The End of an Era - That of The Natural Numbers
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From: Marshall on 9 Jun 2010 23:37 On Jun 9, 11:48 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Aatu Koskensilta wrote: > > Nam Nguyen <namducngu...(a)shaw.ca> writes: > > >> Well then he has yet to demonstrate the formula is true in a false > >> model (where U is empty). > > > To your satisfaction? I doubt that's possible. > > No. Demonstrate using set-membership and 2 complementary predicates in > an empty U. That's purely technical requirements and whether or one is > satisfied is an entirely different matter from the demonstration. This has been done, repeatedly, in this thread. You didn't understand it. Marshall
From: Nam Nguyen on 10 Jun 2010 00:05 Marshall wrote: > On Jun 9, 11:48 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Aatu Koskensilta wrote: >>> Nam Nguyen <namducngu...(a)shaw.ca> writes: >>>> Well then he has yet to demonstrate the formula is true in a false >>>> model (where U is empty). >>> To your satisfaction? I doubt that's possible. >> No. Demonstrate using set-membership and 2 complementary predicates in >> an empty U. That's purely technical requirements and whether or one is >> satisfied is an entirely different matter from the demonstration. > > This has been done, repeatedly, in this thread. You didn't understand > it. Like where? Exactly what post and by whom? (And didn't I post about A = B and C to counter any claim contrary to what I've stating?)
From: Daryl McCullough on 10 Jun 2010 06:56 Nam Nguyen says... > >Marshall wrote: >> On Jun 9, 11:48 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>> Aatu Koskensilta wrote: >>>> Nam Nguyen <namducngu...(a)shaw.ca> writes: >>>>> Well then he has yet to demonstrate the formula is true in a false >>>>> model (where U is empty). >>>> To your satisfaction? I doubt that's possible. >>> No. Demonstrate using set-membership and 2 complementary predicates in >>> an empty U. That's purely technical requirements and whether or one is >>> satisfied is an entirely different matter from the demonstration. >> >> This has been done, repeatedly, in this thread. You didn't understand >> it. > >Like where? Exactly what post and by whom? > >(And didn't I post about A = B and C to counter any claim contrary >to what I've stating?) Yes, you said, essentially, that you are personally using a definition of truth in a model that relates to the definition everyone else uses in the following way: true_nam(M,Phi) = true_everyone-else(M,Phi) and nonempty(M) What everyone else is saying is that the second clause serves no purpose whatsoever. You are certainly free to use whatever definitions you like, but that's a particularly useless definition. So, the claim that people have been making should, to take you into account, be modified as follows: The sentence "Ax x=x" is true in every model, including the empty model, if we use one particular definition of truth in a model. The nice thing about using this definition is that you can then prove that something is true in model M even if you don't know what the domain of M is. For example, you could imagine constructing a model M in which the domain U is "the set of counterexamples to Goldbach's conjecture". We don't know whether such a domain is empty, or not. But we could prove certain facts about the domain, and we could prove that some sentences were true in the domain. I cannot imagine any possible use for your definition of truth in the empty model. -- Daryl McCullough Ithaca, NY
From: Marshall on 10 Jun 2010 10:30 On Jun 10, 3:56 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > The sentence "Ax x=x" is true in every model, including the > empty model, if we use one particular definition of truth in a > model. > > The nice thing about using this definition is that you > can then prove that something is true in model M even > if you don't know what the domain of M is. In database theory, this is called the domain independence property. I never really saw the importance of it, but reading your sentence above, a lightbulb went off, as it were: its value isn't in domain-valued queries but truth valued queries. Marshall
From: Alan Smaill on 10 Jun 2010 14:11
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Alan Smaill wrote: > >> >> You claim that your notion of model is equivalent to Shoenfield's >> notion. Yet Shoenfield follows Tarski's truth definition: >> the negation of a formula is true in a structure if and only if >> the formula is false in the structure. > > You either didn't read it carefully, or did too carefully to the > point of being pedantic and missed what he had said there. You got the bases covered, there. > That's all. > For example, take the condition iii he had in defining (true) model > that I mentioned a few times. > > Nam (responding to William Hughes), May 19 > >> 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). >> >> ... explain why that condition has the phrase "other than =". S. is dealing with FOL with equality, so that the interpretation of the equality predicate is not "user-defined". >> 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? Yes. > Nam, (responding to Jim Burns), May 19 > >> I cited _text book_ definition of model (e.g. condition iii pg 18, >> phrase "other than =", Shoendfield, and other quotes), and >> nobody _including you_ gave a slight reflection on them? Shoenfield's definition is clear; it doesn't need any comment. However, the definition does not stop on p 18. The definition of satisfaction for compound statements (ones with logical connectives) is on p 19, and you need to use the definitions there when seeing what goes on when you are dealing with, for example, negated formulas. > Nam, May 22 > >> The implication of condition iii is that despite all theories T's >> must extend _the_ logical theory T0 = {x=x}, a [true] model of >> T0 will NOT exist until we have in our mind a [true] model of >> a consistent T. No comment (I don't know what the point is). > Allan, May 28 > >> (iii) For each n-ary predicate symbol p of L ..... " > > Your last post on the same subject (condition iii) is about 10 days > older than when I first posted it and not only you didn't respond > to my analysis on the phrase "other than =" but you also snipped it > in your last mentioning. Why? But whether or not the snipping was > intentional, it has cost your reasoning here and prevented you > from seeing that I actually followed Shoenfield's definition even > though (and I did say that many times) he was using _true_ model > in his definition, and I'm talking about false model. The term "false model" is horribly misleading. I'll say empty domain structure, since it looks like that's what you mean. >> By all means don't do the usual thing -- >> but don't then claim that what you have is faithful to >> Shoenfield and Tarski. > > Again, true model and false model both are defined in term > of set-hood and set membership, which would reflect Tarski's > concept of "concrete and factual" truth in the realm of > abstraction and which both Schoenfield and I used for 2 > different cases: the typical case and the atypical one. But your definition is clearly different from their's -- you claim that inconsistent theories have models, they deny that. > Let's revisit the definition again this time we'd use strictly > notation and hopefully you'd see the issue better. > > The language: L = L(c1,c2,blue,non-blue) be a language with 2 > individual constant symbols: c1, c2; and 2 unary predicate ones: > blue, non-blue. Lets define the following: > > U = {1,2}, 1 = {}, 2 = {{}}, pBlue = {1}, pNon-Blue = {2} > > M = { > <'A',U>, > <'c1', 1>, <'c2', 2> > <'blue', pBlue = {1}>, > <'non-blue',pNon-Blue = {2}> > } I already complained about the <'A',U> part, which you claimed had something to do with the universal quantifier. I'll take it that the universe (in Shoenfield's sense) is U. > Let me now present to you very short but technical questions so > that your answers would illustrate the nature of being true/false > in a model-set, whether or not U is empty. > > Q1. Is the _set_ M a complete structure of L? (I.e. Does M > miss any element?) Yes, it gives interpretation for the non-logical syntax. > Q2. Suppose M is a structure of L, is blue(c1) true? Why? Yes, by following definitions. > Q3. Suppose M is a structure of L, is c1 = c2 true? Why? No, from definition of satisfaction of formulas of that shape on p19. > Q4. Let R be a 2-ary "defined symbols" (Shoenfield, pg 6) > in L and be defined as R(x,y) df= (blue(x) \/ non-blue(y)). > Is R(c1,c2) false in M, if M is a structure of L? Why? Since we already saw that blue(c1) is true, so is this statement, using the definition of satisfaction for disjunction on p 19. Note that to work out whether a formula with a defined symbol is satisfied in a structure, you do *not* use clause (iii) on p 17, you must use the unfolded formula -- in this case blue(c1) \/ non-blue(c2). p 6: "We emphasize that defined symbols are *not* symbols of the language, and that definied formulas are *not* formulas of the language. Moreover, when we say anything about a defined formula, we are really talking about the formula of the language which it abbreviates (provided it makes any difference)." So what you cannot do here is to assume that R can be interpreted by whatever subset of U x U you want; its interpretation is defined by the interpretation of the unfolded formula. So far so uncontroversial. But you claimed earlier in the thread to have a model for an inconsistent statement, like blue(c1) /\ ~ blue(c1) (Now, you *can* have a model for blue(c1) /\ non-blue(c1) because the names of the predicates are irrelevant to the logic here.) You can't use a structure with an empty domain, because c1 is interpreted as a 0-ary function into U; and there are *no* such functions if U = {}. What about the case of 0-ary predicates? In that case, whether the universe is empty or not makes no difference to satisfaction of a formula, since the elements of the universe play no role. Yes, you can assign false to every predicate; but the statement A /\ ~ A will evaluate to false using the definitions on p 19, regardless of whether A is true or false. Yet you claim to have a model of A /\ ~ A. Can you show us such a model, using Shoenfield's definitions? Here A is a 0-ary predicate. -- Alan Smaill |