Discrete dualities for some algebras with relations

•We present a unifying system of discrete dualities for spatial reasoning.•Discrete representation theorems are proved for syllogistic structures.•These are connected to quantifiers of restricted scope in first order logic.

In this paper we present a unifying discrete framework for various representation theorems in the field of spatial reasoning. We also show that the universal and existential quantifiers of restricted scope used in first order languages and represented as binary relations in the syllogistic algebras considered by Shepherdson (1956) [1] may be studied in this framework.