Past exam of the year 1 of the computer science course of the University of Oxford / 2005 / Calculus, linear algebra, discrete mathematics, logic and proof, and models of computation / 7 / c

Totality of the binary relation is expressed byWhen $R$ is already an order, this adds comparability and makes it a total order. In fact this first-order sentence alone entails $\rho =\forall xR(x,x)$.

By the definition of semantic entailment, $\tau \models \rho$ means that every first-order structure satisfying $\tau$ also satisfies $\rho$. Let $\mathcal{M}$ be any such first-order structure, and let $a$ be any element of its domain of a first-order structure. The instance of $\tau$ with $x=y=a$ gives $R(a,a)\vee R(a,a)$, which is true only if $R(a,a)$ is true. Since $a$ was arbitrary, $\mathcal{M}\models \forall xR(x,x)$. ThereforeThere is no exception for $x=y$ in the displayed totality axiom; that case is exactly what forces the reflexive relation property.