Mereotopologies with Predicates of Actual Existence and Actual Contact

Abstract

We discuss in this work the importance of some predicates of ontological existence in mereology and in mereotopology especially for systems incorporating time. Tarski showed that mereology can be identified in some sense with complete Boolean algebras with zero 0 deleted. If one prefers to use only first-order language, the first-order theory for Boolean algebras can be used with zero included for simplicity. We extend the language of Boolean algebra with a one-place predicate AE(x), called “actual existence” and satisfying some natural axioms. We present natural models for Boolean algebras with predicate AE(x) motivating the axioms and prove corresponding representation theorems.

Mereotopology is considered as an extension of mereology with some relations of topological nature, like contact. One of the standard mereotopological systems is contact algebra, which is an extension of Boolean algebra with a contact relation C, satisfying some simple and obvious axioms. We consider in this paper a natural generalization of contact algebra as an extension of Boolean algebra with the predicate AE(x) and a contact relation C^a called “actual contact”, assuming for them natural axioms combining C^a and AE. Relational and topological models are proposed for the resulting system and corresponding representation theorems are proved.

I dedicate this paper to my teacher in logic Professor Helena Rasiowa for her 100-th birth anniversary. Professor Rasiowa showed me the importance of algebraic and topological methods in logic and this was her main influence on me.