'When Are Relations Neither True Nor False?
in [Math]
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From: Newberry on 20 Feb 2010 22:08 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? 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. > > 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: Aatu Koskensilta on 21 Feb 2010 04:46 Frederick Williams <frederick.williams2(a)tesco.net> writes: > 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. We can of course introduce whatever terminology we wish. There was nothing profound or even particularly interesting to my trivial observation. -- 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 21 Feb 2010 05:04 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. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Daryl McCullough on 21 Feb 2010 08:26 Nam Nguyen says... > >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. It's hard to tell in a textual medium. Try saying "A horse! A horse! My kingdom for a horse!" out loud. -- Daryl McCullough Ithaca, NY
From: Alan Smaill on 21 Feb 2010 14:29
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. At least that aspect you may have taken the wrong way. -- Alan Smaill |