'When Are Relations Neither True Nor False?
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From: Nam Nguyen on 20 Feb 2010 13:00 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".
From: Nam Nguyen on 20 Feb 2010 13:15 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. Anyone cares to constructively contribute to the classification. Even if just to say that's impossible and explain why it is so.
From: Frederick Williams on 20 Feb 2010 13:18 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). But you now need to correct your: 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. -- ..... A lamprophyre containing small phenocrysts of olivine and augite, and usually also biotite or an amphibole, in a glassy groundmass containing analcime.
From: Nam Nguyen on 20 Feb 2010 13:24 Frederick Williams wrote: > Nam Nguyen wrote: >> > My mistake, let's break that into 2 T's: T1 (with A1) and T2 (with A2). > > But you now need to correct your: > > 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. > Can you elaborate as to what is there to correct, after breaking T into T1 and T2?
From: Frederick Williams on 20 Feb 2010 13:27
Nam Nguyen wrote: > > Frederick Williams wrote: > > Nam Nguyen wrote: > > >> > My mistake, let's break that into 2 T's: T1 (with A1) and T2 (with A2). > > > > But you now need to correct your: > > > > 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. > > > > Can you elaborate as to what is there to correct, after breaking T into > T1 and T2? The thing that I quoted mentions T but you longer have T, you have T1 and T2. -- ..... A lamprophyre containing small phenocrysts of olivine and augite, and usually also biotite or an amphibole, in a glassy groundmass containing analcime. |