For many properties of binary relations, like symmetry, asymmetry, and transitivity, it does not depend on what domain of the relation we are considering. More precisely, given a binary relation $R$ on $A$ and a superset $B$ of $A$, $R$ is symmetric/asymmetric/transitive on $A$ iff it is so on $B$. However, reflexivity does depend on the domain. Why? Also, bonus question, is there a logical characterization of those properties which does not depend on the domain? That is, given a formula $F$ of first-order logic, what syntactic property must it have so that $R$ satisfies $F$ in a domain-independent way?

  • 1
    $\begingroup$ Well, one sufficient syntactic property is that it not explicitly refer to the domain ... $\endgroup$ – Noah Schweber Oct 25 at 22:30
  • $\begingroup$ @NoahSchweber Although density (for any $(a,b)\in R$ with $a\neq b$, there is $c\in A$ with $c\neq a,b$ such that $(a,c),(c,b)\in R$) is also similarly extendible. $\endgroup$ – Arthur Oct 25 at 22:37
  • $\begingroup$ @Arthur That's why I said "sufficient." $\endgroup$ – Noah Schweber Oct 25 at 22:41
  • 1
    $\begingroup$ I don't think this question is remotely as bad as the amount of downvotes would indicate. Would a downvoter care to explain their reasoning? Am I missing something? $\endgroup$ – Z. A. K. Oct 26 at 3:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.