Logics for stable and unstable mereological relations

Abstract

In this paper we present logics about stable and unstable versions of several well-known relations from mereology: part-of, overlap and underlap. An intuitive semantics is given for the stable and unstable relations, describing them as dynamic counterparts of the base mereological relations. Stable relations are described as ones that always hold, while unstable relations hold sometimes. A set of first-order sentences is provided to serve as axioms for the stable and unstable relations, and representation theory is developed in similar fashion to Stone’s representation theory for distributive lattices. First-order predicate logic and modal logic are presented with semantics based on structures with stable and unstable mereological relations. Completeness theorems for these logics are proved, as well as decidability in the case of the modal logic, hereditary undecidability in the case of the first-order logic, and NP-completeness for the satisfiability problem of the quantifier-free fragment of the first-order logic.

Dynamic relational mereotopology: Logics for stable and unstable relations

“…This paper presents the notions of stable and unstable mereotopological relations and several results for representation theory and first-order logics for them. It is an extension of previous work about stable and unstable mereological relations in [12]. These papers feature relational structures with stable and unstable variants of some relations from mereology or mereotopology.…”

“…In most cases, in which such point-free spatial systems are developed ( [14], [12], [23], [22]), a generalization of Stone's representation theory for Boolean algebras and distributive lattices (see [1]) is developed also. The representation theory links the standard definition of these systems with their general definition.…”

“…These systems include the Boolean operations in their language. In the current paper (as well as in [12]) the systems include only relations and no functional operations or constants. Relational systems for mereotopology may have some philosophical motivation which we will not discuss now.…”

“…Stable and unstable mereotopological relations are also defined in [23] and [22], but in systems with richer language which includes the operations and constants from Boolean algebras. Here, as well as in [12], we consider language with only relations and no operations or constants. Thus, both these papers are relational generalizations of [23] and [22].…”

Dynamic Mereotopology: A Point-free Theory of Changing Regions. I. Stable and unstable mereotopological relations

“…In [17] we presented an axiomatic characterization of the dynamic mereological relations ≤ ∃ , O ∃ , U ∃ , ≤ ∀ , O ∀ , U ∀ (see Section 2), considering them not in the context of Boolean algebra, but as a separate relational system in which all relations are treated as primitives. In the representation theorem for this system we used methods quite different from the methods used in this paper and the reason was that in the abstract theory of these relations we can not use the rich theory of Boolean algebras and in a sense we have to imitate it in a very abstract way.…”