The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 12 Jun 2010 11:44 Daryl McCullough wrote: > Nam Nguyen says... >> >> (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. I don't know much about "everyone else" but your statement "the second clause serves no purpose whatsoever" has violated the fundamental basis of Tarski's "factuality" vested in set-membership. (Assuming the 2nd clause being "nonempty(M)"). Your statement has basically discarded Tarski's concept of truth! In fact S. (Shoenfield) did allude to the 2nd clause in "We want to define a formula A to be valid in M if all the meanings of A are true in M", pg. 18. "All the meanings of A are true" requires the relevant pM be non-empty hence U itself can't be empty! [And no, "valid" doesn't have anything to do with being independent of contingency as you had alluded to earlier. Because his "valid" phrase also applies to contingent formulas, such as (n+1) > 0]. > You are certainly free to use whatever > definitions you like, but that's a particularly useless definition. That's why you don't understand it: I don't have my own definition and what I used here is different phrasing of the very definition of model truth based on Tarski's concept of "factual" truths via general set-membership. S. used the same definition but also phrased it in such a way to fit his pg. 19 _specific case_ where U is non-empty. For an analogy, and argument sake, suppose Tarski, had come up with the precise definition that any 3 distinct points would form a triangle, and that by an isosceles triangle he'd mean one with 2 equal angles. Suppose further S. decided to write a math textbook about triangle and it just dawned on him Tarski's definition would have degenerated cases in which the 3 distinct points would be on the same line, and which S. was not interested in! And being a practical author writing a text book, on pg. 19 of this fictitious book, he explicitly stated here in his book we'd consider _only typical_ triangles that have non-parallel line segments, and he rephrased the definition of an isosceles triangle as one having 2 equal sides. And in this case his book stated the formula A = "All triangles having 2 equal angles will have 2 equal sides" is a true formula. What I then stated (in this analogy) was that in the case of degenerated triangles A would be false: simply because there are triangles with 2 equal angles but with 3 different side lengths! Do you now see what happened in the analogy? S.'s rephrasing the definition of an isosceles triangle did not violate Tarski's definition, because in his typical triangle case _and only in this case_ the 2 definitions are equivalent: one definition would be as good as the other and _both_ of them are mutually inference-able and _interchangeable_ ! In other words, S.'s definition in his typical case is still subsumed by Tarski's definition. It's the same situation with the real Shoenfield's "Mathematical Logic". His definition of model would follow precisely Tarski's concept of truth using "factual" set-membership, and the way he phrased it in this particular case wouldn't be applicable to the other case (of the degenerated U being empty). But x=x being false in the degenerated case would follow the same definition of model theoretically being true or false, as much as x=x is being true in the typical case where U is non-empty. It'd would help your side to remember the following points: Point 1. ======== First and foremost, in model truth/falsehood evaluation, the _evaluation using set-membership has to come first_. Any _other_ possible ways to do the evaluation are expendable and are subsumed by being in or not in, in some n-ary predicates (where n > 0), and these other possible ways may or may not be application in different scenarios which would have no consequence to Tarski's concept of "factual" truth or falsehood via set-membership. Point 2. ======== Model truth/falsehood evaluation of a formula A and its ~A would be conducted in a pair of n-ary predicate-sets which are complementary subsets of the set of all n-tuples. Iow, if these 2 predicate-sets are, say, p1M and p2M, then the following say PIC (predicate interpretation condition) must be satisfied: p1M + p2M = {all n-tuples} p1M * p2M = {} where + is set union and * set intersection. For example, let U = {1,2,3}, n-ary be 1-ary, p1M = {{1}}, p2M = {{2},{3}} then the 2 1-ary formulas say blue(c1) and ~blue(c1) must necessarily be symmetrically evaluated w.r.t set-membership of the 2 predicates in accordance to PIC: either of the formulas can be true or false in either predicates, and vice versa. Point 3. ======== If U = {} then, say, p1M and p2M are also empty. And in this case, A and its ~A would still be evaluated using set-membership a la PIC, and both formulas are false since there are no non-empty predicate, and since according to Tarski's concept of "factual" truth and falsehood, if there are no individuals in the predicate set, the predicated statement must be false. I think where you and few others are confused is that ~A is also a predicated statement, just as A, as far as set-membership predicating is concerned. Point 4. ======== Shoendfiel's definition of a formula being "valid" in a model is actually the normal definition of being true in a model. And in this definition, both x=x and Ax[x=x] are interpreted in the identical manner: true when U is non-empty and false otherwise. I also had this somewhere above: [And no, "valid" doesn't have anything to do with being independent of contingency as you had alluded to earlier. Because his "valid" phrase also applies to contingent formulas, such as (n+1) > 0]. > 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. Again, if this is model theoretically truth and falsehood we're talking about then _there's only one definition_ of being true or false: the definition that would conform to Tarski's concept of truth embedded in the 4 points above! > 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. But that would mean you have to acknowledge (and which you and those in your side haven't acknowledged) there would be some formulas whose truth values can't be assigned! Acknowledge that first your continuing talking anything else would be more coherent. > 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, I'm sorry Daryl, what "certain facts" about the domain can we really talk about when we don't know the domain is empty or not (i.e. other than the useless tautological fact that the domain is or isn't empty)? (A domain's being empty or not is a contingency too, right?) It's as if we were saying it'd be an interesting to study life forms in a different universe even though we don't know whether or not there exists another universe! > and we could > prove that some sentences were true in the domain. But why not prove those sentences true in a domain we know for certain is non-empty? > > I cannot imagine any possible use for your definition of > truth in the empty model. Actually, that's Tarski's definition of _falsehood_ in the empty model. It also actually has some use in meta reasoning: it proves that relativity of mathematical reasoning runs deep in the foundation and that even the truth value of the logical axiom x=x isn't absolute, as Marshall would believe. (And no, changing it to Ax[x=x] doesn't change the fact neither truth value is absolute.) (If nothing else, there's alway an inconsistent theory that'd be happy to have this model!)
From: Aatu Koskensilta on 12 Jun 2010 12:10 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Actually, that's Tarski's definition of _falsehood_ in the empty > model. Where in Tarski's writings do we find these odd definitions? Come off it, your peculiar ruminations have nothing to do with the Big T. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 12 Jun 2010 13:25 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Actually, that's Tarski's definition of _falsehood_ in the empty >> model. > > Where in Tarski's writings do we find these odd definitions? Come off > it, your peculiar ruminations have nothing to do with the Big T. > Well, I'd follow the "spirit" of your advise before and say that, for the "where", you just have to go to a public source to read about T. and figure out where he'd directly say or indirectly infer or imply such "odd" definition. What I could only offer you here are a) a caveat that technically arguing doesn't "always" mean scanning the textbook, definition by definitions, digitizing it back to ascii, and automatically posting it on the ng servers; and b) the following example where: U = {1} p1 = <'blue',{1}>, p2 = <'non-blue',{}> and where 'blue' and 'non-blue' are both _explicit_ symbols of the language. Do you see that the 2nd component of p2 is an empty set? Do you understand that there's is a falsehood, involving, say, non-blue(c1) and an empty predicate-set? I guess your peculiar rumination about a purported "rumination" doesn't have anything to do with T.
From: Marshall on 12 Jun 2010 14:39 On Jun 12, 10:25 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Aatu Koskensilta wrote: > > > Where in Tarski's writings do we find these odd definitions? Come off > > it, your peculiar ruminations have nothing to do with the Big T. > > [...] > I guess your peculiar rumination about a purported "rumination" doesn't > have anything to do with T. Maybe, maybe not. Sure you would not definitely assert that Tarski is not a potato chip, right? You've been quite clear that truth is relative, and that the natural numbers are big and scary, so you'd have to also admit that Tarski might be a wafer of fried tuber. In which case, maybe his writings apply to *everything.* But I think it's safe for you to ignore any criticisms of mine (as well as any criticisms from any current or future source) because I might be a couch potato. Marshall
From: Nam Nguyen on 12 Jun 2010 15:52
Marshall wrote: > > But I think it's safe for you to ignore any criticisms of mine Of course, I didn't ignore it; I refuted it that there would be another context where your "potato chip" statement would be false, in the realm of mathematical abstraction, remember? Since then have you had another either better or really more technical example of a FOL absolute formula-truth? |