Talk:Nontransitive relation

Definition

Are you sure that this definition is right? Is the existential quantifier in the right place? Doesn't non-transitive mean "not transitive"? Do you have a reference? --Sam Staton 13:08, 28 August 2007 (UTC)

Yes, I'm sure it's right. The $\exists$ is correct, although it could use some optional parentheses perhaps: $\exists$x($\neg$Rxz). The whole naming scheme is not straightforward. Perhaps not-transitive (i.e. merely putting a $\neg$ in front of the transitive relation) is equivalent or a special case of nontransitivity. I don't have a good reference (just good notes). Perhaps someone closer to a library will build on this stub. Gregbard 13:49, 28 August 2007 (UTC)

I'm also unsure of it. The opposite of transitive would be $\exists x,y,zRxy\wedge Ryz\wedge \neg Rxz$ (well, technically you'd have to use $(\exists xyzRxy\wedge Ryz\wedge \neg Rxz)\vee (\forall wxyz(Rwx\wedge Ryz)\Rightarrow (w=x\wedge y=z))$ in case there are no distinct pairs for which the relation holds.)

I'm also wary of the definite article: there's only one?

As the formula stands, I can't even tell what it means. My usual interpretation would be that for given (!) x, y, and z, if Rxy and Ryz, there is some pair (which may but need not include any of the given members) for which the relation fails. Since it would use given variables, you couldn't have a nontransitive relation as such -- a relation would be (x, y, z)-nontransitive. Clarification, please?

CRGreathouse (t | c) 13:59, 28 August 2007 (UTC)

What's the distinction between "nontransitive" and "intransitive"? I note that the article intransitivity already describes the game Rock, Paper, Scissors, so "nontransitive" is apparently synonymous with "intransitive" (loosely construed), and a redirect/merge may be in order. DavidCBryant 19:56, 28 August 2007 (UTC)

In my notes, nontransitive is different than intransitive. (But that doesn't mean a merge isn't a good idea.)

Nontransitive:(x)(y)(z)((Rxy$\wedge$Ryz)$\to$($\exists$x)($\exists$z)$\neg$Rxz)

Intransitive:(x)(y)(z)((Rxy$\wedge$Ryz)$\to$$\neg$Rxz)

Gregbard 20:39, 28 August 2007 (UTC)

Well, the older article intransitivity defines both kinds of relations you've described. An intransitive, or antitransitive, relation corresponds with your second definition. And your first definition is also discussed, as a relation whose transitive closure is not antisymmetric, or (equivalently) which does not define a partial order. So maybe adding the word "nontransitive" to the other article is the way to go. DavidCBryant 20:59, 28 August 2007 (UTC)

If by "nontransitive" you mean simply "not transitive" then the way you've written it is utterly wrong. When you write ($\exists$x)($\exists$z)$\neg$Rxz) you're necessarily referring to an "x" and a "z" that are not the same as what you called "x" and "z" earlier; you're just saying there exist two elements such that one is not related to the other. You're NOT saying there exist elements x, y, and z such that x is related to y and y to z but x is not related to z. In the notation your using, that would say $\exists$x$\exists$y$\exists$z(Rxy & Ryz & $\neg$Rxz. Those preceeding (x), (y), and (z), meaning for all z, etc...., should not be there at all. Those are not part of any reasonable definition of this concept. Michael Hardy 03:17, 29 August 2007 (UTC)

With respect Michael, you are incorrect. For all x, y, and z it is true that if x has the relation to y and y has the relation to z then there exists an x and a z for which it is not true that they hold the relation, then they have a nontransitive relation. That is exactly what the formula says. Gregbard 05:15, 29 August 2007 (UTC)

Gregbard, for a minute I'll define a relation to be universal if, for all x and y, x R y. Then, according to the definition you supply, every non-universal relation is nontransitive. So it's a very obscure notion. I think the only way you can win this one is to provide a reference.

I can find (via google) some philosophy references that define nontransitive relations as those relations that are neither transitive nor intransitive. I am not sure how standard that definition is, but it is not mentioned here in wikipedia; perhaps it should be. All the best, --Sam Staton 14:38, 29 August 2007 (UTC)

grossly incorrect

The nontransitive relation is defined as:

( x ) ( y ) ( z ) ( ( Rxy $\wedge$ Ryz )$\to$($\exists$x )($\exists$z )~Rxz )

The above is not just badly written; it's horribly incorrect. It says if x is related to y and y to z, then there is some other object, which we will call z (not the same thing as way called "z" after the preceeding word "if") such that x is not related to z. It also say "the" nontransitive relation instead of "an" intransitive relation. The correct definition would say that a (not the!) relation R is nontransitive if there exist x, y, and z such that x is related to y and y to z but x is not related to z. Michael Hardy 03:09, 29 August 2007 (UTC)

Not necessarily. You get to use the word "the" when you have the fundamental form that all nontransitive relations possess. Other notions may be a case of this more general formula which is "the" nontransitive formula. On those, the word "a" is appropriate. The goal of the article I would think is to identify the most general case, as well as others. This formula includes rock paper scissors, and also other cases where there are other z values. I'm not saying that I know for sure, but the concerns raised have not considered the possibility I raised. Your answer to this question is not necessarily as definitive as you seem to believe. I don't know how you would write that "well" in your view. I've researched these thoroughly, so sometime down the road I or someone else will be able to confirm it. Gregbard 03:38, 29 August 2007 (UTC)

Be well, Gregbard 03:38, 29 August 2007 (UTC)

In reviewing these the following seems clear to me. A relation is transitive if A has the relationship to B and B has the relationship to C then A has the relationship to C.

The least thing we need in order for a relation to be NOT-transitive or nontransitive there has to be at least one example where A doesn't have the relation to C. Well that is exactly what the stated formulation for nontransitivity says.

When there is not just one, but all As do not have the relation to all Cs, that is something different. That is intransitive. Well that is exactly what the formula stated for intransitivity says.

I am quite confused by the claim that this is all wrong. Gregbard 05:09, 29 August 2007 (UTC)

Please see Talk:intransitivity where I explain why the above formulation is not a definition of nontransitivity. In particular, the above sentence is satisfied by the four-element Boolean algebra and by any other partial order that is not a linear order. — Carl (CBM · talk) 18:13, 29 August 2007 (UTC)