'When Are Relations Neither True Nor False?
in [Math]
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From: Nam Nguyen on 21 Feb 2010 19:39 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Nam Nguyen wrote: >> >>> Alan Smaill wrote: >>> >>> First of all that's not what Aatu said. What he said: >>> >>> "Philosophical" is not a term of derision. >> I might add: he said this to me, _after_ the fact (i.e. after he had >> dismissed Calvin's short and only note). For the record, I didn't >> "see" his caveat about 'derision' then. Without further information I >> had no choice but thinking his dismissing was based on the word >> "philosophical", which is a bad term about reasoning, without any >> qualifying caveat. > > Calvin didn't present any reasoning, but this notion that > "philosophical" is a "bad term about reasoning" is merely > bizarre. "Philosophical", even without any qualifications, isn't a term > of derision. It's a description, not an evaluation. There's absolutely > first-rate philosophy and philosophical reasoning, and there's > fifth-rate philosophy and philosophical reasoning. > > As for Calvin's comment, I took it to be philosophical because as a > mathematical claim, in classical mathematics or intuitionistic > mathematics, it is simply false, and because such comments are usually > intended to express various philosophical ideas -- you yourself > mentioned musings about absolute undecidability. This comment, about > which we can't really make much anything without further elucidation > from Calvin, I also didn't dismiss; I merely noted that it has no > apparent relevance to Newberry's original post. It may well be relevant, > for all I know, depending on what exactly Calvin had in mind, but its > relevance simply is not apparent. > > As for thread titles, the question of their relevance is not a > "technical" question. It's a matter of convention, interpretation, and > so on. OP has already explained what I had in mind. > > As for rudeness, we may well stipulate that I'm the most rude news > debater ever to walk this earth. In this capacity, I put forth it that > you're a tad overly sensitive, and prone to read into simple and > straightforward comments, observations, words stuff that simply isn't > there. Whatever thought processes led you to your peculiar surmise about > my opinion about the title of G�del's paper from a comment about thread > titles in news I still can't fathom. I certainly could respond your post here point by point and you might respond back, etc... But I also don't feel confidence that would help this "messy" (if not "sour") conversation. So, I'm proposing that we restart the conversation step by step and we'd not argue much further until there are mutual clear understanding where we stand at each step. So, here's what Calvin said in his (only) post: >> The 'continuum hypothesis' is neither true nor false, for example. to which you responded: >> This piece of philosophical reflection -- which stands in need of some >> argument -- has no apparent relevance to Newberry's original post. to which I responded: >> "In need of some argument", yes I'd agree. But not sure about this >> post has no relevance to Newberry's original post, when the title >> of the thread is "When Are Relations Neither True Nor False". I did then and I still do now believe anything related to the essence of the whole thread, its title or its various post conversation is a fair game for discussion (within reasons of course). Of course you might disagree and if so please let me know I won't discuss with you anymore here because apparently we don't share the same philosophy about technical arguments. Other than that, could you kindly explain why you'd think issues related to the first post of a thread is relevant for discussion, while issue related to the tittle of the thread is *not*. (Please note I'm just asking here).
From: Nam Nguyen on 21 Feb 2010 20:58 Newberry wrote: > On Feb 20, 10:15 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Nam Nguyen wrote: >>> Newberry wrote: >>>> On Feb 20, 9:42 am, Frederick Williams <frederick.willia...(a)tesco.net> >>>> wrote: >>>>> Nam Nguyen wrote: >>>>>> Frederick Williams wrote: >>>>>>> Aatu Koskensilta wrote: >>>>>>>> As usually understood it makes no sense to say of a relation that >>>>>>>> it is >>>>>>>> or is not true. >>>>>>> It seems ok to me to take "such and such a relation is false" to mean >>>>>>> that no objects in the domain of discourse have the relation to one >>>>>>> another. For example "x is the mother of y" could be called false in >>>>>>> the domain {Aatu, Fred}. Ok, you might say "not satisfiable" but so >>>>>>> what? >>>>>> For example, given a language L(P1,P2) where P1, P2 are 1-ary symbols, >>>>>> let's consider the following T: >>>>>> A1: P1(x) <-> x=x >>>>>> A2: P2(x) <-> ~P1(x) >>>>>> It's obvious in any model of T, the relation in which A1 is true is >>>>>> a true relation, and the one in which A2 is true is a false relation. >>>>>> Apparently, "satisfiable" isn't even an issue here. >>>>> This is one of those threads that causes me to think "would that the >>>>> contributors could find something more interesting to discuss." >>>> I also thought that we should have rather discussed the substance. But >>>> given all this and concurring that the terminology is less important, >>>> is there perhaps a better way to express my intent rather than "when >>>> are relations neither true nor false" to avoid any potential confusion? >>> First I've already corrected this particular post of mine with: >>> > My mistake, let's break that into 2 T's: T1 (with A1) and T2 (with A2). >>> As for your "original" intention, I also already made a guess and >>> suggested: >>> > (1) When is a relation R such that a particular formula F would >>> > be neither true nor false in it? >>> I think there's a huge _substance_ we could have for furthering the >>> discussion. >>> That's to say, as you've alluded to, if we really care for the >>> substance, instead of the "peripherals". >> One of the substances we could have from the question (1) is the possibility >> of formally classifying formulas of a particular groups: arithmetically >> truth-unassigned-able formulas. > > What do you mean by that? Do you mean defining a class of Goedel > numbers of sentences that do not have a truth value? By that I'd mean the formulas written in L(0,S,+,*,<) that foundationally (in FOL) it's impossible to decide they'd be true or false in any concept that we'd call as the natural numbers > It is probably > possible, but we would be jumping ahead of ourselevs. First we need to > define the semantics i.e. a plausible explanation of why some > sentences (including arithmetic setences) are true, some false and > some neither. But I think we already mentioned the explanation, when we talked about the definitions of being true/false, of relations, incomplete relations, in the context of model-definition. Would you like for me to review the plausibility of that explanation? > >> Anyone cares to constructively contribute to the classification. Even if just >> to say that's impossible and explain why it is so.- Hide quoted text - >> >> - Show quoted text - >
From: Newberry on 21 Feb 2010 21:43 On Feb 16, 9:24 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Feb 15, 10:49 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Aatu Koskensilta wrote: > >>> Sentences, statements, propositions, claims, are what > >>> we usually take to be true or false, not relations. > >> Right. But this seems very pedantic to me. Under the circumstances > >> people would understand that it's a relation that would make formulas > >> asserting something about the relation as true or false. (E.g. the formula > >> xRy could be read as x is related to y). > > > It is the difference between asking if "2<3" is true or not, and > > asking if "<" is true or not. Is it pedantic to point out that it is > > wacky to ask if "<" is true or not? I would say not. > > Really, Marshall, do you understand what "Under the circumstances" or > "given certain contexts" mean? My guess is you didn't because nobody > here has asked the kind of wacky question like "is '<' true or not?", > *without leaving a slightest clue what that question is about*! > > I already explained to AK the circumstance in which the title-question > would make sense. You either didn't read that explanation, or simply > didn't understand and ignore it. Here is my explanation: > > >> In constructing a model if you, for argument sake, _incompletely_ > >> define a relation, say, symbolized by '<', as: > >> > >> {e0,e1), (e1,e3), ...} > >> > >> Then although you can determine the truth value of some formulas, > >> isn't it true some other formulas would be in the category of being > >> neither true nor false in this incomplete model, technically speaking? > > A mistake _some_ people tend to make is failing to remember that defining > model is _different_ from defining an instance of a model. The former is > just model definition, while the later is model construction! > > If you define a model in such a way that the truth value a formula is in > the non-LEM state, then that's nonsensical definition. On the other hand > in constructing a model (especially those _complex_ models about certain > properties of infinity), you might end up having an actual incomplete > relation and in which case some formulas would have to be neither true > nor false. And *in light of possible incomplete model cases*, Newberry's > question (in the thread title), about which scenarios of model > construction would you have an incomplete relation where some predicates > about the relation is neither true nor false, is a valid question. > > Am I exaggerating to cover a not-so-clear-cut thread-title? Hardly. > You could google on "absolute undecidability" Well, I googled this, I found this paper Peter Koellner, "On the Question of Absolute Undecidability." The very first sentence, "The incompleteness theorems show that for every sufficiently strong consistent formal system of mathematics there are mathematical statements undecided relative to the system" is false or at least misleading. Goedel's theorems do not apply to systems with gaps. So if a sentence is neither true nor false then I do not know if calling it it "undecidable" conveys what is going on. We usualy use the term "undecidable" in the context of sentences that are true but underivable. > to see some links > about Godel's view on the subject together with some related mentioning > of CH. (It's these information that I think Calvin's mentioning CH > in this thread is valid, not "philosophical" as AK suspected." > > In summary, there's nothing "wacky" about the fact that there are > constructed relations in which some formulas would be neither true > nor false.
From: Nam Nguyen on 21 Feb 2010 22:44 Newberry wrote: > On Feb 16, 9:24 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Am I exaggerating to cover a not-so-clear-cut thread-title? Hardly. >> You could google on "absolute undecidability" > > Well, I googled this, I found this paper Peter Koellner, "On the > Question of Absolute Undecidability." The very first sentence, "The > incompleteness theorems show that for every sufficiently strong > consistent formal system of mathematics there are mathematical > statements undecided relative to the system" is false or at least > misleading. Goedel's theorems do not apply to systems with gaps. > So if > a sentence is neither true nor false then I do not know if calling it > it "undecidable" conveys what is going on. That's why there's always the prefix "absolute" when we talk about this particular issue. Also, I think there are more than just 1 technical context where "undecidable" could be very meaningful. For example, in our context here, "undecidable" means it's undecidable as to which _exact_ relation (set) is the underlying one, when talking about a particular formula F as being true/false. > We usualy use the term > "undecidable" in the context of sentences that are true but > underivable. That's right. We _usually_ do. But that doesn't mean the word would be applicable only in that context.
From: Nam Nguyen on 21 Feb 2010 23:30
Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Alan Smaill wrote: >> >>> You may regard him as being rude. >>> You may also be over-reacting, by taking the comment about philosophy >>> as negative in itself. >> I don't think I've been overacting. >> >> There's nothing wrong with AP's "correcting mathematics" in the other >> thread on the ground that it must be limited by the actual physical >> universe, _as a philosophy_ (AP's philosophy). It's only in mathematical >> reasoning context that it's ridiculous to claim mathematics is wrong >> on the ground of such philosophy. > > If you believe that, then presumably you think that mathematicians like > Brouwer or Poincare were ridiculous when they contested received > mathematical opinion on whether every mathematical statement must be > true or false; that's the sort of example presumably in > the background here. > >> The point being is yes there's "Mathematical Philosophy" and there are cases >> we have to take some _technical philosophical_ stand on mathematical issues, >> but dismissing people's technical arguments, proposals,... on the ground of >> being "philosophical" _without_ elaboration is at minimum double-talking: >> after all dismissing here is really _discrediting_. > > Theer are two separate issues -- whether introducing philosophy > is in itself being dismissive, and whether your point was treated seriously > or not. It's only the first I was talking about. > > When Aatu says that the mention of philosophy is not dismissive, > I see no reason to think he is lying to us. For the record I didn't accuse anyone as lying. I might have accused him of being unfair in single-handedly deciding when relevancy exist or not in technical conversations and in the process leaving people no fair chance to defend or express their views; and he might believe he has his own valid reasons to do so, and I might not agree with the reasons (and might still protest them). But that's not the same as accusing him of lying. > At least that > aspect you may have taken the wrong way. > Since I didn't go to that lying-accusation path, I have to disagree with you. |