Foundations of the Theory of Parthood: A Study of Mereology (Trends in Logic, 54)

Foreword to the Polish Edition
	References
Foreword to the English Edition
	References
Contents
Symbols
List of Models
List of Diagrams
1 An Introduction to the Problems of the Theory of Parthood
	1.1 Parts as Fragments
	1.2 The Problem of the Transitivity of the Concept Being a Part
	1.3 The Problem of the Existence of an ``Empty Element\'\'
		1.3.1 Non-degenerate Structures
		1.3.2 The Zero and Unity Elements of a Structure
		1.3.3 Structures with Unity
		1.3.4 The Absence of an ``Empty Element\'\'
		1.3.5 The Existence of an ``Empty Element\'\'
	1.4 Leśniewski\'s Mereology
		1.4.1 Collective Classes in Leśniewski\'s Sense
		1.4.2 Axioms of Leśniewski\'s Mereology
		1.4.3 Collective Sets in Leśniewski\'s Sense
	1.5 The Problem of ``Existential Involvement\'\' in the Theory of Parts
	1.6 ``Quasi-parts-or-wholes\'\'—Theories Without Antisymmetry
	References
2 ``Existentially Neutral\'\' Theories of Parts
	2.1 Fundamental Relations and Their Properties
		2.1.1 The Relations of Being a Part and Being an Ingrediens
		2.1.2 Three Theories in One
		2.1.3 The Absence of an ``Empty (Zero) Element\'\'
		2.1.4 The Relation of Is Exterior to
		2.1.5 The Relation of Overlapping
		2.1.6 Four Auxiliary Operators
		2.1.7 The Relation of Proper Overlapping
	2.2 The Concept of an Atom
	2.3 Supplementation Principles
		2.3.1 The Weak Supplementation Principle
		2.3.2 The Strong Supplementation Principle
		2.3.3 The Supplementation Principle for the Relation Proper Overlapping. Comparison of Supplementation Principles
	2.4 Extensionality Principles
		2.4.1 Definition of Extensionality Principles
		2.4.2 Extensionality Principles with Respect to the Relations Overlapping and Is Exterior to
		2.4.3 Extensionality Principles with Respect to the Relation Is a Part of
	2.5 Dependencies Between Theories
		2.5.1 Definition of the Concept of a Theory
		2.5.2 A Lattice of Theories
	2.6 Mereological Sum
		2.6.1 Definition and Basic Properties of Mereological Sum
		2.6.2 The Uniqueness of Mereological Sum
	2.7 The Monotonicity Principle for Mereological Sum
	2.8 Basic Differences Between the Relations of Supremum and Mereological Sum
	2.9 Supplementation Principles and Connections Between the Relations …
	2.10 Greatest Lower Bound Versus Mereological Sum
	2.11 The Choice of a Proper Theory of Parthood
	2.12 Mereological Fusion
		2.12.1 Another Definition of a Collective Class. The Relation Is a Fusion of
		2.12.2 Identity of Mereological Sum and Fusion
	2.13 Theories with an ``Empty Element\'\'
	References
3 ``Existentially Involved\'\' Theories  of Parts
	3.1 Mereological Strictly Partially-Ordered Sets
	3.2 Simons\' Minimal Extensional Mereology
	3.3 The Classes MEM+(‡) and MEM+(‡)
	3.4 The Existence of Algebraic and Mereological Sums for Pairs
		3.4.1 The Unconditional Existence of Algebraic and Mereological Sums for Pairs
		3.4.2 The Conditional Existence of Algebraic and Mereological Sums for Pairs
		3.4.3 Conditional Mereological Lattices
		3.4.4 Operations in Minimal Closure Mereology
		3.4.5 Operations in Semi-conditional Mereological Lattices
		3.4.6 Operations in the Second Kind of Semi-conditional Mereological Lattice
	3.5 Extensions of Theories MEM+(‡) and MEM+(‡) via Conditions for the Existence of Sums for Pairs
	3.6 ``Super-Supplementation\'\' Principles
		3.6.1 Definitions and Fundamental Properties of the Principles
		3.6.2 Polarised Strict Partial Orders ``Plus\'\'
		3.6.3 Unity in Structures from the Class sPPOSp
		3.6.4 The Theory ``MEM Plus\'\' (Equals to sPPOSp)
		3.6.5 Extensions of Theory sPPOSp via Conditions for the Existence of Mereological Sums for Pairs
	3.7 Grzegorczykian Mereological Structures
		3.7.1 A Problem of Elementary Mereology
		3.7.2 Description of Grzegorczykian Mereological Structures
		3.7.3 Unity in Structures from the Class GMS
		3.7.4 The Theory GMS Versus the Theory of Grzegorczykian Lattices
		3.7.5 The Theory GMS Versus the Theory of Boolean Lattices
		3.7.6 Operations in the Class GMS Versus Operations in the Class of Grzegorczykian Lattices and in the Class of Boolean Lattices
	3.8 Two Weak Axioms of the Existence of Mereological Sums
	3.9 Classical Mereological Structures
		3.9.1 ``Classical Mereology\'\' with the Primitive Relation
		3.9.2 Operations in Mereological Structures
		3.9.3 Some Weaker Theories than the Theory LMS
		3.9.4 The Generalised Operations of Mereological Sum and Product
		3.9.5 ``Classical Mereology\'\' with the Primitive Relation sqsubseteq
		3.9.6 Mereological Structures and Complete Boolean Lattices (Complete Boolean Algebras)
	3.10 The Case of Finite Structures
	References
4 Theories Without the Assumption  of Transitivity
	4.1 Introduction
	4.2 The First Two Axioms (Adopted Instead of Transitivity)
	4.3 Maximally Closed Transitive Sets
	4.4 The Third Axiom
		4.4.1 Definition
		4.4.2 Some Auxiliary Facts
		4.4.3 An Equivalent Version of Axiom (A3)
		4.4.4 A Stronger Version of Axiom (A3)
	4.5 Two Versions of the Fourth Axiom …
	4.6 ``A Partial\'\' Monotonicity Principle\'\'
	4.7 Mereological Sums in Structures Without Transitivity
		4.7.1 Definitions and Basic Properties
		4.7.2 Mereological Sum Versus Supremum
	4.8 The Fifth Axiom
	4.9 Mereological Sums for Axioms (A1)–(A5)
	4.10 Existentially-Involved Theories
		4.10.1 Axioms (‡SUM) and (‡SUM)
		4.10.2 Axiom (c) and its Various Versions
		4.10.3 Axioms (cpairsup) and (cparSUM)
		4.10.4 Axioms (pairsup) and (parSUM)
		4.10.5 Leśniewski\'s Axiom
		4.10.6 Weak Axioms of the Existence of a Mereological Sum
		4.10.7 Axiom (SSP+)
	References
Appendix A Logic and Set Theory
A.1  Logical Symbolism
A.2 Fundamental Features of Set Theory
A.2.1 Axioms of Set Theory
A.2.2 Properties of the Sum and Product of a Family of Sets
A.2.3 Cartesian Products and Relations
A.2.4 Functions, Partial Functions and Indexed Families of Sets
A.3 Binary Relations
A.3.1 Operations on Binary Relations
A.3.2 Fundamental Properties of Binary Relations
A.3.3 ``Reflexivisation\'\', ``Irreflexivisation\'\' and ``Asymmetricisation\'\' of Binary Relations
Appendix B Algebra
B.1 Strict Partial Orders
B.1.1 Fundamental Concepts and Definitions
B.1.2 Chains and the Kuratowski–Zorn Lemma
B.2 Partial Orders
B.2.1 Fundamental Concepts and Definitions
B.2.2 Bounds and Distinguished Elements
B.2.3 Suprema and Infima
B.2.4 Chains as Linearly-Ordered Subsets
B.3 Polarisation
B.3.1 Polarised Partial Orders
B.3.2 Semi-polarisation in Partial Orders
B.3.3 Semi-polarisation in the Class POS
B.4 Lattices
B.4.1 Lattices as Partially Ordered Sets
B.4.2 Lattices with Unity
B.4.3 Lattices with Zero
B.4.4 Bounded Lattices
B.4.5 Lattices as Algebras
B.4.6 Distributive Lattices
B.4.7 Complementation in Bounded Distributive Lattices
B.5 Boolean Lattices
B.5.1 Definition
B.5.2 Boolean Algebras
B.5.3 Properties of Boolean Operators
B.6 Grzegorczykian Lattices
B.6.1 Definitions
B.6.2 Grzegorczykian Lattices Versus Boolean Lattices
B.7 Completeness of Structures
B.7.1 Complete Partial Orders—Complete Lattices
B.7.2 Complete Boolean Lattices (Boolean Algebras). Tarski\'s Theorem
B.7.3 Complete Grzegorczykian Lattices Versus Complete Boolean Lattices
B.8 Atoms, Atomicness and Atomisticness
B.8.1 General Definition of an Atom
B.8.2 Atoms in Partially-Ordered Sets with Zero
B.8.3 Atoms in Non-degenerate Partially Ordered Sets Without Zero
B.8.4 Atomistic and Atomic Partial Orders
B.9 Quasi-orders
Appendix  List of Featured Formulas
Index of Names and Terms