'When Are Relations Neither True Nor False?
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From: Nam Nguyen on 20 Feb 2010 12:36 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.
From: Frederick Williams on 20 Feb 2010 12:42 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." -- ..... A lamprophyre containing small phenocrysts of olivine and augite, and usually also biotite or an amphibole, in a glassy groundmass containing analcime.
From: Jesse F. Hughes on 20 Feb 2010 12:43 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > 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. What does "in any model of T, the relation in which A1 is true is a true relation" *mean*? (Let's ignore, for now, the fact that there are no models of T.) -- Quincy (age 5): Baba, play some [computer games]. Mama: Quincy, if you want [Baba] to live, don't make those suggestions. Quincy: Make those suggestions. Got it.
From: Nam Nguyen on 20 Feb 2010 12:48 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. My mistake, let's break that into 2 T's: T1 (with A1) and T2 (with A2).
From: Newberry on 20 Feb 2010 12:52
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 relatons neither true nor false" to avoid any potential confusion? |