'When Are Relations Neither True Nor False?
in [Math]
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From: Nam Nguyen on 17 Feb 2010 00:24 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" 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 17 Feb 2010 01:53 Nam Nguyen 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 Sorry, that should have been "If you define model in such a way ..." > 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" 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: Aatu Koskensilta on 17 Feb 2010 03:45 Newberry <newberryxy(a)gmail.com> writes: > On Feb 15, 8:43�pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > >> No logic in any technical sense is involved in the observation that >> on an ordinary understanding it makes no sense to say of a relation >> that it is or is not true. Sentences, statements, propositions, >> claims, are what we usually take to be true or false, not relations. > > Are you saying that (x)Px can be true or false but (x)(y)Pxy cannot? What gives you this peculiar idea? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 17 Feb 2010 04:53 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Am I exaggerating to cover a not-so-clear-cut thread-title? Thread titles are irrelevant. > Hardly. You could google on "absolute undecidability" 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." "Philosophical" is not a term of derision. Musings about absolute undecidability are most assuredly philosophical. It seems you think I have said something about the validity of such musings. Why? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Newberry on 18 Feb 2010 00:00
On Feb 17, 12:45 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Newberry <newberr...(a)gmail.com> writes: > > On Feb 15, 8:43 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > >> No logic in any technical sense is involved in the observation that > >> on an ordinary understanding it makes no sense to say of a relation > >> that it is or is not true. Sentences, statements, propositions, > >> claims, are what we usually take to be true or false, not relations. > > > Are you saying that (x)Px can be true or false but (x)(y)Pxy cannot? > > What gives you this peculiar idea? You are a smart guy. Surely you can come up with a more substantive objection than "As usually understood it makes no sense to say of a relation that it is or is not true." > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |