Duality between logics and equivalence relations

Abstract:

Assuming $\omega$ is the only measurable cardinal, we prove: (i) Let $\sim$ be an equivalence relation such that $\sim = { \equiv _L}$ for some logic $L \leqslant {L^{\ast }}$ satisfying Robinson’s consistency theorem (with ${L^{\ast }}$ arbitrary); then there exists a strongest logic ${L^ + } \leqslant {L^{\ast }}$ such that $\sim = { \equiv _{{L^ + }}}$; in addition, ${L^ + }$ is countably compact if $\sim \ne \cong$. (ii) Let $\dot \sim$ be an equivalence relation such that $\sim = { \equiv _{{L^0}}}$ for some logic ${L^0}$ satisfying Robinson’s consistency theorem and whose sentences of any type $\tau$ are (up to equivalence) equinumerous with some cardinal ${\kappa _\tau }$; then ${L^0}$ is the unique logic $L$ such that $\sim = { \equiv _L}$; furthermore, ${L^0}$ is compact and obeys Craig’s interpolation theorem. We finally give an algebraic characterization of those equivalence relations $\sim$ which are equal to ${ \equiv _L}$ for some compact logic $L$ obeying Craig’s interpolation theorem and whose sentences are equinumerous with some cardinal.