoenfield. Clearly Shoenfield himself had ruled out empty universes, and clearly also he insists that constants correspond to elements of the universe. If you want a different notion, then just be clear that you *are* proposing something different (and then it can be asked what advantages there might be). > It'd would also help if you notice that for a non-empty U, an n-ary > predicate is always of a different type than that of an (n+1)-ary > predicate. In fact, the 2 predicates can't even be equal, given that > the individuals in U are treated here as "urelements". But why they, > including the 0-ary function value for an individual constant symbol, > all in a sudden become equal here? > > Because that's the nature of the empty set {}, and because that's what > we mean in the meta level by the word "degenerated". > > Hope this has helped you to understand the essence of Shoendfield's > wording on the definition of structure, model of a language, which > would be used for theories in general, including the degenerated theory, > for each language. That's not *his* usage at all. And it leads to contradictions with basic principles, like Tarski's recursive definition of truth, and important claims, such as the completeness theorem. -- Alan Smaill |