Vacuous transitive relations

Question

We know that transitive relations are those such that for every u, v, w, if uRv and vRw then uRW.

My question, which comes specifically from a modal logic context is, is a model where we have uRv and also wRq considered transitive, too? Since the antecedent of the conditional is not met, I am inclined to think that a such a model is indeed transitive, albeit vacuously.

Thanks.

Answer 1

I think you mean for the frame to have exactly four nodes $\{u,v,w,q\}$ and accessibility relation $R=\{(u,v), (w,q)\}.$ You are correct, this is a transitive frame because the definition of transitivity holds vacuously. It's as good as it holding in any other way and everything we know about transitive frames (e.g. that $\square A\to \square \square A$ holds at every node) holds in this frame.

Answer 2

If there are no triplets u,v,w such that uRv and vRw, then yes the relation is vacuously transitive.

You’re statement is quite different though, just because you found some 4 elements u, v, w, and q, where uRv and wRq, doesn’t mean that we don’t also have vRw.

Maybe it’s best to think of it as a game, you try to make a transitive relation then I come along and if I get to pick what we call u, what we call v, and what we call w, so if I can do that in a way that makes uRv and vRw but not uRw, then your relation isn’t transitive

Answer 3

Considering the implication $p$ implies $q$, its truth value is $1$ (True) whenever the antecedent $p$ is False. This is known as the law of the excluded middle. This is the reason that in the absence of any transitive pair, a relation is classified as transitive.

Answer 4

In the absence of any transitive triplets, a relation is termed as a vacuously transitive relation. The examples include the null relation and every singleton relation.