CRS and Guarded Logics: a fruitful contact

Back and forth between algebra and model theory. Algebra and model theory are complementary stances in the history of logic, and their interaction continues to spawn new ideas, witness the interface of First-Order Logic and Cylindric Algebra. This chapter is about a more specialized contact: the flow of ideas between algebra and modal logic through ‘guarded fragments’ restricting the range of quantification over objects. Here is some general background for this topic. For a start, the connection between algebra and model theory is rather tight, since we can view universal algebra as the equational logic part of standard first-order model theory. As an illustration, van Benthem [Ben,88] has a purely model-theoretic proof of Jónsson’s Theorem characterizing the equational varieties with distributive lattices of congruence relations, a major tool of algebraists. Deeper connections arise in concrete cases with categorial dualities, such as that between BAOs and the usual relational models of modal logic. An important example is the main theorem in Goldblatt and Thomason [Gol-Tho,74] characterizing the elementary modally definable frame classes through their closure under taking generated sub-frames, disjoint unions, p-morphic images, and anti-closure under ultrafilter extensions. Its original proof goes back and forth between algebras and frames, in order to apply Birkhoff’s characterization of equational varieties.