# Why, exactly, does reflexivity depend on the domain of the relation?

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?

• Well, one sufficient syntactic property is that it not explicitly refer to the domain ... – Noah Schweber Oct 25 at 22:30
• @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. – Arthur Oct 25 at 22:37
• @Arthur That's why I said "sufficient." – Noah Schweber Oct 25 at 22:41
• 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? – Z. A. K. Oct 26 at 3:27