Although Aristotle (Metaphysics, Book IV, Chapter 2) was perhaps the first person to consider the part-whole relationship to be a proper subject matter for philosophic inquiry, the Polish logician Stanislow Lesniewski [15] is generally given credit for the first formal treatment of the subject matter in his Mereology.1 Woodger [30] and Tarski [24] made use of a specific adaptation of Lesniewski's work as a basis for a formal theory of physical things and their parts. The term 'calculus of individuals' was (...) introduced by Leonard and Goodman [14] in their presentation of a system very similar to Tarski's adaptation of Lesniewski's Mereology. Contemporaneously with Lesniewski's development of his Mereology, Whitehead [27] and [28] was developing a theory of extensive abstraction based on the two-place predicate, 'x extends over y\ which is the converse of 'x is a part of y\ This system, according to Russell [22], was to have been the fourth volume of their Pήncipia Mathematica, the never-published volume on geometry. Both Lesniewski [15] and Tarski [25] have recognized the similarities between Whitehead's early work and Lesniewski's Mereology. Between the publication of Whitehead's early work and the publication of Process and Reality [29], Theodore de Laguna [7] published a suggestive alternative basis for Whitehead's theory. This led Whitehead, in Process and Reality, to publish a revised form of his theory based on the two-place predicate, 'x is extensionally connected with y\ It is the purpose of this paper to present a calculus of individuals based on this new Whiteheadian primitive predicate. (shrink)

Mereotopology is an extension of mereology with some relations of topological nature like contact. An algebraic counterpart of mereotopology is the notion of contact algebra which is a Boolean algebra whose elements are considered to denote spatial regions, extended with a binary relation of contact between regions. Although the language of contact algebra is quite expressive to define many useful mereological relations and mereotopological relations, there are, however, some interesting mereotopological relations which are not definable in it. Such are, for (...) instance, the relation of n-ary contact, internal connectedness and some others. To overcome this disadvantage, we introduce a generalisation of contact algebra, replacing the contact with a binary relation between finite sets of regions and a region, satisfying some formal properties of Tarski consequence relation. The obtained system is called sequent algebra, considered as an algebraic counterpart of a new mereotopology. We develop the topological representation theory for sequent algebras showing in this way certain correspondence between point-free and point-based models of space. As a by-product, we show how one logical relation in nature notion, Tarski consequence relation, may have also certain spatial meaning. (shrink)

The notion of contact algebra is one of the main tools in the region-based theory of space. It is an extension of Boolean algebra with an additional relation C, called contact. Standard models of contact algebras are topological and are the contact algebras of regular closed sets in a given topological space. In such a contact algebra we add the predicate of internal connectedness with the following meaning—a regular closed set is internally connected if and only if its interior is (...) a connected topological space in the subspace topology. We add also a ternary relation \ meaning that the intersection of the first two arguments is included in the third. In this paper the extension of a Boolean algebra with \, contact and internal connectedness, satisfying certain axioms, is called an extended contact algebra. We prove a representation theorem for extended contact algebras and thus obtain an axiomatization of the theory, consisting of the universal formulas, true in all topological contact algebras with added relations of internal connectedness and \. (shrink)