Foundations of the Theory of Parthood: A Study of Mereology (Trends in Logic, 54) 3030365328, 9783030365325

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Trends in Logic 54

Andrzej Pietruszczak

Foundations of the Theory of Parthood A Study of Mereology

Trends in Logic Volume 54

TRENDS IN LOGIC Studia Logica Library VOLUME 54 Editor-in-Chief Heinrich Wansing, Department of Philosophy, Ruhr University Bochum, Bochum, Germany Editorial Board Arnon Avron, Department of Computer Science, University of Tel Aviv, Tel Aviv, Israel Katalin Bimbó, Department of Philosophy, University of Alberta, Edmonton, AB, Canada Giovanna Corsi, Department of Philosophy, University of Bologna, Bologna, Italy Janusz Czelakowski, Institute of Mathematics and Informatics, University of Opole, Opole, Poland Roberto Giuntini, Department of Philosophy, University of Cagliari, Cagliari, Italy Rajeev Goré, Australian National University, Canberra, ACT, Australia Andreas Herzig, IRIT, University of Toulouse, Toulouse, France Wesley Holliday, UC Berkeley, Lafayette, CA, USA Andrzej Indrzejczak, Department of Logic, University of Lódz, Lódz, Poland Daniele Mundici, Mathematics and Computer Science, University of Florence, Firenze, Italy Sergei Odintsov, Sobolev Institute of Mathematics, Novosibirsk, Russia Ewa Orlowska, Institute of Telecommunications, Warsaw, Poland Peter Schroeder-Heister, Wilhelm-Schickard-Institut, Universität Tübingen, Tübingen, Baden-Württemberg, Germany Yde Venema, ILLC, Universiteit van Amsterdam, Amsterdam, Noord-Holland, The Netherlands Andreas Weiermann, Vakgroep Zuivere Wiskunde en Computeralgebra, University of Ghent, Ghent, Belgium Frank Wolter, Department of Computing, University of Liverpool, Liverpool, UK Ming Xu, Department of Philosophy, Wuhan University, Wuhan, China Jacek Malinowski, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warszawa, Poland Assistant Editor Daniel Skurt, Ruhr-University Bochum, Bochum, Germany Founding Editor Ryszard Wojcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland The book series Trends in Logic covers essentially the same areas as the journal Studia Logica, that is, contemporary formal logic and its applications and relations to other disciplines. The series aims at publishing monographs and thematically coherent volumes dealing with important developments in logic and presenting significant contributions to logical research. Volumes of Trends in Logic may range from highly focused studies to presentations that make a subject accessible to a broader scientific community or offer new perspectives for research. The series is open to contributions devoted to topics ranging from algebraic logic, model theory, proof theory, philosophical logic, non-classical logic, and logic in computer science to mathematical linguistics and formal epistemology. This thematic spectrum is also reflected in the editorial board of Trends in Logic. Volumes may be devoted to specific logical systems, particular methods and techniques, fundamental concepts, challenging open problems, different approaches to logical consequence, combinations of logics, classes of algebras or other structures, or interconnections between various logic-related domains. Authors interested in proposing a completed book or a manuscript in progress or in conception can contact either [email protected] or one of the Editors of the Series.

More information about this series at http://www.springer.com/series/6645

Andrzej Pietruszczak

Foundations of the Theory of Parthood A Study of Mereology

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Andrzej Pietruszczak Department of Logic Nicolaus Copernicus University Toruń, Poland Translated by Matthew Carmody

The research and publication of this work was funded as part of the Ministry of Science and Higher Education’s “National Programme for the Development of the Humanities” (project number 0047/NPRH5/H21/84/2017) during 2017–2019. ISSN 1572-6126 ISSN 2212-7313 (electronic) Trends in Logic ISBN 978-3-030-36532-5 ISBN 978-3-030-36533-2 (eBook) https://doi.org/10.1007/978-3-030-36533-2 Revised and extended edition of Podstawy teorii części, © Wydawnictwo Naukowe Uniwersytetu Mikołaja Kopernika 2013. Published by The Nicolaus Copernicus University Scientific Publishing House, Toruń, Poland. All Rights Reserved. Translation of the Polish language edition by Matthew Carmody. © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword to the Polish Edition

Mereology arose as a theory of collective sets (or mereological sums). It was formulated by the Polish logician Stanisław Leśniewski (1927, 1928, 1929, 1930, 1931). Collective sets are certain wholes composed of parts and the concept of being a collective set itself can be defined with the help of the concept of being a part. Mereology may therefore be considered as a theory of “the relation of a part to the whole” (from the Greek: leqo1, meros, “part”). Leśniewski’s mereology was formulated in a specific way, distancing it from the standard formulations. The theory was, so to speak, built on top of another system of Leśniewski’s which he called “ontology” (see, e.g., Pietruszczak 2000b, Chap. I). His theory can, however, be translated into the language of structure theory. In such a form, it is characterised by an axiom (Leśniewski’s fourth), which postulates the existence of an element of a given structure which is to be the mereological sum (collective set) of an arbitrarily chosen non-empty group of elements of that structure, where we allow that the group is infinite.1 Mereological sums postulated by this axiom are, in general, what we might call ad hoc objects, and it is for this reason that the axiom has stirred up such a controversy. For example, it is hard to acknowledge the existence of a material object composed of the moon and the heart of the author of this book. And what should we say about the object that would be my left and right arms? Structures for theories of parts in which Leśniewski’s fourth axiom holds we will call classical mereological structures. Classical mereology is the name we shall give to the theory of such structures. It is known that classical mereology corresponds to the theory of complete Boolean algebras (otherwise known as complete Boolean lattices). This is to say that the class of all classical mereological structures coincides with the class of structures which arise from complete and non-degenerate Boolean algebras after the removal of the zero from each of them. Hence, it also follows that 1

Expressed thus, we have a generalisation of Leśniewski’s theory, even though Leśniewski’s (fourth) axiom is very strong and cannot be “universally” applied. We say more on this in Remark I.5.1. On the role and application of Leśniewski’s axioms, see (e.g.) (Pietruszczak 2000b, Chap. II.5).

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from each mereological structure to which is added an appropriate “zero element” we obtain a non-degenerate complete Boolean algebra. We believe, however, that the only methodologically proper description of mereological structures is one in which we do not appeal to the concept of a zero (because this “zero” is outside of these structures). There are many axiomatisations satisfying this condition (cf. Leonard and Goodman 1940; Pietruszczak 2000a,b, 2005; Tarski 1956a,b). In the literature, some authors also apply the label “mereology” to those theories of the concept being a part in which Leśniewski’s axiom of the existence of mereological sums is not in force. Although they are in keeping with the etymology of the term ‘mereology’, we believe that they create what might be called a “terminological confusion”. Where a weaker theory is under analysis, we should prefix it with a suitable adjective to indicate what kind of “mereology” it is. We find, for example, Peter Simons (1987) doing this in his examination of “Minimal Extensional Mereology”. In this book, however, we shall be examining “existentially neutral” theories. Such theories will not have any axioms postulating the existence of mereological sums. Mereological sums postulated by various existential axioms—not just the aforementioned axiom adopted by Leśniewski’s—will also in some cases come across as ad hoc objects. In “existentially neutral” theories, it will not be the case, for example, that each pair of objects which are parts of a third object must have a mereological sum (which would be a part of the third object). It is possible to prove the existence of only those mereological sums which it is possible to obtain exclusively via the definition of and fundamental properties of the concept of being a part. Precisely such theories will be examined in Chap. 2. Chapter 1 is a short introduction to the problems of the theory of parts. We will discuss fundamental concerns of both a philosophical and formal nature. In the first two chapters, we will also be concerned with the fundamental formal properties of various theories in which the antisymmetry of the concept being a part or a whole will not be admitted. We wish to show that there is nothing “odd” about such theories from a theoretical point of view. The majority of the results proven there are also well-known as holding in “normal theories” (i.e. those in which the concept being a part or a whole is antisymmetric). Chapter 3 is devoted to “existentially involved” theories of parts. We will examine a great number of such theories and show in detail the relationships that obtain between them. Amongst other things, we will analyse in which theories the relation is a part of is polarised and in which all existing suprema of groups of objects are also mereological sums. Next, we will examine Simons’ “minimal extensional mereology” and its various extensions. We will then consider the so-called principle of “super-supplementation”. The larger part of Chap. 3 is devoted to a detailed presentation of Grzegorczykian mereological structures. They are models of Grzegorczyk’s elementary theory from (Grzegorczyk 1955). Although this theory dates from 1955, it and its models have never been given a proper examination. The theory of these structures is weaker than classical mereology. It corresponds to the theory of Boolean algebras, whereas classical mereology corresponds to the theory of complete Boolean algebras. Therefore, in the theory of Grzegorczykian mereological

Foreword to the Polish Edition

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structures, we are supposed to have guaranteed only the existence of mereological sums for finite groups of arbitrarily chosen elements of a given structure (lacking so-called completeness). In addition, in the case of such structures, the only methodologically proper description of them is one in which we do not appeal to the concept of a zero (as found in a Boolean algebra) coming from outside the structure. The rest of Chap. 3 concerns mereological structures with so-called weak sum-existence axioms and classical mereological structures. One of the fundamental principles of mereology and other theories connected with it is the transitivity of the concept being a part. This property is often called into question in the literature. In Chap. 4, we present a general analysis which should satisfy both transitivity’s proponents and opponents. We will introduce a new concept of the local transitivity of a given relation in such a way that each transitive concept is also locally transitive. We will assume only that the concept being a part has the latter property. We will show that even without full transitivity one can define the concept of a mereological sum of objects, this being of course a fundamental mereological concept. We will formulate proposals for a general approach to the concept of being a part of the whole. Some will bring with them existential assumptions whereas others will be “existentially neutral”. Through the addition of the transitivity of the relation is a part of, we will obtain an axiomatisation of either the “existentially neutral” theories mentioned above or the “existentially involved” ones. The book has two appendices. The first is given over to set theory and the logical concepts used in this book. The second concerns algebra. Amongst other things, we discuss there in detail (and definitely for the first time) Grzegorczykian lattices closely connected to Grzegorczykian mereological structures. We will also examine the relationship between such lattices and Boolean lattices. The majority of the results established in this book have either not appeared in my earlier book (Pietruszczak 2000b) or appear here as important developments of the work found there. Andrzej Pietruszczak Department of Logic Nicolaus Copernicus University Toruń, Poland Acknowledgements I would like to extend my warm and sincere thanks to the reviewers of this book, Professors Marek Nasieniewski and Marcin Tkaczyk, for their valuable comments which helped shape the final version of this book. I would also like to extend similar thanks to the proofreader, Katarzyna Czerniejewska, for her close and careful reading of a text bristling with so many logical formulas. Her suggestions and corrections have definitely ensured a friendlier book for the reader. Finally, I would like to express my thanks to the Editor in Chief of the Nicolaus Copernicus University Press, Elżbieta Kossarzecka, for her help and kindness.

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References Grzegorczyk, A. (1955). The system of Leśniewski in relation to contemporary logical research. Studia Logica, 3, 77–95. http://dx.doi.org/10.1007/BF02067248 Leonard, H. S., & Goodman, N. (1940). The calculus of individuals and its uses. Journal of Symbolic Logic, 5(2), 45–55. http://dx.doi.org/10.2307/2266169 Leśniewski, S. (1927) O podstawach matematyki (On the foundations of mathematics). Przegląd Filozoficzny, 30, 164–206. Leśniewski, S. (1928). O podstawach matematyki (On the foundations of mathematics). Przegląd Filozoficzny, 31, 261–291. Leśniewski, S. (1929). O podstawach matematyki (On the foundations of mathematics). Przegląd Filozoficzny, 32, 60–101. Leśniewski, S. (1930). O podstawach matematyki (On the foundations of mathematics). Przegląd Filozoficzny, 33, 77–105. Leśniewski, S. (1931). O podstawach matematyki (On the foundations of mathematics). Przegląd Filozoficzny, 34, 142–170. Pietruszczak, A. (2000a). Kawałki mereologii (Pieces of mereology). In J. Perzanowski & A. Pietruszczak (Eds.) Logika i filozofia logiczna. FLFL 1996–1998 (Logic and Logical Philosophy) (pp. 357–374). Toruń: The Nicolaus Copernicus Univesity Press. Pietruszczak, A. (2000b). Metamereologia (Metamereology). Toruń: The Nicolaus Copernicus Univesity Press. English version (2018): Metamereology. Toruń: The Nicolaus Copernicus Univesity Scientific Publishing House. https://doi.org/10.12775/3961-4 Pietruszczak, A. (2005). Pieces of mereology. Logic and Logical Philosophy, 14(2), 211–234. English version of Pietruszczak (2000a). https://doi.org/10.12775/LLP.2005.014 Simons, P. (1987). Parts. A Study in Ontology. Oxford: Oxford University Press. http://dx.doi.org/ 10.1093/acprof:oso/9780199241460.001.0001 Tarski, A. (1956a). Fundations of the geometry of solids. In J. H. Woodger (Ed.), Logic, Semantics, Metamathematics. Papers from1923 to 1938 (pp. 24–29). Oxford: Oxford University Press. English version of (Tarski, 1929). Tarski, A. (1956b). On the foundations of Boolean algebra. In J. H. Woodger (Ed.), Logic, Semantics, Metamathematics. Papers from 1923 to 1938 (pp. 320–341). Oxford: Oxford University Press. English version of (Tarski, 1935).

Foreword to the English Edition

The English edition is revised and extended version of my book (Pietruszczak 2013). Alongside corrections to a number of minor errors of content, it contains a greater amount of commentary and many of the key claims have had their proofs filled out. Where possible, the bibliography contains the original English versions of the works cited in the book and not their Polish translations (as was the case in (Pietruszczak 2013)). In 2018, the English version of my earlier book (Pietruszczak 2000) was published. When making reference to the results contained therein, I shall cite both versions in the following way: (Pietruszczak 2000, 2018). In Chap. 3, I have added a historical note on the unpublished notes made by Jan F. Drewnowski of lectures given by Stanisław Leśniewski in 1922/1923 at the Warsaw University. In these lectures, Leśniewski outlined a version of a theory of mereology not known from his published work. He made use of the concept of a collective class (which turns out to be the concept of a supremum restricted to empty sets), a concept he did not make use of elsewhere. Despite this, Leśniewski worked out a theory definitionally equivalent to those which he analysed in work from 1916 and over the period 1927–1931. I will prove this using the results obtained in this book. In his notes, Drewnowski replaced one of Leśniewski’s axioms with an essentially weaker one and showed that one may still obtain an equivalent theory. This can also be proven using the results in this book. I would like to thank Professor Kordula Świętorzecka of the Cardinal Wyszyński University in Warsaw for making Drewnowski’s notes available to me.

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There is an important difference in the section on set theory in Appendix A.2.1 between the Polish and English editions. In the Polish edition, the problem of whether one needs to adopt the formula (ASet) as an axiom of the set theory I present is left open. In this edition, I acknowledge that, in a set theory with the primitive predicate ‘is a set’, such an axiom must be present. The adoption of this axiom has a significant influence on the whole theory of sets developed in Appendix A.2.1. Andrzej Pietruszczak Department of Logic Nicolaus Copernicus University Toruń, Poland Acknowledgements I would like to thank the Ministry of Science and Higher Education (Poland) for the funding the preparation and publication of this work as part of the National Programme for the Development of the Humanities 2017–2019 (grant no. 0047/NPRH5/H21/84/2017). I would also like to thank Dr. Matthew Carmody for translating the original Polish version of this book into English. Finally, I would like to express my thanks to Professor Marek Nasieniewski for his valuable comments which helped shape the final version of this book.

References Pietruszczak, A. (2000). Metamereologia (Metamereology). Toruń: The Nicolaus Copernicus Univesity Press. English version (2018): Metamereology. Toruń: The Nicolaus Copernicus Univesity Scientific Publishing House. https://doi.org/10.12775/3961-4 Pietruszczak, A. (2013). Podstawy teorii części (Foundations of the Theory of Parthood). Toruń: The Nicolaus Copernicus Univesity Press. Pietruszczak, A. (2018). Metamereology. Toruń: The Nicolaus Copernicus Univesity Scientific Publishing House. English version of (Pietruszczak, 2000b). https://doi.org/10.12775/3961-4

Contents

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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 . . . . . . . . . . . . . . . . . . . . .

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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 of Mereological Sum and Supremum . . . . . . . . . 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 . . . . . . . .

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Conditional Mereological Lattices . . . . . . . . . . . . . Operations in Minimal Closure Mereology . . . . . . 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 s PPOSp . . . . . . 3.6.4 The Theory “MEM Plus” (Equals to s PPOSp) . . . . 3.6.5 Extensions of Theory s PPOSp 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 GMS1 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 v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.6 Mereological Structures and Complete Boolean Lattices (Complete Boolean Algebras) . . . . . . . . . . 3.10 The Case of Finite Structures . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . u

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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. Supplementation Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 “A Partial” Monotonicity Principle. An Equivalent Version of Axiom ðA4w Þ . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . ) and (‡SUM ) . . . . . . . . . . . . . . . . . . 4.10.1 Axioms (‡SUM £ 4.10.2 Axiom (c9u) and Its Various Versions . . . . . . . . 4.10.3 Axioms (c9pair sup) and (c9par SUM) . . . . . . . . . . . 4.10.4 Axioms (9pair sup) and (9par SUM) . . . . . . . . . . . . 4.10.5 Leśniewski’s Axiom . . . . . . . . . . . . . . . . . . . . . . 4.10.6 Weak Axioms of the Existence of a Mereological Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.7 Axiom (SSP+) . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents

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. . . . . . . . .

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. . . . . . . . .

. . . . . . . . .

153 153 154 155 156 156 157 160 164

. . . . . 166 . . . . . . . . . . . .

. . . . . . . . . . . .

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. . . . . . . . . . . .

. . . . . . . . . . . .

170 171 171 174 176 178 178 179 179 182 184 185

. . . . . 185 . . . . . 186 . . . . . 188

Appendix A: Logic and Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Appendix B: Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 List of Featured Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Index of Names and Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

u

Symbols

U 2

8 ,  R1  R2 u

n

v :() _ ¼ idU 0 )

u

u

: ^ 9 6 ¼ fx : uðxÞg :¼ hx; yi  62

Relation is a part of Universe of discourse Set-theoretic predicate ‘is an element of’ (‘belongs to’, ‘is a member of’) Truth-connective of negation Truth-connective of conjunction Existential quantifier Symbol of non-identity Set of elements satisfying a condition uðxÞ Symbol of identity for definitions Ordered pair of x and y Operation of Cartesian product of sets Set-theoretic predicate ‘in not an element of’ (‘does not belong to’; ‘is not a member of’) Component of ; relation is not a part of Universal quantifier Truth-connective of material biconditional Operation of relative product of relations Relative product of relations R1 and R2 Relative n-product of Relation is an ingrediens of Truth-connective of material biconditional for definitions Truth-connective of inclusive disjunction Symbol of identity Relation of identity on a set U Set-theoretic operation of sum of two sets Component of v; relation is not an ingrediens of Truth-connective of material conditional

xv

Symbols

1

( x) 0

£  U

u



v  Uþ n þ /  P S † \

Structure with a universe U and a relation Cardinality of sets Unity of a given structure Description operator: the unique x such that Zero of a given structure Empty set Empty object Universe of discourse with empty object Strict partial order in U   Set-theoretic sum of the relation and identity Partial order Universe of discourse without empty object (or without zero) Set-theoretic operation of difference of two sets (or the relative complementation) Restriction of  to non-empty (or non-zero) objects Set-theoretic operation of product of two sets Strict partial order Set-theoretic sum of the relations  and idU Schematic letter for general names Schematic letter for general names Quasi-order and the relation is a quasi-part-or-whole of Asymmetricisation of ., i.e. the set-theoretic u

u

hU; i Card

u

xvi

^

 ( SPOS POS QOS hU; vi  P I E O op[S] S S

op[S] O[S]

difference of the relations . and . Set-theoretic predicate ‘is a subset of’ Set-theoretic predicate ‘is a proper subset of’ Class of strict partially-ordered sets Class of partially-ordered sets Class of quasi-ordered sets Structure with a universe U and a relation v Relation is exterior to Relation of overlapping Parts-function Ingredienses-function Exterior-function Overlapping-function Family of sets which is the image of a set S determined by a function op Set-theoretic sum of the image of a set S determined by a function op Set-theoretic sum of the image of a set S determined by the function O

Symbols

T T

op[S] O[S]

G At C+(W) C+(W)+(W′) SPPOS PPOS C  Cfin sum sup inf fu SMPOS MEM MEM 91 u CMM Un CMM + t SPPOSp SMPOSp SPPOSp1

– MEMp TMS LMS SGMS GMS GMS1 Uþ

xvii

Set-theoretic product of the image of a set S determined by a function op Set-theoretic product of the image of a set S determined by the function O Relation of proper overlapping Set of mereological atoms Class of structures from C satisfying condition (W) Class of structures from C satisfying conditions (W) and (W′) Class of polarised strict partial orders Class of polarised partial orders Class of non-degenerate structures from C Class of finite structures from C Relation is a mereological sum of Relation of supremum with respect to v Relation of infimum with respect to v Relation is a mereological fusion of Class of mereological strictly partially-ordered sets Minimal Extensional Mereology Class of non-degenerate structures belonging to MEM Quantifier ‘there is exactly one’ Operation of mereological product of two elements Minimal Closure Mereology Relation of underlapping Class of non-degenerate structures belonging to CMM Operation of algebraic sum of two elements Operation of mereological sum of two elements Class of polarised strict partial orders “plus” Operation of mereological relative complementation Class of mereological strictly partially-ordered sets “plus” Subclass of SPPOSp of structures with unity Operation of mereological complementation Minimal Extensional Mereology plus Class of Tarskian mereological structures Class of Leśniewskian mereological structures Class of (strict) Grzegorczykian mereological structures Class of (non-strict) Grzegorczykian mereological structures Class of Grzegorczykian mereological structures with unity Universe of discourse without zero

xviii

Uo GMS1 SGMS1

Universe of discourse with zero Class of (non-strict) Grzegorczykian mereological structures with unity Class of (strict) Grzegorczykian mereological structures with unity Generalised operations of mereological sum Generalised partial operations of mereological product Class of finite structures from class A Class of finite structures from class B Family of subsets of the set U which are closed with respect to Over-parts-function Over-ingredienses-function Subfamily of CTU of maximal sets with respect to inclusion Set of maximal elements in S with respect to Set of minimal elements in S with respect to Relation of overlapping in M of MCTU Relation of proper overlapping in M of MCTU Relation is exterior to in M of MCTU The only set from MCTU which contains x and y, where y x The only set from MCTU that includes I(x), where PðxÞ 6¼ [ Overlapping-function for M of MCTU Set-theoretic sum of the image of a set S determined by the function OM Non-standard relation is a mereological sum of in structures without transitivity of Relation of underlapping in M of MCTU Range of a relation R Set-theoretic predicate ‘is a set’ Set-theoretic predicate ‘is not a subset of’ Set-theoretic predicate ‘is a family of sets’ Operation of the set-theoretic sum of a family of sets Non-ordered pair of x and y Singleton of x Non-ordered triple of x, y, z Finite set of x1 ; . . .; xn Power set of X Set obtained from X by use (Arepw ) Image of X once “transformed’’ by s Image of X once “transformed’’ by s

MCTU u u

max ðSÞ min ðXÞ M GM

u

Mxy

u u

u

t u Afin Bfin CTU

Symbols

Mx OM S OM[S]

u

SUM UnM rngðRÞ Set * F S fx; yg fxg fx; y; zg fx1 ; . . .; xn g 2X fy : 9x2X wðx; yÞg fsðxÞ : x 2 Xg s½X

Symbols

fy 2 X : uðyÞg U E TS

S S s½X sðxÞ Tx2X s½X T sðxÞ Tx2X X

hx; y; zi hx1 ; x2 ; . . .; xn i s½X; Y s½X  Y domðRÞ fhx; yi 2 X  Y : vðx; yÞg F:X!Y FðxÞ F:x sðxÞ xRy xRyRz ^

R Rn R

R R/ hU; i hU;  i fl

xix

Subset of X to which belong elements satisfying a condition uðyÞ Extension of a concept S in a universe U Operation of the set-theoretic product of a non-empty family of sets Set-theoretic sum of a family s½X Set-theoretic sum of a family fsðxÞ : x 2 Xg Set-theoretic product of a family s½X Set-theoretic product of a family fsðxÞ : x 2 Xg Set-theoretic product of a family F of sets included in X Ordered pair of x, y and z n-Tuple of x1 , …, xn Image of X  Y once “transformed’’ by s Image of X  Y once “transformed’’ by s Domain of a relation R Relation designated by a formula vðx; yÞ Function from X to Y The value of a function F at x F maps x to sðxÞ Instead of ‘hx; yi 2 R’ Instead of ‘x R y ^ y R z’ Converse of a relation R Relative n-product of R Reflexivisation of R Irreflexivisation of R Asymmetricisation of R Strict partially-ordered set with a universe U and a relation  Partially-ordered set with a universe U and a relation  Irreflexivisation of  , i.e. the set-theoretic difference of the relations  and idU ; as well the asymmetricisation of  , i.e. the set-theoretic difference of the ^

POS1 POS0 POS01 max  ðSÞ min  ðSÞ sup  inf 

relations  and  Class of partially-ordered sets with unity Class of partially-ordered sets with zero Class of bounded partially-ordered sets Set of all maximal elements in S Set of all minimal elements in S Relation of supremum with respect to  Relation of infinium with respect to 

xx

Symbols

?

Relation of separation with respect to  (Second) relation of separation with respect to  Restriction of to U þ Class of lattices Lattice Operation of algebraic sum of two elements Operation of algebraic product of two elements Lattice with unity Class of lattices with unity Lattice with zero Class of lattices with zero Lattice with zero and unity Class of bounded lattices Operation of complementation Class of all Boolean lattices Set which is used in defining the difference operation Binary difference operation Class of Grzegorczykian lattices Class of Grzegorczykian lattices with unity Operation of supremum Operation of infimum Class of complete Boolean lattices Class of complete Grzegorczykian lattices Set of atoms Congruence with respect to .

?þ L L ¼ hU;  i þ  L ¼ hU;  ; 1i L1 L ¼ hU;  ; 0i L0 hU;  ; 0; 1i L01  BL Rxy  GL GL1 sup inf CBL CGL At

List of Models

Model 2.4 Model 2.5 Model 2.6 Model 2.7

Model 2.8 Model 3.1 Model 3.2 Model 3.3 Model 3.4

Model 3.5

Model 3.6 Model 3.7

u

..

30

..

36

..

38

.. ..

38 39

.. .. ..

54 57 68

.. ..

69 69

..

72

u

20

u

u

Model 2.3

..

u

Model 2.2

(ext þ), (9sup£ ) and (9sum) hold, but (6 90) and (WSP) do not hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (WSP) and (9fu) hold, but (2.3.3), (SSP), (ext ) and (ext ) do not hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (ext ) and (Usum ) hold, but (ext þ) and (SSP) do not hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (ext ), (ext þ) and (Usum ) hold, but (SSP) and fu * sum do not hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (ext þ) and (∃ ) hold, but (WSP) do not hold . . . . . . . . . . . (WSP) and (ext þ) hold, but (ext ) does not hold . . . . . . . . (WSP), (SSP), (c9u), (c9pair sup) (9pair sup) and (9sup£ ) hold, but (‡£ ), ð‡Þ, (c9pair fu) (c9pair SUM), (9pair SUM), (WSPþ), (SSPþ) and (9sum) do not hold . . . . . . . . . . . . . . (y), (‡£ ) and ð‡Þ hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (SSP) holds, but (c9u) and ðc9pair supÞ does not hold . . . . . (SSP) and (‡£ ) hold, but (c9u), ðSSPþÞ and ðWSPþÞ do not hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (SSP) holds, but (c9u), (‡£ ) and (‡) do not hold . . . . . . . . . (SSP), (WSP), (c9u), ðc9pair supÞ, (c9pair fu), (c9pair sum), ðSSPþÞ, ðWSPþÞ, ðw1 9sumÞ and ðw2 9sumÞ hold, but (9pair sum), (9pair fu), (9pair sup) and (91v ) do not hold . . . . . . (WSP), (SSP), (c9u), ðWSPþÞ, ðSSPþÞ, ðc9pair supÞ, (c9pair fu), (c9pair sum), (‡£ ), (‡), ðw1 9sumÞ and ðw2 9sumÞ hold, but (9pair sup), (9pair fu), (9pair sum) and (91v ) do not hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The simplest non-degenerate structure from GMS . . . . . . . . ðSSPþÞ and ðw1 9sumÞ hold, but ðw2 9sumÞ does not hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . u

Model 2.1

.. 85 . . 113 . . 125

xxi

xxii

Model 3.8 Model 4.1 Model 4.2 Model 4.3

Model 4.4 Model 4.5 Model 4.6 Model 4.7

List of Models

(ext ), ðUsum Þ, (Ufu ), (WSP) and ð9fuÞ hold, but (SSP) and ð9sumÞ do not hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A1)–(A5) hold, but ðA3s Þ does not hold . . . . . . . . . . . . . . . The tree structure satisfies axioms (A1)–(A5) and ðA3s Þ . . . The structure satisfies axioms (A1)–(A5) and ðA3s Þ hold, but does not satisfy “quasi-transitivity” conditions from Remark 4.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A1), (A2), ðA3s Þ and ðA4w Þ hold but (WSP), (SSP), (A4) and (A5) do not . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A1)–(A4), ðA3s Þ and (SSPþ) hold, but (A5) and (c9u@4 ) do not hold. Furthermore, sum * SUM . . . . . . . . . . . . . . . . (A1)–(A5), ðA3s Þ, (c9u@2 ) and (c9u@4 ) hold, but (c9u@1 ) and (c9u) do not hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A1)–(A4), ðA3s Þ and (SSPþ) hold, but (A5) and (SSPþ@2 ) do not hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 134 . . 164 . . 164

. . 166 . . 168 . . 170 . . 181 . . 187

List of Diagrams

Diagram 2.1 Diagram 3.1 Diagram 3.2

Diagram 3.3

Diagram 3.4

Diagram 3.5 Diagram 3.6 Diagram 3.7

The lattice of “existentially neutral” theories . . . . . . . . . . . The lattice of theories related to sentences (WSP), (SSP), (c9u), ðc9pair supÞ, (c9pair sum), (9pair sup) and (9pair sum) . . The lattice of theories related to sentences (WSP), (SSP), (c9u), (‡£ ), (‡), (c9pair sup), (c9pair sum), (9pair sup) and (9pair sum) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The lattice of non-degenerate theories related to sentences (WSP), (SSP), (c9u), (‡), ðc9pair supÞ, (c9pair sum), (9pair sup) and (9pair sum) . . . . . . . . . . . . . . . . . . . . . . . . . . The lattice of theories related to sentences (WSP), (SSP), (c9u), (‡£ ), (c9pair sup), (c9pair sum), (9pair sup), (9pair sum), (WSPþ) and (SSPþ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The lattice of “existentially involved” theories . . . . . . . . . The lattice of classes of finite structures related to existentially involved theories . . . . . . . . . . . . . . . . . . . . The lattice of classes of finite structures related to existentially involved theories (after simplifications) . . .

..

42

..

77

..

87

..

88

. . 103 . . 127 . . 148 . . 149

xxiii

Chapter 1

An Introduction to the Problems of the Theory of Parthood

1.1 Parts as Fragments In everyday speech, the expression ‘part’ is usually understood as having the sense of the expressions ‘fragment’, ‘bit’, and so forth. Thus understood, the relation of a part to the whole has two properties: (a) no object is its own part; (b) there are not two objects such that the first could be a part of the second and the second a part of the first. Thanks to condition (a), we have no difficulty in interpreting the phrase ‘two objects’ in condition (b). It can be seen that it concerns “two different” objects. Sentences (a) and (b) say respectively that the relation of part to whole is irreflexive and antisymmetric. In order to abbreviate these and other properties of the concept being a part of , we will symbolise ‘x is a part of y’ with ‘x  y’ and likewise with other variables. We are therefore treating the symbol ‘’ as a two-argument predicate. With respect to an arbitrary universe of discourse U (this being a non-empty distributive set), the irreflexivity and antisymmetry of the concept being a part may be expressed in the form of the following two sentences1 : 

¬∃x∈U x  x,



¬∃x,y∈U x = y ∧ x  y ∧ y  x .

(irr  ) (antis )

The conjunction of (irr  ) and (antis ) is logically equivalent to a sentence which says that the relation of part to whole is asymmetric2 :   ¬∃x,y∈U x  y ∧ y  x . (as ) 1 See

Footnote 18 in Appendix A. is well-known that a relation is asymmetric iff it is irreflexive and antisymmetric. All necessary information on the subject of relations (including dependencies between the properties of a given relation) may be found in Appendix A.3.

2 It

c Springer Nature Switzerland AG 2020  A. Pietruszczak, Foundations of the Theory of Parthood, Trends in Logic 54, https://doi.org/10.1007/978-3-030-36533-2_1

1

2

1 An Introduction to the Problems of the Theory of Parthood

In general, the relation of part to whole is acyclical in the sense expressed by the following schema (where n > 0):   ¬∃x1 ,...,xn ∈U x1  x2 ∧ · · · ∧ xn  x1 .

(1.1.1)

Observe that we may obtain condition (as ) from schema (1.1.1) and, as a consequence, conditions (irr  ) and (antis ) too. In order not to multiply terms, the same symbol ‘’ will be used to signify the extension of the concept being a part in the context of a particular universe U . Thus understood,  is a binary relation on U that may be defined as follows:  := {x, y ∈ U × U : x is a part of y}. In accordance with the convention detailed in Appendix A.3.1, to shorten our formalisations, instead of writing ‘x, y ∈ ’ (resp. ‘x, y ∈ / ’) we will write ‘x  y’ (resp. ‘x  y’) . This convention allows us to treat such conditions as (irr  ), (antis ), (as ) and (1.1.1) in a two-fold way: as ones in which a predicate appears; and as ones in which the name of a relation appears (omitting set-theoretic vocabulary). In the first case, by leaving out the restriction ‘∈ U ’ from the quantifiers, conditions (irr  ), (antis ) etc. can be written in a first-order language. Schema (1.1.1), however, can be formally expressed in the language of a weak second-order logic (allowing quantification over the natural numbers): ∀n>0 ¬∃x∈U x n x,

(ac )

where for an arbitrary positive natural number n we define the relation n by the following inductive conditions: 1 =  and for an arbitrary k > 0, k+1 = k ◦ .3 Put simply, condition (ac ) is the counterpart of schema (1.1.1) under which fall an infinite number of the following formulas: (irr  ), (as ) and those below:   ¬∃x,y,z∈U x  y∧y z∧z x ,   ¬∃x,y,z,u∈U x  y ∧ y  z ∧z  u ∧u  x ,  ¬∃x,y,z,u,v∈U x  y ∧ y  z ∧ z  u ∧ u  v ∧ v  x

etc.

In the literature on the subject, the custom has spread of using the expression ‘proper part’ rather than ‘part’. The term ‘part’ here takes on a new meaning, in which it has wider range of use. A part of a given object is now understood to be the object itself or any given part in the ordinary sense of that term. A part of a given object that differs from that object itself is named a proper part. Therefore, in conditions (a) and (b) the term ‘proper part’ would appear in place of ‘part’. It follows

that for binary relations R1 and R2 on U the binary relation R1 ◦ R2 on U may be defined by the following condition: x R1 ◦ R2 y ⇐⇒ ∃z∈U (x R1 z ∧ z R2 y).

3 Recall

1.1

Parts as Fragments

3

directly from the new meaning of ‘part’ that every object is its own (improper) part. If we understand ‘two objects’ in the sense of “two different” objects, then, under the new meaning of ‘part’, condition (b) is satisfied: that is, we have the antisymmetry of the altered concept being a part. Stanisław Le´sniewski did not change the ordinary meaning of the word ‘part’. Instead, he adopted the word ‘ingredjens’ in his works during the years 1927–1931, a word which did not exist in Polish at that time. It seems that he felt the need for a new term. We will also employ his neologism but spell it thus: ‘ingrediens’.4 An ingrediens of a given object is therefore it itself or any part of it in the ordinary sense of the word ‘part’. Once again, for the sake of abbreviation, let us understand that the expression ‘x is an ingrediens of y’ is to be symbolised by ‘x y’ and similarly for its use with other variables. We may therefore treat the symbol ‘ ’ as a two-place predicate, whose sense with respect to an arbitrary universe U of discourse may be expressed by the following definition:   ∀x,y∈U x y :⇐⇒ x  y ∨ x = y .

(df )

Furthermore, the reflexivity and antisymmetry of the concept being an ingrediens with respect to U may be expressed in the form of the following two sentences: 

∀x∈U x x ,



¬∃x,y∈U x = y ∧ x y ∧ y x .

(r ) (antis )

Condition (r ) follows directly from the reflexivity of the identity predicate and condition (antis ) from condition (antis ) along with the symmetry of identity. As in the case of the concept being a part, in order not to multiply terms, we will use the symbol ‘ ’ to signify the extension of the relational concept being an ingrediens in the context of a given universe U . Thus understood, is a binary relation on U defined as follows:

:= {x, y ∈ U × U : x is and ingrediens y}. This is the “reflexivisation” of the relation  in the sense that is the set-theoretic sum of the relation  and idU := {x, y ∈ U × U : x = y}, i.e., :=  Y idU (see point A.3.3 in Appendix A). Once again, instead of writing ‘x, y ∈ ’ (resp. 4 The

broadening of the extension of the word ‘part’ can sometimes lead to “philosophical misunderstandings”. The strange-sounding term ‘ingrediens’ is in this case an “ally”, as it reminds us that it is an “artificial” concept. Let us also note here that a misunderstanding arises from the choice of ‘ingredient’ as the translation of ‘ingredjens’ in the English translations of Le´sniewski’s works (cf. Le´sniewski 1991). The word ‘ingredient’ has a ready translation in Polish as ‘składnik’ and Le´sniewski deliberately distanced himself from this word. It is therefore better to use the word ‘ingrediens’ in translation, thus preserving continuity with its (philosophico-logical) use in contemporary Polish.

4

1 An Introduction to the Problems of the Theory of Parthood

‘x, y ∈ / ’) we will write ‘x y’ (resp. ‘x  y’). This convention generates the same consequences as were earlier found in the case of the symbol ‘’.

1.2 The Problem of the Transitivity of the Concept Being a Part Le´sniewski (1927) understood the relation of part to whole as asymmetric, (as ), and transitive; i.e., that an arbitrary part of some part of a given object is also a part of that given object. Formally:   ∀x,y,z∈U x  y ∧ y  z =⇒ x  z .

(t )

Observe that conditions (irr  ) and (t ) entail every instance of schema (1.1.1) and, as a consequence, condition (as ) too. Furthermore, on the strength of (t ), the relation is an ingrediens of is also transitive, i.e.: ∀x,y,z∈U (x y ∧ y z =⇒ x z).

(t )

If we accept that the relation  is transitive, then it strictly partially orders the set U ; that is, we have the strictly partially-ordered set U, . In support of the transitivity of the concept of being a part, the following example is often given: my left arm is a part of my body, from which it follows that my left hand is part of my body. Nicholas Rescher (1955), however, shows that the transitivity of the relation of part to whole is in other cases problematic. He provides the following counterexample: a nucleus is part of a cell and a cell is part of an organ, but the nucleus is not part of an organ. This is so, at least, if we consider a part to be a direct functional constituent of a whole, for a nucleus is not a part of an organ. Another example of the same with a military flavour: a given platoon is part of a given company and the company part of a given battalion, but no platoon is a (composing) part of a battalion. Peter Simons (1987, pp. 107–108) has observed, however, that the concept of a part understood transitively corresponds to spatiotemporal inclusion and in that sense a nucleus is part of an organ. Simons claims that the fact that the word ‘part’ has an additional meaning does not undermine the mereological concept of a part, because there is no reason to think that the mereological concept should embrace all the meanings of the word ‘part’ but only those that are basic and most important. The transitivity of the relation is a part of does not therefore present special difficulties when referring to spatiotemporal relations, including events. In the literature one can find various interesting positions on the issue of the transitivity of the concept of being a part (see, e.g., Johansson 2005/2006; Lyons 1977; Rescher 1955; Varzi 2005/2006). Several theories of the relation being a part of without the assumption of transitivity will be presented in Chap. 4.

1.3

The Problem of the Existence of an “Empty Element”

5

1.3 The Problem of the Existence of an “Empty Element” 1.3.1 Non-degenerate Structures We say that a structure U,  is degenerate iff U is a singleton. We can express this with the help of the set-theoretic operator ‘Card’, which gives the cardinal number of a set, or the number of its elements: a structure U,  is degenerate iff Card U = 1. We say that a structure U,  is non-degenerate iff it is not degenerate iff the universe U has at least two elements iff Card U > 1. Such structures therefore satisfy the following elementary condition: ∃x,y∈U x = y.

(n-d)

1.3.2 The Zero and Unity Elements of a Structure We say that a given element of U is the unity element (resp. the zero element) of the structure U,  iff it is the least (resp. greatest) element in U, . For convenience, let us both abbreviate ‘unity element’ to just ‘unity’ and ‘zero element’ to just ‘zero’; and introduce the two predicates ‘is the unity’ and ‘is the zero’ regardless of whether a unity or a zero exists:   ∀x∈U x is the unity :⇐⇒ ∀u∈U u x ,   ∀x∈U x is the zero :⇐⇒ ∀u∈U x u .

(df unity) (df zero)

It follows from (antis ) that there can be at most one unity (resp. zero) in a given structure (see point B.2.2 in Appendix B).

1.3.3 Structures with Unity We say that U,  is a structure with unity iff it satisfies the following condition: ∃x∈U ∀u∈U u x.

(∃1 )

Of course, every degenerate structure has the unity. In a structure U,  with unity we signify that unity by ‘1’, i.e.: 1 := ( x) ∀u∈U u x.

(df 1)

ι

6

1 An Introduction to the Problems of the Theory of Parthood

It then follows directly from the definition that: ∀u∈U u 1,

(1.3.1)

∀x∈U (x = 1 ⇐⇒ ∀u∈U u x).

(1.3.2)

1.3.4 The Absence of an “Empty Element” We say that U,  is a structure with zero iff it satisfies the following condition: ∃x∈U ∀u∈U u x.

(∃0 )

Of course, every degenerate structure has the zero, which is identical to its unity: Card U = 1 =⇒ ∃x∈U ∀u∈U u x u. In the structures we will be considering, the zero will only exist in degenerate structures (cf. condition (0) below). Such an element is obviously both the least and greatest element (unity) in a given structure. In a structure U,  with zero we signify that zero by ‘0’, i.e.: 0 := ( x) ∀u∈U x u.

(df 0)

ι

It then follows directly from the definition that: ∀u∈U u 0,

(1.3.3)

∀x∈U (x = 0 ⇐⇒ ∀u∈U x u),

(1.3.4)

Card U = 1 =⇒ 0 = 1.

(1.3.5)

If we assume that the universe of discourse is composed of spatial objects, spatial regions and spatiotemporal events, then we may exclude from our analysis the existence of an “empty object”, an “empty region” and an “empty event”, which would be parts of any object, region or event. Our situation is therefore not analogous to that of set theory—the theory of distributive sets (classes)—where we assume the existence of the empty set ∅, which is a subset of every distributive set (class). In an algebraic sense, such an “empty element” would correspond to the zero. We will allow the existence of a least element when and only when the universe has exactly one element which is simultaneously the greatest and least element. In that case, we can also say that the single element is the unity of the structure. We do not have to note that single element is “empty”. It is not in any case an interesting case because we are dealing with a degenerate structure.

1.3

The Problem of the Existence of an “Empty Element”

7

The following principle will therefore be in force: condition (n-d) entails that there is no zero, i.e., if a structure is non-degenerate then the structure has no zero. In other words, if a structure has the zero than the structure is degenerate. Formally: ∃x,y∈U x = y =⇒ ¬∃x∈U x is a zero, ∃x∈U x is a zero =⇒ ∀x,y∈U x = y.

(0)

Because it follows from definitions (df ) and (df zero) that the converse implication to the above is an analytic sentence, putting them together we obtain: ∃x∈U x is a zero ⇐⇒ Card U = 1. Once again, let us underline that in the trivial case where Card U = 1, we do not have to consider the sole element of the universe as “empty” (“empty object”, “empty region”, “empty event” etc.). Principle (0) will follow from other principles to be later adopted. For example, it follows if we assume that being an ingrediens of polarises the universe of discourse; i.e., that the following condition is satisfied:    ∀x,y∈U x  y =⇒ ∃z∈U z x ∧ ¬∃u∈U (u z ∧ u y) .

(pol )

This is proven as Fact B.3.2. Principle (0) also follows from the weaker conditions from (pol ).5

1.3.5 The Existence of an “Empty Element” If we consider universes composed of such abstract objects as pieces of information or situations, we can accept the existence of an “empty object” (empty piece of information, empty situation). Such an object would correspond to a tautologous sentence, such as ‘it is raining or it is not raining’. The existence of an “empty element” in is also sometimes accepted in theories concerned with spatial objects. This is done so for convenience when (for example) one wants to have a viable operation of the product of two objects. For example, when objects have no common “non-empty” part, it is agreed that their product is just an “empty object”.

5 For example, it also follows

from the Weak Supplementation Principle adopted by Simons (1987) (see also point 2.3.1). The principle of polarisation (pol ) appears in (Simons 1987) as the Strong Supplementation Principle.

8

1 An Introduction to the Problems of the Theory of Parthood

As should be clear, there is no possibility of building a “unitary theory” which would embrace both cases connected with the existence of an “empty object”.6 Put simply, we must straightaway adopt one of the following approaches: • to adopt principle (0); • to let there be an “empty element” which is a part of all others. The second approach does not require us to create a new theory. It suffices to extend the universe U composed of “non-empty objects” by the single empty object ø which does not belong to U . Then we put U ø := U Y {ø}. This will create a new universe of discourse, within which we define the “artificial relation”   ∀x,y∈U ø x ø y :⇐⇒ (x = ø ∧ y = ø) ∨ x  y , which strictly partially orders it; and, besides, we have: ∀u∈U ø ø u,   ∃u,v∈U u ø v ⇐⇒ u  v . The “reflexivisation” of ø in the set U ø is the relation ø := ø Y idU ø , i.e.:   ∀x,y∈U ø x ø y :⇐⇒ (x ø y ∨ x = y , Notice that the relation ø satisfies the following condition:   ∀x,y∈U ø x ø y ⇐⇒ x = ø ∨ x y . Suppose that x ø y, i.e., x ø y or x = y. In the second case we have x y. In the first case either both x = ø and y = ø or x  y. So, we have x = ø or x y. Conversely, assume that x = ø or x y. In the first case if y = ø then x = y; and so x ø y. If y = ø then x ø y; and so x ø y. Observe also that if we are dealing with a degenerate structure, i.e., where U = {1}, then by adding the empty object ø we obtain the two-element Boolean lattice {1, ø}, {ø, ø, ø, 1, 1, 1}, in which ø is the zero.7 If, for technical reasons, we accept the existence of an empty object, then as a rule we are considering a partially-ordered set U, ≤ with the zero ø. We then put U+ := U \ {ø} and ≤+ := ≤|U+ , i.e., ≤+ is a restriction of the relation ≤ to the set U+ ; and so ≤+ := ≤ Z (U+ × U+ ).8 Both approaches to the question of a zero element rely wholly on the algebraic analyses carried out in Appendix B. We will return to this question in Sect. 2.13. 6 We

explain why in Sect. 2.13. is a special case, as we could also recognise the element u as the zero and so add nothing. 8 By starting from the strictly partially-ordered set U, ≺ with the zero ø and putting  := ≺ Y idU , we can also obtain a partially-ordered set with zero ø. 7 This

1.4

Le´sniewski’s Mereology

9

1.4 Le´sniewski’s Mereology 1.4.1 Collective Classes in Le´sniewski’s Sense We have discussed the topic of collective classes and sets understood in Le´sniewski’s way in greater detail in the first chapter of (Pietruszczak 2000, 2018). We will say more about collective sets in Sect. 1.4.3. Let ‘P’ and ‘S’ be schematic letters representing general names of certain elements of the universe of discourse U . We may introduce the definition of a collective class of Ps given by Le´sniewski in the form of the following sentential schema: x is a collective class of Ps iff every P is an ingrediens of x and each ingrediens of x has a common ingrediens with some P. Written formally, it has the following form: x is a collective  class of Ps :⇐⇒ ∀z (z is a P ⇒ z x) ∧ ∀y y x ⇒ ∃z (z is a P ∧ ∃u (u z ∧ u y)) .

(df ccP)

Obviously, we could have given the above definition using instead the primitive concept being a part, which corresponds to the relation . However, the formalisation would have taken a much more complicated form. Instead, in many formulas and also in the definitions of new auxiliary concepts, the concept being an ingrediens will appear, corresponding to the relation and not to the primitive concept. This may explain why Le´sniewski introduced this “artificial” concept. From (df ccP) two interesting results follow. Regarding the first, it is evident that the concept collective class of Ps is empty if the name represented by the letter P is empty. There is no “aggregate” of Ps if there are no Ps: ∃x x is a collective class of Ps =⇒ ∃z z is a P. If x is a collective class of Ps, then x then has a common ingrediens with some P, since x is an ingrediens of x. Therefore, there exists a P. Regarding the second, it is also evident that the same collective class corresponds to those concepts under which fall the same objects9 :   ∀y y is a S ⇔ y is a P =⇒   ∀x x is a collective class of Ss ⇔ x is a collective class of Ps .

9 Such

concepts we call coextensive (cf. Remark A.2.13).

10

1 An Introduction to the Problems of the Theory of Parthood

However, for some general names S and P the converse implication to the above does not obtain. That is, there are concepts which determine the same collective classes but which are not coextensive (cf., e.g., Pietruszczak 2018, p. 66).10

1.4.2 Axioms of Le´sniewski’s Mereology As noted in Sect. 1.2, Le´sniewski understood the relation of part to whole as asymmetric and transitive (cf. (as ) and (t )). These properties are captured by the first two axioms of his mereology. They say, therefore, that the relation is a part of strictly partially orders the universe of discourse. To put it in a logically equivalent way, we assume that the relation is a part of is irreflexive and transitive. The two remaining axioms of Le´sniewski’s mereology may be formulated with the concept being a collective class of introduced in the previous subsection. It follows from the meaning of the word ‘class’ that the concept being a collective class of Ps can have at most one referent. We will not, however, obtain this from (as ), (t ), (df ) and (df ccP). Le´sniewski simply assumed it, adopting as his third axiom:  ∀x,y x is a collective class of Ps ∧ y is a collective class of Ps  (fccP ) =⇒ x = y . Thus, we can say that Le´sniewski assumed that the concept being a collective class is functional (with respect to its first argument). Since the concept being a collective class of Ps is defined via the concept being an ingrediens, and that concept in turn with the help of the concept being a part, axiom (fccP ) highlights a certain property of that last concept. In Chap. 2 we will show that (fccP ) does not follow from (as ), (t ) and (df ccP). Furthermore, in Sect. 2.6.2 we will give a simpler condition which will concern the relation is a part of and which will be equivalent to (fccP ). As his fourth mereological axiom, Le´sniewski said that if a name represented by the letter ‘P’ is non-empty, then there exists a (unique) object which is the collective class of Ps. Formally: ∃z z is a P =⇒ ∃x x is a collective class of Ps.

(∃ccP )

From an intuitive point of view, this is a very controversial assumption and it is one we have discussed in the introduction.

10 It

is, however, different in the case of distributive classes (cf. Remark A.2.13).

1.4

Le´sniewski’s Mereology

11

1.4.3 Collective Sets in Le´sniewski’s Sense Le´sniewski distinguished the concept collective class of Ps from the concept collective set of Ps. He defined the latter according to the following schema: x is a collective set of Ps iff each ingrediens of x has a common ingredients with some P which is an ingrediens of x. Written formally, the schema takes the following form:  x is a collective set of Ps :⇐⇒ ∀y y x ⇒ ∃z (z is a P ∧  z x ∧ ∃u (u z ∧ u y)) .

(df csP)

We see once again that the concept collective set of Ps is empty if the name represented by the letter ‘P’ is too, i.e.: ∃x x is a collective set of Ps =⇒ ∃z z is a P. Observe that (df csP) allows that the concept collective set of Ps may have more than one referent. This occurs if the name represented by ‘P’ is neither empty nor monoreferential.11 Therefore, to put it vividly, there can be many collective sets of Ps considered as different sets composed from the Ps. Amongst them is the collective class of Ps:   ∀x x is a collective class of Ps =⇒ x is a collective set of Ps . If x is a collective class of Ps, then each P is an ingrediens of x, so the postulated z (which is a P) is also an ingrediens of x. The collective class of Ps is a collective set composed all the Ps, and so there can only be one such set. We therefore have the following identity between the senses of our terms: the collective class of Ps ≡ the collective set of all Ps. Let us note, however, that the expression ‘collective set of all Ps’ does not fit with the syntax of Le´sniewski’s language of mereology. Furthermore, let us observe that each collective set of Ps is also a collective class, namely the collective class of Ps which are ingredienses of that set. Formally:

11 A

name which has exactly one referent we will call monoreferential. A name which has at least two referents we will call polyreferential.

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1 An Introduction to the Problems of the Theory of Parthood

 ∀x x is a collective set of Ps ⇐⇒

 x is a collective class of Ps which are ingredienses of x .

We simply apply (df ccP) to the name ‘P which is an ingrediens of x’.

1.5 The Problem of “Existential Involvement” in the Theory of Parts The fourth of Le´sniewski’s axioms, (∃ccP ), is the strongest of the possible axioms one might adopt postulating the existence of collective classes. For each non-empty concept there is to be a collective class of the referents of that concept. (For empty concepts there are no collective classes by definition.) Nowadays, instead of talking about the collective class of referents of a given concept, we talk instead of the mereological sum of the elements of the extension of that concept. Next, one carries out a generalisation on arbitrary distributive sets, which do not have to be the extensions of any concepts. We therefore speak of a mereological sum of elements of a given distributive set. Remark 1.5.1 Such a generalisation would have been alien to Le´sniewski, if only because he rejected the existence of distributive sets (see, e.g., Pietruszczak 2000, 2018, Chap. I). Furthermore, when generalising, we are looking at arbitrary distributive sets rather than just the extensions of general names. Therefore, if the universe of discourse is infinite, we have an uncountable number of subsets. With general names, however, there can be at most a countable number. In the other direction, we have a problem with the “richness” of general names which yields surprising results. For it has been accepted that two general names may be joined by the connectives ‘and’ and ‘or’ to create a new polyreferential name. We may therefore create a name whose extension is the set {the moon, my heart}. It is a simple matter of connecting the two names: ‘natural satellite of the Earth’, ‘heart of the author of this book’. If the foregoing seems artificial, then observe that, for the “respectable” polyreferential (general) name ‘dog’, it is hard to agree to the existence of a material object which would be the collective class of dogs; that is, the collective set of all dogs (there does of course exist the distributive set, this being the extension of the name ‘dog’). It is then still harder to agree to the existence of a collective set of all objects, this being what one would obtain after applying axiom (∃ccP ) to the universe of the general polyreferential (general) name ‘object’. We see, therefore, that even in its original form, Le´sniewski’s fourth axiom is too strong for general use. We will say more about the role and possible applications of this axiom in Sect. 5 of Chap. II in (Pietruszczak 2000, 2018).  Various existential assumptions exist that are weaker than Le´sniewski’s axiom (∃ccP ). Collective classes postulated by such assumptions may in some cases be

1.5

The Problem of “Existential Involvement” in the Theory of Parts

13

regarded as ad hoc objects. As we have already said, it is hard to accept that there may be a material object that is the mereological sum of the moon and the heart of the author of this book. Problematic also is the existence of the mereological sum of the right and left hands of a person, even though they are parts of his body. “Existentially neutral” theories of parthood are those in which we prove the existence only of those collective classes (mereological sums) which can be obtained exclusively on the strength of the fundamental properties of the relation is a part of. For example, in such theories, it is not the case that any pair of objects, which are parts of a third, have a mereological sum (consider the example of the hands and body of a person). It is obvious that in order to obtain an “existentially neutral” theory of parthood, we need to reject Le´sniewski’s axiom (∃ccP ) or any counterpart of it. What is more, we must not adopt any axiom which postulates the existence of any collective class (mereological sum). Finally, let us observe that an “existential axiom” does not have to explicitly postulate the existence of a collective class (mereological sum). For example, we will show (see Sect. 2.2.9) that the axioms that are sentences (‡∅ ) and (‡) implicitly postulate the existence of mereological sums. “Existentially neutral” theories and “existentially involved” theories will be the subjects of Chaps. 2 and 3 respectively.

1.6 “Quasi-parts-or-wholes”—Theories Without Antisymmetry In the recent literature one can find a number of authors considering a relation that they call the “relation of being a part of”. They do not, however, assume that it is antisymmetric and assume only that it is reflexive and transitive.12 These authors are therefore not concerned with the concepts we have been considering, namely the concept of being a (proper) part, because it is irreflexive, or the concept being an ingrediens, because it is antisymmetric. In the light of its assumed reflexivity, this new concept corresponds more to the concept being an ingrediens. We will therefore introduce the concept being a quasi-part-or-whole and we will signify its extension in the universe U with ‘’. Obviously, we can talk of the “true” relation of being a quasi-part-or-whole only when13 it is not antisymmetric, i.e., when for some x, y ∈ U we have: x = y, x  y and y  x. Let us henceforth assume this to be the case with the relation , because otherwise it would hard to distinguish it from the relation . 12 For

example, see, e.g., (Cotnoir 2010) (where (Varzi 2008) is discussed) and the literature it mentions. 13 We have used here figuratively the word ‘true’ instead of the word ‘proper’, because the latter word could be used in connection with the word ‘part’ whereas here we are concerned with ‘truth’ in the quasi sense.

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1 An Introduction to the Problems of the Theory of Parthood

We may “tease out” of the relation  two further relations: the relation is a (proper) quasi-part of and the relation is a (proper) part of . The first of these can be defined as the “irreflexisivation” of the relation , i.e., as the relation  \ idU .14 Those who study the relation  define the relation is a (proper) part of as the “asymmetricisation” of the relation , because they want a strict partial order in the universe. Let us signify the relation defined this way with ‘ 0, we have: 

I[{x1 , . . . , xn }] = {u ∈ U : u  x1 ∧ · · · ∧ u  xn }.

Therefore, the condition



I[{x1 , x2 }] = ∅ reduces to the condition x1  x2 .



2.1 Fundamental Relations and Their Properties

27

2.1.7 The Relation of Proper Overlapping We say that two objects properly overlap one another (or cross one another) iff they have some common part but neither of them is a part of the other, i.e., that common part is not identical to either of them. Formally, we introduce into U the binary relation  via the following definition:   ∀x,y∈U x  y :⇐⇒ x = y ∧ x  y ∧ y  x ∧ ∃z∈U (z  x ∧ z  y) .

(df )

Obviously, the relation  is symmetric, i.e.:   ∀x,y∈U x  y =⇒ y  x .

(s )

From (df ) and (2.1.7) we have the following connection between these relations:   ∀x,y∈U x  y ⇐⇒ x  y ∧ x  y ∧ y  x .   ∀x, y ∈ U x  y ⇐⇒ x = y ∨ x  y ∨ y  x ∨ x  y .

(2.1.15) (2.1.16)

Therefore  ⊆  and    = ∅. Let us observe that from (as ) and our definitions, we get:   ∀x,y∈U x  y ⇐⇒ x  y ∨ x  y ∨ y  x .

(2.1.17)

If x  y then x  y and x = y. Assume as well that y  x and x  y. Hence y  x and for some z we have: z  x and z  y. Therefore x = z = y, i.e., z  x and z  y. Thus, x  y. For the converse, we apply (2.1.5) and (2.1.15). Furthermore, if y  x, then x  y and x = y, by virtue of (as ) and (irr  ).

2.2 The Concept of an Atom Let U = U,  be a non-degenerate and strictly partially-ordered set in which principle (0) holds.7 Since in that case U has no zero, a given element is an atom iff it has no parts (see Appendix B.8). Just as in Appendix B, let At be the set of all atoms in the structure U,  : At := {x ∈ U : ¬∃u∈U u  x} =: min (U ). Therefore, for an arbitrary x ∈ U we have: 7 Degenerate

structures are of no interest to us, although all the conditions mentioned also hold in such structures if we accept that their single element is an atom.

28

2 “Existentially Neutral” Theories of Parts

x ∈ At ⇐⇒ P(x) = ∅,   ⇐⇒ ∀u∈U u  x ⇒ u = x ,   ⇐⇒ ∀u∈U x  u ∨ x  u . The second equivalence is obtained from (irr  ) and (df ). For the third, if x ∈ At and x  u, then for some v we have v  x and v  u. Hence v = x, i.e., x  u. Conversely, if x ∈ / At then for some u we have u  x. Hence x  u, on the strength of (antis ). Then x  u, by (r ). Furthermore, let us observe that if two objects have at least one part each (i.e., they are not both atoms), then they overlap one another just in case they have at least one common part:   / At ∧ x  y =⇒ ∃z∈U (z  x ∧ z  y) . ∀x,y∈U x, y ∈ If x, y ∈ / At then for some x0 and y0 we have x0  x and y0  y. Furthermore, if x  y then for some z 0 we have: z 0  x and z 0  y. If z 0 = x then x0  z 0 . Hence also x0  y, by (2.1.2). Similarly, if z 0 = y then y0  z 0 . Hence also y0  x, by (2.1.3). If, however, x = z 0 = y, then z 0  x and z 0  y, by (df ). Therefore, thanks to the last two conditions and (antis ), for arbitrary x, y ∈ U we obtain: x  y iff exactly one of the following four conditions holds: • • • •

x, y ∈ At and x = y, x ∈ At and x  y, y ∈ At and y  x, ∃z∈M (z  x ∧ z  y).

Moreover, the properties of atomistic and atomic structures, which are spelled out in Appendix B.8, hold in full in the structure U = U,  .

2.3 Supplementation Principles 2.3.1 The Weak Supplementation Principle The weak supplementation principle comes from (Simons 1987, p. 28) and has the following form:   (WSP) ∀x,y∈U y  x =⇒ ∃z∈U (z  x ∧ z  y) . With the help of the operators P and O principle (WSP) can be written as follows:  ∀x,y∈U y  x =⇒ P(x) ∩ E(y) = ∅,   ∀x,y∈U P(x) ⊆ O(y) =⇒ y  x .

(WSP )

2.3 Supplementation Principles

29

Remark 2.3.1 In the literature, the weak supplementation principle is also found in the following form:   ∀x,y∈U y  x =⇒ ∃z∈U (z  x ∧ z  y) . With the help of the operators P and O the above sentence can be written as follows:  ∀x,y∈U y  x =⇒ I(x) ∩ E(y) = ∅,   ∀x,y∈U I(x) ⊆ O(y) =⇒ y  x . R. Casati and A. C. Varzi, for example, employ it in this form in their (1999, p. 39). We note this so as to avoid any misunderstanding, as we shall stick with the first formulation. We can do this because, in the theory we are presenting, the two versions of the principle are equivalent. To obtain their equivalence, one needs merely (r ), which we obtain directly from (df ). (WSP) entails the second version, by (df ). Conversely, assume that y  x. Then, using the second version, for some z we have z  x and z  y. If it were true that z = x, then we would have a contradiction: x  y and x  y, by virtue of our  assumption and (r ). Therefore, z = x, and so z  x. Let us observe that: Lemma 2.3.1 (Pietruszczak 2000, 2018) (i) (irr  ) follows from (WSP). (ii) (as ) follows from (WSP) and (t ). Proof (i) Assume for a contradiction that for some x we have x  x. Then, by virtue of (WSP), for some z we have: z  x and z  x. From the former, by virtue of (df ) and (df ), we have z  x, which by virtue of (=−) yields a contradiction.  (ii) From (WSP) we have (irr  ). Moreover, (irr  ) and (t ) entail (as ). Therefore, if the transitive relation  satisfies condition (WSP) it strictly partially orders the set U and it is not necessary to assume that either (as ) or (irr  ) is satisfied. We observe this because Simons (1987)—and others, following him (e.g. Chisholm (1992/93); Libardi (1990))—lists (as ), (t ) and (WSP), and so does not recognise the independence of these axioms. Since from (WSP) we have (irr  ), we also obtain that from (WSP) it follows that no object has exactly one part, i.e.:   ∀x,y∈U y  x =⇒ ∃z∈U (z  x ∧ z = y) ,

(2.3.1)

which we may write using the operator P as follows: ∀x∈U Card P(x) = 1.

(2.3.1 )

We will now prove the following lemma which was mentioned in footnote 6:

30

2 “Existentially Neutral” Theories of Parts 12

21

1

2

Model 2.2 (WSP) and (∃fu) hold, but (2.3.3), (SSP), (ext  ) and (ext  ) do not hold

Lemma 2.3.2 (Pietruszczak 2000, 2018) (WSP) entails (∃) and in consequence also (0). Proof Let (a) Card U > 1 and assume for a contradiction that (b) ∀x,y∈U x  y. By virtue of (a) there are x1 , x2 ∈ U such that (c) x1 = x2 . By virtue of (b) there is a y0 such that y0  x1 and y0  x2 . Hence, by (c), either y0  x1 or y0  x2 . In both cases, on the strength of (WSP), there is a z 0 such that z 0  y0 . And this contradicts (b). For (0) see Lemma 2.1.2(i).  In structures with unity satisfying condition (WSP) it is possible to strengthen the above lemma in a certain sense. In such structures every element different from the unity is exterior to another element (also different from the unity). Lemma 2.3.3 The following sentence follows from (WSP):   (∃1 ) =⇒ ∀x∈U x = 1 ⇒ ∃z∈U (z  x ∧ z = 1) .

(2.3.2)

Proof If there is the unity 1 and x = 1, then x  1, by virtue of (df unity). Hence, on the strength of (WSP), for some z we have: z  1 and z  x. Hence z = 1, by  (irr  ) (which follows from (WSP)). Remark 2.3.2 From (WSP) itself the following sentence does not follow:   ∀x∈U ¬ x is the unity =⇒ ∃z∈U z  x .

(2.3.3)

To obtain it, the assumption (∃1 ) is necessary. To see this, there belongs to the class s POS a structure for which U := {1, 2, 12, 21}, whose relation  is depicted in Model 2.2. This model lacks a unity but (WSP) holds in it. We have O(12) = U = O(21), and so E(12) = ∅ = E(21). 

2.3.2 The Strong Supplementation Principle As we have already noted in footnote 5 of Chap. 1, in (Simons 1987) the polarisation principle (pol ) appears as the strong supplementation principle. Using the relation  , we get:   (SSP) ∀x,y∈U x  y =⇒ ∃z∈U (z  x ∧ z  y) .

2.3 Supplementation Principles

31

With the help of the operators I and O principle (SSP) can be written as follows:   ∀x,y∈U x  y =⇒ I(x) ∩ E(y) = ∅ ,   ∀x,y∈U I(x) ⊆ O(y) =⇒ x  y .

(SSP )

The results below show that we are justified in our use of the qualifiers ‘strong’ and ‘weak’ in the context of (strict) partial orders. Lemma 2.3.4 (SSP) entails the following sentence8 :   ∀x∈U ¬ x is the unity =⇒ ∃z∈U (z  x ∧ ¬ z is the unity) .

(2.3.4)

Proof If x is not the unity then, by (df unity), for some y, we have y  x. Hence, there is a z such that z  y and z  x. Hence z = 1, by (df unity), (df ) and (df ).  Lemma 2.3.5 (WSP) follows from (as ), (r ) and (SSP). Proof Assume that y  x. Then, by virtue of (as ), we have x  y. Hence, by (SSP), there is a z such that z  x and z  y. Were it to be true that z = x, we would have a contradiction: x  y and x  y, by virtue of our assumption and (r ). Therefore, z = x, and so z  x.  Remark 2.3.3 With reference to Sect. 2.1.2, let us notice that it is also the case, where our primitive relation is , and  is defined with the help of (df 1 ), that (WSP) follows from (r ), (antis ) and (SSP). To see this, if x  y then x  y and x = y, by (df 1 ). Hence y  x, by (antis ). The rest of the proof is as for Lemma 2.3.5.  Fact 2.3.6 In the class s POS (resp. POS) condition (SSP) does not follow from (WSP) and our definitions. Proof Sentence (2.3.4) follows from (SSP) (see Lemma 2.3.4), but this sentence does not follow from (WSP) (see footnote 8 and Remark 2.3.2). Hence (SSP) does not follow from (WSP). Another proof. Model 2.2 satisfies (WSP). However, we see that in this model we have: 12  21 and I(12) ⊆ O(21). So (SSP) is false in this model.  Remark 2.3.4 The assumption of condition (as ) in Lemma 2.3.5 was unnecessary. (i) We have a structure U,  satisfying (SSP) and (irr  ), which does not satisfy (WSP). We put U := {1, 2} and  := { 1, 2 , 2, 1 }. The relation  is full and the relation  is empty. Condition (SSP) therefore holds but condition (WSP) does not. (ii) There is a structure U,  satisfying (SSP) and (t ) which does not satisfy (WSP). We put U := {1, 2} and  := { 1, 1 , 1, 2 , 2, 1 , 2, 2 }. The relation  is full and the relation  is empty. (SSP) therefore holds but (WSP) does not. 8 Of

course, in addition (2.3.4) does not follow from (WSP) (cf. Remark 2.3.2). Moreover, (2.3.2) follows from (2.3.4); and so (2.3.2) follows also from (SSP).

32

2 “Existentially Neutral” Theories of Parts

(iii) We have incidentally shown that neither (t ) nor (irr  ) follows from (SSP). We obtain the first result thanks to (i), where the structure is non-transitive and the second by appealing to point (ii) where the structure is reflexive. (iv) We therefore have an interesting situation: (irr  ), follows from (WSP) but  without assuming (as ), we cannot derive (WSP) from (SSP). Let us also notice that on the strength of Fact B.3.3 we get: Lemma 2.3.7 (r ) follows from (t ) and (SSP).

2.3.3 The Supplementation Principle for the Relation Proper Overlapping. Comparison of Supplementation Principles In order to carry out a comparison of the supplementation principles, let us introduce one more principle which concerns the relation  of proper overlapping:   ∀x,y∈U x  y =⇒ ∃z∈U (z  x ∧ z  y) .

(SP )

From (SP ) we obtain the following condition, since the relation  is symmetric:   ∀x,y∈U x  y =⇒ (∃z∈U (z  x ∧ z  y) ∧ ∃z∈U (z  y ∧ z  x)) . Now let us note that (cf. Lemma 2.3.5): Lemma 2.3.8 (i) (SP ) follows from (SSP). (ii) (SSP) follows from (WSP) and (SP ). Proof (i) Let x  y. Then from (2.1.15) we have: x  y and x  y. Therefore, by (SSP), there is a z such that z  x and z  y. Since x  y, we have z = x, i.e., z  x. (ii) Suppose that (WSP) and (SP ) hold. Firstly, with the help of the operators P and O and condition (2.1.15) principle (SP ) can be written as follows:   ∀x,y∈U P(x) ⊆ O(y) =⇒ x  y ∨ x  y ∨ y  x ∨ x = y . Hence, by (WSP ), we have:   ∀x,y∈U x  y ∧ P(x) ⊆ O(y) =⇒ x  y ∨ x = y . But from our definitions: x  y ∧ P(x) ⊆ O(y) iff I(x) ⊆ O(y). So we have:   ∀x,y∈U I(x) ⊆ O(y) =⇒ x  y ∨ x = y . That is, we obtain (SSP ).



2.4 Extensionality Principles

33

2.4 Extensionality Principles 2.4.1 Definition of Extensionality Principles Mereological counterparts of the set-theoretic extensionality principle are given by the following sentences for particular relations:   ∀x,y∈U ∃u∈U u  x ∧ ∀z∈U (z  x ⇔ z  y) =⇒ x = y ,   ∀x,y∈U ∀z∈U (z  x ⇔ z  y) =⇒ x = y ,   ∀x,y∈U ∀z∈U (z  x ⇔ z  y) =⇒ x = y ,   ∀x,y∈U ∀z∈U (z  x ⇔ z  y) =⇒ x = y .

(ext  ) (ext  ) (ext  ) (ext  )

Note that in the case of (ext ) we must make an additional assumption in the antecedent of the implication, because x and y could be two (different) atoms. In that case, the second conjunct in the antecedent is true and the consequent false. From (antis ) and (df ), by applying (r ) and (antis ), we get principle (ext  ). Therefore, this extensionality principle is not interesting, because it brings nothing new to our analysis. Lemma 2.4.1 Principle (ext  ) follows from (r ) and (antis ). In consequence, (ext  ) follows from (antis ) and (df ). Moreover, in the light of given that (=−) we obtain: Lemma 2.4.2 Principles (ext  ) and (ext  ) are equivalent. By using the operators P, I, O and E the principles above may be written thus:   ∀x,y∈U ∅ = P(x) = P(y) =⇒ x = y ,   ∀x,y∈U I(x) = I(y) =⇒ x = y ,   ∀x,y∈U O(x) = O(y) =⇒ x = y ,   ∀x,y∈U E(x) = E(y) =⇒ x = y .

(ext P ) (ext I ) (ext O ) (ext E )

Since x could be an atom or the zero, implication (ext P ) is not reversible.9 In the remaining cases, the converse implications are logical truths. So we obtain:   ∀x,y∈U P = ∅ =⇒ (x = y ⇔ P(x) = P(y) ,   ∀x,y∈U x = y ⇐⇒ I(x) = I(y) , 9 Obviously,

it can only be the zero in degenerate structures.

(ext P ) (ext I )

34

2 “Existentially Neutral” Theories of Parts



 ∀x,y∈U x = y ⇐⇒ O(x) = O(y) ,   ∀x,y∈U x = y ⇐⇒ E(x) = E(y) .

(ext O ) (ext E )

We will also be examining “enhanced” extensionality principles that have the following forms10 :   ∀x,y∈U ∃u∈U u  x ∧ ∀z∈U (z  x ⇒ z  y) =⇒ x  y ,   ∀x,y∈U ∀z∈U (z  x ⇒ z  y) =⇒ x  y ,   ∀x,y∈U ∀z∈U (z  x ⇒ z  y) =⇒ x  y ,   ∀x,y∈U ∀z∈U (z  y ⇒ z  x) =⇒ x  y .

(ext  +) (ext  +) (ext  +) (ext  +)

As before, in the case of (ext  +) we must make an additional assumption in the antecedent of the conditional. Note that from (df ), by applying (r ), we get principle (ext  +). Therefore, this enhanced extensionality principle is not interesting, because it brings nothing new to our analysis. Lemma 2.4.3 Principle (ext  +) follows from (r ). In consequence, (ext +) follows only from definition (df ). Given that (=−) we also have: Lemma 2.4.4 Principles (ext  +) and (ext  +) are equivalent. Moreover, it is obvious that: Lemma 2.4.5 Any ordinary version of extensionality principles follows from (antis ) and its enhanced version. Remark 2.4.1 (i) In the following points we will show that, in the class s POS (resp. POS), the enhanced principles are essentially stronger than their “ordinary” counterparts. (ii) With reference to Sect. 1.6, let us observe that rejecting the antisymmetry of the relation  from the “enhanced” principles does not allow us to obtain their “ordinary” counterparts. This is why one talks in this case of “non-extensional mereology”.  By using the operators P, I, O and E, the principles above may be written thus:   ∀x,y∈U ∅ = P(x) ⊆ P(y) =⇒ x  y ,   ∀x,y∈U I(x) ⊆ I(y) =⇒ x  y ,   ∀x,y∈U O(x) ⊆ O(y) =⇒ x  y ,

(ext P +) (ext I +) (ext O +)

 +) is called the Proper Parts Principle. In Simons’ terminology ‘proper part’ corresponds to what we understand here by ‘part’.

10 In (Simons 1987) sentence (ext

2.4 Extensionality Principles

  ∀x,y∈U E(y) ⊆ E(x) =⇒ x  y .

35

(ext E +)

Conditions (t ), (2.1.10 ) and (2.1.6 ) give the converse implications to those appearing in (ext  +), (ext  +) and (ext  +), respectively. Hence, these latter conditions transform themselves into equivalences:   ∀x,y∈U P = ∅ =⇒ (x  y ⇔ P(x) ⊆ P(y) ,   ∀x,y∈U x  y ⇐⇒ I(x) ⊆ I(y) ,   ∀x,y∈U x  y ⇐⇒ O(x) ⊆ O(y) ,   ∀x,y∈U x  y ⇐⇒ E(y) ⊆ E(x) .

(ext P +) (ext I +) (ext O +) (ext E +)

It is not possible to derive the “full” converse implication to the one from (ext P +), because x could be an atom or the zero; that is, it may not have any parts. But we can use (2.1.2 ) to obtain (ext P +). Below, we will examine what we gain from the introduction of the extensionality principles with respect to the relations  and  (resp.  ).

2.4.2 Extensionality Principles with Respect to the Relations Overlapping and Is Exterior to As we have already noted, both extensionality principles with respect to the relations  and  are equivalent. We will begin with the extensionality principles for these two relations because they are essentially stronger than the principle for the relation . Lemma 2.4.6 (i) Conditions (t ) and (SSP) and our definitions entail (ext  +). (ii) Conditions (r ) and (ext  +) and our definitions entail (SSP). (iii) Conditions (r ) and (t ) and our definitions entail the equivalence of conditions (SSP) and (ext  +). Proof Ad (i): ‘O(x) ⊆ O(y)’ follows from (t ) and ‘I(x) ⊆ O(y)’ (see Sect. 2.1.5). Therefore from (t ) and (SSP ) we obtain (ext O +). Ad (ii): ‘I(x) ⊆ O(y)’ follows from (r ) and ‘O(x) ⊆ O(y)’ (see Sect. 2.1.5). Therefore from (r ) and (ext O +) we obtain (SSP ). Ad (iii): Directly from (i) and (ii).  Lemma 2.4.7 (i) Conditions (r ), (antis ), (SSP) and our definitions entail (ext  ). (ii) Conditions (t ), (antis ), (SSP) and our definitions entail (ext  ). Proof Ad (i): Suppose that (r ), (antis ) and (SSP) hold and O(x) = O(y). Then, by (r ), we have both I(x) ⊆ O(y) and I(y) ⊆ O(x). Hence both x  y and y  x, by (SSP ). So x = y, by (antis ).

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2 “Existentially Neutral” Theories of Parts

1

123

23

2

3

Model 2.3 (ext  ) and (Usum ) hold, but (ext  +) and (SSP) do not hold

Ad (ii): Directly by Lemmas 2.4.5 and 2.4.6(i) (or we use Lemma 2.3.7 and (i)).  Lemma 2.4.8 Conditions (irr  ), (t ) and (ext  ) and our definitions entail condition (WSP). Proof Suppose that (irr  ), (t ) and (ext  ) hold and y  x. Then x = y, by virtue of (irr  ). Furthermore, by virtue of (df ) and (2.1.10 ), obtained from (t ), we have O(y) ⊆ O(x). Assume for a contradiction that P(x) ⊆ O(y). On the strength of our assumption, P(x) = ∅. Therefore, applying (2.1.13 ), obtained from (t ), we have O(x) ⊆ O(y). Therefore O(x) = O(y). And hence, by virtue of (ext  ), we have a contradiction: x = y.  From these three lemmas we obtain the following theorem which specifies various relationships between supplementation principles and extensionality principles. Theorem 2.4.9 Let a structure U,  belong to s POS. Then, from our definitions, we can obtain the following results in that structure: (i) Conditions (SSP) and (ext +) are equivalent. (ii) Condition (SSP) entails condition (ext  ). (iii) Condition (ext  ) entails condition (WSP). Taking into account the above theorem, we can strengthen Fact 2.3.6 in the form of the following two facts11 : Fact 2.4.10 In the class s POS, conditions (SSP) and (ext O +) do not follow from (ext  ) and our definitions. Proof To the class s POS belongs the structure in which U := {1, 2, 3, 23, 123}, while Model 2.3 depicts the relation . In this structure, (ext  ) holds. However I(23) ⊆ O(123) but 23  123. Therefore (SSP) does not hold.  Fact 2.4.11 In the class s POS, principle (ext  ) does not follow from (WSP) and our definitions. Proof Model 2.2 belongs to s POS and (WSP) holds in it. However, O(12) = U =  O(21) but 12 = 21. Therefore (ext  ) does not hold. 11 Obviously—as

in the case of Facts 2.1.1, 2.1.3 and 2.3.6—we also have counterparts of Facts 2.4.10 and 2.4.11 for the class POS. We will henceforth not emphasise this.

2.4 Extensionality Principles

37

2.4.3 Extensionality Principles with Respect to the Relation Is a Part of It is evident that the statement that x is a part of y cannot appear in the antecedent of (ext  +), because it does not follow from the antecedent that x = y (for the converse implication to the one appearing in the antecedent might also be true). We have just: Lemma 2.4.12 The following sentence follows from condition (ext  +):   ∀x,y∈U ∅ = P(x)  P(y) =⇒ x  y .

(2.4.1)

Proof From the antecedent of the implication we have x = y. Furthermore, by virtue  of (ext  +), we have x  y. Therefore x  y. Lemma 2.4.13 Conditions (2.4.1), (ext  ) and (df ) entail (ext  +). Proof Assume that (2.4.1) and (ext  ) hold and ∅ = P(x) ⊆ P(y). If P(x)  P(y)  then x  y. If P(x) = P(y) then x = y, by (ext  ). So, in both cases, x  y. From Lemma 2.4.5 and the last two lemmas we have: Corollary 2.4.14 Conditions (antis ) and (df ) entail the equivalence of condition (ext  +) and the conjunction of conditions (ext  ) and (2.4.1). Lemma 2.4.15 Condition (SSP) and our definitions entail (ext  +). Proof Assume that P(x) ⊆ P(y) and for some z 0 we have z 0  x. Therefore we also have z 0  y, i.e., x  y. Furthermore, assume for a contradiction that x  y. Then, by virtue of (SSP), for some z 1 we have: z 1  x and z 1  y. We therefore have z 1 = x, because z 1 = x would entail x  y. Therefore z 1  x. Hence, via our  assumption, we have z 1  y. This, however, contradicts z 1  y.12 By taking into account Lemmas 2.4.12 and 2.4.15 we can strengthen Fact 2.4.10. Fact 2.4.16 In the class s POS, conditions (2.4.1) and (ext  +) do not follow from (ext  ) and our definitions. Proof The structure presented by Model 2.3 belongs to s POS and (ext  ) holds in it. However, ∅ = P(23)  P(123) but 23  123. So, (2.4.1) and (ext  +) do not hold.  Furthermore, we may once again strengthen Fact 2.4.10. Fact 2.4.17 In the class s POS, condition (SSP) does not follow from conditions (ext  ) and (ext  +) and our definitions. 12 Simons

(1987, p. 29) gives another proof of this result.

38

2 “Existentially Neutral” Theories of Parts 234 1234

1

2

3

4

Model 2.4 (ext  ), (ext  +) and (Usum ) hold, but (SSP) and fu  sum do not hold 2

1

Model 2.5 (ext  +) and (∃) hold, but (WSP) do not hold

Proof To s POS belongs the structure which is presented by Model 2.4. In this structure (ext  ) and (ext  +) hold. However I(234) ⊆ O(1234) but 234  1234. There fore, (SSP) does not hold.13 Points (i) and (ii) in the fact below are the strengthenings of Facts 2.1.1 and 2.1.3. Fact 2.4.18 In the class s POS and with our definitions: (i) Condition (0) does not follow from (ext  +). (ii) Condition (∃) does not follow from (ext  +) and (0). (iii) Condition (WSP) does not follow from (ext  +) and (∃). Proof Ad (i): The structure presented by Model 2.1 belongs to s POS and satisfies condition (ext  +), because P(1) = ∅ = P(2). In this structure condition (0) does not hold. Ad (ii): In the model given in the proof of Fact 2.1.3, conditions (0) and (ext  +) are true and (∃) is false. Ad (iii): To s POS belongs the structure which is presented by Model 2.5. In this structure (ext  +) and (∃) are true. However, 1  2 and {1} = P(2) ⊆ {1, 2} =  O(1). Therefore (WSP) is false.14 Fact 2.4.19 In the class s POS, condition (ext  ) does not follow from (WSP), (ext  +) and our definitions.

13 Recall that the simpler Model 2.3 from the class

s POS suffices to show that (SSP) does not follow from (ext  ). Obviously, another model simpler than Model 2.4 would likewise suffice to show that, in the class s POS, condition (SSP) does not follow from (ext  +). 14 The simpler Model 2.1 from POS suffices to show that (WSP) does not follow from (ext +). s 

2.4 Extensionality Principles

39 12

21

1

2

Model 2.6 (WSP) and (ext  +) hold, but (ext  ) does not hold

Proof To s POS belongs the structure presented by Model 2.6. In this structure (WSP) and (ext  +) are true. However, O(12) = U = O(21) and 12 = 21. Therefore  (ext  ) is false. Fact 2.4.20 In the class s POS, condition (ext  ) does not follow from (WSP) and our definitions. Proof Model 2.2 belongs to the class s POS and (WSP) holds in it. However, P(12)  = {1, 2} = P(21) but 12 = 21. Therefore (ext  ) is false.

2.5 Dependencies Between Theories 2.5.1 Definition of the Concept of a Theory If we are being precise, different theories of parts may be identified with appropriate classes of structures of the form U,  . From a more intuitive point of view, however, a given theory may be determined by a properly chosen set of sentences in the language of set theory. We regard these sentences as “axioms” of a given theory. A class of structures identified with a given theory can be—to put it crudely—treated as a “class of models of the axioms” of a given theory. We consider this view of a theory as a set of sentences to be an intuitive one, because we are speaking in the language of set theory in an informal way. The theories under consideration in this chapter could be easily formalised as elementary theories (i.e., of the first-order). To this end, it would suffice to regard the symbols ‘’, ‘’, ‘’, ‘’ and ‘’ as two-place predicates and the remaining symbols we have used as logical constants, individual variables, and brackets. We would omit the restriction ‘∈ U ’ appearing in the quantifiers, leaving just the variables. In this way— sentence (ac ) aside—we would obtain sentences in a first-order language. Sentence (ac ) would be written in a weak second-order language (allowing quantification over the natural numbers). We could, however, replace it with an infinite sequence of elementary sentences falling under the schema (1.1.1). In this framework, the predicates ‘’, ‘’, ‘’, ‘’ and ‘’ would be interpreted in appropriate structures by relations signified by those symbols we have used for

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2 “Existentially Neutral” Theories of Parts

the appropriate predicates. Therefore, the structures that we have been considering would be models of appropriately-constructed first-order theories. In the other direction, the relations , , ,  and  would be elementarily definable in the first-order language specified above. The primitive relation  would simply be elementarily definable by the formula ‘x  y’ and the remaining relations would be elementarily definable by counterparts of the formulas appearing in the right-hand sides of the equivalences taken from (df ), (df ), (df ) and (df ), respectively. In this framework, the class of structures we have been examining in this chapter would be finitely elementarily axiomatisable and all the auxiliary relations we have made use of would be elementarily definable.15 We will not “openly” carry out such a formalisation for a number of reasons. Firstly, where it can be done, it is easily done in the manner described above. Secondly, we will also be investigating theories of parts which either are not in general elementarily axiomatisable or which, even if they are elementarily axiomatisable, will not have built upon them such theories as point-free geometry and topology, to which we will want to our apply various theories of parts. Thirdly, we have already devoted a lot of attention to problems to do with the elementary axiomatisability of various theories in (Pietruszczak 2000, 2018). Let us therefore cross directly to an examination of the dependencies holding between the theories studied in this chapter. The broadest of them is the class s POS. We will adopt the following convention concerning symbols for the classes of structures. The addition to the name of a class C of the name of a condition W creates the name of a class which is a subclass of the class C composed of those structures satisfying the condition. The name of this new class is to be determined according to the following schema: the name of the class C +(name of condition W ); in short, C +(W ). Condition W may be written with the use of the symbols ‘’, ‘’, ‘’ and ‘’. These last three symbols signify the relations determined by definitions (df ), (df ) and (df ), respectively, which will be tacitly assumed in each of the theories (classes of structures) under consideration. To a name of a class of structures built in accordance with the above schema we can add the name of another condition, creating a new name of a class of structures; in short: C +(W )+(W  ); and so on in an obvious inductive fashion.16 Furthermore, the joining of names of the same conditions also creates the name of a class of structures in which all those conditions hold. However, we will use names of the second kind only when some member of the condition defining the class s POS follows from the conditions involved in the name. The class s POS+(pol ) of all polarised strict partial orders will be the narrowest of those considered in this chapter (and therefore the broadest, as a set of sentences). We will henceforth symbolise it as: s PPOS. Thus, we put: 15 Here,

the symbols ‘’, ‘’, ‘’, ‘’ and ‘’ play a double role: as two-place predicates and the names of binary relations (more precisely: variable or binary relations). Obviously, it would be possible to have different symbols for the predicates and the names of relations. 16 It needn’t be a new class of structures. A given class may simply have various different names (through the addition of different names of conditions).

2.5 Dependencies Between Theories s PPOS

41

:= s POS+(pol ) = s POS+(SSP).

Similarly, the class POS+(pol ) of all polarised partial orders will be henceforth symbolised as: PPOS. Thus, we put: PPOS := POS+(pol ) = POS+(SSP). On the strength of Lemma 2.3.1 and Theorem 2.4.9(i), we get: s POS+(WSP) s POS+(ext  )+(WSP) s POS+(ext  +)+(WSP) s PPOS

= (t )+(WSP), = (t )+(ext  )+(WSP), = (t )+(ext  +)+(WSP),

= s POS+(ext  +) = s POS+(ext  +).

Furthermore, for each class of structures C including s POS we have:

C +(ext  ) = C +(ext ), C +(ext  +) = C +(ext  +) = C +(SSP).

2.5.2 A Lattice of Theories In accordance with the comments made in point 2.1.3, the theories of parts that will be of interest to us are those classes of structures in which (0) holds. In order to carry out a full comparison, we have presented in Diagram 2.1 a wider lattice of classes of structures (theories). In this lattice, the partial order ≤ is the relation of inclusion, meaning that: C 1 ≤ C 2 iff C 1 ⊆ C 2 . The strict partial order obtained from ≤ is more visible in this lattice; that is, the relation  satisfying the condition: C 1  C 2 iff C 1  C 2 . In Diagram 2.1, the expression C 1 → C 2 indicates that C 1  C 2 . Obviously, C 1  C 2 iff from the conditions defining the class C 1 follow all the conditions defining the class C 2 (which gives us C 1 ⊆ C 2 ) but not conversely (which therefore gives us C 2  C 1 ). That the inclusions indicated in Diagram 2.1 hold and that they are proper follow from the definitions of the classes there and the lemmas and facts proven in the previous sections of this chapter. It is clear that in each occurrence of ‘(ext  )’ in Diagram 2.1 may be replaced by ‘(ext  )’. Let us adopt the convention that, for an arbitrary class C , the class C ·· is to be composed of all non-degenerate structures from C . We can easily obtain from Diagram 2.1

42

2 “Existentially Neutral” Theories of Parts s POS

s POS+(ext  )

s POS+(0)

s POS+(ext  +)

s POS+(ext  )+(0)

s POS+(∃)

s POS+(ext  +)+(0)

s POS+(ext  )+(∃)

s POS+(WSP)

s POS+(ext  +)+(∃)

(t )+(ext  )+(WSP)

(t )+(ext  +)+(WSP)

s POS+(ext  )

s POS+(ext  )+(ext  +) s PPOS

where s POS

:= (irr  )+(t ) = (as )+(t )

(ext ) = (ext  ) s PPOS

s POS+(WSP)

= (t )+(WSP)

(ext  +) = (ext  +)

:= s POS+(pol ) = s POS+(SSP) = s POS+(ext  +)

Diagram 2.1 The lattice of “existentially neutral” theories

a diagram of all classes of non-degenerate structures. We simply replace occurrences of the class C with the class C ·· and if C = s POS+(. . .) or C = s PPOS+(. . .), then we have C ·· = s POS·· +(. . .) or C ·· = s PPOS·· +(. . .), respectively. If a class C is determined by a set Ax of conditions (its axioms) then the class C ·· is determined by the set Ax extended by condition (n-d) from Sect. 1.3.2. So

C ·· = C +(n-d). Furthermore, let us adopt the convention that, for an arbitrary class C , the class C fin is to be composed of all finite structures from C . In Sect. 1.5 we explained in an introductory way why we are considering the theories in Diagram 2.1 as “existentially neutral” (three of them do not have thesis (0), and so are not theories of parthood). We will return to this issue later.

2.6 Mereological Sum

43

2.6 Mereological Sum 2.6.1 Definition and Basic Properties of Mereological Sum Let ‘P’ be a schematic letter representing a general name of certain elements of the universe of discourse U . The extension of this name is the distributive set of Ps belonging to U , i.e., the set {u ∈ U : u is a P}. In Sect. 1.4.1 we introduced the concept of being a collective class. Following in the footsteps of Tarski’s (1956), instead of writing that x is a collective class of Ps, we will write that x is a mereological sum of all elements of the distributive set {u ∈ U : u is a P}, i.e., the sum of all Ps. We can also go straight to the consideration of arbitrary subsets of the universe without restricting ourselves to the extensions of general names. This allows us to leave schematic letters out of our analysis. Instead of a mereological sum of all Ps, we can speak of the sum of all elements of a given subset of the set U . For an arbitrary element x from U and arbitrary subset S of U , Le´sniewski’s definitional schema, featuring the expression “mereological sum”, may be written as follows: x is a mereological sum of all elements of a set S iff the following two conditions are satisfied: • every element of the set S is an ingrediens of x, • every ingrediens of x has a common ingrediens with some element of S. Obviously, we could have written the above definition with the help of the primitive concept of being a (proper) part instead of the concept of being an ingrediens. Instead of x is a mereological sum of all elements of a set S we can write: x sum S. To put things formally, we are introducing sum as a binary relation included in the Cartesian product U × 2U . We define this relation according the recipe above by putting for arbitrary x from U and S from 2U :   x sum S :⇐⇒ ∀s∈S s  x ∧ ∀y∈U y  x ⇒ ∃s∈S ∃u∈U (u  s ∧ u  y) . (df sum) By applying the relation , we obtain a shorter version of the above definition:   x sum S ⇐⇒ ∀s∈S s  x ∧ ∀y∈U y  x ⇒ ∃s∈S s  y . If, however, we make use of I and O, the definition has the following form: x sum S ⇐⇒ S ⊆ I(x) ⊆



O[S],

(df  sum)

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2 “Existentially Neutral” Theories of Parts

where  the family of sets O[S] is the image of the set S determined by the operator O and O[S] the set-theoretic sum of the family (see Remark 2.1.2). Lemma 2.6.1 The following sentences are consequences of condition (r )17 :   ∀S∈2U ∀x∈U x sum S =⇒ S = ∅ ,

(2.6.1)

¬∃x∈U x sum ∅ , 

(2.6.2) 

∀S∈2U ∀x∈U x ∈ S ⊆ I(x) =⇒ x sum S ,   ∀x∈U x sum U ⇐⇒ x is the unity ,

(2.6.4)

∀x∈U x sum {x},

(2.6.5)

∀x∈U x sum I(x),

(2.6.6)

  ∀x∈U P(x) = ∅ =⇒ x sum P(x) ,   ∀S∈2U ∀x∈U P(x) = ∅ ∧ x sum S =⇒ S = {x} ,    ∀S∈2U ∀x∈U x sum S ⇐⇒ ∅ = S ⊆ I(x) ∧ P(x) ⊆ O[S] .

(2.6.3)

(2.6.7) (2.6.8) (2.6.9)

Proof Suppose that condition (r ) holds in a structure U,  . Ad (2.6.1): Assume that x sum S. Then, by virtue of (df sum), there is a s such that s ∈ S, since x  x. Ad (2.6.2): Directly from (2.6.1). Ad (2.6.3): If x ∈ S ⊆ I(x) and y  x, then from (r ) we have x  y, where x ∈ S. Ad (2.6.4): From (df sum), (2.6.3) and (df unity). Ad (2.6.5): From (r ), {x} ⊆ I(x). So we use (2.6.3) for S := {x}. Ad (2.6.6): From (r ), x ∈ I(x). So we use (2.6.3) for S := I(x). Ad (2.6.7): Obviously, P(x) ⊆ I(x). Furthermore, if y  x then y ∈ P(x) and y  y, by (r ). If, however, y = x then we exploit the fact that for some z 0 we have: z 0 ∈ P(x), z 0  y and z 0  y, by virtue of (r ). Hence x sum P(x). Ad (2.6.8): If P(x) = ∅ and x sum S, then I(x) = {x} and ∅ = S ⊆ I(x), by (df  sum) and (2.6.1). Hence S = {x}. Ad (2.6.9): We obtain the left-to-right direction on the strength of our definitions, including (2.6.1).  For the right-to-left direction: Suppose that (a) ∅ = S ⊆ I(x) and (b) P(x) ⊆ O[S]. Then, by (b), if y  x, then there is s ∈ S such that s  y. If, however, y = x then, by (a), for some s0 we have s0  y. Hence s0  y, by (r ). Therefore x sum S.  Sentence (2.6.2) corresponds to the claim that there is no empty collective class (no empty collective set). reference to Sect. 2.1.2, let us recall that (r ) is a direct consequence of definition (df ) and the reflexivity of identity. Therefore, in our theory we can have condition (r ) without any additional assumptions. This holds for all the sentences in this lemma.

17 With

2.6 Mereological Sum

45

The antecedent of (2.6.3) says that x is the greatest element in a set S (see Appendix B.2.2). It therefore says that the greatest element of a given set is its mereological sum. Sentence (2.6.5) says that the “summation” of an element of x yields x. It would be an error, though, to say that in that case x is a one-element collective set. For if x has parts, then it is also the sum of all its parts, which is what (2.6.7) says. We know that, with (WSP) in place, no object has exactly one part (cf. (2.3.1 )). Remark 2.6.1 For the class s POS, without making further assumptions, we cannot state that in sentences (2.6.3), (2.6.5) and (2.6.6) the element x is the only sum of the sets featuring in those sentences. For example, in Model 2.1 we have: 2 sum {1} and I(1) = {1}. The same goes for (2.6.7) as in Model 2.2 we have: P(12) = {1, 2} = P(21), 12 sum {1, 2} and 21 sum {1, 2} (notice that sentence (ext  ) is false here).  For finite sets the relation sum has an interesting property. Lemma 2.6.2 (Pietruszczak 2000, 2018) It follows from conditions (r ) and (t ) that for arbitrary x, y1 , …, yn ∈ U (n  1) we have: x sum {y1 , . . . , yn } ⇐⇒ x sum {z ∈ U : ∃1in z  yi }. Proof Note that from (r ) and (t ) we have (∗): ∀in yi  x iff ∀in ∀z (z  yi ⇒ z  x) iff ∀z (∃in z  yi ⇒ z  x). “⇒” Suppose that x sum {y1 , . . . , yn }. Then {z ∈ U : ∃1in z  yi } ⊆ I(x), by (∗), since {y1 , . . . , yn } ⊆ I(x). Moreover, by virtue of (2.1.9), we have:   ∀z z  x ⇒ ∃u (∃in u  yi ∧ u  z) , since we have: ∀z (z  x ⇒ ∃in ∃u (u  yi ∧ u  z)). Thus, x sum {z ∈ U : ∃1in z  yi }. “⇐” Suppose that x sum {z ∈ U : ∃1in z  yi }. Then {y1 , . . . , yn } ⊆ I(x), by (∗), since {z ∈ U : ∃1in z  yi } ⊆ I(x). Moreover, by (2.1.10), we have ∀z (z  x ⇒ ∃in yi  z), since ∀z (z  x ⇒ ∃u (∃in u  yi ∧ u  z)). Thus, x sum {y1 ,  . . . , yn }. We will be making regular use of the following property of the relation sum: Lemma 2.6.3 (Pietruszczak 2000, 2018) The following sentence follows from sentences (r ) and (t ):   ∀x,y∈U I(x) ⊆ O(y) =⇒ x sum I(x)  I(y) . Proof Let I(x) ⊆ O(y). Since I(x)  I(y) ⊆ I(x), we need only show that I(x) ⊆  O[I(x)  I(y)]. Let us assume therefore that z  x. Then, from our assumption: z  y. Hence, for some z 0 we have: z 0  z and z 0  y. Thanks to (t ), we have z 0  x. Therefore z 0 ∈ I(x)  I(y)  I(z) ⊆ I(x)  I(y). Furthermore, z 0  z, thanks to (r ). Thus, z ∈ O[I(x)  I(y)]. 

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2 “Existentially Neutral” Theories of Parts

The sentence proven in the lemma below also expresses an essential property of the relation sum.18 Lemma 2.6.4 From condition (t ) follows the following:   ∀S∈2U ∀x∈U x sum S =⇒ ∀y∈U (y  x ⇔ ∃s∈S s  y) , which, if written with the operator O, is:    ∀S∈2U ∀x∈U x sum S =⇒ O(x) = O[S] .  Proof Assume that x sum S, i.e., S ⊆ I(x) ⊆ O[S]. “⇒” Let y  x, i.e., forsome u 0 : u 0  x and u 0  y. Then for some s0 ∈ S we have u 0  s0 , since I(x) ⊆ O[S]. Hence, since u 0  y, we have y  z 0 , by (2.1.10). “⇐” Since S ⊆ I(x), we have ∀s∈S ∀y∈U (y  s ⇒ y  x), by (2.1.10). And this is  logically equivalent to the following: ∀y∈U (∃s∈S y  s ⇒ y  x).

2.6.2 The Uniqueness of Mereological Sum In point 1.4.2 we wrote that, in accordance with the meaning of the word ‘class’, the concept of being a collective class of Ps can have at most one referent and that Le´sniewski adopted an axiom to this effect. In our terminology, the counterpart of his axiom says that if a (distributive) set has a mereological sum it is unique:   ∀S∈2U ∀x,y∈U x sum S ∧ y sum S =⇒ x = y .

(Usum )

We will show that it is only possible to prove (Usum ) in the theory s POS+(ext  ); and, more precisely: Lemma 2.6.5 (Usum ) follows from (ext  ) and (t ).

 Proof Assume that x sum S and y sum S. Then O(x) = O[S] = O(y), thanks to (t ), by virtue of Lemma 2.6.4. Hence x = y, by virtue of (ext  ).  Lemma 2.6.6 (ext  ) follows from (r ), (t ) and (Usum ). Proof Suppose that O(x) = O(y). Then I(x) ⊆ O(y) and I(y) ⊆ O(x), by virtue of (r ). Hence x sum I(x) ∩ I(y) and y sum I(x) ∩ I(y), by Lemma 2.6.3. Therefore  x = y, by virtue of (Usum ). From Lemmas 2.6.5 and 2.6.6 we obtain the following result: 18 The proven sentence says that in the class

the relation sum is included in the s POS (resp. QOS)  relation fu satisfying the following condition: x fu S iff O(x) = O[S]. The relation fu is another explication of the concept of being a collective set. In Sect. 2.12 of this chapter we will show that in the class s PPOS we obtain sum = fu.

2.6 Mereological Sum

47

Theorem 2.6.7 (Pietruszczak 2000, 2018) In the class s POS (resp. QOS) sentences (Usum ) and (ext  ) are equivalent. We therefore have the following equivalences concerning the theories (classes of structures) appearing in Diagram 2.1: s POS+(Usum )

= s POS+(ext  ),

s POS+(Usum )+(ext  +) = s POS+(ext  )+(ext  +).

We think that condition (Usum ) should be binding in any theory worthy of the name “theory of parthood”, if we recognise that (df sum) is a correct definition of collective classes. In Diagram 2.1 we have only three theories: s POS+(ext  ), s POS+(ext  ) +(ext  +) and s PPOS. We will show that the third theory has the greatest claim to the title of an “existentially neutral” theory of parthood. Conditions (Usum ) and (2.6.5) directly entail that {x} is the only singleton whose mereological sum is x, i.e.19 :   ∀x,y∈U x sum {y} ⇒ x = y .

(Ssum )

Since (2.6.5) depends on (r ), we have the following: Lemma 2.6.8 (Ssum ) follows from (Usum ) and (r ). Observe that from Theorem 2.6.7 and Lemma 2.4.8 we can obtain the following: Corollary 2.6.9 In the class s POS, condition (Usum ) entails condition (WSP). A simpler proof of the above corollary can be obtained from Lemma 2.6.10(ii) and the fact that condition (Usum ) entails condition (Ssum ) Lemma 2.6.10 Let us admit  as a primitive relation in the set U and let us use definitions (df sum), (df ) and (df ). Then: (i) (WSP) entails the conjunction of sentences (irr  ) and (Ssum ). (ii) Sentences (irr  ) and (Ssum ) entail sentence (WSP). Proof Ad (i): We get (irr  ) on the basis of Lemma 2.3.1(i). Next, assume for a contradiction that for some x, y ∈ U we have: x sum {y} and x = y. Then y  x, on the basis of (df sum) and (df 1 ). Hence and from (WSP), for some z we have z  x and z  y. We therefore have a contradiction because z  x and x sum {y} entails z  y. Ad (ii): Assume for a contradiction that for some x, y ∈ U : (a) y  x and (b) P(x) ⊆ O(y). We will show that (c): x sum {y}. To see this, if z  x then z  y, by (b). Furthermore, from (a) we have y  x. Hence x  y, by virtue of (r ). Therefore I(x) ⊆ O(y). From (c) and (Ssum ) we have x = y. This and (irr  ) entail a contradiction from (a).  19 Henceforth,

we will omit ‘and from our definitions’ from statements of entailments.

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2 “Existentially Neutral” Theories of Parts

We therefore have the following equivalences relating to the theories (classes of structures) appearing in Diagram 2.1: s POS+(Ssum )

= (t )+(WSP),

= (t )+(ext  )+(WSP), s POS+(ext  +)+(Ssum ) = (t )+(ext  +)+(WSP). s POS+(ext  )+(Ssum )

We also have another lemma which is an enriched version of Lemma 2.6.10. This new Lemma 2.6.12 will come in handy in the next chapter. First we will prove: Lemma 2.6.11 Let  be a transitive relation in a set U and let us use definition (df ) and the operators I and O. Then for arbitrary S ∈ 2U and x, y ∈ U : if S ⊆ I(y) and I(x) ⊆



O[S], then I(x) ⊆

 s∈I(y)

O(s) ⊆ O(y).

   Proof Let S ⊆ I(y) and I(x) ⊆ O[S], i.e., I(x) ⊆ s∈S O(s). Hence I(x)⊆ s∈I(y) O(s). However, we have ∀s∈I(y) O(s) ⊆ O(y),  by virtue of (2.1.10), that follows from  (t ), (df ) and (df ). Therefore, I(x) ⊆ s∈I(y) O(s) ⊆ O(y). Lemma 2.6.12 Let us admit  as a primitive relation in a set U and let us use definitions (df sum), (df 1 ) and (df ). Then: (i) (ii) (iii) (iv)

(r ) and (Ssum ) entail (WSP). (t ) and (Ssum ) entail (WSP). (r ), (t ) and (Ssum ) entail (antis ). (antis ), (t ) and (SSP) entail (Usum ).

Proof Ad (i): Assume for a contradiction that for x, y ∈ U : (a) y  x and (b) P(x) ⊆ O(y). Then (c) y  x and (d) x = y, by virtue of (a) and (df 1 ). We will show that (e): x sum {y}. To see this, if z  x then z  y, by (b). Furthermore, from (c) and (r ) we have x  y. Therefore I(x) ⊆ O(y). From (e) and (Ssum ) we have x = y, which contradicts (d). Ad (ii): Assume for a contradiction that for some x, y ∈ U : (a) y  x and (b) P(x) ⊆ O(y). Then (c) y  x and (d) x = y, by virtue of (a) and (df 1 ). Furthermore, from (a) and (b) we have y  y, i.e., for some z ∈ U : z  y. Hence z  x as well, by virtue of (c) and (t ). Therefore x  y. This and (b) give I(x) ⊆ O(y), which together with (c) shows that x sum {y}. Therefore, by virtue of (Ssum ), we have x = y, which contradicts (d). Ad (iii): Assume that x  y and y  x. We will show that y sum {x}. To this end, take an arbitrary z such that z  y. We have z  x, by (t ). Hence z  x, by virtue of (r ). Furthermore, x sum {x}, by virtue of (r ). Therefore, x = y, by virtue of (Ssum ). Ad (iv): Assume that x sum S and y sum S. We will show that x  y and y  x, which will give us x = y,  by virtue of (antis ). Our assumption gives us: S ⊆ I(x) ⊆  O[S] and S ⊆ I(y) ⊆ O[S]. Hence, in the light of Lemma 2.6.11, we have I(x) ⊆ O(y). Hence x  y, by virtue of (SSP ).

2.6 Mereological Sum

49

  Moreover, we have I(y) ⊆ s∈I(x) O(s), since S ⊆ I(x) and I(y) ⊆ O[S]. Hence, as above, we show that y  x. Thus, we have: x  y and y  x. Hence  x = y, by (antis ).

2.7 The Monotonicity Principle for Mereological Sum Assuming the antisymmetry of the relation , we see that the following monotonicity principle for the relation sum   ∀S1 ,S2 ∈2U ∀x,y∈U S1 ⊆ S2 ∧ x sum S1 ∧ y sum S2 =⇒ x  y

(Msum )

is stronger than the uniqueness of sum, i.e., principle (Usum ). We will now examine (Msum ) and another version of it. Notice that using (df  sum), principle (Msum ) can be written as follows:   O[S1 ] ∧ ∀S1 ,S2 ∈2U ∀x,y∈U S1 ⊆ S2 ∧ S1 ⊆ I(x) ∧ I(x) ⊆   S2 ⊆ I(y) ∧ I(y) ⊆ O[S2 ] =⇒ x  y . It turns out that we can take S1 = S2 and remove the first, third and fourth conjuncts from the antecedent of the above implication. In this way, we will obtain the following new version of the monotonicity principle    ∀S∈2U ∀x,y∈U I(x) ⊆ O[S] ∧ S ⊆ I(y) =⇒ x  y .

 (Msum )

which is equivalent to (Msum ) in the class QOS. Furthermore, we will prove that both versions of the monotonicity principle are equivalent to (SSP) in the class QOS (cf. Pietruszczak 2000, pp. 75–76), and (Pietruszczak 2018, pp. 90–91). Finally, we  ) and the monotonicity principle (m≤ ) from will show the connection between (Msum Appendix B (see Sect. B.3.2). It is easy to see that the following holds:  Lemma 2.7.1 (Msum ) follows from (Msum ).

Proof Assume that S1 ⊆ S2 , x sum S1 and y sum S2 . Then, by (df  sum), we have:   ).  I(x) ⊆ O[S1 ] ⊆ O[S2 ] and S2 ⊆ I(y). Hence x  y, by virtue of (Msum In the class QOS we also have the converse entailment to that given by the lemma above.  Lemma 2.7.2 (Msum ) follows from (r ), (t ) and (Msum ).  Proof Let S ⊆ I(y) and I(x) ⊆ O[S]. Then, in the light of Lemma 2.6.11 (derived from (t )), we have I(x) ⊆ O(y). Hence, by using Lemma 2.6.3 (derived from (r ) and (t )), we have x sum I(x) ∩ I(y). Furthermore, from (r ), we have y sum I(y). Therefore, by virtue of (Msum ), we get x  y. 

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2 “Existentially Neutral” Theories of Parts

 Lemma 2.7.3 (Msum ) follows from (t ) and (SSP).  Proof Let S ⊆ I(y) and I(x) ⊆ O[S]. Then, in the light of Lemma 2.6.11, we  ).  have I(x) ⊆ O(y). Hence x  y, by virtue of (SSP ). Thus, we obtain (Msum

We also have a kind of converse entailment:  ). Lemma 2.7.4 (SSP) follows from (r ) and (Msum

Proof Let I(x) ⊆ O(y). But y ∈ I(y), by (r ). Hence we have I(x) ⊆  ). Thus, we obtain (SSP ). {y} ⊆ I(y). So x  y, by (Msum



O[{y}] and 

From the above lemmas and Lemma 2.4.6(iii) we get: Corollary 2.7.5 From (r ) and (t ) we obtain the equivalence of the following con ). ditions: (SSP), (ext  +), (Msum ) and (Msum Therefore we obtain the following:  Theorem 2.7.6 (SSP), (ext  +), (Msum ), (Msum ) are equivalent in the class QOS. In consequence, they are equivalent also in the class s POS with our definitions.

We therefore have the following identities holding between the classes of structures that interest us: s PPOS

 = s POS+(ext  +) = s POS+(Msum ) = s POS+(Msum ).

As we have just shown in the previous section, the uniqueness of the relation sum, expressed by condition (Usum ), is equivalent to the extensionality principle (ext  ). It is clear that condition (Usum ) follows from (antis ) and (Msum ). We have therefore obtained another proof of the fact that (Usum ) is binding in the class s PPOS. We therefore have: s POS+(Msum ) s POS+(Usum )+(ext  +) s POS+(Usum ).  Remark 2.7.1 Relationships hold between condition (Msum ) and the monotonicity principle (m≤ ) given in Appendix B, and between Lemmas 2.7.3 and B.3.7. The second lemma says that (m≤ ) follows from (t≤ ) and (s-pol≤ ). Let us observe, however, that Fact B.3.5(ii) and Lemma B.3.4(v) say that if a partially-ordered set does not have a zero and is semi-polarised, then it is polarised as well; and, furthermore, ⊥ = . We remember too that the relation ⊥ coincides with the relation  (see Sect. 2.1.4). We will now analyse condition (m≤ ) in the case where ≤ := . Obviously, in degenerate structures this condition trivially holds, by virtue of (r ). Assuming that condition (0) is satisfied, we will concern ourselves just with non-degenerate structures without zero. Under this assumption, since ¬∃u∈U ∀v∈U u  v, we get from (m≤ ) the following:



 S ⊆ I(y) ∧ ¬∃u∈U (u  x ∧ ∀s∈S s  u) =⇒ x    ∀S∈2U ∀x,y∈U S ⊆ I(y) ∧ ∀u∈U (u  x ⇒ ∃s∈S s  u) =⇒ x  ∀S∈2U ∀x,y∈U

 i.e., we obtain condition (Msum ).

 y ,  y , 

2.8 Basic Differences Between the Relations of Supremum and Mereological Sum

51

2.8 Basic Differences Between the Relations of Supremum and Mereological Sum We have written about the relation is a least upper bound (supremum) of in detail in Appendix B.2.3. In the particular case where the relation which partially orders the set U is the relation  (i.e., if we put ≤ :=  in point Appendix B.2.3), we will use the shortened form ‘sup’ of the expression ‘sup≤ ’. Let us recall therefore the definition of the relation sup (included in U × 2U ) writing using ‘’: x sup S :⇐⇒ ∀s∈S s  x ∧ ∀u∈U (∀s∈S s  u ⇒ x  u).

(df sup)

If, however, we use the operator I, the definition above takes the following form: x sup S ⇐⇒ S ⊆ I(x) ∧ ∀u∈U (S ⊆ I(u) ⇒ x  u).

(df  sup)

To begin with, recall that by applying (r ) we obtain: x sup {x},

(2.8.1)

x sup I(x),

(2.8.2)

from (antis ) we get the uniqueness of the relation sup, i.e.:   ∀x,y∈U ∀S∈2U x sup S ∧ y sup S =⇒ x = y ,

(Usup )

and from (r ) and (antis ) we have: ∀x,y∈U (x sup {y} =⇒ x = y).

(Ssup )

Analysing the basic differences between sum and sup, we notice, first of all, that in the class s POS (resp. POS), the “sup-counterparts” of (2.6.2) and (2.6.7) do not have to hold; that is, that the following sentences can be false: ‘¬∃x∈U x sup ∅’ and ‘∀x∈U (P(x) = ∅ ⇒ x sup P(x))’. In Model 2.1 we have: 1 sup ∅ and P(2) = {1}, but ¬ 2 sup {1}. We remember, however, that in Model 2.1, sentence (0) is false, which is supposed to be in force in all the theories of parts of interest to us. Secondly, let us observe that in the class s POS (resp. POS) the “sum-counterparts” of (Ssup ) and (Usup ) do not have to hold; that is, sentences (Ssum ) and (Usum ) can be false. To see this, we know from Theorem 2.6.7 and Lemma 2.6.10 that in the class s POS those last sentences are equivalent to (WSP) and (ext  ), respectively; that is, they are in force only in the classes s POS+(WSP) and s POS+(ext  ).

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2 “Existentially Neutral” Theories of Parts

Thirdly, it is evident that the “sup-counterpart” of the monotonicity principle (Msum ) can be obtained directly from (df sup). Principle (Msum ) holds, however, only in the class s PPOS.20

2.9 Supplementation Principles and Connections Between the Relations of Mereological Sum and Supremum In theories in which condition (WSP) does not hold there are no interesting connections between the relations sum and sup. In such theories we have only a pair of similar sentences: (2.6.5) and (2.6.6); (2.8.1) and (2.8.2). It turns out, however, that both supplementation principles are equivalent to certain connections holding between the relations sum and sup. What’s more, the weak supplementation principle (WSP) is then equivalent to the following: if there exists a mereological sum and a least upper bound of a given set, then they are equal to one another. Lemma 2.9.1 (Pietruszczak 2000, 2018) In the class (t )+(WSP) the following condition holds: ∀S∈2U ∀x,y∈U (x sup S ∧ y sum S =⇒ x = y).

()

Proof Let (a) x sup S and (b) y sum S. From (b) we have S ⊆ I(y). Hence x  y, by (a) and (df  sup). Assume for a contradiction that x = y. Therefore x  y. Hence, by virtue of (WSP), for some v ∈ M we have (c) v  y and (d) v  x. From (b) and (c), by virtue of (df sum), there are z ∈ S and u ∈ U such that u  v and u  z. Since z  x, by virtue of (a), from (t ) we have u  x. Therefore v  x, which contradicts condition (d).  Lemma 2.9.2 In the class s POS (resp. QOS), condition (WSP) follows from (). Proof Assume for a contradiction that y  x and P(x) ⊆ O(y). Then, by virtue of (t ), we have I(y) = I(x) ∩ I(y). Furthermore, by Lemma 2.6.3, we have x sum I(x) ∩ I(y) = I(y). However, y sum I(y), by (r ). Therefore x = y, by virtue of (),  which contradicts (irr  ). From the last two lemmas we obtain: Theorem 2.9.3 In s POS (resp. QOS) conditions (WSP) and () are equivalent. We obtain, therefore, the following equivalence of classes of structures: (t )+(WSP) = s POS+(WSP) = s POS+(). The above result can be strengthened in the class s PPOS. That is, we have the following connection holding between the relations sum and sup. 20 Obviously, to reject

(Ssum ), (Usum ) and (Msum ) in the class s POS, Model 2.1 suffices. To see this, 1  0 and thanks to (2.6.5) and (2.6.7): 0 sum {0} and 1 sum {0}.

2.9 Supplementation Principles and Connections Between the Relations …

53

 Lemma 2.9.4 (Pietruszczak 2000, 2018) (Msum ) entails the following inclusion:

sum ⊆ sup.

(†)

 Proof Let x sum S, i.e., (a) S ⊆ I(x) and (b) I(x) ⊆ O[S]. Take an arbitrary y  ). Taking this such that (c) S ⊆ I(y). Then x  y, since we have (b), (c) and (Msum into account along with (a), by virtue of (df sup), we have x sup S.  Theorem 2.9.5 (Pietruszczak 2000, 2018) (i) (†) follows from (t ) and (SSP). (ii) (SSP) follows from (r ), (t ) and (†). Proof Ad (i): By virtue of Lemmas 2.7.3 and 2.9.4. Ad (ii): For the proof of condition (SSP), assume the inclusion I(x) ⊆ O(y). Then x sum I(x) ∩ I(y), by virtue of Lemma 2.6.3. Hence, by virtue of (†), we have x sup I(x) ∩ I(y). Since I(x) ∩ I(y) ⊆ I(y), from the definition of the relation sup, we have x  y.  We therefore have the following identities of the classes that interest us: s PPOS

= s POS+(†).

Remark 2.9.1 Since sentence () is in force in class (t )+(WSP), then it is also in force in the class s PPOS. In this second class, however, sentence () has a simple proof: if x sum S and y sup S, then x sup S, by (†). Hence x = y, by virtue of  condition (Usup ), which we obtained from (antis ). Remark 2.9.2 (i) We have the following relationship between condition (†) (i.e., the inclusion sum ⊆ sup) and condition (∗) , analysed in Appendix B. Since conditions (∗) and (m≤ ) have similar antecedents in their implications, we may therefore make use of the observation noted in Remark 2.7.1. We may thus reduce condition (∗) in non-degenerate structures (i.e., those without zero) to the following forms: 

 S ⊆ I(x) ∧ ¬∃u∈U (u  x ∧ ∀s∈S s  u) =⇒ x sup   ∀S∈2U ∀x∈U S ⊆ I(x) ∧ ∀u∈U (u  x ⇒ ∃s∈S s  u) =⇒ x sup ∀S∈2U ∀x∈U

 S ,  S ,

so we obtain the inclusion: sum ⊆ sup. (ii) A relationship also obtains between Theorem 2.9.5 and Lemma B.3.9.



Let us now observe that, on the strength of (2.6.1) and (†), in s PPOS we have:   ∀S∈2U ∀x∈U x sum S =⇒ S = ∅ ∧ x sup S .

(2.9.1)

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2 “Existentially Neutral” Theories of Parts

We will show that neither the converse inclusion to (†) nor the converse implication to the one in (2.9.1), which is weaker than it, do not have to hold in s PPOS. We will introduce some conditions which will be of use to us below: sup ⊆ sum,   ∀S∈2U ∀x∈U S = ∅ ∧ x sup S =⇒ x sum S .

(‡) (‡∅ )

Obviously, (‡∅ ) follows from (‡). The converse entailment, however, does not hold. The difference between the above conditions is connected with degenerate structures. Lemma 2.9.6 Condition (‡∅ ) holds in all degenerate structures. Proof If U = {u}, then u sup {u}, u sum {u} and {u} is the single non-empty subset of the set U .  Lemma 2.9.7 (i) If a structure from s POS (resp. POS) has a zero then sup  sum and the structure is non-degenerate. So in s POS+(‡) condition (n-d) holds. (ii) No structure from s POS+(‡) has a zero and so none of these structures is degenerate. Furthermore, we have (see Model 2.7 for a proof ): s POS+(‡)

 s POS·· = s POS+(n-d).

Proof If a structure from s POS (resp. POS) has the zero 0, then 0 sup ∅ (see (B.2.24). However, ¬0 sum ∅, by (2.6.2). Therefore, sup  sum. Furthermore, if a structure is degenerate, i.e., it has only one element, then this element is its zero.  We will now show that conditions (‡) and (‡∅ ) are too strong for s PPOS (resp. PPOS). In the next point we will explain, however, that these conditions “do not suit existentially neutral” theories of parthood. Fact 2.9.8 Condition (‡∅ ), and hence a fortiori (‡), does not follow from the axioms of s PPOS (resp. PPOS). Proof To the class s PPOS belongs the structure from Model 2.7. In this structure, the set {1, 2} has an upper bound equal to 123 but there is no mereological sum of  the elements 1 and 2. Therefore (‡∅ ) is false; a fortiori, so is (‡) too. 123

1

2

3

Model 2.7 (WSP), (SSP), (c∃), (c∃pair sup), (∃pair sup) and (∃sup∅ ) hold, but (‡∅ ), (‡), (c∃pair fu), (c∃pair sum), (∃pair sum), (WSP+), (SSP+) and (∃sum) do not hold

2.10 Greatest Lower Bound Versus Mereological Sum

55

2.10 Greatest Lower Bound Versus Mereological Sum In the previous sections, we have seen how a complicated dependency holds between the relations of mereological sum and least upper bound (supremum). By contrast, the relationship between a mereological sum and a greatest lower bound (infimum) is simpler. To see this, we will show that in the class s POS (resp. POS) the greatest lower bound of a given set is the mereological sum of all lower bounds of that set, i.e., all common ingredients of the elements of that set. We provide more details about the relation is a greatest lower bound of (infimum) in Appendix B.2.3. When the set U is partially ordered by the relation  (i.e., when in Sect. B.2.3 we put ≤ := ) then instead of ‘inf≤ ’, let us adopt the shortened form ‘inf’. Let us then recall the definition of the relation inf expressed with the help of ‘’, i.e., that inf is a binary relation included in U × 2U such that:  x inf S :⇐⇒ x is the greatest element of the set I[S], ⇐⇒ ∀s∈S x  s ∧ ∀u∈U (∀s∈S u  s ⇒ u  x),

(df inf)

  where I[S] the set of all lower bounds of S, i.e., I[S] = {u ∈ U : ∀s∈S u  s} (cf. Remark 2.1.2 on Sect. 2.1.6). From (antis ) we obtain the uniqueness of the relation inf, i.e.:   ∀x,y∈U ∀S∈2U x inf S ∧ y inf S =⇒ x = y .

(Uinf )

For arbitrary x ∈ U and S ∈ 2U , by exploiting the reciprocal definability of the relations of greatest lower bound and least upper bound, (B.2.15), and that the greatest element in a given set is its mereological sum, (2.6.3), we obtain: x inf S :⇐⇒ x is the greatest element of the set  ⇐⇒ x sup I[S],  ⇐⇒ x sum I[S].



I[S], (2.10.1)

 Hence in the particular case for S := ∅, since I[∅] = U , we have (cf. (2.6.4) and (B.2.23)): x inf ∅ ⇐⇒ x sum U, (2.10.2) ⇐⇒ ∀u∈U u  x ⇐⇒ x is the unity . When, however, S := {s1 , . . . , sn } (for n > 0) we have: x inf {s1 , . . . , sn } ⇐⇒ x is the greatest element of the set I(s1 )  · · ·  I(sn ), ⇐⇒ x sup I(s1 )  · · ·  I(sn ), ⇐⇒ x sum I(s1 )  · · ·  I(sn ).

(2.10.3)

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2 “Existentially Neutral” Theories of Parts

2.11 The Choice of a Proper Theory of Parthood The problem of “existentially involved” theories of of parthood was raised in Sect. 1.5. In pursuit of that problem, let us recognise that (2.6.5)–(2.6.7) are the only theses concerning the existence of mereological sums which it is possible to derive in “existentially neutral” theories. Our conviction comes from the following line of reasoning. In accordance with (df sum), in order to show that x sum S, it is necessary inter alia to prove the inclusion S ⊆ I(x). It is possible do this without special assumptions, however, only in the cases where S := {x}, S := I(x) and S := P(x) = ∅. And one must also prove the second inclusion I(x) ⊆ O[S] besides. As we explained earlier, in each theory of parts, the uniqueness of the relation sum (i.e., (Usum )) is a necessary requirement. Therefore, we will take into consideration three theories: s POS+(ext  ), s POS+(ext  )+(ext  +) and s PPOS (remember that condition (SSP) entails (Usum )). The following question now arises: are these theories not too strong for “existentially neutral” theories? The answer to the above question is no. This is so even in the strongest of the three theories, which is s PPOS. For principle (SSP) is existentially neutral. Although it postulates the existence of some object, that object is tied up with the “very being” of the relation is a part of. That (SSP) does not postulate the existence of mereological sums is evident from the fact that (SSP) is equivalent to the extensionality principles  ). And these (ext  +) and (ext  +) and the monotonicity principles (Msum ) and (Msum principles—if one looks at their form—do not postulate the existence of mereological sums. Let us add that in the theory s PPOS the two explications of the concept collective class are equivalent; i.e., we have the following identity of relations: sum = fu. We have discussed this in footnote 18 on Sect. 2.6.2 and we will discuss it in more detail in the next section. Therefore, this theory has the “greatest claim” to the title of being a properly “existentially neutral” theory. Let us now turn our attention to the fact that an “existential axiom” does not have to explicitly postulate the existence of a mereological sum. Examples of such “axioms” are for instance (‡∅ ) and (‡). We will show that they implicitly postulate the existence of mereological sums. Let us consider the structure from Model 2.7. Since 123 sup {1, 2}, 123 sup {2, 3} and 123 sup {1, 3}, the presence in the theory of the sentence (‡∅ ) (resp. (‡)) would force the existence of mereological sums of the following pairs {1, 2}, {2, 3} and {1, 3}. Therefore, by enriching the theory with (‡∅ ), we force its “closure” with these sums. We therefore obtain the structure presented in Model 2.8. We see therefore that the addition of (‡∅ ) (resp. (‡)) as an axiom means that two objects which have some upper bound must automatically have a mereological sum too. In the example we used earlier, the body of a person is the upper bound of his arms (with respect to the relation is an ingrediens of ), and therefore there would have to exist a part of the body which would be the mereological sum of both hands.

2.11 The Choice of a Proper Theory of Parthood

57

123

12

1

13

2

23

3

Model 2.8 (†), (‡∅ ), and (‡) hold

We will say more about the role of (‡) and (‡∅ ) in the theory of parthood in the next chapter. We see, however, that the theory s PPOS+(‡∅ ) does not deserve the title of an “existentially neutral” theory of parthood. We should not be driven just by the lack of an existential quantifier in (‡∅ ) referring directly to mereological sums.

2.12 Mereological Fusion 2.12.1 Another Definition of a Collective Class. The Relation Is a Fusion of In Chap. X of his paper “On the foundations of mathematics”, Le´sniewski (1931) favours another explication of the concept collective class of some objects. Leonard and Goodman (1940) express this explication in set-theoretic language as the relation is a fusion of all elements of a given distributive set. In (Leonard and Goodman 1940) they signify this relation with ‘fu’.21 Le´sniewski (1931) adopts as a primitive concept the relation is exterior to, i.e.  included in U × U . This relation is also a primitive concept in the system presented in (Leonard and Goodman 1940).22 Definition (C) of the expression ‘P is a class of objects a’ from (Le´sniewski 1931, p. 142) in terminology from (Leonard and Goodman 1940) may be written in the following way, as the definition of the relation fu included in U × 2U :   x fu S :⇐⇒ ∀y∈U y  x ⇔ ∀s∈S s  y .

(df fu)

Let us observe that from (irr  ) we obtain: ¬∃x∈U x fu ∅.

(2.12.1)

From x fu ∅ we obtain the false equivalence: x  x ⇔ ∀s∈∅ s  x.

21 In

(Leonard and Goodman 1940), the relation fu is also called the relation of sum-individual.

22 In (Leonard and Goodman 1940), the authors make reference to (Le´sniewski 1931) several times.

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2 “Existentially Neutral” Theories of Parts

Other basic properties of the relation fu are obtained from the basic properties of sum. Using (=−), it suffices to write (df fu) with the help of ‘’ or ‘O’:   x fu S ⇐⇒ ∀y∈U y  x ⇔ ∃s∈S s  y ,  x fu S ⇐⇒ O(x) = O[S].

(df  fu)

Therefore, from Lemma 2.6.4 we get: Lemma 2.12.1 The inclusion sum ⊆ fu follows from (t ). Hence and from (2.6.5), (2.6.6) and (2.6.7) we obtain: ∀x∈U x fu {x},

(2.12.2)

∀x∈U x fu I(x),   ∀x∈U P(x) = ∅ =⇒ x fu P(x) .

(2.12.3) (2.12.4)

Condition (2.12.3) is easily gained in another way. Thanks to the same definition (df ) we have (s ), and hence for S := {x} the left-hand side of definition (df fu) is transformed into a tautology. Lemma 2.12.2 Condition (ext  ) entails the following condition:   ∀x,y∈U ∀S∈2U x fu S ∧ y fu S =⇒ x = y ,

(Ufu )

which talks of the uniqueness of fu. In consequence, (Ufu ) follows from conditions (r ), (t ) and (Usum ).  Proof Assume that x fu S and y fu S, i.e., O(x) = O[S] = O(y), by virtue of (df  fu). Therefore x = y, by virtue of (ext  ). For the rest, we use Lemma 2.6.6.  Lemma 2.12.3 Conditions (t ) and (Ufu ) entail (Usum ). Proof Assume that x sum S and y sum S. Then, by virtue of Lemma 2.12.1, we have x fu S and y fu S. Hence x = y, thanks to (Ufu ).  By applying Theorem 2.6.7 and Lemmas 2.12.2 and 2.12.3 we obtain: Theorem 2.12.4 (Pietruszczak 2000, 2018) In the class s POS (resp. QOS) conditions (Usum ), (ext  ) and (Ufu ) are equivalent. Thus, s POS+(Usum )

= s POS+(ext  ) = s POS+(Ufu ),

QOS+(Usum ) = QOS+(ext  ) = QOS+(Ufu ), s POS+(Usum )+(ext  +)

= s POS+(ext  )+(ext  +) = s POS+(Ufu )+(ext  +).

2.12 Mereological Fusion

59

Obviously, we can also consider a sentence which says that {x} is the only singleton whose mereological fusion is x:   ∀x,y∈U x fu {y} ⇒ x = y .

(Sfu )

However, in accordance with (df  fu), Fact 2.12.5 Sentences (Sfu ) and (ext  ) are (definitionally) equivalent. Proof After expanding (df  fu) for the set {y} we obtain:   x fu {y} ⇐⇒ ∀z∈U z  x ⇔ z  y . Therefore (Sfu ) and (ext  ) have implications with equivalent antecedents.



2.12.2 Identity of Mereological Sum and Fusion From the theorem below and Lemma 2.12.1 it will follow that in the class s PPOS (resp. QOS+(SSP)) we have: sum = fu. Theorem 2.12.6 (Pietruszczak 2000, 2018) (i) The inclusion fu ⊆ sum follows from (t ) and (SSP). (ii) Condition (SSP) follows from (r ), (t ) and the inclusion fu ⊆ sum. Proof (i) Lemma 2.3.7 says that (r ) follows from (t ) and (SSP). Therefore, by  virtue of Lemma 2.4.6, we have (ext +). Assume that x fu S, i.e. (a): O(x) = O[S]. Hence ∀y∈U (∃s∈S s  y ⇒ y  x), which is equivalent to: ∀s∈S ∀y∈U (y  s ⇒ y  x). From this and from (ext  +) we have (b): ∀z∈Z z  x. Furthermore, from (a) we obtain: ∀y∈U (y  x ⇒ ∃s∈S s  y). Hence and from (2.1.9) we have: ∀y∈U (y  x ⇒ ∃s∈S s  y). And from this and from (b) we get: x sum S. (ii) For the proof of (SSP), let us assume the inclusion I(x) ⊆ O(y). Then, by virtueof (2.1.11 ), we have O(x) ⊆ O(y), i.e., O(y) = O(x)  O(y). Hence O(y) = O[{x, y}], i.e., y fu {x, y}. Therefore, by virtue of our assumption, it is also the case that y sum {x, y}. And hence it follows that x  y.  We have therefore got the following equivalences of classes of structures: s PPOS

= s POS+(fu⊆sum) = s POS+(fu=sum).

With reference to the last theorem, the fact below confirms that condition (SSP) is essentially stronger than the conjunction of conditions (ext  +) and (Usum ) (resp. (ext  )).

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2 “Existentially Neutral” Theories of Parts

Fact 2.12.7 The inclusion fu ⊆ sum does not hold in the class s POS+(Usum )+ (ext  +). Proof Model 2.4 belongs  to the class s POS+(Usum )+(ext  +), yet fu  sum. To see this, O(1234) = U = O[U ], i.e. 1234 fu U , although ¬1234 sum U . 

2.13 Theories with an “Empty Element” As we noted in point 1.3.5, when we consider universes composed of such abstract objects as pieces of information or situations, we can admit the existence of an “empty object” (empty information, empty situation). In addition, where theories concerning spatial objects are concerned, one sometimes, for “technical reasons”, admits the existence of a zero which then plays the role of an “empty element”. We will explain why there is no possibility of constructing a “unitary theory” which would embrace both cases below: 1. where principle (0) is in force and the only degenerate element of the structure is not treated as an “empty element”; 2. where there exists an “empty element” which is a part of all others; i.e., which is the zero. In the first case, the relation  is exterior to, defined by condition (df ) and corresponding to the relation ⊥, is irreflexive in all structures under consideration. Therefore, this is also so in each degenerate structure, i.e., one composed of the zero itself. Furthermore, the relation of overlapping , defined by condition (df ), is reflexive. Therefore, the zero of a degenerate structure will overlap itself. In the second case, the relation  is exterior to is supposed to be such that the zero is exterior to each element of the structure, including itself.23 Therefore, the relation  will correspond to the relation , which we have used in Appendix B for partially-ordered sets. Therefore, relation  must satisfy the following condition:   ∀x,y∈U x  y ⇐⇒ ¬∃u∈U (u = 0 ∧ u  x ∧ u  y) . Therefore, for each x ∈ U , x  0, including 0  0 too. Relation  will be the complement of the relation  and should satisfy the following condition:   ∀x,y∈U x  y ⇐⇒ ∃u∈U (u = 0 ∧ u  x ∧ u  y) . Therefore, for each x ∈ U : ¬x  0, including ¬0  0 too. When the second approach is taken for “technical reasons”, then the structure under analysis is not degenerate. Therefore, the set U+ := U \ {0} is created and the relation + :=  |U+ =   (U+ × U+ ) (resp. + :=  |U+ =   (U+ × U+ )) is used. We then have: 23 Similar

to how the empty set ∅ is disjoint from every other set, including itself.

2.13 Theories with an “Empty Element”

61

  ∀x,y∈U+ x  y ⇐⇒ ¬∃u∈U+ (u + x ∧ u + y) ,   ∀x,y∈U+ x  y ⇐⇒ ∃u∈U+ (u + x ∧ u + y) . Therefore, in the context of the set U , we obtain counterparts to our definitions (df ) and (df ). Further examination of such theories will correspond to what we have done in Appendix B for theories with zero. If we are interested in “existentially neutral” theories of parts, then we adopt the analysis of Appendix B.3, and, in particular, of Appendix B.3.3. If, however, we are interested in “existentially involved” theories, then we need just take a ready-made “template” in the form of one of the theories presented in Appendix B such as the theories of Grzegorczykian lattices, Boolean lattices or complete Boolean lattices. If these “templates” are too strong for us as theories of parthood, then we must construct our own theory analogous to those which we will be examining in the opening parts of the next chapter. Such constructions can be carried out by using one of the “templates” or by drawing upon our earlier analyses. We will therefore henceforth not be concerned with them.

References Casati, R., & Varzi, A. C. (1999). Parts and places. Cambridge: The MIT Press. Chisholm, R. M. (1992/93). Spatial continuity and the theory of part and whole. A Brentano study. Brentano Studien, 4, 11–23. Leonard, H. S., & Goodman, N. (1940). The calculus of individuals and its uses. Journal of Symbolic Logic, 5(2), 45–55. http://dx.doi.org/10.2307/2266169 Le´sniewski, S. (1931). O podstawach matematyki (On the foundations of mathematics). Przeglad ˛ Filozoficzny, 34, 142–170. Libardi, M. (1990). Teorie delle parti e dell’intero. Mereolologie estensionali (Theories of parts and the whole. Extensional mereologies). Vol. II of Quaderni, Aprile–Dicembre. Pietruszczak, A. (2000). Metamereologia (Metamereology). Toru´n: The Nicolaus Copernicus Univesity Press. English version (2018): Metamereology. Toru´n: The Nicolaus Copernicus Univesity Scientific Publishing House. https://doi.org/10.12775/3961-4 Pietruszczak, A. (2018). Metamereology. Toru´n: The Nicolaus Copernicus Univesity Scientific Publishing House. English version of (Pietruszczak, 2000). https://doi.org/10.12775/3961-4 Simons, P. (1987). Parts. A study in ontology. Oxford: Oxford University Press. http://dx.doi.org/ 10.1093/acprof:oso/9780199241460.001.0001 Tarski, A. (1929). Les fondements de la géométrie des corps (Foundations of the geometry of solids). Ksiega ˛ Pamiatkowa ˛ Pierwszego Zjazdu Mate (pp. 29–30). Tarski, A. (1956). Fundations of the geometry of solids. In J. H. Woodger (Ed.), Logic, semantics, metamathematics. Papers from 1923 to 1938 (pp. 24–29). Oxford: Oxford University Press. English version of (Tarski, 1929).

Chapter 3

“Existentially Involved” Theories of Parts

Theories bearing this title are those for which at least one defining condition forces the existence of mereological sums. On Sect. 2.6.2 we said that a theory deserving the name ‘theory of parthood’ is one in which condition (Usum ) is in force. In Diagram 2.1 we have just three such theories (classes)1 : s POS+ (Usum ), s POS+ (Usum ) (ext  +) and s PPOS. Theories which are extensions of the third in the list above we will call polarised. It is to such theories that the present chapter is in the main devoted.

3.1 Mereological Strictly Partially-Ordered Sets Let us begin with theories whose defining conditions feature no existential quantifier referring directly to mereological sums. As we explained in Sect. 2.11 the classes s PPOS+ (‡∅ ) (= s POS+ (†)+ (‡∅ )) and s PPOS+ (‡) are such theories. Any strictly partially-ordered set satisfying condition (‡∅ ) will be called a mereological strictly partially-ordered set. The class of all these structures we will henceforth symbolise as: s MPOS. Thus, we put: s MPOS

:= s PPOS+(‡∅ ).

From Fact 2.9.8 we get: s MPOS

 s PPOS.

Furthermore, in the light of Theorem 2.9.5, condition (†) holds in the class s PPOS; i.e., we have sum ⊆ sup. Hence in the class s MPOS the following condition holds: 1 Let

us remember that Theorem 2.6.7 says that, in the class s POS (resp. QOS), sentences (Usum ) and (ext  ) are equivalent.

© Springer Nature Switzerland AG 2020 A. Pietruszczak, Foundations of the Theory of Parthood, Trends in Logic 54, https://doi.org/10.1007/978-3-030-36533-2_3

63

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3 “Existentially Involved” Theories of Parts

  ∀S∈2U ∀x∈U x sum S ⇐⇒ S = ∅ ∧ x sup S .

(sum-sup)

Thus, we have: s MPOS

= s POS+(sum-sup) = s POS+(†)+(‡∅ ).

In the class s MPOS Lemma 2.9.7 can be strengthened: Lemma 3.1.1 In all structures of the class s MPOS the following holds: sup ⊆ sum ⇐⇒ Card U > 1 ⇐⇒ ¬∃x∈U x is the zero. So in s MPOS conditions (‡) and (n-d) are equivalent. Proof We obtain the second equivalence on the strength of ( 0), which holds in (cf. Lemma 2.1.2). The proof of the left-to-right direction in the first equivalence follows from Lemma 2.9.7. For the right-to-left direction, assume that Card U > 1 and x sup S. Then there is no zero, by virtue of ( 0). Hence S = ∅, by virtue of (B.2.24). Therefore, x sum

S, thanks to (‡∅ ).

s PPOS

In accordance with the convention on Sect. 2.5.2, let s MPOS·· be the class of all non-degenerate structures from s MPOS, i.e., s MPOS·· = s MPOS+(n-d). Notice that from Lemma 2.9.6 we get: s MPOS

··

 s MPOS.

Moreover, by applying Lemmas 3.1.1 and 2.9.7 we get: s MPOS

··

= s PPOS+(‡∅ )+(n-d) = s PPOS+(‡).

3.2 Simons’ Minimal Extensional Mereology In (Simons 1987, p. 31) “Minimal Extensional Mereology” is the name given to the theory which is based on axioms (irr  ), (t ), (WSP) and the following:   ∀x,y∈U x  y =⇒ ∃z∈U ∀u∈U (u  z ⇔ u  x ∧ u  y) ,

(c∃ )

which states the conditional existence of an object, this being the “product” of two overlapping objects. Below, we will explain where the significance of this axiom comes from. For the moment, observe that (c∃ ) can be strengthened to the following equivalence:   ∀x,y∈U x  y ⇐⇒ ∃z∈U ∀u∈U (u  z ⇔ u  x ∧ u  y) .

(3.2.1)

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Simons’ Minimal Extensional Mereology

65

To see this, for some z 0 , let us have ∀u∈U (u  z 0 ⇔ u  x ∧ u  y). Then z 0  z 0 ⇔ z 0  x ∧ z 0  y, and applying (r ) and (df ), we have: x  y. Let MEM be the theory (class of structures) satisfying Simons’ axioms. Observe, though, that—by virtue of Lemma 2.3.1(i)—amongst these axioms the sentence (irr  ) is redundant. Therefore MEM := s POS+(WSP)+(c∃ ) = (t )+(WSP)+(c∃ ). Let MEM·· be the class of all non-degenerate structures belonging to MEM (cf. the convention on Sect. 2.5.2). We will now show that axiom (c∃ ) gives us a partial operation of the product of two overlapping elements. This product will be the greatest lower bound of those elements and also the mereological sum of all their common ingredienses (cf. (2.10.3)). Directly from (B.2.29), for S := {x, y} we get: Lemma 3.2.1 The following sentence follows from conditions (r ) and (t ):   ∀x,y,z∈U ∀u∈U (u  z ⇔ u  x ∧ u  y) ⇐⇒ z inf {x, y} . Therefore (c∃ ) is equivalent to the below:   ∀x,y∈U x  y =⇒ ∃z∈U z inf {x, y} ,

(c∃pair inf)

which speaks of the existence of the greatest lower bound of any pair of elements which overlap. In the light of (antis ) and the above lemma, we have:   ∀x,y∈U x  y =⇒ ∃1z∈U ∀u∈U (u  z ⇔ u  x ∧ u  y) ,   ∀x,y∈U x  y =⇒ ∃1z∈U z inf {x, y} . Furthermore, observe that—by virtue of (2.10.3)—for arbitrary x, y, v ∈ U : z inf {x, y} ⇐⇒ z is the greatest element in I(x)  I(y), ⇐⇒ z sup {u ∈ U : u  x ∧ u  y}, ⇐⇒ z sum {u ∈ U : u  x ∧ u  y}.

(3.2.2)

We can therefore define the partial binary operation of product of two elements which overlap one another2 :

: {x, y ∈ U × U : x  y} → U 2 We

are using the term ‘product’ because—by virtue of (3.2.2)—it is the mereological sum of all common ingredienses of x and y. This product must also be at the same time their product in an algebraic sense, i.e., their greatest lower bound.

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3 “Existentially Involved” Theories of Parts

such that for all and only those x and y from U for which x  y we put3 : x y := ( z) ∀z∈U (z  z ⇔ z  x ∧ z  y), = ( z) z inf {x, y}, ι ι

= ( z) z sup {u ∈ U : u  x ∧ u  y}, = ( z) z sum {u ∈ U : u  x ∧ u  y}, = ( z) z is the greatest element in I(x)  I(y).

(df )

ι ι ι

This explains where the significance of Simons’ axiom ‘(c∃ )’ comes from. It says, quite simply, that the product of an arbitrary pair of overlapping elements exists:   ∀x,y∈U x  y =⇒ ∃z∈U z = x y . We therefore see that the product operation must be conditioned by the assumption that x  y. Otherwise, were x  y to hold, we would have I(x)  I(y) = ∅. And only the zero is the least upper bound of an empty set. By virtue of ( 0), the zero exists when and only when the structure is degenerate. And that is ruled out: since x  y, we have x = y. Another explanation of this fact is the following: the sum of an empty set does not exist when the greatest element of an empty set does not either. From the first identity in (df ) we also get:   ∀x,y∈U x  y =⇒ I(x y) = I(x)  I(y) .

(3.2.3)

By exploiting (t ), we also get:   ∀x,y,z,u∈U x  y ∧ x  z ∧ y  u =⇒ z  u ∧ x y  z u .   ∀x,y,z∈U z  x ∧ z  y =⇒ x  y ∧ z  x y .

(3.2.4) (3.2.5)

For (3.2.4): if x  y, x  z and y  u, then there exists a v such that v  x  z and v  y  u. Therefore z  u. Take an arbitrary w such that w  x y. By virtue of (3.2.3), we have w  x and w  y. Hence, by virtue of our assumption, we have w  z and w  u. Hence, by applying (3.2.3) once more, we have w  z u. Given the arbitrariness of w, we have I(x y) ⊆ I(z u). Therefore x y  z u. For (3.2.5): if z  x, u  z and u  y, then u  x and u  y. And hence x  y and u  x y i.e., z  x y. From the second identity in (df ) we obtain the following: the operation is commutative and idempotent, i.e.:   ∀x,y∈U x  y =⇒ x y = y x , ∀x∈U x = x x. 3 The

phrase ‘for all and only those x and y’ we use on the strength of condition (3.2.1).

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67

Furthermore, we obtain in the usual way the result that the operation is associative in the following sense (cf. Remark 2.1.2):   ∀x,y,z∈U x  y =⇒ (z  x y ⇒ (x y) z = x (y z)) ,   ∀x,y,z∈U I[{x, y, z}] = ∅ =⇒ (x y) z = x (y z) . These sentences are equivalent because: x  y and z  x y iff there are x y and u ∈ U such that  u  z and u  x y iff there exists u ∈ U such that u  z, u  x and u  y iff I[{x, y, z}] = {u : u  x ∧ u  y ∧ u  z} = ∅. Since the operation is associative, we may introduce it in the usual way by omitting brackets and admitting for arbitrary y1 , . . . , yn (n > 0) that have common lower bounds (i.e., a common ingrediens): y1 · · · yn := (y1 · · · yn−1 ) yn . By induction, we get: y1 · · · yn inf {y1 , . . . , yn }. Hence, by making use of (2.10.3) and (3.2.3), for arbitrary y1 , . . . , yn (n > 0) that have some common ingrediens, we get: y1 · · · yn sup {u ∈ U : u  y1 ∧ · · · ∧ u  yn }, y1 · · · yn sum {u ∈ U : u  y1 ∧ · · · ∧ u  yn }. An important fact is that the class MEM is included in the class s PPOS . We obtain this from the following lemma and the theorem it entails. Lemma 3.2.2 From (r ), (t ), (WSP), (c∃ ) and (df 1 ) we get (SSP). Proof Let (a) x  y. Then if x  y then the consequence of (SSP) is true, since x  x, by virtue of (r ). Assume therefore that x  y. Then, by virtue of (c∃ ), for some v it is the case that (b) for each z: z  v iff z  x and z  y. Therefore— since v  v—we have (c) v  x and (d) v  y. From (d) and (a) we have v = x, i.e., v  x, by virtue of (c) and (df 1 ). Hence, by virtue of (WSP), for some u we have: (e) u  x and (f) u  v. We will show that (g) u  y. Assume for a contradiction that u  y. Then for some w we have: (h) w  u and (i) w  y. From (h) and (e) we have w  x, by virtue of (t ). Hence and from (i) and (b) we have w  v. Hence and from (h) we get u  v, which contradicts (f). Therefore, (e), (df 1 ) and (g) give us our thesis.

Theorem 3.2.3 In the class MEM condition (SSP) holds. Theorems 2.4.9(ii, iii) and 2.6.7 say, respectively, that in the class s POS, condition (SSP) entails (ext  ), (ext  ) entails (WSP) (Lemma 2.3.5 also says that (WSP) follows from (as ), (r ) and (SSP)) and that sentences (Usum ) and (ext  ) are equivalent. Therefore, applying Theorem 3.2.3, we get:

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3 “Existentially Involved” Theories of Parts

Model 3.1 (SSP) holds, but (c∃ ) and (c∃pair sup) does not hold

12

21

1

2

MEM = s PPOS+(c∃ ) = = s POS+(Usum )+(c∃ ) = s POS+(ext  )+(c∃ ). The fact below shows that the inclusion of the class MEM in the class s PPOS is proper. Fact 3.2.4 (c∃ ) does not follow from the axioms of s PPOS. Proof In Model 3.1 belonging to s PPOS we have 12  21, but there is no element whose ingredienses are elements 1 and 2 and only them. Therefore, 12 and 21 have no product.

From Theorem 3.2.3 and Fact 3.2.4 we get: MEM  s PPOS. We will show that MEM and s MPOS (resp. MEM·· and s PPOS+(‡)) cross one another. Firstly, it is clear that they have a non-empty intersection. Secondly, we have: Fact 3.2.5 (‡∅ ) and, a fortiori, (‡) do not follow from the axioms MEM·· . Proof Model 2.7 belongs to the class MEM·· . In it, though, the set {1, 2} has the least upper bound equal to 123, but there is no mereological sum of the elements 1

and 2. Therefore, (‡∅ ) is false and, a fortiori, (‡). From Fact 3.2.5 we obtain: MEM  s MPOS, MEM··  s MPOS·· . Thirdly, we get: Fact 3.2.6 (c∃ ) does not follow from the axioms of s MPOS·· . Proof Model 3.2 belongs to the class s MPOS·· , but (c∃ ) is false in it, because 12  123 and there is no element whose ingredienses are 1 and 2 and only them. Therefore, 12 and 123 do not have a product.

3.2

Simons’ Minimal Extensional Mereology

Model 3.2 (SSP) and (‡∅ ) hold, but (c∃ ), (SSP+) and (WSP+) do not hold

69 12

123

1

2

3

Model 3.3 (SSP) holds, but (c∃ ), (‡∅ ) and (‡) do not hold

From Fact 3.2.6 we get:  MEM, ·· s MPOS  MEM . s MPOS ··

To finish, we will occupy ourselves with the set-theoretic sum of the classes we have looked at here (in both versions). We will show that the sum is properly included in the class s PPOS (resp. s PPOSp). Let us observe that MEM  s MPOS = MEM  s MPOS·· = MEM··  s MPOS. All degenerate structures belong to MEM and s MPOS. Hence, we have MEM··  ·· s MPOS=MEM  s MPOS ⊆MEM  s MPOS. Furthermore, all non-degenerate structure from s MPOS belong to s MPOS·· , and therefore we also have the converse inclusion. From the above observation we also find that to reject degenerate structures, we must take the sum MEM··  s MPOS·· . Now, let us observe that: MEM  s MPOS  s PPOS, MEM··  s MPOS··  s PPOSp. By virtue of the inclusion we have established earlier, we have “⊆”. Furthermore, Model 3.3, this being the “conjoining” of Models 2.7 and 3.1, shows that the identity does not hold. It is a model from the class s PPOS in which (‡), (‡∅ ) and (c∃ ) are false. With reference to the axiomatisation of the class MEM  s MPOS (= MEM  ·· s MPOS ) we have the following equality: MEM  s MPOS = s PPOS+(c∃ ) ∨ (‡∅ ). This is a simple matter of the disjuncts being closed sentences with no free common variables in them.

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3 “Existentially Involved” Theories of Parts

In the next section we will be concerned with products of the classes under investigation here. We will then analyse other extensions of Minimal Extensional Mereology MEM (resp. MEM·· ).

3.3 The Classes MEM+(‡∅ ) and MEM+(‡) From Lemmas 2.9.6, 2.9.7 and 3.1.1 we get: MEM·· +(‡∅ ) = MEM+(‡)  MEM+(‡∅ ). By virtue of Lemma 2.3.5 and Theorem 3.2.3 we have the following identities: MEM+(‡∅ ) = s MPOS+(c∃ ), MEM+(‡) = s MPOS·· +(c∃ ). From it follows that MEM+(‡∅ ) = MEM  s MPOS, MEM+(‡) = MEM  s MPOS·· . Furthermore, by virtue of Facts 3.2.5 and 3.2.6, we get: MEM+(‡∅ )  MEM, MEM+(‡∅ )  s MPOS, MEM+(‡)  MEM·· , MEM+(‡)  s MPOS·· .

3.4 The Existence of Algebraic and Mereological Sums for Pairs The dependencies between classes of structures (theories) which we have analysed up to now in this chapter and those which we will be examining in the following two sections are presented in Diagrams 3.1 and 3.2, respectively. These diagrams can be connected with Diagram 2.1. For classes of non-degenerate structures, Diagram 3.2 transforms itself into Diagram 3.3. In Diagrams 3.6 and 3.7 we present the situation that obtains between classes of finite structures.

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The Existence of Algebraic and Mereological Sums for Pairs

71

3.4.1 The Unconditional Existence of Algebraic and Mereological Sums for Pairs As we showed on Sect. 3.2, the axiom postulating the existence of a product of two objects must be conditional. It is otherwise, however, with axioms postulating, respectively, the existence of an algebraic sum (least upper bound), a mereological sum and a fusion of a pair of objects. We can give them the following “unconditional” forms: ∀x,y∈U ∃z∈U z sup {x, y},

(∃pair sup)

∀x,y∈U ∃z∈U z sum {x, y},

(∃pair sum)

∀x,y∈U ∃z∈U z fu {x, y}.

(∃pair fu)

Theorem 2.12.6 says that in theories satisfying condition (SSP) the following identity holds: fu = sum (more precisely, condition (SSP) is equivalent to the inclusion fu ⊆ sum and the converse inclusion results from (t ) and our definitions). We therefore obtain: s PPOS+(∃pair sum)

= s PPOS+(∃pair fu).

We also obtain an analogous identity for any extension of the theory s PPOS. For example, we have: MEM+(∃pair sum) = MEM+(∃pair fu).

Fact 3.4.1 (i) (∃pair sup) does not follow from the axioms of MEM. (ii) (∃pair sup) does not follow from the axioms of s PPOS. (iii) (∃pair sum) does not follow from the axioms of MEM+(∃pair sup). (iv) (∃pair sum) does not follow from the axioms of s PPOS+(∃pair sup). Proof Ad (i) It suffices to use Model 3.4, which is composed of two isolated elements, i.e., two elements exterior with respect to each other.4 The model belongs to MEM, but (∃pair sup) is false in it. Ad (ii) From (i), because MEM  s PPOS. Ad (iii) Model 2.7 belongs to the class MEM+(∃pair sup) but it does not satisfy condition (c∃pair sum), because 123 is an upper bound of the pair {1, 2} but this pair does not have a mereological sum in this model.

Ad (iv), (ii) From (iii), because MEM  s PPOS. 4 It

is easy to see that this model arises from the four-element Boolean algebra from which the zero and the unity have been removed. Other models of this type are obtained in the same way from other finite Boolean algebras. For example, from the eight-element Boolean algebras we obtain Model 3.7.

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3 “Existentially Involved” Theories of Parts

Model 3.4 (SSP), (WSP), (c∃ ), (c∃pair sup), (c∃pair fu), (c∃pair sum), (SSP+), (WSP+), (w1 ∃sum) and (w2 ∃sum) hold, but (∃pair sum), (∃pair fu), (∃pair sup) and (∃ 1 ) do not hold

Furthermore, on the strength of (SSP), we have sum ⊆ sup; and so we obtain: Lemma 3.4.2 In the class s PPOS sentence (∃pair sum) entails (∃pair sup). As a consequence of Fact 3.4.1 and Lemma 3.4.2 we have: s PPOS+(∃pair sum)

 s PPOS+(∃pair sup)  s PPOS,

MEM+(∃pair sum)  MEM+(∃pair sup)  MEM, MEM  s PPOS+(∃pair sup), MEM+(∃pair sup)  s PPOS+(∃pair sum). We have the following strengthening of Fact 3.2.4: Fact 3.4.3 (c∃ ) does not follow from the axioms of s PPOS+(∃pair sum). Proof We create a model whose universe U is composed of the following elements: • three open intervals (0, 3), (0, 2) and (1, 3) of the set of real numbers; • any open proper subset of the open interval (1, 2) such that 1 and 2 do not belong to its closure. The relation  is to be the relation  of proper inclusion of subsets of U (so the relation  is the relation ⊆ of inclusion). This model belongs to s PPOS+(∃pair sum). However, (c∃ ) is false in it. To see this, the intervals (0, 2) and (1, 3) overlap, but

they have no product in the model, since (1, 2) does not belong to U .5 From Fact 3.4.3 we get: MEM+(∃pair sum) = s PPOS+(c∃ )+(∃pair sum)  s PPOS+(∃pair sum), MEM+(∃pair sup) = s PPOS+(c∃ )+(∃pair sup)  s PPOS+(∃pair sup), s PPOS+(∃pair sum)

 MEM.

5 In Sect. 3.10—where we investigate the case of finite structures—we will show that in the proof of

this fact we had to take an infinite model, because s PPOSfin +(∃pair sum) = MEMfin +(∃pair sum).

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73

3.4.2 The Conditional Existence of Algebraic and Mereological Sums for Pairs In the literature can be found an extension of Minimal Extensional Mereology entitled “Minimal Closure Mereology” (see, e.g., Casati and Varzi 1999; Simons 1987). It postulates certain conditional existences of a mereological fusion of pairs of elements. We will denote this theory with the label ‘CMM’. The condition placed on objects which are spoken about in axiom (c∃ ) says that they have common lower bounds; i.e., they overlap one another. The condition placed on objects which are the topic of the new axiom will say instead that they have a common upper bound. This condition corresponds to an auxiliary relation of underlapping:   ∀x,y∈U x Un y :⇐⇒ ∃z∈U (x  z ∧ y  z) .

(df Un)

Obviously, the relation Un is reflexive, symmetric and it includes the relation , by virtue of (r ). Therefore, two objects are in the relation Un iff they are ingredienses of some object, i.e., iff they have common upper bounds. To build Minimal Closure Mereology, one expands the theory MEM with the following axiom:   ∀x,y∈U x Un y =⇒ ∃z∈U ∀u∈U (u  z ⇔ u  x ∨ u  y) ,

(c∃pair fu)

Thus, we put: CMM := MEM+(c∃pair fu). The lemma below explains the significance of this new axiom. Lemma 3.4.4 On the basis of (df fu) we obtain:   ∀x,y,z∈U z fu {x, y} ⇐⇒ ∀u∈U (u  z ⇔ u  x ∨ u  y) . So (c∃pair fu) is equivalent to the below:   ∀x,y∈U x Un y =⇒ ∃z∈U z fu {x, y} .

(c∃pair fu )

Thus, (c∃pair fu) also postulates the existence of a mereological fusion of any pair of elements which underlap. By once again applying Theorem 2.12.6, from Lemma 3.4.4 we get: Lemma 3.4.5 In the class s PPOS we have:   ∀x,y,z∈U ∀u∈U (u  z ⇔ u  x ∨ u  y) ⇐⇒ z sum {x, y} . Therefore, in the class s PPOS sentence (c∃pair fu) is equivalent to the following:

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3 “Existentially Involved” Theories of Parts

  ∀x,y∈U x Un y =⇒ ∃z∈U z sum {x, y} .

(c∃pair sum)

As a consequence of Lemmas 3.4.4 and 3.4.5 we have: s PPOS+(c∃pair fu)

= s PPOS+(c∃pair fu ) = s PPOS+(c∃pair sum).

We also obtain an analogous identity for any extension of the theory s PPOS. For example, we have: CMM := MEM+(c∃pair fu) = MEM+(c∃pair fu ) = MEM+(c∃pair sum). It is interesting that the conjunction of (c∃pair sum) and (∃pair sup), which postulate the conditional existence of a mereological sum and the unconditional existence of a least upper bound, respectively, reduce to a postulate (∃pair sum) of the unconditional existence of a sum. Lemma 3.4.6 In the class s PPOS sentence (∃pair sum) is equivalent to the conjunction of (c∃pair sum) and (∃pair sup). Proof Left-to-right: from Lemma 3.4.2. Right-to-left: for arbitrary x, y ∈ U , on the basis of (∃pair sup), there exists a z such that z sup {x, y}. Hence x Un y. So from

(c∃pair sum) there exists a v such that v sum {x, y}.6 By the above lemmas we get: s PPOS+(∃pair sum)

= s PPOS+(c∃pair sum)+(∃pair sup), = s PPOS+(c∃pair sum)  s PPOS+(∃pair sup),

and MEM+(∃pair sum) = CMM+(∃pair sup) = CMM  MEM+(∃pair sup). Fact 3.4.7 (∃pair sup), and hence a fortiori (∃pair sum), do not follow from the axioms of MEM+(c∃pair sum). Proof Model 3.4 belongs to the class MEM+(c∃pair sum) but it does not satisfy con

dition (∃pair sup). In the light of Fact 3.4.7 we get: MEM+(∃pair sum)  MEM+(c∃pair sum) = CMM, MEM+(c∃pair sum)  MEM+(∃pair sup), s PPOS+(∃pair sum)  s PPOS+(c∃pair sum), s PPOS+(c∃pair sum)

 s PPOS+(∃pair sup).

that v = z, because we have (Usup ) and sum ⊆ sup, from (SSP). One could also use (), which is equivalent to (WSP).

6 Note

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The Existence of Algebraic and Mereological Sums for Pairs

75

Fact 3.4.8 (c∃pair sum), and hence a fortiori (∃pair sum), do not follow from the axioms of MEM+(∃pair sup). Proof Model 2.7 belongs to the class MEM+(∃pair sup) but it does not satisfy condition (c∃pair sum), because 123 is an upper bound of the pair {1, 2} but this pair does not have a mereological sum in this model.

In the light of Fact 3.4.8 we get: MEM+(∃pair sum) = MEM+(c∃pair sum)+(∃pair sup)  MEM+(∃pair sup), MEM+(∃pair sup)  MEM+(c∃pair sum), s PPOS+(∃pair sum)  s PPOS+(∃pair sup), s PPOS+(∃pair sup)  s PPOS+(c∃pair sum).

3.4.3 Conditional Mereological Lattices By analogy with (c∃pair fu) and (c∃pair sum), we will also introduce a postulate which speaks of the existence of an algebraic sum (i.e., a least upper bound) of any pair of elements which underlap.   ∀x,y∈U x Un y =⇒ ∃z∈U z sup {x, y} .

(c∃pair sup)

By adding (c∃pair sup) as an axiom to the theory MEM we are dealing with a certain type of conditional lattices. Such lattices are “mereological” because—as we remember—in the theory MEM both the mereological principles (WSP) and (SSP) are in force. We will not give the theory MEM+(c∃pair sup) any special label. On the strength of (SSP), we have sum ⊆ sup; and so we obtain: Lemma 3.4.9 In the class s PPOS sentence (c∃pair sum) entails (c∃pair sup). From Lemma 3.4.9 and Facts 3.4.7 and 3.4.8, respectively, we obtain directly: Fact 3.4.10 (i) (∃pair sup), and hence a fortiori (∃pair sum), does not follow from the axioms of MEM+(c∃pair sup). (ii) (c∃pair sum), and hence a fortiori (∃pair sum), does not follow from the axioms of MEM+(c∃pair sup). We therefore have the follow inclusions: MEM+(∃pair sum)  CMM  MEM+(c∃pair sup) ⊆ MEM, MEM+(∃pair sum)  MEM+(∃pair sup)  MEM+(c∃pair sup) ⊆ MEM. Below, we will show that all the inclusions appearing above are proper.

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3 “Existentially Involved” Theories of Parts

Because Models 3.4 and 2.7 used in the proofs of Facts 3.4.7 and 3.4.8, respectively, are non-degenerate, we also have: MEM·· +(∃pair sum)  CMM··  MEM·· +(c∃pair sup) ⊆ MEM·· , MEM·· +(∃pair sum)  MEM·· +(∃pair sup)  MEM·· +(c∃pair sup) ⊆ MEM·· , MEM·· +(c∃pair sum)  MEM·· +(∃pair sup), MEM+(∃pair sup)  MEM+(c∃pair sum). For classes of non-degenerate structures we also obtain all proper inclusions. Furthermore, we will strengthen Fact 3.4.1(ii): Fact 3.4.11 (c∃pair sup) do not follow from the axioms of s PPOS. Proof Model 3.1 belongs to s PPOS but it does not satisfy condition (c∃pair sup), because 12 and 21 are upper bounds of the pair {1, 2} but this pair has no least upper bound in this model.

Thus, we obtain:  s PPOS+(c∃pair sum)  s PPOS+(c∃pair sup)  s PPOS, s PPOS+(∃pair sum)  s PPOS+(∃pair sup)  s PPOS+(c∃pair sup)  s PPOS.

s PPOS+(∃pair sum)

Because Models 3.4, 2.7 and 3.1 used in the proofs of Facts 3.4.7, 3.4.8 and 3.4.11, respectively, are non-degenerate, we also have: +(∃pair sum)  s PPOS·· +(c∃pair sum)  s PPOS·· +(c∃pair sup)  s PPOS·· , ·· ·· ·· ·· s PPOS +(∃pair sum)  s PPOS +(∃pair sup)  s PPOS +(c∃pair sup)  s PPOS ,

s PPOS

··

+(c∃pair sum)  s PPOS·· +(∃pair sup), ·· ·· s PPOS +(∃pair sup)  s PPOS +(c∃pair sum).

s PPOS

··

The fact below says that the class MEM+(‡) is not included in the class MEM+ (c∃pair sup) and, furthermore, MEM+(c∃pair sup) is a proper supertheory of the theory MEM; i.e., the class MEM+(c∃pair sup) is a proper subclass of the class MEM. Fact 3.4.12 Sentence (c∃pair sup) does not follow from the axioms of MEM+(‡). Proof Let IN be the set of all natural numbers and IN·· := IN \ {0}. We build a model whose universe U is composed of the pairs 0, 0, 0, 1, 1, 0 and  n1 , i, where n > 1 and i ∈ {0, 1}.7 Let  be the smallest transitive relation in U × U satisfying the following conditions: •  n1 , 0   m1 , 0 iff m < n, for arbitrary m, n ∈ IN·· , 1 , 1   n1 , 0, for each n ∈ IN·· , •  n+1 7 The

universe U of the model of the class MEM in which (c∃pair sup) is false must be infinite, because the proof of Fact 3.10.1 shows that MEMfin +(c∃pair sup) = MEMfin .

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77

s PPOS

MEM

s PPOS+(c∃pair sup)

MEM+(c∃pair sup)

s PPOS+(∃pair sup)

CMM

s PPOS+(∃pair sum)

MEM+(∃pair sup)

s PPOS+(c∃pair sum)

MEM+(∃pair sum) where s PPOS := s POS+(pol ) = s POS+(SSP) MEM := s POS+(WSP)+(c∃ ) = (t )+(WSP)+(c∃ ) = s PPOS+(c∃ ) CMM := MEM+(c∃pair fu) = MEM+(c∃pair sum) MEM+(∃pair sum) = MEM+(∃pair fu) = CMM+(∃pair sup) = CMM  MEM+(∃pair sup) s PPOS+(∃pair sum) s PPOS+(c∃pair fu)

= s PPOS+(∃pair fu) = s PPOS+(c∃pair sum)  s PPOS+(∃pair sup)

= s PPOS+(c∃pair fu ) = s PPOS+(c∃pair sum)

Diagram 3.1 The lattice of theories related to sentences (WSP), (SSP), (c∃ ), (c∃pair sup), (c∃pair sum), (∃pair sup) and (∃pair sum)

• 0, 0   n1 , 0 and 0, 1   n1 , 0, for each n ∈ IN·· . Obviously, the model belongs to s POS and satisfies the conditions: (WSP), (c∃ ) and (‡). All the elements of the form  n1 , 0 are upper bounds of the elements 0, 0 and 0, 1. However, those elements do not have a least upper bound.

We therefore obtain from Fact 3.4.12: MEM+(c∃pair sup)  MEM. The results obtained in this chapter so far concerning classes of structures are portrayed in Diagram 3.1. For classes of non-degenerate structures, Diagram 3.1 transforms itself into a new diagram by replacing s PPOS and MEM with s PPOS·· and MEM·· , respectively. On the basis of (df sup) and (df Un) it is possible to strengthen axiom (c∃pair sup) to the following equivalence:   ∀x,y∈U x Un y ⇐⇒ ∃z∈U z sup {x, y} .

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3 “Existentially Involved” Theories of Parts

This equivalence yields a partial binary operation + of algebraic sum of elements which have common upper bounds8 : + : {x, y ∈ U × U : x Un y} → U such that, for all and only those x and y from U , for which x Un y we put: x + y := ( z) z sup {x, y}.

(df +)

ι

Therefore, the operation + is commutative and idempotent, i.e.:   ∀x,y∈U x Un y =⇒ x + y = y + x , ∀x∈U x = x + x. Furthermore, we obtain in a standard way the result that the operation + is associative in the following sense:   ∀x,y,z∈U x Un y =⇒ (z Un x + y ⇒ (x + y) + z = x + (y + z)) ,   ∀x,y,z∈U ∃u∈U {x, y, z} ⊆ I(u) =⇒ (x + y) + z = x + (y + z) . The antecedents of the implications above are equivalent: x Un y and z Un x + y iff there exist x + y and u ∈ U such that z  u and x + y  u iff there exists u ∈ U such that z  u, x  u and y  u. Furthermore, from (df sup) we get:   ∀x,y∈U x Un y =⇒ x  x + y ∧ y  x + y .

(3.4.1)

Since the operation + is associative, we drop the brackets in the standard way and write, for arbitrary y1 , . . . , yn (n > 0) that have common upper bounds: y1 + · · · + yn := (y1 + · · · + yn−1 ) + yn . By induction, we obtain: y1 + · · · + yn sup {y1 , . . . , yn }. Furthermore, from condition (B.2.27), we have: z sup {y1 , . . . , yn } ⇐⇒ z sup {u ∈ U : u  y1 ∨ · · · ∨ u  yn }, ⇐⇒ z sup I(y1 )  · · ·  I(yn ).

8 The

operation + can also be introduced in the class s PPOS+(c∃pair sup). Structures of this class with the operation + are conditional semi-lattices.

3.4

The Existence of Algebraic and Mereological Sums for Pairs

79

Thus: y1 + · · · + yn = ( z) z sup I(y1 )  · · ·  I(yn ). ι

In the class MEM+(c∃pair sup) we therefore have two partial operations defined: • the partial binary operation +, defined by (df +) for all and only those elements of U that have common upper bounds; • the partial binary operation , defined by (df ) for all and only those elements of U that overlap one another. Regarding this operation , we can repeat everything we said for the theory MEM. Finally for this point, let us observe that from (t ) and (3.4.1) we get:   ∀x,y∈U x Un y =⇒ O(x)  O(y) ⊆ O(x + y) .

(3.4.2)

Remark 3.4.1 Model 2.7 shows that the inclusion in (3.4.2) is not reversible in the class MEM+(c∃pair sup). In this model: 1 Un 2, O(1)  O(2) = {1, 2}, 1 + 2 = 123 and O(123) = {1, 2, 3, 123}. This model has a connection with Model B.1 and therefore it shows that in the class MEM+(c∃pair sup) it is not possible to prove socalled partial distributivity in any form.

3.4.4 Operations in Minimal Closure Mereology As we know, the theory under discussion in this point, signified by ‘CMM’, is the theory MEM extended by axiom (c∃pair fu), as well CMM = MEM+(c∃pair sum) (see Lemmas 3.4.4 and 3.4.5). This theory is equivalent to the theory CEM, “Extensional Closure Mereology”, defined as follows (see, e.g., Casati and Varzi 1999): CEM := s PPOS+(c∃ )+(c∃pair fu). Indeed, on the basis of Theorem 3.2.3, (SSP) holds in MEM. Lemma 2.3.5 says, however, that (as ), (r ) and (SSP) entail (WSP). Thus, we have: CMM = CEM. Since in CMM we have fu = sum and (Usum ) holds, by applying Lemmas 3.4.4 and 3.4.5 we get:   ∀x,y∈U x Un y =⇒ ∃1z∈U z fu {x, y} , ∀x,y,z∈U (z fu {x, y} ⇐⇒ z sum {x, y}).

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3 “Existentially Involved” Theories of Parts

In CMM we can define the partial operation  of mereological sum of elements which have common upper bounds9 :  : {x, y ∈ U × U : x Un y} → U such that for all and only those x and y from U for which x Un y: x  y := ( z) z sum {x, y}.

(df )

ι

In all theories satisfying condition (SSP) we have: sum ⊆ sup and sum = fu. So: x  y = ( z) z sup {x, y}, = ( z) z fu {x, y}.

(3.4.3)

ι ι

So the operation  corresponds to the operation + defined on Sect. 3.4.3 for the theory MEM+(c∃pair sup). Therefore, the partial operation  in CMM satisfies the following conditions:   ∀x,y∈U x Un y =⇒ O(x  y) = O(x)  O(y) ,   ∀x,y,z∈U x Un y =⇒ (z  x  y ⇔ I(z) ⊆ O(x  y) .

(3.4.4) (3.4.5)

For (3.4.4): since x  y fu {x, y}. For (3.4.5): if x Un y, z  x  y and u  z, then u  x  y, by (t ) and (3.4.1). We have x  y sum {x, y}. Conversely, if x Un y and I(z) ⊆ O(x  y), then we apply (SSP). Since x  y sup {x, y}, then—analogously as for the operation + from the theory MEM+(c∃pair sup)—we show that the operation  is commutative, idempotent and associative. We will inductively generalise condition (3.4.4) to an arbitrary finite number of elements y1 , . . . , yn (n > 0) that have a common upper bound: O(y1  · · ·  yn ) = O(y1 )  · · ·  O(yn ).

(3.4.6)

This allows us to generalise our earlier results. Firstly, for arbitrary y1 , . . . , yn (n > 0) having common upper bounds: y1  · · ·  yn fu {y1 , . . . , yn }. And from this and the identity sum = fu we have: y1  · · ·  yn sum {y1 , . . . , yn }.

(3.4.7)

Furthermore, by virtue of Lemma 2.6.2, we obtain: operation  can also be entered in the class s PPOS+(c∃pair sum). Structures of this class with the operation  are also conditional semi-lattices.

9 The

3.4

The Existence of Algebraic and Mereological Sums for Pairs

81

y1  · · ·  yn sum I(y1 )  · · ·  I(yn ). From conditions (3.4.4), (3.4.5) and (SSP) we obtain:   ∀x,y,z∈U x Un y =⇒ (z  x  y ∧ z  x ⇒ z  y) .

(3.4.8)

Let x Un y, z  x  y and z  y. Then thanks to (SSP), for some u: u  z and u  y. Hence, by virtue of (3.4.4) and (3.4.5), we have u  x. Hence z  x, by virtue of (t ). By drawing upon (3.4.8) and (SSP), for arbitrary x, y, z, u ∈ M, we obtain:   ∀x,y,z,u∈U x Un y ∧ x Un z ∧ u  x  y ∧ u  x  z ∧ y  z =⇒ u  x . (3.4.9) Let x Un y, x Un z, u  x  y, u  x  z and u  x. Then there exists a w such that w  u and w  x. On the strength of (t ) we have w  x  y and w  x  z. Hence, by virtue of (3.4.8), we have w  y and w  z, i.e., y  z. Now let us note that, by virtue of (3.4.3), we have:   ∀x,y,z∈U x Un y ∧ x  z ∧ y  z =⇒ x  y  z .

(3.4.10)

All structures from the class CMM are partially distributive in a sense to be introduced below.10 Before formulating the distributivity condition of the operation  with respect to the operation , let us note that, by virtue of (3.4.1), we have:   ∀x,y,z∈U x Un y ∧ x Un z =⇒ x  y  x  z ,

(3.4.11)

i.e., if x Un y and x Un z, then there exist objects x  y, x  z and (x  y) (x  z). To formulate the distributivity condition of the operation with respect to the operation  we will make use of condition (3.4.4). Theorem 3.4.13 (i) If x Un y and x Un z , then  (x  y) (x  z) =

x  (y z) if y  z, x if y  z.

(ii) If y Un z and either x  y or x  z, then ⎧ ⎪ ⎨(x y)  (x z) if x  y and x  z, x (y  z) = x y if x  y and x  z, ⎪ ⎩ x z if x  y and x  z.

10 We

do not want to use the expression ‘conditional distributivity’ because it is used for “full” lattices. Here we are dealing with “full” distributivity in “partial” lattices.

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3 “Existentially Involved” Theories of Parts

Proof Ad (i) Since x  x  y and x  x  z, we have (∗): x  (x  y) (x  z), by virtue (3.2.3). Let us now consider two cases. (A) y  z: (x  y) (x  z)  x  y and (x  y) (x  z)  x  z, by (3.2.3) and (3.4.1). Therefore (x  y) (x  z)  x, by (3.4.9). (B) y  z: We have y  x  y and z  x  z. Hence y z  (x  y) (x  z), by (3.2.4). From this and (∗) we have x  (y z)  (x  y) (x  z), by (3.4.10). Assume for a contradiction that (x  y) (x  z)  x  (y z). Hence, by (SSP), there is a u such that u  (x  y) (x  z) and u  x  (y z). From the first result we have: u  x  y and u  x  z. From the second, by (3.4.4), we obtain u  x and u  y z. Thus—by applying (3.4.8)—we have u  y and u  z. Hence we have u  y z, which contradicts u  y z. (ii) Let us assume that y Un z. We have (∗): If x  y, then x y  x and x y  y  y  z. Hence x y  x (y  z). Furthermore, we have (∗∗): if x  z, then x z  x and x z  z  y  z. Hence x z  x (y  z). We are investigating three cases: (A) x  y and x  z: assume for a contradiction that x (y  z)  x y. Then, by (SSP), there is a u such that u  x (y  z) and u  x y. Therefore u  x, u  y  z and, by virtue of (3.2.5), we have u  y. Hence, by (3.4.8), we get u  z. Therefore, we have a contradiction: x  z. (B) x  y and x  z: analogously as in (A). (C) x  y and x  z: From (∗) and (∗∗), by (3.4.10), we have (x y)  (x z)  x (y  z). Conversely, assume for a contradiction that x (y  z)  (x y)  (x z). Then, by (SSP), there is a u such that u  x (y  z) and u  (x y)  (x z). Therefore u  x, u  y  z, u  x y and u  x z. By virtue of (3.2.5) we have u  y and u  z. Hence, by (3.4.4), we get u  y  z, which contradicts u  y  z.

3.4.5 Operations in Semi-conditional Mereological Lattices Structures belonging to MEM+(∃pair sup) we may also call semi-conditional mereological lattices. In these structures we have two defined operations: • the partial binary operation , defined by (df ) for all and only those elements of U which overlap one another; • the binary relation +, defined by (df +) for the whole of U × U .11 Because only one of the two operations is conditional, we say “semi-conditional”. Regarding these operations, we can repeat everything we said about them in MEM and MEM+(c∃pair sup).

11 The

operation + can also be introduced in the class s PPOS+(∃pair sup). Structures of this class with the operation + are semi-lattices.

3.4

The Existence of Algebraic and Mereological Sums for Pairs

83

3.4.6 Operations in the Second Kind of Semi-conditional Mereological Lattice Also structures belonging to MEM+(∃pair sum) we may call semi-conditional mereological lattices. As we have shown in point 3.4.2: MEM+(∃pair sum) = MEM+(∃pair fu) = CMM  MEM+(∃pair sup). In MEM+(∃pair sum) we have two defined operations: • the partial binary operation , defined by (df ) for all and only those elements of U which overlap one another; • the binary operation , defined by (df ) for the whole of U × U .12 Regarding these operations, we can repeat everything we said about them in the theories CMM and MEM+(∃pair sup) with this difference: that by adding (∃pair sum) as an axiom to the theory MEM we obtain a theory of a special kind of semi-conditional mereological lattice in which the “full” operation  yields a mereological sum of an arbitrary pair of elements. Furthermore, strengthened versions of conditions (3.4.4), (3.4.5) and (3.4.8)– (3.4.11) obtain. We are bypassing the restrictions connected with the relation Un, but the proofs remain the same, because axiom (∃pair sum) allows us to carry them across without those restrictions:   ∀x,y∈U O(x  y) = O(x)  O(y) ,   ∀x,y,z∈U z  x  y ⇔ I(z) ⊆ O(x  y) ,   ∀x,y,z∈U z  x  y ∧ z  x =⇒ z  y ,

  ∀x,y,z,u∈U u  x  y ∧ u  x  z ∧ y  z =⇒ u  x ,   ∀x,y,z∈U x  z ∧ y  z =⇒ x  y  z , ∀x,y,z∈U x  y  x  z .

(3.4.12) (3.4.13) (3.4.14) (3.4.15) (3.4.16) (3.4.17)

In addition, all the structures from MEM+(∃pair sum) are partially distributive in the sense introduced in the theorem below. This is a strengthening of Theorem 3.4.13, which is carried out by bypassing the restrictions connected with the relation Un. The proof of the strengthened theorem is identical to the proof of Theorem 3.4.13, because condition (∃pair sum) allows it.13 Theorem 3.4.14 (i) For arbitrary x, y, z ∈ U , we have operation  can also be entered in the class s PPOS+(∃pair sum). Structures of this class with the operation  are semi-lattices. 13 What was said in footnote 10 in the context of the theory CMM also applies to here the strengthened theorem. 12 The

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3 “Existentially Involved” Theories of Parts

 (x  y) (x  z) =

x  (y z) if y  z, x if y  z.

(ii) If x  y or x  z, then ⎧ ⎪ ⎨(x y)  (x z) if x  y and x  z, x (y  z) = x y if x  y and x  z ⎪ ⎩ x z if x  y and x  z

3.5 Extensions of Theories MEM+(‡∅ ) and MEM+(‡) via Conditions for the Existence of Sums for Pairs We will now occupy ourselves with cases where we will add to the theory MEM+(‡∅ ) one of the conditions (c∃pair sum), (∃pair sum). To begin with, notice that: MEM+(‡∅ ) = s MPOS+(c∃ ), MEM+(‡) = s MPOS·· +(c∃ ). Moreover, we have: CMM+(‡∅ ) = s MPOS+(c∃ )+(c∃pair sum), CMM+(‡) = s MPOS·· +(c∃ )+(c∃pair sum). Now, notice that in structures from the class s MPOS·· the relations sup, sum and fu are equal. Indeed, from condition (SSP) we have sum = fu and sum ⊆ sup; but condition (‡) says that sup ⊆ sum. Furthermore, in structures from the class s MPOS these relations are the same for all non-empty sets; the same holds for pairs. Indeed, (‡∅ ) says that if x sup S and S = ∅, then x sum S. So in the class s PPOS, condition (∃pair sup) follows from (∃pair sum); and condition (c∃pair sum) follows from (c∃pair sup) and (‡∅ ) (resp. (‡)). Thus, in the class s MPOS we do not distinguish between algebraic sums and mereological sums for pairs and we obtain the following identities: s MPOS+(c∃pair sup) s MPOS+(∃pair sup)

= s MPOS+(c∃pair sum) = s MPOS+(c∃pair fu), = s MPOS+(∃pair sum) = s MPOS+(∃pair fu).

Thanks to the above, in the class MEM+(‡∅ ) we once again do not distinguish between algebraic sums and mereological sums for pairs and we obtain the following identities:

3.5

Extensions of Theories MEM+(‡∅ ) and MEM+(‡) via Conditions . . .

Model 3.5 (WSP), (SSP), (c∃ ), (WSP+), (SSP+), (c∃pair sup), (c∃pair fu), (c∃pair sum), (‡∅ ), (‡), (w1 ∃sum) and (w2 ∃sum) hold, but (∃pair sup), (∃pair fu), (∃pair sum) and (∃ 1 ) do not hold

85

1

2

MEM+(‡∅ )+(c∃pair sup) = MEM+(‡∅ )+(c∃pair sum) = CMM+(‡∅ ), MEM+(‡∅ )+(∃pair sup) = MEM+(‡∅ )+(∃pair sum) = MEM+(‡∅ )+(∃pair fu). By adding either (c∃pair sum) or (∃pair sum) to MEM+(‡), however, by virtue of Lemma 3.1.1, we dispose of degenerate models. Therefore, we will get— analogously to those identities and inclusions given above and below—identities and inclusions that differ in just this way: that ‘(‡)’ appears instead of ‘(‡∅ )’, and where in the name of a class we do not have the symbol ‘(‡)’ we have instead the superscript ‘·· ’. Let us recall that the model on which we based the proof of Fact 3.4.12 belongs to the class MEM+(‡). Since this model falsifies (c∃pair sup), then it also falsifies (c∃pair sum), (∃pair sup) and (∃pair sum). This shows that CMM+(‡∅ )  MEM+(‡∅ ), MEM+(‡∅ )+(∃pair sum)  MEM+(‡∅ ), MEM+(‡∅ )  MEM+(c∃pair sup). Hence, it follows that MEM+(‡∅ ) is not also included in any of the classes CMM, MEM+(∃pair sum) and MEM+(∃pair sup). Furthermore, we obtain: Fact 3.5.1 (∃pair sup), and hence a fortiori (∃pair sum), does not follow from the axioms of CMM+(‡), and so from the axioms of CMM+(‡∅ ). Proof Model 3.5 belongs to CMM+(‡), but (∃pair sup) is false in it, because the set {1, 2} has no least upper bound.

Thus, from the above fact we obtain: CMM+(‡∅ )  MEM+(∃pair sup), CMM+(‡∅ )  MEM+(∃pair sum). Furthermore, MEM+(‡∅ )  MEM+(∃pair sup) and MEM+(‡∅ )  MEM+(∃pair sum). Analogous results hold for the theory with condition (‡). Also thanks to the above fact we get: MEM+(‡∅ )+(∃pair sum)  CMM+(‡∅ ). An analogous result holds for the theory with condition (‡).

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3 “Existentially Involved” Theories of Parts

To finish, we have: Fact 3.5.2 (‡∅ ) does not follow from the axioms of MEM+(∃pair sum), and so not from the axioms of CMM either. Proof We create a model whose universe U is composed of the set IN of natural numbers and all its non-empty subsets. The relation  is to be the relation  of proper inclusion of subsets of U (so the relation  is the relation ⊆ of inclusion). This model belongs to the class MEM+(∃pair sum). However, (‡∅ ) is false in it. If, for example, S is a family of all singletons of even numbers, then IN sup S and {1}  IN, but {1} is exterior to each singleton from S. Therefore, IN is not a

mereological sum of the family S.14 Therefore, from Fact 3.5.2 we get: MEM+(‡∅ )+(∃pair sum)  MEM+(∃pair sum), CMM+(‡∅ )  CMM, MEM+(∃pair sum)  MEM+(‡∅ ), and none of the classes MEM+(∃pair sup), MEM+(c∃pair sum) and MEM+(c∃pair sup) is included in the class MEM+(‡∅ ). The results obtained in this chapter so far concerning classes of structures are portrayed in Diagrams 3.2 and 3.3.

3.6 “Super-Supplementation” Principles 3.6.1 Definitions and Fundamental Properties of the Principles Of great interest is the strengthening of supplementation principles, both strong and weak, to super-supplementation principles (or “enhanced” supplementation principles; or supplementation principles “plus”). Once again, we have two versions of such a principle—a weak and a strong one:    ∀x,y∈U y  x =⇒ ∃z∈U z  x ∧ z  y ∧ ∀u∈U (u  x ∧ u  y ⇒ u  z) , (WSP+)    ∀x,y∈U x  y =⇒ ∃z∈U z  x ∧ z  y ∧ ∀u∈U (u  x ∧ u  y ⇒ u  z) . (SSP+) Obviously, the “normal” versions are logically entailed by the “super” versions. 14 In

Sect. 3.10—where we investigate the case of finite structures—we will show that in the proof of this fact we had to take an infinite model, because CMMfin +(‡∅ ) = CMMfin and MEMfin +(∃pair sum)+(‡∅ ) = MEMfin +(∃pair sum).

3.6

“Super-Supplementation” Principles

87 s PPOS

MEM  s MPOS

MEM+(∃pair sup)

MEM

s MPOS

MEM+(c∃pair sup)

MEM+(‡∅ )

·· s MPOS

CMM

MEM+(∃pair sum)

MEM+(‡)

CMM+(‡∅ ) MEM+(∃pair sum)+(‡∅ )

CMM+(‡)

MEM+(∃pair sum)+(‡) where s PPOS

:= s POS+(pol ) = s POS+(SSP)

s PPOS

:= s PPOS+(‡∅ )

MEM := s POS+(WSP)+(c∃ ) = (t )+(WSP)+(c∃ ) = s PPOS+(c∃ ) CMM := MEM+(c∃pair fu) = MEM+(c∃pair sum) CMM+(‡∅ ) = MEM+(c∃pair sum)+(‡∅ ) = s MPOS+(c∃ )+(c∃pair sum) MEM+(∃pair sum) = MEM+(∃pair fu) = CMM  MEM+(∃pair sup) MEM+(‡∅ ) = MEM  s MPOS = s MPOS+(c∃ ) Diagram 3.2 The lattice of theories related to sentences (WSP), (SSP), (c∃ ), (‡∅ ), (‡), (c∃pair sup), (c∃pair sum), (∃pair sup) and (∃pair sum)

From Lemma 2.3.5 we obtain (see Remarks 2.3.3 and 2.3.4): Lemma 3.6.1 (WSP+) follows from (SSP+), (as ) and (r ). With reference to Remark 2.3.1 one can observe that: Remark 3.6.1 (i) On the basis of (r ), the weak super-complementation principle (WSP+) has the following equivalent form:

  ∀x,y∈U y  x =⇒ ∃z∈U z  x ∧ z  y ∧ ∀u∈U (u  x ∧ u  y ⇒ u  z) . On the basis of (df ), sentence (WSP+) entails the first conjunct of the conjunction in the consequent of the implication in the new version. Furthermore, if u  x and u  y, then u = x, i.e., u  x, on the basis of our assumptions and (r ). Therefore, we are making use of (WSP+). Conversely, let us assume that y  x. Then, on the

88

3 “Existentially Involved” Theories of Parts ·· s PPOS

MEM··  s MPOS··

MEM·· +(∃pair sup)

MEM··

·· s MPOS

MEM·· +(c∃pair sup)

MEM+(‡)

CMM·· MEM·· +(∃pair sum)

CMM+(‡)

MEM+(∃pair sum)+(‡) Diagram 3.3 The lattice of non-degenerate theories related to sentences (WSP), (SSP), (c∃ ), (‡), (c∃pair sup), (c∃pair sum), (∃pair sup) and (∃pair sum)

basis of this new version, for some z we have: z  x and z  y. Hence z = x, i.e., z  x, by virtue of our assumptions and (r ). Furthermore, if u  x and u  y, then we are making use of (df ) and this assumed new version. (ii) It is evident from the above reasoning that we still have two equivalent versions of (WSP+):    ∀x,y∈U y  x =⇒ ∃z∈U z  x ∧ z  y ∧ ∀u∈U (u  x ∧ u  y ⇒ u  z) ,    ∀x,y∈U y  x =⇒ ∃z∈U z  x ∧ z  y ∧ ∀u∈U (u  x ∧ u  y ⇒ u  z) .

We swap only one occurrence of ‘’ for ‘’ in the consequent of (WSP+). 15

Fact 3.6.2 (i) (WSP+) and (2.1.3) entail the following principle :   ∀x,y∈U y  x =⇒ ∃z∈U ∀u∈U (u  z ⇔ u  x ∧ u  y) .

(WRP)

So (WRP) follows from (WSP+), (t ), (df ) and (df ). (ii) On the basis of (r ), the principle (WRP) has the following equivalent form:   ∀x,y∈U y  x =⇒ ∃z∈U ∀u∈U (u  z ⇔ u  x ∧ u  y) .

(WRP )

(iii) (WRP) and (r ) entail (WSP+). So (WSP+) follows from (WRP), (df ) and (df ). 15 In some of the literature, (WRP) is actually called the ‘weak remainder axiom’; see, e.g., (Cotnoir and Varzi 2018) and references therein.

3.6

“Super-Supplementation” Principles

89

Proof Ad (i) Firstly, directly from (WSP+) we obtain that for arbitrary x, y ∈ U such that y  x we have ∃z∈U ∀u∈U (u  x ∧ u  y ⇒ u  z). Secondly, from (WSP+) and (2.1.3) we obtain that for arbitrary x, y ∈ U such that y  x we have ∃z∈U ∀u∈U (u  z ⇒ u  x ∧ u  y). Ad (ii) As in Remark 3.6.1. Ad (iii) Obvious.

The strong super-supplementation principle we can also call the superstrong supplementation principle (see, e.g., Pietruszczak 2018; Gruszczy´nski and Pietruszczak 2014).16 Note that: Fact 3.6.3 (i) (SSP+) and (t ) entail the following remainder principle17 :   ∀x,y∈U x  y =⇒ ∃z∈U ∀u∈U (u  z ⇔ u  x ∧ u  y) .

(RP)

So (RP) follows from (SSP+), (t ), (df ) and (df ). (ii) (RP) and (r ) entail (SSP+). So (SSP+) follows from (RP),(df ) and (df ). (iii) (SSP+) and (RP) are equivalent in the class QOS. Proof Ad (i) Firstly, directly from (SSP+) we obtain that for arbitrary x, y ∈ U such that x  y we have ∃z∈U ∀u∈U (u  x ∧ u  y ⇒ u  z). Secondly, from (SSP+) and (t ) we obtain that for arbitrary x, y ∈ U such that x  y we have ∃z∈U ∀u∈U (u  z ⇒ u  x ∧ u  y). Ad (ii) Obvious. Ad (iii) Directly from (i) and (ii).

Fact 3.6.3 shows that (SSP+) informally says that if x is not an ingrediens of y, then we can find an ingrediens z of x which is not only exterior to y but also is the greatest amongst the ingredienses of x that are exterior to y. Fact 3.6.2 shows that (WSP+) informally says that if y is a part of x, then we can find a part z of x which is not only exterior to y but is also the greatest amongst the parts of x exterior to y. In both cases, this z may be recognised as the relative component of y in x (i.e., z = x − y in (Cotnoir and Varzi 2018), pp. 25–26 and p. 94, respectively; in the case (SSP+) see the formal analysis in the next point).

3.6.2 Polarised Strict Partial Orders “Plus” Let s PPOSp (“s PPOS plus”) be the theory arising out of s PPOS by swapping axiom (SSP) for its “plus version”, i.e., (SSP+). So we put: s PPOSp

16 In

:= s POS+(SSP+).

Sect. 3.7 we will show that principle (SSP+) has a connection with axiom M4 used by Grzegorczyk (1955). 17 In some of the literature, (RP) is actually called the ‘remainder axiom’ or ‘complementation’; see, e.g., (Varzi 2016; Cotnoir and Varzi 2018) and references therein.

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3 “Existentially Involved” Theories of Parts

First we will show that from condition (SSP+) we obtain the partial operation of mereological relative complementation; i.e., on condition that x  y, there will exist the mereological relative component of y with respect to x. This operation is partial, because if x  y then nothing will “arise” out of the relative component of y in x. For we do not have an “empty element”, which would “give back this nothing”. With condition x  y obtaining, however, x  y should be the greatest element of the non-empty set {z ∈ U : z  x ∧ z  y} ; that is, it should also be its least upper bound and mereological sum. With the partial operation of mereological relative complementation in hand, we will prove sentence (c∃ ) from the theory MEM. We therefore be able to construct a partial operation which, taking condition x  y to be satisfied, assigns to objects x and y their product x y.18 In any structure U,  from s PPOSp, for arbitrary x, y ∈ U let us define a set (which will be of use for the definition of the partial operation  ): R yx := {u ∈ U : u  x ∧ u  y}. Lemma 3.6.4 For arbitrary x, y ∈ U : (i) (ii) (iii) (iv)

R yx = ∅ ⇐⇒ x  y. If x  y then there exists exactly one z such that z is the greatest in the set R yx . If x  y then there exists exactly one z such that z ∈ R yx and z sup R yx . If x  y then there exists exactly one z such that z sum R yx .

Proof Ad (i) If x  y and u ∈ R yx , then u  y, by (t ). Hence, by virtue of (r ), we have a contradiction: ¬u  y. Whereas if x  y then R yx = ∅, by virtue of (SSP+). Ad (ii) If x  y then the set R yx has a greatest element, by virtue of (SSP+). In the light of (antis ), there can be at most one such element. Ad (iii) From (ii) and (B.2.13).

Ad (iv) The greatest element in the set R yx is its mereological sum. If U,  is non-degenerate, then we can define in U the partial operation mereological relative complementation by putting for all x, y ∈ U such that x  y 19 : x  y := ( z) z is the greatest element in R yx , = ( z) (z ∈ R yx ∧ z sup R yx ), = ( z) z sup R yx ,

(df )

ι ι ι

= ( z) z sum R yx . ι

18 The

results described above concerning the partial operations of mereological relative complementation and product make reference to facts established in Sect. B.6, which concerns Grzegorczykian lattices from (Grzegorczyk 1955). We derive these results in an analogous way (with an insignificant modification). Such operations are not introduced in (Grzegorczyk 1955). 19 We will find two such elements x and y when and only when U,  is non-degenerate.

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91

We obtain these identities from (B.2.13), Lemma 3.6.4, (Usup ) and (Usum ), respectively. Lemma 3.6.5 For arbitrary x, y ∈ U such that x  y: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

x  y  x and x  y  y. ∀z∈U (z  x  y ⇐⇒ z ∈ R yx ⇐⇒ z  x ∧ z  y). ∀z∈U (z  x  y ⇐⇒ ∃u∈U (u  x ∧ u  y ∧ u  z). x  y ⇐⇒ x = x  y ⇐⇒ x  x  y. x  y ⇐⇒ Rxx  y = ∅ ⇐⇒ x  x – y. If x  y, then x  (x  y)  y and x  (x  y)  x. If y  x then y = x  (x  y). ∀z∈U (z  x ∧ z  x  y =⇒ z  y). If x  y then ∀z∈U (z  x ∧ z  x  (x  y) =⇒ z  x  y).

Proof Across the whole proof we choose arbitrary x, y ∈ U such that x  y. Ad (i) Since x  y belongs to R yx , we have x  y  x and x  y  y. Ad (ii) Assume that z  x  y. Then z  x, by virtue of (i) and (t ). If z  y were the case, then there would exist a u such that u  z and u  y. Hence u  x  y, by virtue of (t ) and our assumptions. Therefore x  y  y, which contradicts (i). The converse follows from the definition of the operation of mereological relative complementation. Ad (iii) From Lemma 2.6.4, since x  y sum R yx . Ad (iv) By virtue of (i) and (antis ): x  x  y iff x = x  y. If x  y then x ∈ R yx , by virtue of (r ). Hence x  x  y, since x  y is the greatest in R yx . Conversely, if x = x  y then x ∈ R yx ; so x  y. Ad (v) We have: x  y iff x  x  y iff Rxx  y = ∅, by (iv) and Lemma 3.6.4(i), respectively. Ad (vi) Let x  y. Then, by (iv), x  x  y, i.e., we have defined x  (x  y). Firstly, assume for a contradiction that x  (x  y)  y. Then there exists a z such that (a) z  x  (x  y) and (b) z  y. From (a) and (ii) we have (c) z  x and (d) z  x  y. From (b) and (c) we have z ∈ R yx . Hence z  x  y, which contradicts (d). Secondly, x  (x  y) ∈ Rxx  y , and therefore x  (x  y)  x. The equality x = x  (x  y) is ruled out by the first case and the assumption that x  y. Ad (vii) Let y  x. Then x  y in addition, by virtue of (r ). Therefore, (vi) gives x  (x  y)  y. Were it the case that x  (x  y) = y, then y  x  (x  y), by virtue of (antis ). Hence, by virtue of (ii), either y  x or y  x  y. Both cases are, however, ruled out by our assumptions and (i). Ad (viii) Assume that z  x and z  y. Then, by virtue of (SSP+), there exists a u such that u  z and u  y. Therefore, by virtue of (t ), u  x too. Hence we therefore have u  x  y. We have therefore got z  x  y. Ad (ix) Let x  y. Then, by virtue of (iv), x  x  y and we have defined

x  (x  y). We can therefore apply (viii) with x := x and y := x  y. We may now obtain the following condition that will come in handy later:   ∀x,y,z∈U x  y ∧ z  y =⇒ x  z ∧ x  y  x  z .

(3.6.1)

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3 “Existentially Involved” Theories of Parts

To see this, by (t ), if x  y and z  y, then also x  z. So there are x  y and x  z. Were it the case that x  y  x  z, then—by virtue of (SSP)—there would exist a u such that (a) u  x  y and (b) u  x  z. By virtue of (a) and Lemma 3.6.5(ii) we have: u  x and u  y. Hence u  z. Hence, by virtue of (b) and Lemma 3.6.5(iii), there does not exist a v such that v  x, v  z and v  u. We therefore have a contradiction, as u  x, u  z and u  u. Applying Lemma 3.6.5, we prove: Lemma 3.6.6 In the class s PPOSp the following condition holds:   ∀x,y∈U x  y ∧ x  y =⇒ x  (x  y) inf {x, y} . Proof If x  y and x  y, then x  x  y, by Lemma 3.6.5(v). Moreover, from Lemma 3.6.5(vi), x  (x  y) is a lower bound of the set {x, y}. We will show that it is the greatest lower bound of that set. Assume for a contradiction that for some u we have: u  x, u  y and u  x  (x  y). Hence, by virtue of (SSP+), there exists a z such that z  u and z  x  (x  y). Therefore, by virtue of(t ), z  x and z  y. Hence z  y, by virtue of (r ). Applying Lemma 3.6.5(ix), we get z  x  y. Hence, by virtue of Lemma 3.6.5(ii), we have a contradiction z  y.

From Lemma 3.6.6 we get: Theorem 3.6.7 In the class s PPOSp sentences (c∃pair inf) and (c∃ ) hold. Proof Suppose that x  y. If x  y then x inf {x, y}. If x  y then x  (x  y) inf {x, y}, in the light of Lemma 3.6.6. Moreover, we apply Lemma 3.2.1.

Since (SSP+) entails (SSP) and (WSP+), from the above theorem we obtain: s PPOSp

⊆ s PPOS+(c∃ )+(WSP+) ⊆ s POS+(c∃ )+(WSP+).

We will later prove that these both inclusions are not proper, i.e., we have the equality of these three classes. Furthermore, we obtain: s PPOSp

 MEM.

Firstly, we use the fact that MEM = s PPOS+(c∃ ). So, by Lemma 3.6.6, we have s PPOSp ⊆ MEM, since (SSP+) entails (SSP). Secondly, Model 2.7 belongs to the class MEM. In this structure, we have 1  123 and 123  1, but there is no relative component of 1 in 123. Therefore, (SSP+) does not hold. Just as in the case of theory MEM on Sect. 3.2, we can define on the Cartesian product U × U a partial operation of mereological product such that for arbitrary x and y such that x  y we have: x y = ( z) z inf {x, y}, = ( z) z sum {u ∈ U : u  x ∧ u  y}. ι ι

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“Super-Supplementation” Principles

93

As before, x y is the greatest element of the set I(x)  I(y), which is equal to I(x y) (see (3.2.3)). So we also have: x y = ( z) z sum I(x)  I(y). ι

Furthermore, the operation now has the following property for arbitrary x, y ∈ U such that x  y : ⎧ ⎪ ⎨x x y= y ⎪ ⎩   x (x y)

if x  y, if y  x, in the remaining cases.

(3.6.2)

Obviously, in those remaining cases, x  y holds, and therefore x  y is defined. Furthermore, since we have assumed that x  y, then x  x  y, by Lemma 3.6.4(v); and so x  (x  y) is also defined. Notice that the second case includes also the situation in which y  x. In that case, x  y and x  y. Therefore, via Lemma 3.6.5(vii), we have y = x  (x  y). Earlier, we obtained the inclusion s PPOSp  MEM. We will now show that PPOSp ⊆ s MPOS. This is connected with the lemma below. s Lemma 3.6.8 (Pietruszczak 2000, 2018) (i) In the class s PPOSp conditions (‡∅ ) and (sum-sup) hold. (ii) In the class s PPOSp·· we have the identity sup = sum Proof Ad (i) For (‡∅ ): Assume that (a) x sup S, (b) S = ∅ and, for a contradiction, that (c) ¬x sum S. From (a) and (c) it follows that there exists a u such that (d) u  x and (e) ∀z∈S z  u. Assume for a contradiction that u = x. Then from z (a), (b) and (e), there exists a z 0 ∈ S such that z 0  x and z 0  x, which contradicts (r ). Hence u = x. So (f) u  x, by (d) and (antis ). From (f), by virtue of (SSP+), there exists a y such that (g) y  x, (h) y  u and for an arbitrary w: w  x and w  u entails w  y. From (a) and (e) it follows that for any z ∈ S we have z  x and z  u. From this and (i) we have ∀z∈S z  y, i.e., x  y, by virtue of (a). Therefore x = y, by virtue of (g) and (antis ). And this contradicts the conjunction: (d), (h) and (r ). For (sum-sup): The simple implication holds in the class s PPOS. We have, though, s PPOSp  s PPOS. The converse implication is given by (‡∅ ). Ad (ii) Since (SSP+) entails (SSP) and (‡∅ ) holds, we have s PPOSp ⊆ s MPOS. We therefore apply Lemma 3.1.1.

By the above lemma we see that structures from s PPOSp we can also call mereological strictly partially-ordered sets “plus” and the class of all these structures we can symbolise as: s MPOSp. From the above lemmas we have (cf. Sect. 3.3 of this chapter):

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3 “Existentially Involved” Theories of Parts s PPOSp ·· s PPOSp

⊆ MEM  s MPOS = MEM+(‡∅ ) = s MPOS+(c∃ ), ⊆ MEM  s MPOS·· = MEM+(‡) = s MPOS·· +(c∃ ).

The following fact shows that the above inclusions are proper. However—as we will show in Sect. 3.10 of this chapter—in the case where we restrict ourselves to finite structures, we obtain the identity of the classes. Fact 3.6.9 (WSP+), and hence a fortiori (SSP+), does not follow from the axioms of MEM+(‡)+(∃ 1 ). +

Proof Let IE+ be a set of all positive even numbers. Let E := 2IE \ {∅}. We now build a model whose universe is to be the family of sets composed of: • the singletons {0} and {1}; • all sets from the family E; • all sets of the form X  Y , where X ∈ 2{0,1} \ {∅} and Y ∈ E. The relation  in our model is to be the relation of proper inclusion for sets from U , that is, the relation  is to be the relation of inclusion of all sets from U . The set {0, 1}  IE+ is the unity in our model, i.e., it includes all sets in the universe. Therefore, as standard, we put 1 := {0, 1}  IE+ . V ERIFICATION OF CONDITION (c∃ ). Let x  y, i.e., ∃u∈U (u ⊆ x ∧ u ⊆ y). Then, obviously, x  y ∈ U , as the product x  y in the model is also the greatest lower bound of the pair {x, y}. Therefore, we can take x y := x  y. V ERIFICATION OF CONDITION (WSP). Let y  x. We are looking for a z such that z  x and z  y. If x \ y = {0, 1}, i.e., y ∈ E and x = {0, 1}  Y , where Y ∈ E, then we can take {0} or {1} as our z. In the remaining cases, when x \ y belongs to U , we will take x \ y as the set z. V ERIFICATION OF CONDITION (SSP). We see that our model belongs to the class MEM. Hence, by virtue of Theorem 3.2.3, it also satisfies condition (SSP). V ERIFICATION OF CONDITION (‡), i.e. sup ⊆ sum. To begin with, observe that if S = {{0}, {1}}, then S does not have a least upper bound. Furthermore, if S = {x} then x sum S. Take arbitrary x ∈ U and S ⊆ U such that x sup S. Since our model lacks the zero, then ∅ = S ⊆ I(x) and for arbitrary u ∈ U : if S ⊆ I(x) then x  u. We will show that then for each y  x there exist z ∈ S and u ∈ U such that u ⊆ z and u ⊆ y, i.e., z  y. We will now look at various possible cases. Let S ⊆ E. Then x = S ⊆ IE+ and it is evident that x sum S. Let 1 ∈ S. Since 1 is the greatest element in S, then x = 1 and also x sum S. Therefore, let us henceforth accept that 1 ∈ / S and S  E. Let IE+ ∈ S and to S belong some z 0 of the form {0, 1}  e0 ∈ S, where e0 ∈ E \ {IE+ }. Then x = 1, because 1 is the only upper limit of the pair {IE+ , z 0 }. Take an arbitrary y ∈ U , i.e., y ⊆ 1. If y ⊆ IE+ then y  IE+ . Similarly, if y ⊆ z 0 . So, take a y such that y  IE+ and y  z 0 . Therefore, y has the form v  e, where v ∈ 2{0,1} \ {∅} and e  e0 . Let s be a singleton from S such that s ⊆ v. Then {s}  z 0 and {s}  y, i.e., z 0  y.

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“Super-Supplementation” Principles

95

Therefore, we may henceforth accept that either IE+ ∈ / S or for each e ∈ E \ / S. {IE } we have {0, 1}  e ∈ Let S ⊆ {{0}, {1}}  E, S  E and S  {{0}, {1}}. Then x = bx  ex , where bx = {s ∈ {{0}, {1}} : s ∈ S} and ex = {e ∈ E : e ∈ S}. Assume that y  x. We will consider the three possible cases. (1) If y ∈ {{0}, {1}}, then y ⊆ bx , i.e., y ∈ S and y  y. (2) If y ∈ E, then y ⊆ {e ∈ E : e ∈ S}. Let n ∈ y, i.e., {n}  y. Then there is an en ∈ S such that n ∈ en , i.e., {n}  en . Hence en  y. (3) If y = b y  e y , for some b y ∈ 2{0,1} \ {∅} and e y ∈ E, then b y ⊆ bx and e y ⊆ {e ∈ E : e ∈ S}. Hence b y ∈ S and b y  y, i.e., b y  y. Therefore, in each case, we have x sum S. Let S ⊆ {{0}, {1}}  E  {{0, 1}  e : e ∈ E \ {IE+ }} and S  {{0}, {1}}  E. Then there exists in S a set z 0 of the form {0, 1}  e0 , where e0 ∈ E \ {IE+ }. In this case we have x = {0, 1}  {e ∈ E : e ∈ S ∨ {0, 1}  e ∈ S}. Suppose that y  x. We will consider the three possible cases. (1) If y ∈ {{0}, {1}}, then y  z 0 , i.e., y  z 0 . (2) If y ∈ E, then y ⊆ {e ∈ E : e ∈ S ∨ {0, 1}  e ∈ S}. Let n ∈ y, i.e., {n}  y. Then there is an en ∈ E \ {IE+ } such that n ∈ en and either z n := en ∈ S or z n := {0, 1}  en ∈ S. We have {n}  z n . Hence z n  y. (3) If y = b y  e y , for some b y ∈ 2{0,1} \ {∅} and e y ∈ E, then e y ⊆ {e ∈ E : e ∈ S ∨ {0, 1}  e ∈ S}. To finish, we proceed as before, just swapping ‘y’ for ‘e y ’. Therefore, in each case, we have x sum S. S ⊆ {{0}, {1}}  E  {b  e : b ∈ 2{0,1} \ {∅} ∧ e ∈ E \ {IE+ }} and S  {{0}, 2{0,1} \ {1}}  E. Then there exists in S a set z 0 of the form b0  e0 , where b0 ∈ + bx  ex , where bx = {b ∈ {∅} and e0 ∈ E \ {IE }. In this case we have x = 2{0,1} \ {∅} : b ∈ S  S ∨ ∃e∈E b  e ∈ S} and ex = {e ∈ E : e ∈ S ∨ ∃b b  e ∈ S}. Suppose that y  x. We will consider the three possible cases. (1) If y ∈ {{0}, {1}}, then y ⊆ bx , i.e., y ∈ S and y  y. (2) If y ∈ E, then y ⊆ {e ∈ E : e ∈ S ∨ ∃b b  e ∈ S}. Let n ∈ y, i.e., {n}  y. Then there is an en ∈ E such that n ∈ en and either z n := en ∈ S or for some b, z n := b  en ∈ S. We have {n}  z n . Hence z n  y. (3) If y = b y  e y , for some b y ∈ 2{0,1} \ {∅} and e y ∈ E, then e y ⊆ {e ∈ E : e ∈ S ∨ {0, 1}  e ∈ S}. To finish, we proceed as before, just swapping ‘y’ for ‘e y ’. Therefore, in each case, we have x sum S. D EMONSTRATION OF THE FALSITY OF CONDITION (WSP+). We have IE+  1, but in U there does not exist a z such that z  IE+ and for arbitrary u ∈ U : if u  z and u  z, then u  z. To see this, the only elements exterior with respect to IE+ are the singletons {0} and {1}, but neither of them satisfies condition (WSP+). D EMONSTRATION OF THE FALSITY OF CONDITION (SSP+). Lemma 3.6.1 says that (WSP+) follows from (as ), (r ) and (SSP+). Since our model falsifies (WSP+), it also falsifies (SSP+).

+

We have therefore obtained the following: s PPOSp  MEM+(‡∅ ) = s MPOS+(c∃ ), ·· ·· s PPOSp  MEM+(‡) = s MPOS +(c∃ ).

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3 “Existentially Involved” Theories of Parts

3.6.3 Unity in Structures from the Class s PPOSp We will now take a look at the problem of the existence of a unity in structures from s PPOSp. The issue is that a unity may not exist in finite structures from that class. This is shown by both Models 3.4 and 3.5. Fact 3.6.10 (∃ 1 ) does not follow from the axioms of s PPOSp. More precisely, there is already a structure in the class s PPOSpfin without unity. Let s PPOSp1 be the subclass of the class s PPOSp of structures with unity. On the strength of Fact 3.6.10 we have: s PPOSp1

 s PPOSp.

In all non-degenerate structures from s PPOSp1, we can define a partial operation of mereological complementation for objects distinct from the unity. That is to say, if x = 1 then 1  x, by virtue of (antis ), since x  1. Therefore, by virtue of (SSP+), one may apply the operation of mereological relative complementation to get 1  x. We take an arbitrary x distinct from 1 and therefore: – x := 1

 x.

(df –)

So for any x distinct from 1, we say that – x is the mereological complement of x. From the definition of the concepts we have used we get:   ∀x∈U x = 1 =⇒ – x sum {u ∈ U : u  x} .

(3.6.3)

If x = 1 then 1  x, i.e., 1  x sum {u ∈ U : u  1 ∧u  x}. Therefore, we use the fact that ∀u∈U u  1. Below, we will make use of the following fact several times, which we obtain from (3.6.3) and Lemma 2.6.4.   ∀x∈U x = 1 =⇒ O(– x) = {y ∈ U : ∃z∈U (z  x ∧ y  z)} .

(3.6.4)

Consider: O(– x) = O[{u ∈ U : u  x}] = {y ∈ U : ∃z∈{u∈U :ux} y  z} = {y ∈ U : ∃z∈U (z  x ∧ y  z)}. We will show that:   (3.6.5) ∀x∈U x = 1 =⇒ – x = 1 . Assume for a contradiction that x = 1 and – x = 1, i.e., 1 sum {u ∈ U : u  x} and O(1) = {y ∈ U : ∃z∈U (z  x ∧ y  z)}, by virtue of (3.6.3) and (3.6.4). Since x ∈ O(1), for some z we have a contradiction: x  z and z  x. The partial unitary operation of mereological complementation is therefore defined on the set {u ∈ U : u = 1} and takes values in this set. Furthermore, we have:

3.6

“Super-Supplementation” Principles



97

  ∀x∈U x = 1 =⇒ x  – x ,



∀x∈U x = 1 =⇒ (y  x ⇔ y  – x) ,   ∀x,y∈U x = 1 = y =⇒ (x  y ⇔ – y  – x) ,   ∀x∈U x = 1 =⇒ – – x = x .

(3.6.6) (3.6.7) (3.6.8) (3.6.9)

For (3.6.6): If x = 1, then O(– x) = {y ∈ U : ∃z∈U (z  x ∧ y  z)}, by (3.6.4). Were it the case that x  – x, then we would have a contradiction. For (3.6.7): If x = 1 and y  x, then y  – x, by (3.6.3). Conversely, if y  – x, then O(y) ⊆ O(– x). Thus, x ∈ / O(y), since x ∈ / O(– x), by (3.6.6). For (3.6.8): suppose that x = 1 = y. Then: x  y iff O(x) ⊆ O(y) (by (ext  +) and (t )) iff E(y) ⊆ E(x) iff I(– y) ⊆ I(– x) (by (3.6.7)) iff – y  – x.20 For (3.6.9): by virtue of (3.6.6) and (3.6.7), for any x = 1 we have x  – – x. Applying this to – x, we get – x  – – – x. Hence – – x  x, by virtue of (3.6.8). Therefore x = – – x, by virtue of (antis ).

3.6.4 The Theory “MEM Plus” (Equals to s PPOSp) Let MEMp (“MEM plus”) be the theory arising out of MEM by swapping axiom (WSP) for its “plus version”, i.e., (WSP+). So we put: MEMp := s POS+(WSP+)+(c∃ ). Since (WSP), and so in consequence also (WSP+), entails (irr  ), we have: MEMp = (t )+(WSP+)+(c∃ ). Notice that Lemma 3.6.1 and Theorem 3.6.7 show that: s PPOSp

⊆ MEMp.

We will show that the converse inclusion obtains too. This will result from the following theorem. Theorem 3.6.11 (SSP+) holds in the class MEMp. Proof Assume that x  y. If x  y then, thanks to (r ), we can take z := x. Assume therefore that x  y. Then, by virtue of (c∃ ), there exists an element x y. Observe that x y  x. From the definition, we have: x y  x and x y  y; but were it the case that x = x y then it would be the case that x  y, which contradicts our main assumption. 20 We

and 1

can also get the simple implication from (3.6.1). If x = 1 = y and x  y, then 1  x, 1  y  y  1  x.

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3 “Existentially Involved” Theories of Parts

Therefore, by virtue of (WSP+), there exists a z such that z  x, z  x y and for an arbitrary u: if u  x and u  x y, then u  z. Hence z  y, by virtue of (3.2.5). Furthermore, if u  x and u  y, then it is ruled out that u = x.

From the above theorem we obtain the inclusion MEMp ⊆ s PPOSp. So s PPOSp

= MEMp.

The results obtained so far in this chapter concerning classes of structures of theories are portrayed in Diagram 3.4. By changing the name of a given class in this diagram to the name of a corresponding class of non-degenerate structures we get a diagram of classes of non-degenerate structures.

3.6.5 Extensions of Theory s PPOSp via Conditions for the Existence of Mereological Sums for Pairs We will now occupy ourselves with cases where we will add to the theory s PPOSp one of the conditions (c∃pair sum), (∃pair sum). Notice that, in the light of Lemma 3.6.8, in structures from the class s PPOSp— as in structures from the class s MPOS (see Sect. 3.5)—the relations sup, sum and fu are equal. Thus, in the class s PPOSp we do not distinguish between algebraic sums and mereological sums for pairs and we obtain the following identities: s PPOSp+(c∃pair sup)

= s PPOSp+(c∃pair sum) = s PPOSp+(c∃pair fu)

s PPOSp+(∃pair sup) = s PPOSp+(∃pair sum) = s PPOSp+(∃pair fu).

By adding either (c∃pair sum) or (∃pair sum) to s PPOS·· , we dispose of degenerate models. Therefore, we will get—analogously to those identities and inclusions given above and below—identities and inclusions that differ in just this way: that ‘(‡)’ appears instead of (‡∅ )’, and where in the name of a class we do not have the symbol ‘(‡)’ we have instead the superscript ‘·· ’. To begin with, let us note that: Fact 3.6.12 (∃pair sup) does not follow from the axioms of s PPOSp·· +(c∃pair sum). Proof Model 3.5 belongs to s PPOSp·· +(c∃pair sum), but (∃pair sup) is false in it, because the set {1, 2} has no least upper bound.

We have therefore obtained the following results: s PPOSp+(∃pair sum)

= s PPOSp+(∃pair sup)  s PPOSp+(c∃pair sup) = s PPOSp+(c∃pair sum),

3.6

“Super-Supplementation” Principles

and s PPOSp

··

99

+(c∃pair sum)  s POS+(∃pair sup).

Fact 3.6.13 (c∃pair sup) does not follow from the axioms of s PPOSp·· .21 Proof Let RO be a family of all regular open sets included in the interval (0, 2) of real numbers, RO+ := RO \ {∅} and RO+ := {X \ {1} : X ∈ RO+ }.22 We put: \{1}

U := RO+  {1}}  RO+ , \{1}  := {x, y ∈ U × U : x  y}, i.e.,  is the relation of inclusion in the family U and : x  y iff x  y = ∅. We will show that in U,  condition (SSP+) holds. To this end, take arbitrary x, y ∈ U such that x  y and consider the various cases. If x  y = ∅, then x satisfies the appropriate condition. We therefore assume in addition that x  y = ∅. x, y ∈ RO+ : then x satisfies the appropriate condition. x ∈ RO+ and y = {1}: then x \ {1} satisfies the appropriate condition. \ RO+ and y ∈ RO+ : then for some z ∈ RO+ we have 1 ∈ z and x = x ∈ RO+ \{1} z \ {1}. If 1 ∈ y, then we take z − y. If instead 1 ∈ / y, then we take (z − y) \ {1}, since z − y ∈ RO+ . \ RO+ : then for some z ∈ RO+ we have 1 ∈ z and y = x ∈ RO+ and y ∈ RO+ \{1} z \ {1}. We take x − z. + + x, y ∈ RO+ −1 \ RO : then for some z, w ∈ RO we have 1 ∈ z  w, x = z \ {1} and y = w \ {1}. We take z − w. Sentence (c∃pair sup) is false in the structure, since for any x ∈ RO+ such that 1∈ / x, the sets {1} and x have upper bounds but have no least upper bound.

We have therefore obtained the following results: s PPOSp+(c∃pair sum)

= s PPOSp+(c∃pair sup)  s PPOSp,

 MEM+(c∃pair sup)+(‡∅ ) = CMM+(‡∅ ), PPOSp  s POS+(c∃pair sup). s

s PPOSp+(c∃pair sum)

Of course, Fact 3.6.13 also shows that sentences (c∃pair sum), (∃pair sum) and (∃pair sup) do not follow from the axioms of s PPOSp, because these sentences are stronger than (c∃pair sup). Hence it follows that the class s PPOSp is also not included in the classes in Diagrams 3.1 and 3.2 which are included in the classes s PPOS+(c∃pair sup) and MEM+(c∃pair sup), respectively. Moreover, we also get an analogous result 21 This

fact cannot be proven by using finite models, as we will show in Sect. 3.10. a topological space U, Int a set X is regularly open iff X = IntCl(X ), where Cl(X ) = U \ Int(U \ X ), i.e., X is equal to the interior of its closure. The family of regularly open sets in U, Int creates a complete Boolean algebra with the operations X · Y := X  Y , X + Y := IntCl(X  Y ) and −X := Int(U \ X ) = U \ Cl(X ), in which ∅ is the zero and U is the unity. We can therefore put X − Y := X · −Y . We have X − Y = ∅ iff X ⊆ Y . 22 In

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3 “Existentially Involved” Theories of Parts

for classes of non-degenerate structures. So, for example, the class s PPOSp·· is also not included in the classes in Diagram 3.3 which are included in the class MEM·· +(c∃pair sup). We establish the proof of the fact below with the help of an infinite model which is an appropriate “compilation” of the models used in Facts 3.5.2 and 3.6.9.23 Fact 3.6.14 (WSP+), and hence a fortiori (SSP+), does not follow from the axioms of MEM+(∃pair sum)+(‡)+(∃ 1 ). Proof Let IE be a set of all positive even numbers and let IO be a set of all odd IO numbers. Let E := 2IE \ {∅} and O := 2fin \ {∅}, i.e., O is a family of all non-empty finite subsets of the set IO. We now build a model whose universe U is to be the family of sets composed of: • all sets of the family O; • all sets of the family E; • all sets of the form X  Y , where X ∈ O  {IO} and Y ∈ E. The relation  in the model is to be the relation of proper inclusion of the sets from U ; that is, the relation  is to be the relation of inclusion of sets from U . The set of natural numbers IN (= IO  IE) is the unity in the model; i.e., it contains all sets in the universe. As is standard, we put 1 := IN. V ERIFICATION OF CONDITION (c∃ ). Let x  y, i.e., ∃u∈U (u ⊆ x ∧ u ⊆ y). It is then obvious that x  y ∈ U , as is x  y inf {x, y} in the model. We can therefore let x y := x  y. V ERIFICATION OF CONDITION (∃pair sum). For all x, y ∈ U , we have x  y ∈ U , and also x  y sum {x, y}. V ERIFICATION OF CONDITION (WSP). Let y  x. We are looking for a z such that z  x and z  y. If the set x \ y belongs to U then we take it as our z. This will always be the case except for those cases in which x \ y = IO, where Y ∈ E, a y ∈ E. In those cases, we take an arbitrary set from O as our z. C ONDITION (SSP). We see that the model belongs to the class MEM. Hence, by virtue of Theorem 3.2.3, it also satisfies condition (SSP). V ERIFICATION OF CONDITION (‡), i.e., the inclusion sup ⊆ sum. We take arbitrary x ∈ U and S ⊆ U such that x sup S. Since the model lacks the zero, we have ∅ = S ⊆ I(x) and for any arbitrary u ∈ U we obtain: if S ⊆ I(x) then x  u. We will show that for each y  x there are z ∈ S and u ∈ U such that u ⊆ z and u ⊆ y, i.e., z  y. We will now look at the various cases. Let S ⊆ E. Then x = S ∈ E and x sum S. The case is similar if S is a finite family included in O. Then x = S ∈ O and x sum S.24 Let 1 ∈ S. Since 1 is the greatest element in S, then x = 1 and also x sum S. We therefore henceforth accept that 1 ∈ / S and that S is not a finite family included in O. 23 The

fact below is obviously a strengthening of Fact 3.6.9. However, the model used in the proof of Fact 3.6.9 is simpler than the one used below. 24 If S is an infinite family included in O, then S has no least upper bound.

3.6

“Super-Supplementation” Principles

101

Let IE ∈ S and to S belong some z 0 of the form IO  e0 ∈ S, where e0 ∈ E \ {IE}. Then x = 1, since 1 is the only upper bound of the pair {IE, z 0 }. Take any y ∈ U , i.e., y ⊆ 1. If y ⊆ IE then y  IE. The case is similar if y ⊆ z 0 . We take, therefore, a y such that y  IE and y  z 0 . So y has the form v  e, where v ∈ O and e  e0 . Let s be a singleton from O such that s ⊆ v. Then s  z 0 and s  y, i.e., z 0  y. Therefore, we may henceforth accept that either IE ∈ / S or for each e ∈ E \ {IE} we have IO  e ∈ / S. Let S ⊆ O  E, S  E and S  O. Then x = ox  ex , where ox = (S  O) and ex = (S  E). Suppose that y  x. We will now examine the three possible an on ∈ S  O cases. (1) If y ∈ O then y ⊆ ox . Let n ∈ y, i.e., {n}  y. Then there is such that n ∈ on , i.e., {n}  on . Hence on  y. (2) If y ∈ E then y ⊆ (S  E). Let n ∈ y, i.e., {n}  y. Then there is an en ∈ S such that n ∈ en , i.e., {n}  en . Hence en  y. (3) If y = o y  e y , for some o y ∈ O  {IO} and e y ∈ E, then o y ⊆ ox and e y ⊆ (S  E). We proceed as in the cases (1) and (2). Therefore, in each case we have x sum S. Let S ⊆ O  E  {IO  e : e ∈ E \ {IE}} and S  O  E. Then to S belongs some set z 0 of the form IO  e0 , where e0 ∈ E \ {IE}. In this case we have x = IO  {e ∈ E : e ∈ S ∨ IO  e ∈ S}. Suppose that y  x. We will examine the three possible cases. (1) If y ∈ O then we proceed as we did before. (2) If y ∈ E then y ⊆ {e ∈ E : e ∈ S ∨ IO  e ∈ S}. Let n ∈ y, i.e., {n}  y. Then there is an en ∈ E \ {IE} such Hence that n ∈ en and either z n := en ∈ S or z n := IO  en ∈ S. We have {n}  z n . z n  y. (3) If y = o y  e y , for some o y ∈ O  {IO} and e y ∈ E, then e y ⊆ {e ∈ E : e ∈ S ∨ IO  e ∈ S}. We proceed as we did before, just changing ‘y’ to ‘e y ’. Therefore, in each case we have x sum S. Let S ⊆ O  E  {b  e : b ∈ O  {IO} ∧ e ∈ E \ {IE}} and S  O  E. Then to S belongs some set z 0 of the form o0  e0 , where o0 ∈ O  {IO} and e0 ∈ E \ {IE}. In this case we have x = ox  ex , where ox = {b ∈ O  {IO} : b ∈ O  S ∨ ∃e∈E b  e ∈ S} and ex = {e ∈ E : e ∈ S ∨ ∃b b  e ∈ S}. Suppose that y  x. We will examine the three possible cases. (1) If y ∈ O, then y ⊆ ox and we proceed as we did before. (2) If y ∈ E, then y ⊆ {e ∈ E : e ∈ S ∨ ∃b b  e ∈ S}. Let n ∈ y, i.e., {n}  y. Then there is an en ∈ E such that n ∈ en and either z n := en ∈ S or for some b we have z n := b  en ∈ S. We get {n}  z n . Hence z n  y. (3) If y = o y  e y , for some o y ∈ O  {IO} and e y ∈ E, then e y ⊆ {e ∈ E : e ∈ S ∨ IO  e ∈ S}. We proceed as we did before, just changing ‘y’ to ‘e y ’. Therefore, in each case we have x sum S. D EMONSTRATION OF THE FALSITY OF CONDITION (WSP+). We have IE  1, but in U there does not exist a z such that z  IE and for any u ∈ U : if u  z and u  z, then u  z. To see this, the only elements in U that are exterior with respect to IE are sets from O, but none of them satisfies the condition. D EMONSTRATION OF THE FALSITY OF CONDITION (SSP+). Lemma 3.6.1 says that (WSP+) follows from (as ), (r ) and (SSP+). Since the model falsifies condition (WSP+), it also falsifies (SSP+).

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3 “Existentially Involved” Theories of Parts

We therefore obtain the following results:  MEM+(∃pair sum)+(‡∅ ), MEM+(∃pair sum)+(‡∅ )  s PPOSp, s PPOSp+(∃pair sum)

MEM+(∃pair sum)+(‡)  s PPOSp. Thus, all classes including the class MEM+(‡∅ )+(∃pair sum) are also not included in s PPOSp. As in the case of Facts 3.5.2 and 3.6.9—as we will show in Sect. 3.10—in the case where we restrict ourselves to finite structures, we obtain the appropriate identities of classes. By virtue of Lemma 3.6.8, we can axiomatise the product of the classes s PPOSp  MEM+(‡∅ )+(∃pair sum) in the following way: s PPOSp

 MEM+(‡∅ )+(∃pair sum) = s PPOSp+(∃pair sum) = s POS+(c∃ )+(∃pair sum)+(WSP+).

The results obtained so far concerning classes of structures of theories are portrayed in Diagram 3.4 (cf. also in Diagram 3.1). As we have already said, by changing the name of a given class in this diagram to the name of a corresponding class of non-degenerate structures (i.e., with the superscript ‘·· ’ or by changing ‘(‡∅ )’ to ‘(‡)’) we get a diagram of classes of non-degenerate structures. Many of the results shown in the diagram can also be established using only finite models (including those we have already proposed.)

3.7 Grzegorczykian Mereological Structures 3.7.1 A Problem of Elementary Mereology One of the theories presented in (1955) Grzegorczyk calls “elementary mereology”. He connects it with another theory presented in that same work, which he calls “a theory of Boolean algebras with zero-element but without unity-element” (Grzegorczyk 1955, p. 92). Since the second theory does not guarantee the existence of a unity, then—in our opinion—it does not deserve the title of an elementary theory of Boolean algebra. It is for this reason that, in Appendix B.6, we have named models of this second theory Grzegorczykian lattices. We will now analyse the first of the two theories, one that does deserve the title of elementary mereology. Grzegorczyk’s choice of name was clearly guided by a connection that holds between classical mereology and the theory of complete Boolean algebras, a connection which he made plain in the following way25 : 25 Classical

mereological structures will be discussed in Sect. 3.9.

3.7

Grzegorczykian Mereological Structures

103 s PPOS

MEM  s MPOS

MEM+(∃pair sup)

MEM

s MPOS

MEM+(c∃pair sup)

MEM+(‡∅ )

CMM

MEM+(∃pair sum)

CMM+(‡∅ )

s PPOSp

MEM+(∃pair sum)+(‡∅ ) s PPOSp+(c∃pair sum)

s PPOSp+(∃pair sum)

Diagram 3.4 The lattice of theories related to sentences (WSP), (SSP), (c∃ ), (‡∅ ), (c∃pair sup), (c∃pair sum), (∃pair sup), (∃pair sum), (WSP+) and (SSP+)

A classical mereological structure is that and only that relational structure which arises from a non-degenerate complete Boolean lattice (Boolean algebra) through the removal of the zero and the “irreflexivisation” of the relation ≤. Remark 3.7.1 This fits with the case we have considered where the primary concept is the irreflexive relation . If instead we took the primary relation to be the reflexive relation , then we would write: A classical mereological structure is that and only that relational structure which arises from a non-degenerate complete Boolean lattice (Boolean algebra) through the removal of the zero.26



It is clear that Grzegorczyk considered there to be a certain analogy between Boolean algebra and mereology—and such an analogy assuredly can be drawn. But is it a complete one? Let us introduce this question by considering some history. First of all, there was the theory of Boolean algebras, expressed in the form of a certain number of so-called Boolean identities, which expressed relationships holding between three operations: sum, product and complement; and two elements: zero and unity. The sum of a finite number of elements is always their least upper bound and the product their greatest lower bound. Added are then the requirements that the sum and product also exist for an arbitrary number of elements (i.e., pos-

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3 “Existentially Involved” Theories of Parts

sibility infinitely many). Such a generalised sum (resp. product) corresponds to the least upper (resp. greatest lower) bound of those elements. It suffices to satisfy one of the requirements because the one would then entail the other.27 In this way, the theory of complete Boolean algebras emerged, which could not be expressed in the form of an elementary theory. In sum: the theory of complete Boolean algebras arose as a result of the addition to the Boolean identities condition (complsup ) (see Sect. B.7) guaranteeing the existence of a least upper bound for an arbitrary subset of the universe (formally: ∀S∈2U ∃x∈U x sup S). With mereology, the picture is different. Originally, Le´sniewski’s theory was expressed in a specific language.28 In his (1929; 1956a), Tarski expressed Le´sniewski’s mereology in the language of algebraic structures as a non-elementary theory with a primitive concept corresponding to the relation . Tarski assumed of that relation that it was transitive and that the universe of discourse U satisfied the following condition: (∃1 sum) ∀S∈2U \{∅} ∃1x∈U x sum S, i.e., that for each non-empty subset there exists exactly one element which is the mereological sum of the elements of that set. Remark 3.7.2 (i) By appealing to his work on complete Boolean lattices, Tarski (1935, 1956b) proved that the reflexivity and antisymmetry of the relation  follows from (t ) and (∃1 sum).29 Structures of the form U,  satisfying two conditions (t ) and (∃1 sum) we shall call Tarskian mereological structures. The class of all such structures we signify with ‘TMS’.30 This class is equal to the class of all partiallyordered sets satisfying condition (∃1 sum), i.e., TMS := (t )+(∃1 sum) = POS+(∃1 sum). It is obvious that from (∃1 sum) we obtain: ∀S∈2U \{∅} ∃x∈U x sum S.

(∃sum)

27 We have the reciprocal definability of the greatest lower and the least upper bounds. It is, however, essential to assume that the least upper (resp. greatest lower) bound should exist for an arbitrary set of elements. A greatest lower (resp. least upper) bound of some infinite set may, however, correspond to the least upper (resp. greatest lower) bound of a finite set of elements. We do not, therefore, have the reciprocal definability of the operations of sum and product, because they are operative only on finite sets of elements. 28 Le´sniewski’s original mereology was presented in Sect. 1.4. 29 An elementary proof of this fact can be found (for example) in (Pietruszczak 2000, p. 125); cf. also (Pietruszczak 2018, pp. 155–156). 30 In (Pietruszczak 2000, 2018), this theory is signified by ‘TS’.

3.7

Grzegorczykian Mereological Structures

105

Furthermore, (Usum ) follows from (t ) and (∃1 sum). Since (r ) follows from (t ) and (∃1 sum), then from this and (df sum) there does not exist a mereological sum of the empty set.31 It is also clear that from (Usum ) and (∃sum) we obtain (∃1 sum). We therefore have the following identity: TMS = (t )+(Usum )+(∃sum) = POS+(Usum )+(∃sum). (ii) Although axioms (∃1 sum) and (∃sum) are conditional, we have not highlighted this in our symbolisation of them, because the need to accept the condition assumed in them is obvious. Otherwise, were we to accept them in their unconditional form, the theory would be contradictory, as then it would follow from them that there simultaneously exists and does not exist the sum of the empty set. (iii) If we accept  as a primitive relation, then we obtain condition (r ) from (df ). If we base our axiomatisation of the mereology on (Usum ), (∃sum) or (∃1 sum), then we will have to assume that the relation strictly partially orders the universe of discourse. To express this, Le´sniewski adopted conditions (t ) and (as ).32 With reference to his and Tarski’s works, we may express classical mereology in Le´sniewski’s sense as a theory of structures of the form U,  which are strictly partially- ordered sets satisfying two conditions (Usum ) and (∃sum). All such structures we shall call Le´sniewskian mereological structures and their class we signify with ‘LMS’, i.e.33 : LMS := s POS+(Usum )+(∃sum). Since (r ) follows from (df ) and hence the mereological sum of the empty set does not exist, then (Usum ) follows from (df ). From this and the observations made in (i) we obtain the following identity: LMS = s POS+(∃1 sum). We will be taking a closer look at structures from LMS in Sect. 3.9. (iv) Since the classes of structures POS and s POS are elementarily definitionally equivalent, it follows from the identities given above that TMS and LMS are also elementarily definitionally equivalent. The elementarily definitional equivalence in this case is one in the sense of (Szmielew 1981, p. 8) (cf. (Szmielew 1983)). We discuss this issue in detail in (Pietruszczak 2000, pp. 103–105) or (Pietruszczak 2018, pp. 129–131).

31 The

problem here was that (∃1 sum) postulates the uniqueness of the relation sum only for nonempty sets, but (Usum ) for all sets. Since, however, the relation sum does not overlap ∅, condition (Usum ) is also satisfied. 32 Obviously, instead of the last condition we could use (irr ) whilst also admitting any of the  conditions that define the class s POS. 33 In (Pietruszczak 2000, 2018), this theory is signified by ‘MS’.

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3 “Existentially Involved” Theories of Parts

Axioms (∃1 sum) and (∃sum) are modelled on schema (∃ccP ), which guarantees the existence of a collective class of Ss, if some S exists. One can see that (∃sum) guarantees the “completeness”34 of classical mereology and recognises it as its “immanent feature”. Observe that (∃sum) guarantees the existence of a mereological sum of the universe U . It is the upper bound of the universe U . Hence we obtain: Lemma 3.7.1 (∃ 1 ) follows from (∃sum). Assuming (Usum ), it follows that there is only one sum of the universe. We will get the same result from (antis ) (there exists at most one upper bound of the universe.) In both cases, the sum of the universe is the unity, i.e., 1 sum U . We may therefore recognise this as a “repeated feature” of classical mereology. If we remove condition (complsup ) from the axioms for the theory of complete Boolean algebras, then we obviously get the axioms for the elementary theory of Boolean algebras.35 If, however, we remove condition (∃1 sum) from the axioms for classical mereology, we only get an elementary theory of transitive relations and not some sort of elementary mereology. And so it is at this point that the analogy between Boolean algebra and mereology breaks down. Let us, however, add that in his (1935; 1956b), Tarski—influenced by his research into mereology—formulated an axiomatisation of the theory of complete Boolean lattices composed of two axioms. The first was (t ) and the second corresponds to our (∃1 sum): it is condition ().36 Tarski showed the connection between his axiomatised theory of complete Boolean algebras and classical mereology in one of his footnotes.37 We have discussed this connection in detail in (Pietruszczak 2000, pp. 91–92; Pietruszczak 2018, pp. 113–114). Let us now observe that, when we remove () from Tarski’s axiomatisation of complete Boolean lattices once again there remains just an axiomatisation of transitive relations. However, there is a fundamental difference, as far as mereology is concerned. In the case of Boolean algebras, there is an “archetype” of the elementary theory to which we add condition (complsup ) to obtain the theory of complete algebras. The axiomatisation in (Tarski 1935) is the only other axiomatisation of the latter theory. In the case of mereology, there is no such “archetype” of it in its elementary version. It is undoubtedly to his credit that Grzegorczyk managed to find such an elementary version of mereology which, when axiom (∃1 sum) is added to it, becomes classical mereology. His version does not, however, guarantee the existence of a unity.38 However, the addition of condition (∃ 1 ). does guarantee this. 34 Another

of Le´sniewski’s axiomatisations guarantees the uniqueness of this collective class. We use the word ‘completeness’ in a metaphorical sense because there is no mereological sum for the empty set. 35 The first set of axioms arises simply through the addition of condition (compl sup ) to the second set. 36 We discussed this issue in detail there. 37 In (Tarski 1956b), this is footnote 1 on pp. 333–334. 38 We will show this later when we enter into a more detailed discussion of Grzegorczyk’s theory.

3.7

Grzegorczykian Mereological Structures

107

In order to maintain the analogy between his “template” of elementary mereology and the elementary theory of Boolean algebra, Grzegorczyk removed from the latter the possibility of obtaining the existence of a unity.39 He was mistaken, though, in writing (Grzegorczyk 1955, p. 94): “mathematicians [. . . ] would call mereology, from a purely formal point of view, the theory of Boolean algebra of elements different from zero.” Firstly, there is no “Boolean algebra of elements different from zero”. Secondly, classical mereology is the name we are giving to the theory of structures that arise from complete Boolean algebras when the zero has been removed from them. Thirdly, each Boolean algebra has a unity, but in Grzegorczyk’s elementary theory we cannot prove the existence of a unity. Therefore, the order in which theories connected with mereology arise is different to the order for Boolean algebra, where we went from an elementary theory to a non-elementary theory. In the case of mereology, we depart from a nonelementary theory. In addition, there exists an elementary theory which greatly resembles Le´sniewski’s original mereology. It may even be said that it yet more greatly resembles it than Tarski’s non-elementary theory. That is to say, from a formal point of view, Le´sniewski’s theory uses the axiom schema (∃ccP ). We can capture this schema with the help of a schema for certain axioms that are expressed in a firstorder language with identity. This language will have certain two-place predicates: ‘’, ‘’, ‘’ and ‘’, the first of them being primitive and satisfying axioms (as ) and (t ), as in Le´sniewski’s original theory. The remaining three will satisfy counterparts of conditions (df ), (df ) and (df ). Instead of schema (∃ccP ) we say that for an arbitrary formula ϕ(u) of this theory (with at least one free variable ‘u’) we have the following axiom:

    ∃u ϕ(u) =⇒ ∃x ∀u ϕ(u) ⇒ u  x ∧ ∀y y  x ⇒ ∃u (ϕ(u) ∧ u  y) . The formula after the quantifier ‘∃x ’ says that “x is a collective class of objects satisfying ϕ(u).” Therefore, we may call this x the “mereological sum of all ϕs”, by analogy with the “mereological sum of all Ss” which is given by schema (∃ccP ). By putting this in the language of the theory of structures, we may adopt an infinite number of axioms of the form below (for each formula ϕ(u)) instead of axiom (∃1 sum): ∃u∈U ϕ(u) =⇒ ∃1x∈U x sum {u ∈ U : ϕ(u)}. They say that for each formula satisfied in U there exists in U a mereological sum of the extension of that formula. Taking ‘u = u’ as the formula ϕ(u), we obtain the thesis ‘∃x ∀u u  x’, which guarantees the existence of a unity in each universe along with a mereological sum of that universe.

39 In this way arose those structures which we have called Grzegorczykian lattices and which are the subject of detailed discussion in Appendix B.6. All Grzegorczykian lattices with unity are, however, Boolean lattices.

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The elementary theory sketched above is extensively discussed in Chapter VI of (Pietruszczak 2000, 2018). We will not detail it further in this book as we will not be applying it anywhere. The theory sits between Grzegorczyk’s theory of mereological structures and the theory of classical mereological structures. It does perhaps deserve the name of elementary mereology after all. In (Pietruszczak 2000, 2018) we also provide details of an elementary theory of complete Boolean lattices, which corresponds to the elementary version of mereology above.

3.7.2 Description of Grzegorczykian Mereological Structures The primitive concept of the elementary theory which in (Grzegorczyk 1955) is called “elementary mereology” is the two-place predicate ‘ingr’ to which corresponds the symbol ‘’.40 The first three axioms, M1–M3, say that ‘ingr’ is reflexive, antisymmetric and transitive, respectively. The first of the axioms, M1, turns out in any case to be superfluous in Grzegorczyk’s theory. Grzegorczyk also adopts an axiom M4 from which it follows that the predicate ‘ingr’ is polarised (it corresponds to our (pol )). We know, however, that from its transitivity and polarisation follows its reflexivity (see Lemma 2.3.7). Below—in accordance with the basic principle that runs through this book— we will treat Grzegorczyk’s theory as a theory of certain structures, but not as an elementary theory. Since our primitive concept is the relation , we will investigate structures of the form U,  which belong to s POS and which in addition satisfy the axioms M4–M6 introduced by Grzegorczyk under the definition of (df ) we have accepted.41 Later, it will turn out that axiom M6 is also superfluous; and so, in Grzegorczyk’s original theory there are two superfluous axioms: M1 and M6. Models of Grzegorczyk’s elementary mereology we shall call (strict) Grzegorczykian mereological structures. Let s GMS be the class of all such structures, i.e.: s GMS

:= s POS+{M4, M5, M6}.

Obviously, in accordance with (Grzegorczyk 1955), we can adopt the relation  as the primitive concept and examine structures of the form U,  satisfying axioms M1–M6 (though, as just noted, M1 and M6 are superfluous). Let GMS be the class of all such reflexive structures, i.e.:

40 Although in (Grzegorczyk 1955) the definition of ‘ingr’ is given with the help of a condition corresponding to our (df ), the truth is that it provides only an “intuitive” explanation of that concept. Otherwise, the axiom stating that ‘ingr’ is reflexive would clearly be superfluous, as it would follow from that definition. Furthermore, axioms for the predicate ‘is a part of’, to which corresponds the symbol ‘’, would also be missing. 41 The adoption of definition (df ) fits with Grzegorczyk’s intentions (see the previous footnote). Obviously, one can treat the presentation of Grzegorczyk’s theory as a description of an elementary theory whose models we would consider as structures (cf. point 2.5.1).

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GMS := POS+{M4, M5, M6}. Axioms M4–M6 have the same form for both classes s GMS and GMS. The classes POS and s POS are elementarily definitionally equivalent in the sense of (Szmielew 1981, p. 8). This means that the following two facts obtain, in which the assignments of structures are injective. • From an arbitrary structure U,  from s POS we get a structure U,  from POS in which the relation  is elementarily defined with the help of the formula corresponding to (df ). • From an arbitrary structure U,  from POS we get a structure U,  from s POS in which the relation  is elementarily defined with the help of the formula corresponding to (df 1 ). Hence the classes s GMS and GMS are also elementarily definitionally equivalent under the assignment of structures described above. We will now turn to the analysis of Grzegorczyk’s axioms M4–M6. Condition M4 has the following form:

 ∀x,y∈U x  y =⇒ ∃z∈U z  x ∧ ∀u∈U (u  z ⇒ u  y) ∧

   ∀w∈U w  x ∧ ∀u∈U (u  w ⇒ u  y) ⇒ w  z .

(M4)

This is a different but equivalent way of writing condition (SSP+). A simple logical transformation gives us:

 ∀x,y∈U x  y =⇒ ∃z∈U z  x ∧ ¬∃u∈U (u  z ∧ u  y) ∧

   ∀w∈U w  x ∧ ¬∃u∈U (u  w ∧ u  y) ⇒ w  z . Hence, thanks to (df ), we obtain (SSP+) in an equivalent way:   ∀x,y∈U x  y =⇒ ∃z∈U [z  x ∧ z  y ∧ ∀w∈U (w  x ∧ w  y ⇒ w  z)] . Thus, we obtain: s POS+(M4)

= s PPOSp = MEMp.

Therefore, from Lemma 3.6.8 it follows that in the class s GMS (resp. GMS) sentence (sum-sup) holds:   ∀S∈2U ∀x∈U x sum S ⇐⇒ S = ∅ ∧ x sup S .

(sum-sup)

Moreover, by virtue of Lemma 3.6.8(ii), the identity sup = sum holds in the class ·· ·· s GMS (resp. GMS ) of non-degenerate structures.

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Remark 3.7.3 From (M4) we also get (SSP), i.e. polarisation. On the strength of Lemma 2.3.7, condition (r ) follows from (t ) and (SSP). This yields a simplification of the original axioms from (Grzegorczyk 1955) through the loss of axiom M1 from them.

Axiom M5 has the following form in Grzegorczyk’s theory:   ∀x,y∈U ∃z∈U x  z ∧ y  z ∧ ∀u∈U (x  u ∧ y  u ⇒ z  u) .

(M5)

From (df sup) we have that (M5) says that, for arbitrary x and y form U there is the least upper bound of the set {x, y}. Therefore, (M5) and (∃pair sup) are equivalent. On the basis of (sum-sup), condition (∃pair sum) holds too. The final axiom M6 has the following form:

∀x,y,u∈U (u  x ∧ u  y) =⇒   ∃z∈U z  x ∧ z  y ∧ ∀u∈U (u  x ∧ u  y ⇒ u  z) .

(M6)

A logical transformation gives us the following equivalent form:

∀x,y∈U ∃u∈U (u  x ∧ u  y) =⇒   ∃z∈U z  x ∧ z  y ∧ ∀u∈U (u  x ∧ u  y ⇒ u  z) . After applying (df ) we obtain:

  ∀x,y∈U x  y =⇒ ∃z∈U z  x ∧ z  y ∧ ∀u∈U (u  x ∧ u  y ⇒ u  z) . Now, by applying (df inf), we see that axiom (M6) says that for arbitrary x and y in U such that x  y, there exists the greatest lower bound for the set {x, y}. Therefore, (M6) is equivalent to (c∃pair inf). We also know (cf. Lemma 3.2.1) that in structures from the class s POS (resp. POS) conditions (c∃pair inf) and (c∃ ) are equivalent. These remarks on (M6) show, quite simply, that it is inessential. Fact 3.7.2 (M6) follows from the axioms of the class s POS+(M4) Proof We see that (M6) is equivalent to (c∃pair inf). But, thanks to Theorem 3.6.7,

(c∃pair inf) follows from the axioms of the class s PPOSp (= s POS+(M4)). Moreover, we get: s GMS

= s POS+{M4, M5} = s PPOSp+(M5) = MEMp+(M5).

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From Lemma 3.6.8 we get the result that in the class GMS sentence (‡∅ ) holds and from this and from axiom (M5) we get sentence (∃pair sum). We therefore establish that s GMS is identical to s PPOSp  MEM+(∃pair sum)+(‡∅ ). And therefore, taking into account the considerations from the previous section, we have: s GMS

= s PPOSp  MEM+(∃pair sum)+(‡∅ ) = s PPOSp+(∃pair sum) = MEMp  MEM+(∃pair sum)+(‡∅ ) = MEMp+(∃pair sum) = s POS+(c∃ )+(∃pair sum)+(WSP+).

Analogous identities also hold for non-degenerate structures. We may swap the name ‘s PPOSp+(∃pair sum)’ for ‘s GMS’ in Diagram 3.4. These changes are presented in Diagram 3.5. We also have GMS = POS+(c∃ )+(∃pair sum)+(WSP+) = POS+(c∃ )+(∃pair sum)+(SSP+). We obtain these two identities via Lemma 3.6.1 and Theorem 3.6.11, respectively, applied to the class POS. In the class s GMS (resp. GMS) we can define the partial binary operation  of mereological relative complementation by using condition (df ) for arbitrary x, y ∈ U such that x  y. We also have the partial binary operation which for arbitrary x, y ∈ U such that x  y satisfies condition (3.6.2). All the facts proven for the class s PPOSp (= MEMp) hold for these two operations. Finally, thanks to (M4) and (M5), which yield (sum-sup) and (∃pair sum), we have the binary operation  of the mereological sum of a pair of elements. Conditions (3.4.14)–(3.4.17) hold for the operations and . Furthermore, all structures from the class s GMS, like all structures from the class MEM+(∃pair sum), are partially distributive in the sense of Theorem 3.4.14. To finish this point let us prove the “partial” de Morgan law for the operation of mereological relative complementation. Let us take arbitrary x, y, z ∈ U such that x  y and z  x y. Then either z  x or z  y, and: ⎧ ⎪ ⎨z  x,  z (x y) = z  y, ⎪ ⎩  (z x)  (z  y),

if z  x and z  y if z  x and z  y if z  x and z  y

(3.7.1)

If x  y and z  x y, then z  (x y) is defined and one of the three cases above obtains because: z  x y iff z  x and z  y. In the first case, since x y  x, we have z  x  z  (x y), by virtue of (3.6.1). Suppose that z  (x y)  z  x. Then, by virtue of (SSP), there is a u such that (a) u  z  (x y) and (b) z  z  x. By virtue of (a) and Lemma 3.6.5(ii), we have: u  z and u  x y. Hence u  x

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or u  y. By virtue of (b) and Lemma 3.6.5(iii), there does not exist a v such that v  z, v  x and v  u. We therefore have u  y, because otherwise there would be a contradiction: u  z, u  x and u  u. This, however, leads to contradiction: z  y, u  z and u  y. We argue in a similar way in the second case. In the third case, since x y  x, we have z  x  z  (x y), by virtue of (3.6.1). Similarly, we get: z  y  z  (x y). Hence (z  x)  (z  y)  z  (x y), by (3.4.10). Suppose that z  (x y)  (z  x)  (z  y). Then, by virtue of (SSP), there exists a u such that (a) u  z  (x y) and (b) z  (z  x)  (z  y). By virtue of (a) and Lemma 3.6.5(ii), we have: u  z and u  x y. Hence u  x or u  y. By virtue of (b) (3.4.4), we have (c) z  z  x and (d) u  z  y. From (c) and Lemma 3.6.5(iii), there does not exist a v such that v  z, v  x and v  u. From (d) and Lemma 3.6.5(iii), there does not exist a v such that v  z, v  y and v  u. We therefore have a contradiction because—as before—both the other cases lead to one. For the second version of the “partial” de Morgan law for the operation of mereological relative complementation, we must prove that, for arbitrary x, y, z ∈ U such that z  x and z  y: (z  x)  (z  y) =⇒ z  x  y.

(3.7.2)

If z  x and z  y, then z  x and z  y are defined. If, furthermore, z  x  z  y, then there is a u such that u  z  x and u  z  y. Hence u  z, u  x and u  y. So u  x  y, by (3.4.4). Therefore, z  x  y. Take arbitrary x, y, z ∈ U such that z  x, z  y and z  x  z  y. Then z  x  y and: (3.7.3) z  (x  y) = (z  x) (z  y). If z  x and z  y, then z  x and z  y are defined. If, furthermore, z  x  z  y, then (z  x) (z  y) and z  (x  y) are defined. For an arbitrary u we have: u  z  (x  y) iff u  z, u  x and u  y iff u  z  x and u  z  y iff u  (z  x) (z  y). Hence we have: z  (x  y) = (z  x) (z  y), by (r ) and (antis ).

3.7.3 Unity in Structures from the Class GMS We will now concern ourselves with the problem of the existence of a unity in structures from the class s GMS (resp. GMS). Fact 3.7.3 (∃ 1 ) does not follow from the axioms of theory s GMS (resp. GMS). In other words, in this class there is a structure without unity. As a consequence, (∃sum) does not follow either.

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Model 3.6 The simplest non-degenerate structure from GMS

IN Proof Let U := 2fin \ {∅} and  be the relation of proper inclusion in U . Then U,  belongs to s GMS, because the set-theoretic sums, products and differences of finite set are themselves finite. So (irr  ), (t ) and (M4)–(M6) are also satisfied. This structure does not have a unity. Obviously,  is in this model the relation of inclusion in U and the pair U,  belongs to the class GMS and is also a structure without unity.

Henceforth we will omit subscripts from the labels for the class s GMS. So, the label ‘GMS’ could equally refer to strict or reflexive Grzegorczykian mereological structures. Let GMS1 be the subclass of the class GMS of structures with unity. On the strength of Fact 3.7.3 we have: GMS1  GMS.

Fact 3.7.4 In all finite structures from the class GMS sentence (∃sum) holds. So (∃ 1 ) holds, too. Proof Let us take an arbitrary non-empty set S included in the finite universe U . Then S = {s1 , . . . , sn }, for some n  1. By virtue of (M5), s1  s2  · · ·  sn exists. We have s1  s2  · · ·  sn sum S, by virtue of (3.4.7). For S = U , the resulting mereological sum is the unity.

In the class GMS there are finite structures. For example, • degenerate structures, which arise from two-element Boolean algebras with the zero removed; • non-degenerate structures which arise from four-element Boolean algebras with the zero removed (this is depicted in Model 3.6); • non-degenerate structures which arise from eight-element Boolean algebras with the zero removed (this is depicted in Model 2.8). The connection between Grzegorczykian mereological structures, Grzegorczykian lattices and Boolean lattices will be presented in points 3.7.4 and 3.7.5. Since GMS1 ⊆ s PPOSp = MEMp1, we may define a conditional operation of complementation (to unity) for objects that differ from unity. In keeping with the comments on Sect. 3.6.3, for an arbitrary x not identical to 1 we put: – x := 1

 x.

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Obviously, everything that holds in the class s PPOSp1 (= MEMp1) holds for this unary operation too. Furthermore, we have the following two “partial” de Morgan laws for complementation. For the first version: take arbitrary x, y ∈ U such that x  y and x y = 1. Then x = 1 or y = 1, and: ⎧ ⎪ ⎨– x, –(x y) = – y, ⎪ ⎩ – x  – y,

if x = 1 and y = 1 if x = 1 and y = 1 if x = 1 = y

(3.7.4)

We put z := 1 in (3.7.1). From our assumptions we get 1  x y. For the second version, we need for arbitrary x, y ∈ U such that x = 1 = y: – x  – y =⇒ x  y = 1 .

(3.7.5)

We put z := 1 in (3.7.2). From our assumptions we get 1  x and 1  y. For the second version of de Morgan’s law: take arbitrary x, y ∈ U such that x = 1 = y and – x  – y. Then x  y = 1 and: –(x  y) = – x – y.

(3.7.6)

We put z := 1 in (3.7.3). From our assumptions we get 1  x and 1  y.42

3.7.4 The Theory GMS Versus the Theory of Grzegorczykian Lattices Since each Grzegorczykian lattice has zero, we will use the expression ‘U, ≤, 0’ to emphasise that the element 0 is the zero in the structure U, ≤ (see Appendix B.6). The theorem below says that from each non-degenerate Grzegorczykian lattice with the zero removed we may obtain a structure from GMS. Theorem 3.7.5 Let U, ≤, 0 be an arbitrary non-degenerate Grzegorczykian lattice. Let U+ := U \ {0} and  := ≤  (U+ × U+ ); that is,  is the restriction of ≤ to the set U+ . Then: (i) ⊥+ = . (ii) For arbitrary S ∈ 2U+ \ {∅} and x ∈ U+ x sup≤ S ⇐⇒ x sup S, x inf≤ S ⇐⇒ x inf S. 42 In

(Pietruszczak 2000, 2018, p. 86 or p. 107), other proofs of the last three conditions are given. They were carried out, however, for the class LMS of classical mereological structures.

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(iii) (Grzegorczyk 1955)43 U+ ,  belongs to GMS and for arbitrary S ∈ 2U+ \ {∅} and x ∈ U+ x sup S ⇐⇒ x sum S. (iv) If U, ≤, 0 has the unity 1 (i.e., U, ≤, 0 is a Boolean lattice from BL), then 1 is the unity of U+ ,  (i.e., U+ ,  belongs to GMS1). Proof (i) Obvious. (ii) If x sup≤ S and S = ∅, then x ∈ U+ . (iii) Since U, ≤, 0 satisfies (B4), also satisfies (B.6.4). Since, however, in U+ ,  we have ⊥+ = , and from this we therefore get (SSP+). Furthermore, (B5) gives us (B5 ). From this and from (ii) we have (∃pair sup), which is equivalent to (M5). What is more, having (SSP+), we also have sum ⊆ sup and (‡∅ ). (iv) If 1 is the greatest under ≤ then it is also the greatest under .

We also have the following theorem which is, in a way, the converse of the preceding one. Theorem 3.7.6 Let U,  belong to GMS and let 0 be an arbitrary element which does not belong to U . Let U o := U  {0} and define in U o the binary relation ≤ :=  ({0} × U o ), i.e., for arbitrary x, y ∈ U o : x ≤ y :⇐⇒ x  y ∨ x = 0. Then: (i)  = ≤  (U × U ), ⊥+ =  and ∀x∈U o 0 ≤ x. (ii) For arbitrary S ∈ 2U \ {∅} and x ∈ U x sum S ⇐⇒ x sup S ⇐⇒ x sup≤ S ⇐⇒ x sup≤ (S  {0}), x inf≤ S ⇐⇒ x inf S. (iii) (Grzegorczyk 1955)44 U o , ≤, 0 is a non-degenerate Grzegorczykian lattice from GL in which zero is 0. (iv) If U,  has the unity 1 then U o , ≤, 0, 1 is a Boolean lattice from BL in which 1 is the unity. Proof (i) Obvious. (ii) Since we have (SSP+), we also have sum ⊆ sup and (‡∅ ). The remaining equivalences are obvious. (iii) On the strength of the definition, the relation ≤ is reflexive, transitive and antisymmetric, and 0 is the zero in U, ≤. Therefore, this structure satisfies B1–B3 and B7. Furthermore, since in U+ ,  we have ⊥+ = , then (SSP+) is equivalent to (B.6.4), which is in turn equivalent to (B4). On the strength of (ii), (M5) entails (B5 ), and this is equivalent to (B5). 43 In

(Grzegorczyk 1955), only this third part of this theorem appears, and without any proof. was said in the previous footnote applies here in the same way.

44 What

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(iv) On the strength of (iii) and the definition of the relation ≤, U o , ≤, 0 is a Boolean lattice with the unity 1. Theorem B.6.11 says, however, that a given structure is a Grzegorczykian lattice with unity iff it is a Boolean lattice.

We see that “whole weight” of the proof of point (iv) of the above theorem rests on the theorem proven in Appendix B which gives the connection between Grzegorczykian lattices with unity and Boolean lattices. This theorem does not appear in (Grzegorczyk 1955), because there, Grzegorczykian lattices are treated as Boolean algebras (Boolean lattices). By drawing on the last two theorems, we obtain: Corollary 3.7.7 For an arbitrary structure U,  composed of a non-empty set U and a binary relation  the following conditions are equivalent: (a) The pair U,  belongs to GMS . (b) For some (equivalently, for any) 0 which does not belong to U , for U o := U  {0} and for ≤ :=  ({0} × U o ), the structure U o , ≤, 0 is a Grzegorczykian lattice with the zero 0. (c) For some non-degenerate Grzegorczykian lattice G, ≤, 0 with the zero 0 we have U = G \ {0} and  = ≤  (U × U ). Proof “(a) ⇒ (b)” From Theorem 3.7.6(iii). “(b) ⇒ (c)” G := U o , ≤ := ≤ and 0 := 0. Obviously, U = G \ {0} = U o \ {0} and  = ≤  (U × U ) = ≤  (U × U ) = ( ({0} × U o ))  (U × U ). “(c) ⇒ (a)” From Theorem 3.7.5(iii).

Remark 3.7.4 The equivalence “(a) ⇔ (c)” may be more plainly expressed in the following way, which is modelled on the form of words we used to in Remark 3.7.1 when talking about classical mereological structures from TMS: A structure belonging to GMS is that and only that relational structure which arises from some non-degenerate Grzegorczykian lattice after the zero has been removed. The equivalence “(a) ⇔ (b)” may be expressed as follows: A structure belonging to GMS is that and only that relational structure which yields a Grzegorczykian lattice through the addition of the zero.



By drawing on the earlier elementary definitional equivalence of the classes and GMS, we obtain the following theorem from Theorems 3.7.5 and 3.7.6.

s GMS

Theorem 3.7.5 Let U, ≤, 0 be an arbitrary non-degenerate Grzegorczykian lattice from GL. Let U+ := U \ {0} and  := (≤ \ idU )  (U+ × U+ ) = (≤  (U+ × U+ )) \ idU+ , i.e.,  is a restriction of  to the set U+ . Then condition (ii) holds along with the versions of conditions (i), (iii) and (iv) below from Theorem 3.7.5: (i)  = ≤  (U+ × U+ ) and ⊥+ = . (iii) U+ ,  belongs to s GMS and for arbitrary S ∈ 2U+ \ {∅} and x ∈ U+ x sup S ⇐⇒ x sum S.

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(iv) If U, ≤, 0 has the unity 1 (i.e., if it is a Boolean lattice from BL), then 1 is the unity of U+ , . Theorem 3.7.6 Let U,  belong to s GMS and let 0 be arbitrary element which does not belong to U . Let U o := U  {0} and define in the set U o the binary relation ≤ :=  ({0} × U o ). Then conditions (i)–(iii) from Theorem 3.7.6 hold along with the version of condition (iv) below: (iv) If U,  has the unity 1 then U o , ≤, 0, 1 is a Boolean lattice from BL in which 1 is the unity. For structures from the class s GMS we have the following version of Corollary 3.7.7, which was formulated for the (non-strict) class GMS.45 Corollary 3.7.7 For an arbitrary structure U,  composed of a non-empty set U and a binary relation  the following conditions are equivalent: (a) U,  belongs to s GMS . (b) The relation  is irreflexive and for some (equivalently, for any) 0 which does not belong to U , for U o := U  {0} and for ≤ :=  idU  ({0} × U o ), the structure U o , ≤, 0 is a Grzegorczykian lattice with the zero 0. (c) For some non-degenerate Grzegorczykian lattice G, ≤, 0 with the zero 0 we have U = G \ {0} and = (≤  (U × U )) \ idU . Proof “(a) ⇒ (b)” From Theorem 3.7.6 (iii). “(b) ⇒ (c)” We take G := U o , ≤ := ≤ and 0 := 0. Obviously U = G \ {0} = o U \ {0} and = (≤  (U × U )) \ idU = (≤  (U × U )) \ idU = ((  idU  ({0} × U o )  (U × U )) \ idU , since the relation , by assumption, is irreflexive.

“(c) ⇒ (a)” From Theorem 3.7.5 (iii). Remark 3.7.5 The equivalence “(a) ⇔ (c)” may be more plainly expressed in the following way, which is modelled on the form of words we used to on Sect. 3.7, when talking about classical mereological structures: A structure belonging to s GMS is that and only that relational structure which arises from some non-degenerate Grzegorczykian lattice after the zero has been removed and the relation has been “irreflexivised”. The equivalence “(a) ⇔ (b)” may be expressed as follows: A structure belonging to s GMS is that and only that relational structure with an irreflexive relation which yields a Grzegorczykian lattice through the addition of the zero and the “reflexivisation” of the relation.

45 Observe



that, in the corollary below, we must add the assumption of the irreflexivity of the relation  in condition (b).

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3.7.5 The Theory GMS1 Versus the Theory of Boolean Lattices From the theorems obtained in the previous section we also obtain some conclusions concerning the class GMS1 (resp. s GMS1). From Theorem 3.7.5 we have directly: Theorem 3.7.8 Let U, ≤, 0, 1 be an arbitrary non-degenerate Boolean lattice from BL. Let U+ := U \ {0} and  := ≤  (U+ × U+ ), i.e.,  is a restriction of ≤ to the set U+ . Then conditions (i) and (ii) from Theorem 3.7.5 hold along with the following: (iii) The structure U+ ,  belongs to GMS1, 1 is the unity and for any S ∈ 2U+ \ {∅} and x ∈ U+ x sup S ⇐⇒ x sum S. From Theorem 3.7.6 we get: Theorem 3.7.9 Let U,  belong to GMS1 and have the unity 1, and let 0 be an arbitrary element which does not belong to U . Let U o := U  {0} and define in U o the binary relation ≤ :=  ({0} × U o ). Then conditions (i) and (ii) from Theorem 3.7.6 hold along with the following: (iii) U o , ≤, 0, 1 is a Boolean lattice from BL in which 1 is the unity. For structures from GMS1 we have the following version of Corollary 3.7.7: Corollary 3.7.10 For an arbitrary structure U,  composed of a non-empty set U and a binary relation  the following conditions are equivalent: (a) U,  belongs to GMS1 and has the unity 1. (b) For some (equivalently, for any) 0 which does not belong to U , for U o := U  {0} and for ≤ :=  ({0} × U o ), the structure U, ≤, 0, 1 is a Boolean lattice. (c) For some non-degenerate Boolean lattice B, ≤, 0, 1 with the zero 0 and the unity 1 we have U = B \ {0},  = ≤  (U × U ) and 1 = 1. Proof “(a) ⇒ (b)” From Theorem 3.7.9. “(b) ⇒ (c)” (b) As in the case of the proof of Corollary 3.7.7. “(c) ⇒ (a)” From Theorem 3.7.8.



Remark 3.7.6 The equivalence “(a) ⇔ (c)” in the theorem above may be more plainly expressed in the following way: A structure belonging to GMS1 is that and only that relational structure which arises from some non-degenerate Boolean lattice once the zero has been removed. The equivalence “(a) ⇔ (b)” may in turn be expressed thus: A structure belonging to GMS1 is that and only that relational structure which yields a Boolean lattice through the addition of the zero.



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119

We also have corresponding results for the class s GMS1. Firstly, from Theorem 3.7.5 (resp. 3.7.8) we get: Theorem 3.7.8 Let U, ≤, 0, 1 be an arbitrary non-degenerate Boolean lattice. Let U+ := U \ {0} and  := (≤ \ idU )  (U+ × U+ ) = (≤  (U+ × U+ )) \ idU+ , i.e.,  is a restriction of  to the set U+ . Then conditions (i) and (ii) from Theorem 3.7.5 hold along with the following: (iii) The structure U+ ,  belongs to s GMS1, 1 is its unity and for all S ∈ 2U+ \ {∅} and x ∈ U+ x sup S ⇐⇒ x sum S. Secondly, from Theorem 3.7.6 (resp. 3.7.9) we obtain: Theorem 3.7.9 Let U,  belong to s GMS1 and have the unity 1, and let 0 be arbitrary element which does not belong to U . Let U o := U  {0} and define in U o the binary relation ≤ :=  ({0} × U o ). Then conditions (i)–(iii) from Theorem 3.7.9 hold. For structures from s GMS1 the counterpart of Corollary 3.7.10 is: Corollary 3.7.10 For an arbitrary structure U,  composed of a non-empty set U and a binary relation  the following conditions are equivalent: (a) U,  belongs to s GMS1 and has the unity 1. (b) The relation  is irreflexive and for some (equivalently, for any) 0 which does not belong to U , for U o := U  {0} and for ≤ :=  idU  ({0} × U o ), the structure U, ≤, 0, 1 is a Boolean lattice. (c) For some non-degenerate Boolean lattice B, ≤, 0, 1 with zero 0 and the unity 1 we have U = B \ {0}, = (≤  (U × U )) \ idU and 1 = 1. Proof “(a) ⇒ (b)” From Theorem 3.7.9 . “(b) ⇒ (c)” As in the proof of Corollary 3.7.7 . “(c) ⇒ (a)” From Theorem 3.7.8 .



Remark 3.7.7 The equivalence “(a) ⇔ (c)” in the theorem above may be more plainly expressed in the following way: A structure belonging to s GMS1 is that and only that relational structure which arises from some non-degenerate Boolean lattice once the zero has been removed and the relation ordering the lattice has been “irreflexivised”. The equivalence “(a) ⇔ (b)” may in turn be expressed thus: A structure belonging to s GMS1 is that and only that relational structure with an irreflexive relation which yields a Boolean lattice through the addition of the zero and the “reflexivisation” of the relation.



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3.7.6 Operations in the Class GMS Versus Operations in the Class of Grzegorczykian Lattices and in the Class of Boolean Lattices Let U,  (resp. U, ) belong to GMS (resp. s GMS) and let 0 be an arbitrary element which does not belong to U . Let U o := U  {0} and define in U the binary relation ≤ :=  ({0} × U o ), i.e., as in the case of Theorem 3.7.6 (resp. 3.7.6 ). By virtue of this theorem. U o , ≤, 0 is a non-degenerate Grzegorczykian lattice in which the zero is 0. In this lattice, as in Appendix B.6, we introduce into U o × U o three binary operations: difference −, product · and sum + x − y := ( z) z sup≤ {u ∈ U : u ≤ x ∧ u  y}, x · y := ( z) z inf≤ {x, y} = x − (x − y), x + y := ( z) z sup≤ {x, y}, ι ι ι

where  = ⊥+ := |U . Hence, by Theorem 3.7.6 (resp. 3.7.6 ) and the definitions of the operations  , and  in the class GMS (resp. s GMS), for all x, y ∈ U we obtain: x  y =⇒ x  y = x − y, x  y =⇒ x y = x · y, x  y = x + y. Therefore, in their domains, the operations  , and  coincide with the operations −, · and +, respectively. If, however, U,  (resp. U, ) belongs to GMS1 (resp. s GMS1) and 1 is its unity, then—by virtue of Theorems 3.7.6(iv) (resp. 3.7.6 (iv)) and B.6.11—the structure U o , ≤, 0, 1 is a Boolean lattice with the unity 1. Furthermore, the unary operation in U o satisfying the following condition for each x ∈ U o : −x = 1 −x is the complementation operation in the Boolean lattice. Therefore, by virtue of the definition of the operator of complementation in the class GMS1 (resp. s GMS1) and the previous comments on the operation of difference, we have:   ∀x∈U x = 1 =⇒ – x = −x . Therefore, the operation of complementation in the structures from GMS1 (resp. s GMS1) in its domain of definition also coincides with the operation of complementation defined in Boolean lattices corresponding to them. We now take an arbitrary non-degenerate Grzegorczykian lattice U, ≤, 0 with the zero 0. We put U+ := U \ {0} and define in the set U+ the binary relation  := ≤  (U+ × U+ ) (resp.  := (≤ \ idU )  (U+ × U+ )), i.e., just as in Theorem 3.7.5 (resp. 3.7.5 ). On the strength of this theorem, U+ ,  (resp. U+ , ) is a mereological Grzegorczykian lattice from GMS (resp. s GMS). As earlier, we

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Grzegorczykian Mereological Structures

121

show that in their domains, the operations  , and  overlap with the operations −, · and +, respectively. If, however, the Grzegorczykian lattice U, ≤, 0 has the unit 1, i.e., is a Boolean lattice, then Theorem 3.7.5(iv) (resp. 3.7.5 (iv)) says that 1 is the unity in U+ ,  (resp. U+ , ). Once again—just as above—we show that the operation of complementation in structures from the class GMS1 (resp. s GMS1) in their domains coincides with the operation of complementation defined in structures from the class BL that correspond to them. In this way, we get the following theorems concerning the connection between mereological Grzegorczykian structures with unity and Boolean algebras. The first two of them are counterparts of Theorems 3.7.8 and 3.7.8 . Theorem 3.7.11 Let U, +, ·, −, 0, 1 be an arbitrary Boolean algebra and ≤ a relation defined by the condition: x ≤ y :⇔ x + y = y. Let U+ := U \ {0} and  := ≤  (U+ × U+ ), i.e.,  is a restriction of ≤ to the set U+ . Then conditions (i)–(iii) from Theorem 3.7.8 hold. Theorem 3.7.11 Let U, +, ·, −, 0, 1 be an arbitrary Boolean algebra and ≤ a relation defined by the condition: x ≤ y :⇔ x + y = y. Let U+ := U \ {0} and  := (≤ \ idU )  (U+ × U+ ), i.e.,  is a restriction of  to the set U+ . Then conditions (i)–(iii) from Theorem 3.7.8 hold. The two following theorems are continuations of Theorems 3.7.9 and 3.7.9 . Theorem 3.7.12 Let U,  belong to GMS1 and have the unity 1 and let 0 be arbitrary element which does not belong to U . Let U o := U  {0} and define in the set U o the binarny relation ≤ :=  ({0} × U o ). Then conditions (i)–(iii) from Theorem 3.7.9 hold along with: (iv) in the non-degenerate Boolean algebra U o , +, ·, −, 0, 1 obtained from U o , ≤, 0, 1, for arbitrary x, y ∈ U o we have: ⎧ ⎪ ⎨x  y, x + y = x, ⎪ ⎩ y,

if x, y ∈ U if y = 0 if x = 0 

x·y=

x y, 0,

⎧ ⎪ ⎨– x, − x = 0, ⎪ ⎩ 1,

if x ∈ U \ {1} if x = 1 if x = 0

if x, y ∈ U and x  y in the other case

As a consequence, by defining in U o with the help of the identities from (iv) the three operations +, · and −, we get from U,  the non-degenerate Boolean algebra U o , +, ·, −, 0, 1, which is the same algebra that we get from the lattice U o , ≤, 0, 1. Proof (iv) Firstly, from U o , ≤, 0, 1 we get a Boolean algebra and by introducing the operations (df +), (df ·) and (df −), with the help of (df +), (df ·) and

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(df −), and putting for arbitrary x, y ∈ U o : x + y := ( z) z sup≤ {x, y}, x · y := ( z) z inf≤ {x, y} and −x := ( z) (x · z = 0 ∧x + z = 1). Thanks to (B.5.12), for any x ∈ U o we have: −x sup≤ {u ∈ U : x · u = 0}. We obtain the identities by applying points (i) and (ii), the properties of the operations +, · and − given in Appendix B, the definitions (df ), (df ) and (df ), and the following result: for any x ∈ U \ {1}, we have – x := 1  x. Secondly, define in U o the three operations +, · and − with the help of the identities. We know that from U,  we get the Boolean lattice U o , ≤, 0, 1 and from it the Boolean algebra U o , +, ·, −, 0, 1 with identical operations to ours. Therefore, our operations also define a Boolean algebra.

ι

ι

ι

Theorem 3.7.12 Let U,  belong to s GMS1 and have the unity 1, and let 0 be an arbitrary element which does not belong to U . Put U o := U  {0} and define in the set U o the binary relation ≤ :=  ({0} × U o ). Then conditions (i)–(iv) from Theorem 3.7.12 hold. As a consequence, by defining in U o with the help of the identities from (iv) the three operations +, · and −, we get from U,  the non-degenerate Boolean algebra U o , +, ·, −, 0, 1, which is the same algebra that we get from the lattice U o , ≤, 0, 1. By making use of the standard connection between Boolean lattices and Boolean algebras, we also have the following continuations of Corollaries 3.7.10 and 3.7.10 . Corollary 3.7.13 For an arbitrary structure U,  composed of a non-empty set U and a binary relation , conditions (a)–(c) from Corollary 3.7.10 and the condition below are equivalent: (d) For some non-degenerate Boolean algebra A, +, ·, −, 0, 1 with the zero 0 and the unity 1, we have U = A \ {0},  = ≤  (U × U ) and 1 = 1, where for x, y ∈ A, the relation ≤ is defined by: x ≤ y iff x + y = y. Corollary 3.7.13 For an arbitrary structure U,  composed of a non-empty set U and a binary relation , conditions (a)–(c) from Corollary 3.7.10 and the condition below are equivalent: (d) For some non-degenerate Boolean algebra A, +, ·, −, 0, 1 with the zero 0 and the unity 1, we have U = A \ {0}, = (≤  (U × U )) \ idU and 1 = 1, where for all x, y ∈ A,the relation ≤ is defined by: x ≤ y wtw x + y = y.

3.8 Two Weak Axioms of the Existence of Mereological Sums We will now examine two versions of a strengthening of condition (c∃pair sum) and with them a weakening of condition (∃sum) which was introduced in (Gruszczy´nski and Pietruszczak 2014):

3.8

Two Weak Axioms of the Existence of Mereological Sums

∀S∈2U \{∅} (∃u∈U S ⊆ I(u) =⇒ ∃x∈U x sum S),   ∀S∈2U \{∅} ∀y,z∈S y Un z =⇒ ∃x∈U x sum S .

123

(w1 ∃sum) (w2 ∃sum)

The first of the above sentences says that each non-empty set that has an upper bound also has a mereological sum. The second says that if every pair of elements of a given non-empty set has an upper bound, then this set has a mereological sum. Since the antecedent of the implication in (w1 ∃sum) is stronger than the antecedent of the implication in (w2 ∃sum), we get the following: Lemma 3.8.1 (i) (w1 ∃sum) follows from (w2 ∃sum). (ii) (c∃pair sum) follows from (w1 ∃sum). We have, however, the following facts. Fact 3.8.2 In the class s POSfin , sentences (c∃pair sum) and (w1 ∃sum) are equivalent. Proof Assume that (c∃pair sum) holds. Take an arbitrary S = ∅ such that for some u we have S ⊆ I(u). In finite structures we have S = {z 1 , . . . , z n }, n > 0, i.e., for arbitrary i, j  n there is z i  z j which is the mereological sum of {z i , z j }, by virtue of (c∃pair sum). We have z i  z j  u, and hence, as standard, we sum the finite number of sums to get z 1  · · ·  z n . This is the mereological sum of the set S, by virtue of (3.4.7).

Lemma 3.8.3 The following conditions are equivalent: • (∃sum), • the conjunction of (∃ 1 ) and (w1 ∃sum), • the conjunction of (∃ 1 ) and (w2 ∃sum). Proof By virtue of Lemma 3.7.1, (∃sum) entails (∃ 1 ). Conversely, as condition (∃ 1 ) is satisfied, each non-empty set S satisfies the antecedents of (w1 ∃sum) and (w2 ∃sum). Therefore, S satisfies their consequents. Hence, if (∃ 1 ) and (w1 ∃sum) (resp. (w2 ∃sum)) hold, then (∃sum) holds too. Moreover, we use Lemma 3.8.1(i).

Lemma 3.8.4 (∃ 1 ) follows from (∃pair sum) and (w2 ∃sum). Proof By virtue of (∃pair sum), for arbitrary y, z ∈ U we have y Un z. Hence, by

(w2 ∃sum), there is an x ∈ U such that x sum U ; so also U ⊆ I(x). From Lemmas 3.8.3 and 3.8.4 we get46 : Corollary 3.8.5 (∃sum) is equivalent to the conjunction of (∃pair sum) and (w2 ∃sum). by drawing on this corollary, we will prove that LMS = s PPOS+(∃pair sum)+ (w2 ∃sum), which entails GMS+(w2 ∃sum) = LMS (see Theorem 3.9.6).

46 Below,

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3 “Existentially Involved” Theories of Parts

Lemma 3.8.6 (‡∅ ) follows from the axioms of the class s POS+(WSP)+(w1 ∃sum) (resp. QOS+(WSP)+(w1 ∃sum)). Proof Let S = ∅ and x sup S. Then S ⊆ I(x). Hence, thanks to (w1 ∃sum), there is a y such that y sum S. However, x = y, by virtue of () (see Theorem 2.9.3), i.e., x sum S.

Of greater interest than that lemma is the following theorem: Theorem 3.8.7 (Gruszczy´nski and Pietruszczak 2014) (SSP+) follows from the axioms of the class s PPOS+(w1 ∃sum) (resp. QOS+(SSP)+(w1 ∃sum)). Proof Let x  y and we put S0 := {z ∈ U : z  x ∧ z  y}. Then ∅ = S0 ⊆ I(x), by (SSP). From this and (w1 ∃sum) there is a z 0 such that z 0 sum S0 . To begin with, observe that z 0  x. Assume for a contradiction that z 0  x. Then, thanks to (SSP), there is a u such that u  z 0 and u  x. From this and (df sum) there are v ∈ S0 and w ∈ U such that w  v  x and w  u  x. Therefore, we have a contradiction: u  x and u  x, by (t ). We will now show that z 0  y. Assume for a contradiction that z 0  y. Then there is a u such that u  z 0 and u  y. From this and (df sum) there are v ∈ S0 and w ∈ U such that w  v  y and w  u  y. Therefore, we have a contradiction: w  y, w  y and w  y, thanks to (t ) and (r ).

Finally, if u  x and u  y, then u ∈ S0 , i.e., u  z 0 , since z 0 sum S0 . Since MEMp = s PPOSp and s PPOS ⊆ MEM ⊆ MEMp, we get from the theorem above the following: s PPOS+(w1 ∃sum)

= s PPOSp+(w1 ∃sum) = MEM+(w1 ∃sum) = MEMp+(w1 ∃sum).

Similarly, we get: s PPOS+(w2 ∃sum)

= s PPOSp+(w2 ∃sum) = MEM+(w2 ∃sum) = MEMp+(w2 ∃sum).

We have shown that (w1 ∃sum) follows from (w2 ∃sum). We will now show that the converse implication does not hold and do so with “various additions”. Fact 3.8.8 (w2 ∃sum) does not follow from the axioms of s PPOSp+(w1 ∃sum). Proof Model 3.7 belongs to s PPOSp+(w1 ∃sum), but does not satisfy (w2 ∃sum), because the set {1, 2, 3} satisfies its antecedent but does not have a mereological

sum (this set doesn’t even have an upper bound).47 We also have a stronger result than the one above. We can only derive it, however, with the help of infinite structures. 47 Model

3.7 arose from Model 2.8 through the removal of the unity. Obviously, we had to use model without unity in the light of Lemma 3.8.3.

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Two Weak Axioms of the Existence of Mereological Sums

125

Model 3.7 (SSP+) and (w1 ∃sum) hold, but (w2 ∃sum) does not hold

12

1

13

2

23

3

Fact 3.8.9 (∃ 1 ) and (w2 ∃sum) do not follow from the axioms of GMS+(w1 ∃sum). Proof We use model from the proof of Fact 3.7.3. It belongs to GMS and has no unity. Moreover, if S from 2U \ {∅} has an upper bound, then the family S is finite, i.e., the sum S is non-empty and finite set, and so belongs to U . Therefore S sum S. So (w1 ∃sum) holds in this model. However, this model does not satisfy (w2 ∃sum), because taking S := U , for example, the antecedent of the conditional is satisfied but there is no sum of the universe.

Fact 3.8.10 (w1 ∃sum) does not follow from the axioms of MEM+(∃pair sum)+(‡). Proof If (w1 ∃sum) were to follow from the axioms of MEM+(∃pair sum)+(‡), then (SSP+) would also follow, because it follows from the axioms of s PPOS+(w1 ∃sum), by virtue of Theorem 3.8.7. We would therefore obtain a contradiction with Fact 3.6.14, which says that (SSP+) does not follow from the axioms

of MEM+(∃pair sum)+(‡). Fact 3.8.11 (∃ 1 ) and (∃pair sup) do not follow from the axioms of the class

s PPOSp+(w2 ∃sum).

Proof Models 3.4 and 3.5 belong to s PPOSp+(w2 ∃sum), but (∃ 1 ) and (∃pair sup) do not hold in these models.

Fact 3.8.12 (w1 ∃sum) does not follow from the axioms of the class GMS1. Proof Build a model whose universe U is composed of: IN • sets from the family A := 2fin \ {∅}, • sets from the family B created from all non-empty open subsets of the set IR of real numbers, • sets from the family C := {xa  xb : xa ∈ A ∧ yb ∈ B ∧ xa  xb }.

The relation  is to be the relation of proper inclusion in U (i.e.,  is to be the inclusion U ). Obviously, IR is the unity of this model. We now check that (M5) holds. If x, y ∈ A, then x  y ∈ A. If x, y ∈ B, then x  y ∈ B. If x, y ∈ C, then x  y ∈ B  C. If x ∈ A and y ∈ B, then x  y ∈ B  C. If x ∈ A and y ∈ C, then x  y ∈ C. If x ∈ B and y ∈ C, then x  y ∈ B  C. Therefore, in each case we can put x  y := x  y. We now check that (M4) holds. Assume that x  y. Then in particular cases we create the mereological relative complement x  y. If x ∈ A and y ∈ U , then x  y := x \ y ∈ A. If x, y ∈ B, then x  y := Int(x \ y) ∈ B. If x, y ∈ C, then

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3 “Existentially Involved” Theories of Parts

x  y := Int(x \ y) ∈ B  C. If x ∈ B and y ∈ A, then x  y := x \ y ∈ B. If x ∈ C and y ∈ A, then x  y := x \ y ∈ B  C. If x ∈ B and y ∈ C, then x  y := Int(x \ y) ∈ B. If x ∈ C and y ∈ B, then x  y := Int(x \ y) ∈ B. Let S0 be the family of singletons in IN. Each set of B including the set IN is an upper limit of the family S0 . None of them is, however, a least upper bound of this family and therefore none of them is a mereological sum either.

We therefore obtain: GMS+(w2 ∃sum)  s PPOSp+(w2 ∃sum)  s PPOSp+(w1 ∃sum), s PPOSp+(w1 ∃sum)

 CMM+(‡), GMS+(w2 ∃sum)  GMS+(w1 ∃sum)  GMS, MEM+(∃pair sum)+(‡)  MEMp+(w1 ∃sum) = s PPOSp+(w1 ∃sum), s PPOSp+(w1 ∃sum) = MEMp+(w2 ∃sum)  MEM+(∃pair sup), s PPOSp+(w2 ∃sum) = MEMp+(w2 ∃sum)  GMS. The final fact follows from the penultimate one, because GMS ⊆ MEM+(c∃pair sup). Furthermore, for the class GMS1 we have: GMS+(w2 ∃sum)  GMS1  GMS, GMS+(w1 ∃sum)  GMS1, PPOSp+(w s 2 ∃sum)  GMS1,

GMS1  GMS+(w1 ∃sum), GMS1  s PPOSp+(w2 ∃sum).

We have portrayed the results obtained up to this point concerning structures of theories in Diagram 3.5. Once again, by changing the name of a given class for the name of the corresponding class of non-degenerate structures (i.e., with the superscript ‘·· ’) we will get a diagram for non-degenerate structures. Our examination of the class LMS of mereological structures in Le´sniewski’s sense will not mean that Diagram 3.5 has to change, because we can prove that LMS = s GMS+(w2 ∃sum) (cf. footnote 46 and Theorem 3.9.6).

3.9 Classical Mereological Structures 3.9.1 “Classical Mereology” with the Primitive Relation  In accordance with our discussion in Sect. 3.7.1, we understand classical mereology in Le´sniewski’s sense to be expressible as a theory of structures of the form U,  which are strictly partially-ordered sets satisfying the conditions (Usum ) and (∃sum). In Sect. 3.7.1 we denoted this class of such structures by ‘LMS’, i.e., we put: LMS := s POS+(Usum )+(∃sum).

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Classical Mereological Structures

127

s PPOS

MEM  s MPOS

MEM

s MPOS

MEM+(c∃pair sup)

MEM+(‡∅ )

MEM+(∃pair sup)

CMM

MEM+(∃pair sum)

CMM+(‡∅ )

MEM+(∃pair sum)+(‡∅ )

s PPOSp

s PPOS+(w1 ∃sum)

s GMS

s PPOS+(w2 ∃sum)

s GMS+(w1 ∃sum)

where s PPOSp = MEMp s GMS

s GMS1

s GMS+(w2 ∃sum) = LMS

= s PPOSp  MEM+(∃pair sum)+(‡∅ ) = s PPOSp+(∃pair sum)

s PPOS+(w1 ∃sum) = s PPOSp+(w1 ∃sum) = MEM+(w1 ∃sum) = MEMp+(w1 ∃sum) s PPOS+(w2 ∃sum)

= s PPOSp+(w2 ∃sum) = MEM+(w2 ∃sum) = MEMp+(w2 ∃sum)

Diagram 3.5 The lattice of “existentially involved” theories

In Remark 3.7.2, we showed that one could equivalently adopt Tarski’s axiom (∃1 sum) instead of conditions (Usum ) and (∃sum); i.e., we have48 : LMS = s POS+(∃1 sum). Lemma 3.7.1 says that (∃ 1 ) follows from (∃sum). Therefore, all mereological structures have a unity. As is standard, we denote it with ‘1’. We have: 48 We

are not drawing on here the following result proven by Tarski, namely that for structures of the form U,  conditions (r ) and (antis ) follow from (t ) and (∃1 sum). For we are looking at structures of the form U, , and (r ) follows from (df ). Therefore, we immediately get the result that there is no mereological sum for the empty set.

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3 “Existentially Involved” Theories of Parts

1 = ( x) x sum U. ι

In the proofs of Theorems 3.9.6, 3.9.8 and 3.9.10 to follow we give several equivalent definitions of the class LMS. Furthermore, on the strength of Lemma 3.8.3, the following identities hold: LMS = s POS+(Usum )+(∃ 1)+(w1 ∃sum) 

= s POS+(Usum )+(∃ 1)+(w2 ∃sum). 

If we show that structures from LMS are polarised, i.e., that they satisfy (SSP), then we will able to make use of Theorem 3.8.7. In that way, we will be able to get a result that says that (SSP+) also holds in LMS.49 Theorem 3.9.1 Condition (SSP) follows from (r ), (t ), (Usum ) and (∃sum). In consequence (SSP) follows from (t ), (Usum ), (∃sum) and (df ). Proof Assume that (r ), (t ), (Usum ), (∃sum) hold. Obviously, from (df ) we have (r ), and from (df ) and (t ) we have (t ). Assume for a contradiction that (SSP) does not hold, i.e., for some x0 , y0 ∈ U we have (a) x0  y0 and (b) I(x0 ) ⊆ O(y0 ). Since I(y0 ) ∪ {x0 } = ∅, then—by virtue of (∃sum)—for some z 0 we have z 0 sum I(y0 ) ∪ {x0 }. Therefore: (c) ∀y∈I(y0 ) y  z 0 , (d) x0  z 0 and (e) ∀z (z  z 0 ⇒ ∃u ((u  y0 ∨ u = x0 ) ∧ u  z)), i.e., ∀z (z  z 0 ⇒ (x0  z ∨ ∃u∈I(y0 ) u  z)). We will show that it follows from (b) that condition x0  z entails ∃u∈I(y0 ) u  z, i.e., we will get (f): ∀z (z  z 0 ⇒ ∃u∈I(y0 ) u  z). To see this, let x0  z. Then, by (df ), for some x1 we have x1  x0 and x1  z. By (b) we have x1  y0 . From this and from (df ), for some x2 we have: x2  x1 and x2  y0 . But by (t ) we have x2  z. From this and from (r ) we obtain x2  z. Therefore x2  y0 and x2  z. From (c) and (f) it follows that z 0 sum I(y0 ), but y0 sum I(y0 ). Therefore from (Usum ) we have y0 = z 0 . But this contradicts (a) and (d), because from them it follows that y0 = z 0 .

From the above theorem we get the following: Corollary 3.9.2 (Pietruszczak 2000, 2018) Condition (SSP) follows from the axioms of the class LMS. From Theorems 3.9.1 and 3.8.7 we get the following: Corollary 3.9.3 (Pietruszczak 2000, 2018) Condition (SSP+) follows from the axioms of the class LMS.

49 This is in any case perhaps only the only way of establishing this result, even if we were not to involve ourselves with s PPOS+(w1 ∃sum). This is shown by the fact that, with (SSP) holding, condition (SSP+) can be derived from a weaker condition than (∃sum).

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129

Therefore, from Theorems 3.9.1, 2.4.9 and 2.6.7 (the last two say that (SSP) entails (Usum )) it follows that all and only those polarised strict partial orders which satisfy condition (∃sum) are mereological structures. The “power” of axiom (∃sum) is shown by the fact that, by applying it in conjunction with the “weak” principle (WSP), we can derive the “strong” principle (SSP). To this end, we use Theorem 3.9.1 and the lemma below: Lemma 3.9.4 (Pietruszczak 2000, 2018) Condition (Usum ) follows from conditions (r ), (t ), (WSP) and (∃sum). Proof Assume that (A) x sum S and (B) y sum S. We will show that from this and from (WSP) it follows that (C) x  y and (D) y  x and x  y. (C): ∅ = S ⊆ I(x) ∩ I(y), so for some z ∈ S we have z  x and z  y. (D): Indirectly. If y  x, then by virtue of (WSP) there is a u such that (a) u  x and (b) u  y and (b) u  y. By virtue of (a) and (A), there is a z such that z  u. Therefore, for some w we have (c) w  z and (d) w  u. In the other direction, (B) entails (e) z  y. From (c) and (e) we have (f) w  y. Therefore, (d) and (f) yield u  y, which contradicts (b). We show that x  y in a similar way. Assume for a contradiction that x = y. From this and (D) we obtain (D ) y  x. By virtue of (∃sum), for some s we have (α) s sum {x, y}. Therefore, y  s and x  s. From this and (D ) we have x = s, i.e., (β) x  s. From this and (WSP), for some u we have (γ ) u  s and (δ) u  x. From (α) and (γ ): x  u or y  u. From this and (δ) it follows that (ε) y  u. Therefore, there is a w such that (ζ ) w  y and (η) w  u. From (B) and (ζ ) for some z ∈ S we have z  w. And hence there is a v such that (θ ) v  z and (ι) v  w. From (ι) and (η) we have v  u. Therefore,

x  u, which contradicts (δ). Applying Lemma 3.9.4 and Theorem 3.9.1 we obtain: Corollary 3.9.5 Condition (SSP) follows from (r ), (t ), (WSP) and (∃sum). Because, in accordance with Theorem 2.12.6, in polarised strict partial orders sum = fu, from this and the previous considerations it follows that all and only those polarised strict partial orders which satisfy the condition below on the existence of fusions are mereological structures ∀S∈2U \{∅} ∃x∈U x fu S.

(∃fu)

Our results so far on the various possible axiomatisations of mereological structures may be summed up by the theorem below: Theorem 3.9.6 (Pietruszczak 2000, 2018) The following identities of classes of structures hold:

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LMS := s POS+(Usum )+(∃sum) = s POS+(∃1 sum) = s PPOS+(∃sum) = s PPOS+(∃fu) = s PPOSp+(∃sum) = s PPOSp+(∃fu) = s POS+(Ssum )+(∃sum) = (t )+(WSP)+(∃sum) = s PPOS+(∃pair sum)+(w2 ∃sum) = s GMS+(w2 ∃sum). Furthermore, we obtain a series of further identities because of the following: • In place of condition (Usum ) we can put conditions (ext  ), (Ufu ) and (Sfu ), which are equivalent to it in the class s POS.50  • In place of condition (SSP) we can put conditions (ext  +), (ext  +), (Msum ), (Msum ) and (†), which are equivalent to it in the class s POS.51 • In place of condition (∃sum) we can put the following two conjunctions of conditions (∃ 1 ) and (w1 ∃sum), and (∃ 1 ) and (w2 ∃sum).52 Proof 1. We have already established the first identity (the non-definitional one). 2. LMS = s PPOS+(∃sum): This follows from Theorems 3.9.1, 2.4.9 and 2.6.7. 3. LMS = s PPOSp+(∃sum): From Corollary 3.9.3 we have LMS ⊆ s PPOSp+ (∃sum). Moreover, s PPOSp ⊆ s PPOS, so we use LMS = s PPOS+(∃sum). 4. s POS+(Ssum )+(∃sum) = (t )+(WSP)+(∃sum): From Lemma 2.6.10. 5. (t )+(WSP)+(∃sum) ⊆ LMS: By virtue of Lemmas 2.3.1(i) and 3.9.4. 6. LMS ⊆ s POS+(Ssum )+(∃sum): Condition (Usum ) entails (Ssum ). 7. LMS = s PPOS+(∃fu): From Theorems 3.9.1, 2.12.6, 2.4.9 and 2.6.7. 8. s PPOS+(∃fu) = s POS+(SSP)+(∃pair sum)+(w2 ∃sum): From Corollary 3.8.5. 9. LMS ⊆ s GMS+(w2 ∃sum): From Corollary 3.9.3.

10. s GMS+(w2 ∃sum) ⊆ s POS+(SSP)+(∃pair sum)+(w2 ∃sum): Obvious. As mentioned on Sect. 3.8, with respect to the identity LMS = s GMS+(w2 ∃sum) Diagram 3.5 equally concerns the class of mereological structures. One can see from Diagram 3.5 that in all broader classes of structures than LMS it is already the case that being a mereological sum of elements of a given non-empty set coincides with being a least upper bound of elements of that set (cf. Lemma 3.6.8 and Corollary 3.9.3), i.e., we we have:   ∀S∈2U ∀x∈U x sum S ⇐⇒ S = ∅ ∧ x sup S .

(sum-sup)

Remark 3.9.1 (Historical) For Le´sniewski’s original system, the above result was proven by Tarski in two formulations: the first when of the relation sup is defined 50 See

Theorem 2.12.4 and Fact 2.12.5. Corollary 2.7.5 and Theorem 2.9.5, respectively. 52 See Lemma 3.8.3. 51 See

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with the help of (df sup); and the second when the relation is defined with using the condition corresponding to condition (B.2.28). Le´sniewski writes about this in (1930, p. 87, theses (b) and (c), respectively).

The fact that (sum-sup) is thesis of LMS is not, however, shown by the fact that it is permissible to take the pair of conditions (Usup ) and ∀S∈2U \{∅} ∃x∈U x sup S.

(∃sup∅ )

instead of the pair of axioms (Usum ) and (∃sum). Condition (Usup ) follows from the axioms of the class s POS, and below we will show that condition (∃sup∅ ) itself does not suffice for the axiomatisation of the class LMS (see Fact 3.9.9). We have, however, the following continuation of Theorem 3.9.6. Lemma 3.9.7 (c∃pair inf) follows from the axioms of s POS+(∃sup∅ ). Proof Suppose that x  y, i.e., {u ∈ U : u  x ∧ u  y} = ∅. Then, by (∃sup∅ ), there is a z ∈ U such that z sup {u ∈ U : u  x ∧ u  y}. So there is a z ∈ U such

that z inf {x, y}. Thus, we obtain (c∃pair inf). Theorem 3.9.8 The following identities of classes of structures hold: LMS = s POS+(sum-sup)+(∃sup∅ ) = s POS+(SSP+)+(‡∅ )+(∃sup∅ ) = s POS+(SSP+)+(∃sup∅ ) = s POS+(WSP+)+(∃sup∅ ). Proof By Lemma 3.6.8, in the class s PPOSp condition (sum-sup) holds. So, by Theorems 2.9.5 and 3.9.6 and Corollary 3.9.3, we obtain LMS = s PPOS+ (∃sum)= s POS+(SSP)+ (∃sum)= s POS+(†)+(∃sum)= s PPOSp+(∃sum)= s POS+(sum-sup)+(∃sum)= s POS+ (sum-sup)+(∃sup∅ ) = s PPOS+(‡∅ )+ (∃sup∅ ). Moreover, we have s PPOSp = MEMp = s POS+(WSP+)+(c∃pair inf) and, by Lemma 3.9.7, (c∃pair inf) holds in s POS+(∃sup∅ ). So we see that LMS =

s PPOSp+(∃sum) = s PPOSp+(∃sup∅ ) = s POS+(WSP+)+(∃sup∅ ). Fact 3.9.9 (i) Sentences (∃sum) and (‡∅ ) do not follow from the axioms of the class s PPOS+(∃sup∅ ). (ii) Sentences (Usum ), (WSP) and (SSP) do not follow from the axioms of the class s POS+(∃sup∅ ). (iii) LMS  s POS+(SSP)+(∃sup∅ )  s POS+(∃sup∅ ). Proof (i) Model 2.7 belongs to s PPOS+(∃sup∅ ). In this model (∃sum) does not hold, because, e.g., the set {1, 2} does not have a mereological sum. Similarly, 123 sup {1, 2}, but ¬ 123 sum {1, 2}, i.e., (‡∅ ) does not hold either. (ii) Model 2.1 belongs to s POS+(∃sup∅ ). In this model sentence (Usum ), (WSP) and (SSP) do not hold.

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(iii) From Theorem 3.9.8 we have LMS = s POS+(SSP+)+(∃sup∅ ) ⊆ s POS +(SSP)+(∃sup∅ ) ⊆ s POS+(∃sup∅ ). By virtue of (ii), Model 2.1 from the class s POS+(∃sup∅ ) does not belong to the class s POS+(SSP)+(∃sup∅ ), yet by virtue of (i), Model 2.7 from the class s POS+(SSP)+(∃sup∅ ) does not belong to the class LMS.

3.9.2 Operations in Mereological Structures By virtue of (∃sum) in all mereological structures we can—just as in structures form the class MEM+(∃pair sum)—introduce the operation of sum by putting for arbitrary x, y ∈ U : x  y := ( u) u sum {x, y}. ι

Furthermore, since in LMS condition (SSP+) holds (cf. Corollary 3.9.3), condition (c∃ ) also holds (cf. Theorem 3.6.7). We can, therefore, define in these structures the following operations: mereological relative complementation  and product , just as in class s PPOSp. Since all mereological structures have a unity, we can therefore also define in them the conditional operation of complementation for objects distinct from the unity, just as in class s PPOSp1:  ∀x∈U x = 1 =⇒ – x := 1

x



.

The above operation has all the properties listed in the discussion of classes s GMS and s GMS1, respectively. As has already been noted in footnote 42, the above facts have been proven (Pietruszczak 2000, 2018, Chap. II) only for mereological structures. We therefore see that they already hold in weaker theories. There is another route to the above results. Firstly, thanks to (∃sum) and (2.6.1), we have:

   I[S] = ∅ ⇐⇒ ∃x∈U x sum I[S] , (3.9.1) ∀S∈2U where have:



I[S] = {y ∈ U : ∀s∈S y  s}. In the particular case where S := {y, z} we   ∀y,z∈U y  z ⇐⇒ ∃x∈U x sum {u ∈ U : u  y ∧ u  z} .

Therefore, in accordance with our analysis in Sect. 3.2, we can define the partial operation . We can also use the fact that sum ⊆ sup, from which we get:   ∀y,z∈U y  z =⇒ ∃x∈U x sup {u ∈ U : u  y ∧ u  z} . We bear in mind that the reciprocal definability of relations sup and inf, (c∃pair inf) holds. Therefore, condition (c∃ ) also is satisfied.

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Having condition (c∃ ) we can once again “extend” Theorems 3.9.6 and 3.9.8: Theorem 3.9.10 The following identities of classes of structures hold: LMS = s POS+(Ufu )+(c∃ )+(∃fu) = s POS+(Ssum )+(c∃ )+(∃fu) = (t )+(WSP)+(c∃ )+(∃fu) = s POS+(WSP)+(c∃ )+(∃fu). Proof By virtue of Corollary 2.6.9, in the class s POS condition (Usum ) (resp. (Ufu )) entails condition (WSP). By virtue of Theorem 3.2.3, conditions (t ), (WSP) and (c∃ ) also entail (SSP). We therefore apply Theorem 3.9.6.

3.9.3 Some Weaker Theories than the Theory LMS The addition of condition (c∃ ) to the identities in the theorem above would be, however, superfluous, contrary to what Simons says in (1987, pp. 37–41). This is because the class s POS+(WSP)+(∃fu) is associated with a certain “misunderstanding” which concerns “Classical Extensional Mereology” as formulated by Simons. Simons adopts elementary counterparts of conditions (WSP) and (∃fu) as axioms for his theory. The elementary counterpart of (∃fu) is a certain sentential schema which we obtain from (∃fu) by “expanding” definition (df fu) written with the help of ‘’ (see Sect. 2.12) and by replacing the set-theoretic formula ‘s ∈ S’ with the sentential schema ‘Fs’.53 Simons considers that the group of conditions he adopts axiomatises Le´sniewski’s mereology in an elementary fashion. He is wrong, though, as his conditions entail Breitkopf’s third axiom from (1978). This is a matter we have discussed in detail in (Pietruszczak 2000, 2018, Chap. IV, Sect. 8). Here, we will show that the axioms of classical extensional mereology are even weaker than the axioms of the class s POS+(Ufu )+(∃fu). Fact 3.9.11 (i) s POS+(Ssum )+(∃fu) = s POS+(WSP)+(∃fu) = (t )+(WSP)+ (∃fu). (ii) LMS  s POS+(WSP)+(∃fu). Proof (i) From the facts given in Chap. 2 we obtain: s POS+(Ssum ) = s POS+ (WSP) = (t )+(WSP). (ii) By virtue of Corollary 2.6.9, s POS+(Usum )+(∃fu) ⊆ s POS+(WSP)+(∃fu). Furthermore, by Theorem 2.12.4, s POS+(Usum ) = s POS+(Ufu ). Model 2.2 belongs to s POS+(WSP)+(∃fu), but condition (Ufu ) is false in it. To see this, in this model (WSP) holds in an obvious way. Sentence (∃fu) is also true 53 It is possible to substitute various “particular” formulas for the schema ‘Fs’, such as ‘s = x ∨ s = y’ etc. In (Simons 1987) an elementary counterpart of condition (as ) is needlessly adopted, since it follows from (t ) and (WSP).

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Model 3.8 (ext  ), (Usum ), (Ufu ), (WSP) and (∃fu) hold, but (SSP) and (∃sum) do not hold

123

12

13

23

1

2

3

in it. Firstly, by virtue of (2.12.2), for each x ∈ U we have x fu {x}. Secondly, for an arbitrary set of at least two elements S included in U , we have O(S) = U = O(12) = O(21), i.e., 12 fu S (and 21 fu S). Therefore, each non-empty set of the universe has a fusion. In Model 2.2, conditions (Usum ) and (Ufu ) do not hold, because 12 sum {1, 2}, 21 sum {1, 2}, 12 fu {1, 2} and 21 fu {1, 2}.

Remark 3.9.2 (Pietruszczak 2000, 2018) Simons’ classical extensional mereology is not even extensional; i.e., the sentences (ext  ), (ext  ) and (ext  ) are not theses of it. For example, they are false in Model 2.2, where P(12) = {1, 2} = P(21) and O(12) = U = O(21), but 12 = 21.

Model 2.2 shows that Simons’ classical extensional mereology is a very weak theory. It is even weaker than the theory s POS+(Ufu )+(∃fu), which is substantially weaker than the theory LMS. Fact 3.9.12 (Pietruszczak 2000, 2018) (i) (∃sum) does not follow from the axioms of the class s POS+(Ufu )+(∃fu). (ii) LMS  s POS+(Ufu )+(∃fu)  s POS+(WSP)+(∃fu). Proof Observe that, by Theorem 2.12.4, we have: s POS+(Usum ) = s POS+(Ufu ) = s POS+(ext  ). (i) We will show that Model 3.8 belongs to the class s POS+(Ufu )+(∃fu), but (∃sum) is false in it. In Model 3.8 condition (ext  ) holds, because O(1) = {1, 12, 13, 123}, O(2) = {2, 12, 23, 123}, O(3) = {3, 13, 23, 123}, O(12) = U \ {3}, O(13) = U \ {2}, - O(23) = U \ {1}, O(123) = U . Furthermore, the identities above show that if S is a subset of the universe with at least two elements, then: O(S), / O(S), 13 fu S ⇐⇒ 2 ∈ 23 fu S ⇐⇒ 1 ∈ / O(S), 123 fu S ⇐⇒ O(S) = U. 12 fu S ⇐⇒ 3 ∈ /

Form this and (2.12.2) it follows that in Model 3.8, (∃fu) is true.

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In Model 3.8 sentence (∃sum) is false. For example, in this model there is no unity and the universe therefore does not have a mereological sum (this is also the case for an arbitrary set to which belong at least two of the three elements: 12, 13 and 23). (ii) Earlier, we showed that (Ufu ) and (∃fu) follow from the axioms of the class LMS; i.e., we have LMS ⊆ s POS+(Ufu )+(∃fu), and (i) shows that there is no converse inclusion. As we recall, (Ufu ) entails (WSP). Hence s POS+(Ufu )+(∃fu) ⊆ s POS+(WSP)+ (∃fu). Model 2.2 shows, however, that the converse inclusion does not hold.

The theories s POS+(Ufu )+(∃fu) and s POS+(WSP)+(∃fu) are therefore examples of non-polarised existentially-involved theories and the second of them isn’t even extensional. They show that in the identities LMS = s POS+(Ufu )+(∃sum) = s POS+(SSP)+(∃fu) it is not possible to change either (∃sum) for (∃fu) or (SSP) for (WSP).54 Remark 3.9.3 Models 2.2 and 3.8 are also models of Simons’ elementary version of classical extensional mereology. This is so because an arbitrary non-empty “whole” of elements is finite and therefore “described” by the elementary formula ‘s = x1 ∨

· · · ∨ s = xn ’, which we can use as ‘Fs’ instead of the set {x1 , . . . , xn }.

3.9.4 The Generalised Operations of Mereological Sum and Product Thanks to conditions (Usum ) and (∃sum) we can introduce in mereological structures two generalised operations of mereological sum and mereological product. The for mer is the function : 2U \ {∅} → U which assigns to every non-empty set the mereological sum of its elements: 

S := ( x) x sum S. ι

∀S∈2U \{∅}

(df



)

From the definition above and from the fact that (sum-sup) and sum = fu hold in LMS, we get:  1 = U,  x  y = {x, y},

(3.9.2) (3.9.3)

ι

(3.9.4)

∀x,y∈U  ∀S∈2U \{∅} S = ( x) x sup S = ( x) x fu S,    ∀S∈2U \{∅} ∀u∈U u  S ⇐⇒ ∃s∈S u  s , ι

54 We

remember that the following identity holds: LMS = s POS+(WSP)+(∃sum).

(3.9.5)

136

∀S∈2U \{∅} O(



3 “Existentially Involved” Theories of Parts

S) =



(3.9.5 )

O[S].

Furthermore, we get:    ∀S∈2U \{∅} ∀u∈U u  S ⇐⇒ ∀y∈U (y  u ⇒ ∃s∈S y  s) ,

   ∀S∈2U \{∅} I( S) = u ∈ U : I(u) ⊆ O( S) .

(3.9.6) (3.9.6 )

  If u  S and y  u,then y  S, by virtue of (t ). Hence ∃s∈S y  s, by virtue of  (df sum) and (df ). Conversely, if  ∀y∈U (y  u ⇒ ∃s∈S y  s) then I(u) ⊆ O( S), by virtue of (3.9.5 ). Hence u  S, by (SSP). We have the following generalisation of condition (3.4.15)55 :    ∀S∈2U ∀x,y∈U ∀s∈S y  x  s ∧ I(S) = ∅ =⇒ y  x .

(3.9.7)

Let ∀s∈S y  x  s and y  x. Then, by (SSP), there is a z such that z y and z  x. By (t ) we have ∀s∈S z  x  s. Hence, by (3.4.14), ∀s∈S z  s, i.e., I(S) = ∅. By virtue of (3.9.4) we also have generalisation of condition (3.4.16):    ∀S∈2U \{∅} ∀x∈U ∀s∈S s  x =⇒ S  x .

(3.9.8)

Thanks  to condition (3.9.1) we can introduce the following partial operation of product : 2U → U by putting: 

I[S] = ∅ =⇒



S := ( x) x sum ι

∀S∈2U



 I[S] .

(df



)

 Condition (3.9.1) says that in thedomain of the operator is just the family of sets {S ∈ 2U : I[S] = ∅}. If exists S, then we will call it the product of all elements of the set S. From ( 0) we know that so long as the universe U is not a singleton, then there will be in it no element that is an ingrediens of all elements of the set U . Therefore—excepting the case of a “trivial” structure—the product of all elements of the set U does not exist.    We allow the case where  S = ∅. Then I[∅] = {I(s) : s ∈ ∅} = ∅ = U . Therefore, ∅ exists and ∅ := ( x) x sum U =: U = 1. By exploiting the connection between the reciprocal definability of the greatest lower bound and the least upper bound, we get, for any S ∈ 2U and x ∈ U : ι

x inf S ⇐⇒ x sup {y ∈ U : ∀s∈S y  s}. Therefore, by drawing on the identity sum = sup for non-empty sets, we get:

Obviously, 55 Observe



that



I[S] = ∅ =⇒



 S = ( x) x inf S . ι

∀S∈2U

I[{x, y}] = ∅ iff x  y, therefore:



I[∅] =

  {I(z) : z ∈ ∅} = ∅ = U = ∅.

(3.9.9)

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   ∀x,y∈U x  y =⇒ x y = {x, y} .

(3.9.10)

We have the following generalisations of (3.2.3) for an arbitrary S ∈ 2U :  S  s, I[S] = ∅ =⇒ ∀s∈S     ∀x∈U ∀s∈S x  s ⇐⇒ I[S] = ∅ ∧ x  S ,    I[S] = ∅ =⇒ I( S) = {I(s) : s ∈ S}. 

(3.9.11) (3.9.12) (3.9.13)

   For (3.9.11): If I[S] = ∅ then S exists and so we apply(3.9.9), because S  x  s, then x ∈ I[S] and S exists. is a lower bound of S. For (3.9.12): If ∀ s∈S     Since S sum I[S], then x  S (this also follows from the fact that S inf S and x is a lower bound of S). Conversely, we use (t ) and (3.9.11). For (3.9.13): Directly from (3.9.12). In mereological structures hold counterparts of the distributivity conditions for complete lattices. This result is a generalisation of Theorem 3.4.14. To prove it, we will need the following: ∀S∈2U ∀x∈U x ∈



I[{x  s : s ∈ S}].

For any s ∈ S: x  x  s (for S = ∅ we have



I[∅] =



(3.9.14) ∅ = U ).

Theorem 3.9.13 For arbitrary S ∈ 2U and x ∈ U :   I[S] = ∅ =⇒ x = {x  s : s ∈ S},    I[S] = ∅ =⇒ x  S = {x  s : s ∈ S},   ∃s∈S s  x =⇒ x S = {x s : s ∈ S ∧ s  x}.

(3.9.15) (3.9.16) (3.9.17)

  Proof For (3.9.15):  Suppose that I[S] = ∅. Then, by (3.9.14), {x  s: s ∈ S} exists and x  {x  s: s ∈ S}. Conversely, by virtue of (3.9.11), ∀s∈S {x  s : s ∈ S}  x  s. Hence {x  s : s ∈ S}  x, by  (3.9.7). Finally, we use (antis ). For (3.9.16): Suppose that I[S]  = ∅. Then  S exists.  Furthermore, by virtue of (3.9.14), {x  s : s ∈ S} exists. We have also S  ({x  s : s ∈ S}). There  fore x  S  ({x  s : s ∈ S}), by (3.4.16). Assume for a contradiction that    ({x  s : s ∈ S})  x  S. Then, by (SSP), there is a u such that u  ({x  s:   x  S. From the former, we have: ∀s∈S u  x  s. From the latter: s ∈ S}) and u  u  x and u  S, by (3.4.12).  Hence, by (3.4.14), we have ∀s∈S u  s, which gives us our contradiction: u  S.  For (3.9.17): By virtue of our assumption, S = ∅, i.e., S exists. Take an arbitrary u ∈ S such that   u  x. Then we have x u  u S. Hence x u  x S. Furthermore, {x s : s ∈ U ∧s  x}   x S, by (3.9.8). Conversely, assume for a contradiction that x  S  {x s : s ∈ S ∧ z  x}. Then, by (SSP), there is a usuch that u  x S and u  {x s : s ∈ S ∧ s  x}. Therefore u  x, u  S and ∀s∈S (s  x ⇒ u  x s), by (3.9.5). By virtue of (3.2.5)

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we have ∀s∈S (s   x ⇒ u  s). Hence, by (3.9.5), we get u  contradicts: u  S.

 {z ∈ S : z  x}, which

3.9.5 “Classical Mereology” with the Primitive Relation  For certain “purely technical” reasons, classical mereology is often presented with the reflexive relation , i.e., as a theory of structures of the form U, . This will be the class TMS := (t )+(∃1 sum), which was defined in Remark 3.7.2. This is the approach we shall take in this point. Theorem 3.9.14 (Pietruszczak 2000, 2018) In the class TMS conditions (r ), (antis ), (Usum ), (∃sum) and (SSP) hold.56 Proof For (∃sum): This follows in an obvious way from (∃1 sum). For (r ): Take an arbitrary x ∈ U . From (∃sum) we have a y such that y sum {x}. From (df sum) we get (a) x  y and (b) ∀u (u  y ⇒ u  x). Hence, it follows that x  x. Applying (df ), we get ∃v v  x, i.e., I(x) = ∅. Therefore, by virtue of (∃sum), there is a z such that z sum I(x). Now, from (df sum) we have ∀u (u  x ⇒ u  z) and ∀u (u  z ⇒ ∃v (v  x ∧ v  u)). Hence (c) ∀v (v  x ⇒ ∃v (v  x ∧ v  u)). This suffices for us to say—by applying (df sum)—that x sum I(x). We will show that y sum I(x) too. Hence, by virtue of (∃1 sum), we will get x = y, which—by virtue of (a)—gives us x  x. Firstly, by virtue of (a) and (t ), ∀u (u  x ⇒ u  y) hold. Secondly, by virtue of (b) and (df ), we have ∀u (u  y ⇒ ∃w (w  x ∧ w  u)). From this and (c) we get ∀u (u  y ⇒ ∃v,w (v  x ∧ v  w ∧ w  u)). Therefore, ∀u (u  y ⇒ ∃v,w,u  (v  x ∧ u   v ∧ u   w ∧ w  u)). Hence, by virtue of (t ) and (df ), we have ∀u (u  y ⇒ ∃v (v  x ∧ v  u)). From both of the above results it follows that y sum I(x). For (Usum ): In Remark 3.7.2 we showed that this follows from (r ) and (∃1 sum). For (antis ): From (r ) we have x sum {x}. Assume that x  y and y  x. We will show that then y sum {x}. Take an arbitrary z such that z  y. Then, by virtue of (t ), we have z  x. In addition, by virtue of (r ), z  z holds. So z  x. Therefore x = y, by virtue of (Usum ). For (SSP): This follows from (r ), (t ), (Usum ) and (∃sum), by Theorem 3.9.1.

It is obvious that (∃1 sum) follows from (Usum ) and (∃sum). So in the light of Theorem 3.9.14 we get: Corollary 3.9.15 The identities below hold: TMS = POS+(∃1 sum) = POS+(Usum )+(∃sum) = (t )+(Usum )+(∃sum). 56 That,

in the class TMS, conditions (r ) and (antis ) hold was proven by Tarski by applying Theorem B.7.4. We will give an elementary proof of this result.

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139

The classes of structures POS and s POS are elementarily definitionally equivalent in the sense of Szmielew (1981, p. 8), under the assignment of the structures described on Sect. 3.7.2. Hence—by virtue of the identities above—it follows that the classes of structures LMS and TMS are also elementarily definitionally equivalent under the same assignment of structures. Therefore, from the identities above along with Corollary 3.9.15 and Theorems 3.9.6, 3.9.8 and 3.9.10: Corollary 3.9.16 The following identities hold: TMS = POS+(SSP)+(∃sum) = POS+(SSP)+(∃fu) = POS+(Ssum )+(∃sum) = POS+(Ssum )+(c∃ )+(∃fu) = POS+(Ufu )+(∃sum) = POS+(Ufu )+(c∃ )+(∃fu) = POS+(WSP)+(∃sum) = POS+(WSP)+(c∃ )+(∃fu) = POS+(SSP)+(∃pair sum)+(w2 ∃sum) = GMS+(w2 ∃sum) = POS+(sum-sup)+(∃sup∅ ) = POS+(SSP)+(‡∅ )+(∃sup∅ ) = POS+(SSP+)+(∃sup∅ ) = POS+(WSP+)+(∃sup∅ ). However, in some of the above cases, instead of a complete axiomatisation of the class POS, one may take just condition (t ) or both (t ) and (antis ). For example, Lemma 2.3.7 says that (r ) follows from (t ) and (SSP). Furthermore, we have the following: Lemma 3.9.17 (Pietruszczak 2000, 2018) Adopting  as a primitive relation in a set U , use (df sum), (df 1 ) and (df ). Then (Ssum ) and (∃sum) entail (r ). Proof Take an arbitrary x from U . By (∃sum) there is a y such that y sum {x}. From this and (df sum) we have x  y, and from (Ssum ) we have x = y. Therefore x = x.

Theorem 3.9.18 (Pietruszczak 2000) The following identities hold: TMS = (t )+(antis )+(SSP)+(∃sum) = (t )+(antis )+(SSP)+(∃fu) = (t )+(Ufu )+(∃sum) = (t )+(Ssum )+(∃sum) = (r )+(t )+(Ssum )+(c∃ )+(∃fu) = (t )+(WSP)+(∃sum) = (r )+(t )+(WSP)+(c∃ )+(∃fu) = (t )+(antis )+(sum-sup)+(∃sup∅ ) = (t )+(antis )+(SSP)+(‡∅ )+(∃sup∅ ) = (r )+(t )+(Ufu )+(c∃ )+(∃fu) = (t )+(antis )+(SSP)+(∃pair sum)+(w2 ∃sum). Proof On the strength of Lemma 2.6.12 we have: • (t )+(Ssum ) = (t )+(WSP); • (r ), (t ) and (Ssum ) entail (antis ); • (antis ), (t ) and (SSP) entail (Usum ). Furthermore, Lemmas 2.3.7, 2.12.2, 2.12.3 and 3.2.2 say in turn that:

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from (t ), (SSP) we get (r ); from (r ), (t ) and (Usum ) we get (Ufu ); from (t ) and (Ufu ) we get (Usum ); from (r ), (t ), (WSP), (c∃ ) and (df 1 ) we get (SSP).

1. Everywhere from (t ) and (SSP) we get (r ), by strength of Lemma 2.3.7. 2. The identities in the first band: by virtue of Corollary 3.9.16 and Lemma 2.3.7. 3. TMS = (t )+(Ufu )+(∃sum): From Lemma 2.12.3, (t )+(Ufu )+(∃sum) ⊆ (t )+ (Usum )+(∃sum) = TMS. Moreover, by virtue of Corollary 3.9.15 and Lemma 2.12.2, we have the converse inclusion. 4. TMS = (t )+(Ssum )+(∃sum): By virtue of Lemma 3.9.17, from (Ssum ) and (∃sum) we have (r ). Moreover, by Lemma 2.6.12, from (r ), (t ) and (Ssum ) we have (antis ). Therefore, we apply Corollary 3.9.16. 5. (t )+(Ssum )+(∃sum) = (t )+(WSP)+(∃sum): From 4 and Lemma 2.6.12. 6. (r )+(t )+(WSP)+(c∃ )+(∃fu) ⊆ TMS: From (r ), (t ), (WSP) and (c∃ ) we get (SSP), by Lemma 3.2.2. Hence sum = fu. Furthermore, we apply Lemma 2.6.12. The converse implication has already been proven. 7. TMS = (r )+(t )+(Ssum )+(c∃ )+(∃fu): From 5 and Lemma 2.6.12. 8. (r )+(t )+(Ufu )+(c∃ )+(∃fu) ⊆ TMS: From (t ) and (Ufu ) we get (Usum ). From (r ) and (Usum ) we have (Ssum ) (see Lemma 2.6.8). Therefore, we make use of 7. 9. Theorem 2.9.5 says that (r ) and (t ) we hame the equivalence of the conditions (SSP) and (†).

Remark 3.9.4 (Historical) (i) In his notes from Le´sniewski’s lectures at the University of Warsaw in 1922/1923, Jan F. Drewnowski wrote that at the time Le´sniewski presented mereology in a form that looks as follows in our set-theoretic notation.57 The primary notion of Le´sniewski’s theory was the relation . Structures of the form U,  belong to POS. The relation  Le´sniewski defined by condition (df 1 ). Surprisingly, Le´sniewski defined the concept of collective class as the concept of supremum in his lectures but limited the latter to non-empty sets. Let us replace this concept of collective class with the relation Kl which Le´sniewski defined by putting for arbitrary x from U and S from 2U : x Kl S :⇐⇒ S = ∅ ∧ ∀s∈S s  x ∧ ∀ y∈U (∀s∈S s  y ⇒ x  y).

(df Kl)

So we have: ∀S∈2U ∀x∈U (x Kl S ⇐⇒ S = ∅ ∧ x sup S),

(Kl-sup)

where the relation sup is defined by (df sup).58 For the relation Kl Le´sniewski adopted the following axiom (his axiom A3): ´ etorzecka notes are unpublished. I read them courtesy of Professor Kordula Swi˛ of the Cardinal Wyszy´nski University in Warsaw. 58 Thus, we see that Le´sniewski defined the concept of collective class differently than in (1927; 1928) (cf. Definition (df ccP)). 57 Drewnowski’s

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Classical Mereological Structures

∀S∈2U \{∅} ∃x∈U x Kl S.

141

(∃Kl∅ )

So, condition (∃Kl∅ ) corresponds to condition (∃sup∅ ). Furthermore, Le´sniewski adopted condition (SSP+) as axiom A4. Therefore, Le´sniewski operated with the class POS+(SSP+)+(∃Kl∅ ). Now we show that in such Le´sniewski’s theory, however, we obtain Kl = sum, where the relation sum is defined by (df sum). By Lemma 3.6.8, in the class POS+ (SSP+)+(∃Kl∅ ) condition (sum-sup) holds. So, by (Kl-sup), for arbitrary x from U and S from 2U we have: x sum S iff S = ∅ and x sup S iff x Kl S. Therefore, by (∃Kl∅ ), the relation sum satisfies condition (∃sum). Obviously, if in the class TMS the relation Kl is defined by (df Kl), then—in virtue of (sum-sup) and (Kl-sup)—condition (∃Kl∅ ) holds (cf. Corollary 3.9.16). We see, therefore, that Le´sniewski’s theory from 1922 is definitionally equivalent to the theory of the class TMS. (ii) In his notes, Drewnowski noted that in Le´sniewski’s 1922 theory the weaker axiom (WSP+) can be adopted instead of the axiom (SSP+). This is indeed the case, as we will now show. The primary notion of Drewnowski’s theory was the relation . Structures of the form U,  belong to s POS. The relation  Le´sniewski defined by condition (df ). Now let is notice two facts. Firstly, s PPOSp = MEMp = s POS+(WSP+)+ (c∃pair inf). Secondly, we can obtain condition (c∃pair inf) in the theory of the class s POS+(∃Kl∅ ). Just repeat the proof of Lemma 3.9.7 using (∃Kl∅ ) and (Kl-sup) instead of (∃sup∅ ). We see, therefore, that s POS+(SSP+)+(∃Kl∅ ) = s POS+ (WSP+)+(∃Kl∅ ). Similarly we have POS+(SSP+)+(∃Kl∅ ) = POS+(WSP+)+

(∃Kl∅ ).

3.9.6 Mereological Structures and Complete Boolean Lattices (Complete Boolean Algebras) The theorem below says that from each non-degenerate complete Boolean lattice whose zero has been removed we obtain a structure from TMS. This is a sort of supplementation of Theorem 3.7.8 which was given for the classes BL and GMS1. Theorem 3.9.19 (Pietruszczak 2000, 2018) Let U, ≤, 0, 1 be an arbitrary nondegenerate complete Boolean lattice from CBL. We put U+ := U \ {0} and  := ≤  (U+ × U+ ), i.e.,  is the restriction of the relation ≤ to the set U+ . Then conditions (i) and (ii) from Theorem 3.7.5 hold along with the following: (iii) the structure U+ ,  belongs to TMS, 1 is the unity and for arbitrary S ∈ 2U+ \ {∅} and x ∈ U+ we have x sup S iff x sum S. Proof We use Theorem 3.7.8. We only need show that condition (∃sum) holds as well. It follows from the assumption that the structure is a complete Boolean lattice and from the equivalence above.

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Remark 3.9.5 In (Pietruszczak 2000, 2018) another proof of Theorem 3.9.19 was given which did not rest on the connections between Grzegorczykian lattices and structures from the class GMS or between Boolean structures and structures from the class GMS1.

As in the case of classes GMS and GMS1, we have the following theorem for the class TMS, which is in a certain sense the reverse of the above. It is a supplementation of Theorem 3.7.9 for Boolean lattices and structures from the class GMS1. Theorem 3.9.20 (Pietruszczak 2000, 2018) Let U,  belong to TMS and have the unity 1 and let 0 zero be an arbitrary element which does not belong to U . Let U o := U  {0} and define in the set U o a binary relation ≤ :=  ({0} × U o ). Then conditions (i) and (ii) from Theorem 3.7.6 hold along with the following: (iii) The structure U o , ≤, 0, 1 is a complete Boolean lattice from CBL in which 1 is the unity. Proof By Theorem 3.7.9 we can obtain a structure belonging to BL. It is a complete lattice with respect to condition (∃sum) and the equivalences above.

Remark 3.9.6 In (Pietruszczak 2000, 2018) we proved Theorem 3.9.20 in various ways. One of them was by applying Tarski’s Theorem B.7.4. However, unlike here, none of the ways we used appealed to the connection between Grzegorczykian lattices and structures from GMS, and the connection between Boolean structures and structures from GMS1.

Using these last two theorems we get: Corollary 3.9.21 For an arbitrary structure U,  composed of a non-empty set U and a binary relation  the following conditions are equivalent: (a) The pair U,  belongs to TMS. (b) For some (equivalently, for any) 0 which does not belong to U , for U o := U  {0} and for ≤ :=  ({0} × U o ), the structure U o , ≤, 0 is a complete Boolean lattice with the zero 0. (c) For some non-degenerate Boolean lattice B, ≤, 0 with the zero 0 we have U = B \ {0} and  = ≤  (U × U ). Proof “(a) ⇒ (b)” From Theorem 3.9.20. “(b) ⇒ (c)” ) As in the proof of Corollary 3.7.7. “(c) ⇒ (a)” From Theorem 3.9.19.



Remark 3.9.7 The equivalence “(a) ⇔ (c)” can be more plainly expressed in the following way, corresponding to what was said in Remark 3.7.1: A structure from TMS is that and only that structure which arises from a non-degenerate Boolean lattice after the zero has been removed. The equivalence “(a) ⇔ (b)” may also be put more simply:

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Classical Mereological Structures

A structure from TMS is that and only that structure which yields a nondegenerate Boolean lattice with the addition of the zero.

143



By using the elementary definitional equivalence of the classes LMS and TMS discussed earlier, the following theorem results from Theorems 3.9.19 and 3.9.20. Theorem 3.9.19 Let U, ≤, 0 be an arbitrary non-degenerate complete Boolean lattice from CBL. Put U+ := U \ {0} and  := (≤ \ idU )  (U+ × U+ ), i.e.,  is a restriction of the relation  to the set U+ . Then conditions (i) and (ii) from Theorem 3.7.5 hold along with the following: (iii) The structure U+ ,  belongs to LMS, 1 is its unity and for any S ∈ 2U+ \ {∅} and x ∈ U+ x sup S ⇐⇒ x sum S. Theorem 3.9.20 Let U,  belong to LMS and let 0 be an arbitrary element which does not belong to U . Let U o := U  {0} and define in U o the binary relation ≤ :=  ({0} × U o ). Then conditions (i)–(iii) of Theorem 3.9.20 hold. For structures from LMS we have the following version of Corollary 3.9.21, which was formulated for TMS.59 Corollary 3.9.21 For an arbitrary structure U,  composed of a non-empty set U and a binary relation  the following conditions are equivalent: (a) U,  belongs to LMS. (b) The relation  is irreflexive and for some (equivalently, for any) 0 which does not belong to U , for U o := U  {0} and for ≤ :=   idU  ({0} × U o ), the structure U o , ≤, 0 is a complete Boolean lattice with the zero 0. (c) For some non-degenerate complete Boolean lattice B, ≤, 0 with the zero 0 we have U = B \ {0} and = (≤  (U × U )) \ idU . Proof “(a) ⇒ (b)” From Theorem 3.9.20 . “(b) ⇒ (c)” As in the proof of Corollary 3.7.7 . “(c) ⇒ (a)” From Theorem 3.9.19 .



Remark 3.9.8 The equivalence “(a) ⇔ (c)” can be more plainly expressed in the following way, corresponding to what was said on Sect. 3.7: A structure from LMS is that and only that structure which arises from a complete non-degenerate Boolean lattice after the zero has been removed and the relation has been “irreflexivised”. The equivalence “(a) ⇔ (b)” may also be put more simply: A structure from z LMS is that and only that structure with an irreflexive relation which yields a complete Boolean lattice with the addition of the zero and the “reflexivisation” of the relation. 59 Observe



once again that, in the corollary below, we must add the assumption of the irreflexivity of  to condition (b).

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In point 3.9.2, we introduced in classical mereological structures counterparts of the Boolean operations. In connection with this, we can supplement Theorems 3.7.11 (resp. 3.7.11 ) and 3.7.12 (resp. 3.7.12 ) which show the connection between Boolean algebras and Grzegorczykian mereological structures with unity. The theorems we will now establish differ from them only in that they will address complete Boolean algebras. Theorem 3.9.22 Let U, +, ·, −, 0, 1 be an arbitrary complete Boolean algebra and ≤ a relation defined by the following condition: x ≤ y :⇔ x + y = y. Put U+ := U \ {0} and  := ≤  (U+ × U+ ), i.e.,  is the restriction of the relation ≤ to the set U+ . Then conditions (i)–(iii) of Theorem 3.9.19 hold. Theorem 3.9.22 Let U, +, ·, −, 0, 1 be an arbitrary complete Boolean algebra and ≤ a relation defined by the following condition: x ≤ y :⇔ x + y = y. Put U+ := U \ {0} and  := (≤ \ idU )  (U+ × U+ ), i.e.,  is the restriction of the relation  to the set U+ . Then conditions (i)–(iii) of Theorem 3.9.19 hold. The two following theorems are converse theorems to those above and continuations of Theorems 3.7.12 and 3.7.12 . The first one is obtained from Theorems 3.9.20 and 3.7.12. Theorem 3.9.23 Let U,  belong to TMS and have the unity 1, and let 0 be an arbitrary element which does not belong to U . Let U o := U  {0} and define in U o a binary relation ≤ :=  ({0} × U o ). Then conditions (i)–(iii) from Theorem 3.7.12 hold along with: (iv) In the complete Boolean algebra U o , +, ·, −, 0, 1 obtained from the complete Boolean lattice U o , ≤, 0, 1 the Boolean operations satisfy the conditions of Theorem 3.7.12(iv). In consequence, by defining in U o the three operations +, · and − with the help of the identity from Theorem 3.7.12(iv), we obtain from U,  a complete Boolean algebra U o , +, ·, −, 0, 1, the same as that which we obtain from the lattice U o , ≤, 0, 1. Theorem 3.9.23 Let U,  belong to LMS and have the unity 1, and let 0 be an arbitrary element which does not belong to U . Let U o := U  {0} and define in the set U o a binary relation ≤ :=  ({0} × U o ). Then conditions (i)–(iv) from Theorem 3.9.23 hold. In consequence, by defining in U the three operations +, · and − via the identities from Theorem 3.7.12(iv), we obtain from U,  a complete Boolean algebra U o , +, ·, −, 0, 1, the same as that which we obtain from the lattice U o , ≤, 0, 1. By making use of the standard connection between Boolean lattices and Boolean algebras, we have the following continuations of Corollaries 3.9.21 and 3.9.21 .

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145

Corollary 3.9.24 For an arbitrary structure U,  composed of a non-empty set U and a binary relation , conditions (a)–(c) from Corollary 3.9.21 are equivalent along with the following: (d) For some complete non-degenerate Boolean algebra A, +, ·, −, 0, 1 with the zero 0 and the unity 1 we have U = A \ {0},  = ≤  (U × U ) and 1 = 1, where the relation ≤ is defined for all x, y ∈ A: x ≤ y iff x + y = y. Corollary 3.9.24 For an arbitrary structure U,  composed of a non-empty set U and a binary relation  conditions (a)–(c) from Corollary 3.9.21 are equivalent along with the following: (d) For some complete non-degenerate Boolean algebra A, +, ·, −, 0, 1 with the zero 0 and the unity 1 we have U = A \ {0}, = (≤  (U × U )) \ idU and 1 = 1, where the relation ≤ is defined for all x, y ∈ A: x ≤ y iff x + y = y.

3.10 The Case of Finite Structures A consideration of finite structures is interesting both from a theoretical and from a philosophical point of view (we can, after all, recognize that there is a finite number of so-called specific objects). We will show that when we consider classes of finite structures, then in Diagram 3.5 (and others), in some cases, the strict inclusion of classes A  B is translated into the identity Afin = B fin . This obviously happens only when the sharp inclusion cannot be proven with the help of finite models. That we have used an infinite model to prove the strict inclusion of the classes A  B does not mean that it is not possible to use a finite model. Our infinite model “suggests”, however, that it must be infinite; i.e., it “suggests” a proof of the identity Afin = B fin . Since that “suggestion” does not suffice as a proof, however, we must provide a proper one. We will first present our results in Diagram 3.6, in which the arrows from Diagram 3.5 have been replaced with identities ‘=’ (for classes of finite structures). Next, using these identities, we will simplify Diagram 3.6, the result of which is Diagram 3.7. Let us begin with the top of Diagram 3.5, i.e., with the first of the cases of dependencies between classes of structures we have considered, where we used infinite models. We will prove that in the case of finite structures Fact 3.4.12 corresponds to the following: Fact 3.10.1 In the class MEMfin condition (c∃pair sup) holds.

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Proof Assume that the universe is finite and that x Un y. Therefore, there exists at least one upper bound of the pair {x, y} and there is a finite number of such upper bounds. Let z 1 , . . . , z n be all the upper bounds of the pair {x, y}. Amongst them is also the product z 1 z 2 · · · z n , which exists by virtue of (c∃ ). This product is, however, the least upper bound of the pair {x, y}.

We therefore have the following identity: MEMfin = MEMfin +(c∃pair sup). Hence, since MEM+(c∃pair sup)+(‡∅ ) = CMM+(‡∅ ), it is also the case that: MEMfin +(‡∅ ) = CMMfin +(‡∅ ). We will now show that in the case of finite structures Fact 3.5.2 corresponds to the following: Fact 3.10.2 In the class CMMfin condition (‡∅ ) holds. Proof Assume that ∅ = S ⊆ U and x sup S. Then S ⊆ I(x) and for some n > 0, z 1 , . . . , z n we have S = {z 1 , . . . , z n }. Since {z 1 , z 2 } ⊆ I(x), then by virtue of (c∃pair sum) there exists a z 1,2 such that z 1,2 sum {z 1 , z 2 }. Obviously, z 1,2  x. Similarly, by virtue of (c∃pair sum), there exists a z 1,2,3 such that z 1,2,3 sum {z 1,2 , z 3 }. Obviously z 1,2,3  x. We create inductively z 1,...,n such that z 1,...,n sum {z 1,...,n−1 , z n } such that z 1,...,n sum {z 1,...,n−1 , z n }. We have z 1,...,n sum {z 1 , . . . , z n } (this is proven in the same way as condition (3.4.7)). Since sum ⊆ sup, z 1,...,n sup {z 1 , . . . , z n }.

Therefore x = z 1,...,n ; so, x sum S. From the fact above it follows that in the class MEMfin +(∃pair sum) condition (‡∅ ) holds. We therefore get: CMMfin = CMMfin +(‡∅ ), MEMfin +(∃pair sum) = MEMfin +(∃pair sum)+(‡∅ ). The remaining proper inclusions from Diagram 3.2 can be established with the finite Models 2.7 and 3.1–3.5. It is a lot harder to prove the counterpart of Facts 3.6.9 and 3.6.14 for the class of finite structures. Theorem 3.10.3 In the class MEMfin +(‡∅ ) conditions (WSP+) and (SSP+) hold. Proof Assume that U,  is a strict partially-ordered set which satisfies conditions (c∃ ), (SSP) and (‡∅ ); so and (WSP) too. We will show that for the falsification of condition (WSP+) the set U must be infinite.

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147

Let us accept, therefore, that there exist x and y such that y  x (so x = y) and:   ∀z∈U z  x ∧ z  y ⇒ ∃u∈U (u  x ∧ u  y ∧ u  z) .

(§)

By virtue of (WSP), there exists a z 0 such that z 0  x and z 0  y. From this and from (§) there exists a u 0 such that u 0  x, u 0  y and u 0  z 0 , so also u 0 = z 0 and u 0  z 0 . By virtue of (SSP) there exists a w0 such that w0  u 0 and w0  z 0 . We also have w0  x and w0  y. Let D yx := {u ∈ U : u  x ∧ u  y}. We have D yx = ∅. Furthermore, each element of the set D yx is exterior to each element of the set I(y). Observe that x is not the least upper bound of the set D yx . If it were so, then—in virtue of (‡∅ )—there would also be a mereological sum of this set and there is not, but y is exterior to each element of the set D yx . Since, however, D yx ⊆ I(x), then there exists an x0 such that D yx ⊆ I(x0 ) and x  x0 ; and so x = x0 and x  x0 . Since x  x0 , then there exists an element x0 x, by (c∃ ). Observe that x0 x  x. If it were the case that x0 x = x, then it would also be the case that x  x0 , and this we have ruled out. From the fact above, from (§) and from the inclusion D yx ⊆ I(x0 x) we get / D yx . Therefore, by virtue of (c∃ ), there the following: y  x0 x; and so x0 x ∈ exists an element (x0 x) y. We have (x0 x) y  x. Observe that (x0 x) y  y. If it were the case that (x0 x) y = y, then it would also be the case that y  x0 x. However, y = x0 x, because z 0  x0 x and z 0  y. Therefore, it would be the case that y  x0 x. This is, however, ruled out, because if follows from x0 x  x and from (WSP) that there would exist a z 1 such that z 1  x and z 1  x0 x. Hence, in addition z 1  y, because y  x0 x. Therefore, z 1 would be an element of the set D yx , and we would arrive at a contradiction: z 1  x0 x. Therefore, x0 x is not the least upper bound of the set D yx . To see this, if it were so, then in accordance with (‡∅ ), that set would also have a mereological sum; but this is not so, because (x0 x) y  x0 x and (x0 x) y is exterior to each element of the set D yx . Since, however, D yx ⊆ I(x0 x), then from (df sup) there is an x1 such that D yx ⊆ I(x1 ) and x0 x  x1 ; so x0 x = x1 and x0 x  x1 . Since x0 x  x1 , therefore, from (c∃ ) there exists an element x1 (x0 x). Observe that x1 (x0 x)  x0 x. Were it the case that x1 (x0 x) = x0 x, then it would be the case that x0 x  x1 , and this we have ruled out. From the above result, from (§) and from the inclusion D yx ⊆ I(x1 (x0 x)) we get y  x1 (x0 x). Therefore, by virtue of (c∃ ), there exists an element (x1 (x0 x)) y. We have (x1 (x0 x)) y  x. Observe that (x1 (x0 x)) y  y. Were it the case that (x1 (x0 x)) y = y, then it would be the case that y  x1 (x0 x). However, y = x1 (x0 x), because z 0  x1 (x0 x) and z 0  y. Therefore, y  x1 (x0 x) would be true.

148

3 “Existentially Involved” Theories of Parts s PPOSfin

MEMfin  s MPOSfin

MEMfin

s MPOSfin

MEMfin +(c∃pair sup)

MEMfin +(‡∅ )

MEMfin +(∃pair sup)

CMMfin

MEMfin +(∃pair sum)

CMMfin +(‡∅ )

MEMfin +(∃pair sum)+(‡∅ )

s PPOSpfin

s PPOSfin +(w1 ∃sum)

s GMSfin

s PPOSfin +(w2 ∃sum)

s GMSfin +(w1 ∃sum)

s GMS1fin

s GMSfin +(w2 ∃sum) = LMSfin

Diagram 3.6 The lattice of classes of finite structures related to existentially involved theories

This is, however, ruled out because of the following: that x1 (x0 x)  x0 x and from (WSP) there would exist a z 2 such that z 2  x0 x and z 2  x1 (x0 x). Hence also z 2  y, because y  x1 (x0 x). Therefore, z 2 would be an element of set D yx ; i.e., we would obtain the contradiction: z 2  x1 (x0 x). Therefore, x1 (x0 x) is not a least upper bound of the set D yx . Were it so, then, in accordance with (‡∅ ), it would also be the mereological sum of that set; and this is not so, because (x1 (x0 x)) y  x1 (x0 x) and (x1 (x0 x)) y is exterior to each element of the set D yx . We have twice repeated almost exactly the same line of reasoning in order to show that an “infinite chain of reasonings” arises, as a result of which we obtain, for example, the following infinite sequence of different elements: x, x0 , x1 , x2 , . . . ; x, x0 x, x1 x0 x, . . . ; x, y, x0 x y, x1 x0 x y, . . . . In order to better understand the construction of an infinite model from the proof of Fact 3.6.9, let us observe that the above “chains of reasonings” are not the only

3.10

The Case of Finite Structures

149 s PPOSfin

MEMfin  s MPOSfin

MEMfin

MEMfin +(∃pair sup)

s MPOSfin

s PPOSpfin

s PPOSfin +(w2 ∃sum)

LMSfin

Diagram 3.7 The lattice of classes of finite structures related to existentially involved theories (after simplifications)

“source” of infinite sequences. Below, we will introduce another source (we will only give one “run-through”, because the others can be obtained in an analogous fashion). Take the set I(y)  {z 0 }. Obviously, x is an upper bound of that set. It is not, however, the least upper bound. If it were, it would also be its sum and it is not, because w0  x, but w0 is exterior to each element of the set I(y)  {z 0 }. Therefore, from (df sup) there exists a y1 such that I(y)  {z 0 } ⊆ I(y1 ) and x  y1 ; so x = y1 and x  y1 . Since x  y1 , then from (c∃ ) there exists an element y1 x. Observe that y  y1 and z 0  y1 , because y  y1 , z 0  y1 and z 0  y. We have also y  y1 x and z 0  y1 x, since y  y1 x, z 0  y1 x and z 0  y. We say that in all finite structures of the class MEM+(‡∅ ), (WSP+) holds. Therefore, all axioms of the class MEMp are true. Therefore, (SSP+) is also true in them, because it follows from the axioms, by virtue of Theorem 3.6.11.

From the last theorem we get: s PPOSpfin = MEMpfin = MEMfin +(‡∅ ), GMSfin = MEMpfin +(∃pair sum) = MEMfin +(∃pair sum)+(‡∅ ).

Furthermore, by virtue of Fact 3.8.2, Lemma 3.8.6 and Theorem 3.8.7 we have: s PPOSfin +(w1 ∃sum)

= MEMpfin +(c∃pair sum) = MEMpfin .

To finish, observe that, on the basis of Fact 3.7.4 and our analysis of classical mereological structures we get:

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3 “Existentially Involved” Theories of Parts

GMSfin = s PPOSfin +(w1 ∃sum) = s PPOSfin +(w2 ∃sum) = LMSfin . The remaining proper inclusions we get with the help of the finite Models 3.5 and 3.7.

References Breitkopf, A. (1978). Axiomatisierung einiger Begriffe aus Nelson Goodmans the structure of appearance (Axiomatization of some concepts from Nelson Goodman’s the structure of appearance). Erkenntnis, 12, 229–247. Casati, R., & Varzi, A. C. (1999). Parts and places. Cambridge: The MIT Press. Cotnoir, A. J., & Varzi, A. C. (2018). Mereology. Complete draft, 17 March 2018. http://www. columbia.edu/~av72/Mereology-index.pdf. Gruszczy´nski, R., & Pietruszczak, A. (2014). The relations of supremum and mereological sum in partially ordered sets. In C. Calosi & P. Graziani (Eds.), Mereology and the sciences. Parts and wholes in the contemporary scientific context. Synthese library “Studies in epistemology, logic, methodology, and philosophy of science” (Vol. 371). Berlin: Springer. https://doi.org/10.1007/ 978-3-319-05356-1_6. Grzegorczyk, A. (1955). The system of Le´sniewski in relation to contemporary logical research. Studia Logica, 3, 77–95. https://doi.org/10.1007/BF02067248. Le´sniewski, S. (1927). O podstawach matematyki (On the foundations of mathematics). Przeglad ˛ Filozoficzny, 30, 164–206. Le´sniewski, S. (1928). O podstawach matematyki (On the foundations of mathematics). Przeglad ˛ Filozoficzny, 31, 261–291. Le´sniewski, S. (1930). O podstawach matematyki (On the foundations of mathematics). Przeglad ˛ Filozoficzny, 33, 77–105. Pietruszczak, A. (2000). Metamereologia (Metamereology). Toru´n: The Nicolaus Copernicus University Press. English version (2018): Metamereology. Toru´n: The Nicolaus Copernicus University Scientific Publishing House. https://doi.org/10.12775/3961-4. Pietruszczak, A. (2018). Metamereology. Toru´n: The Nicolaus Copernicus University Scientific Publishing House. English version of (Pietruszczak 2000b). https://doi.org/10.12775/3961-4. Simons, P. (1987). Parts. A study in ontology. Oxford: Oxford University Press. https://doi.org/10. 1093/acprof:oso/9780199241460.001.0001. Szmielew, W. (1981). Od geometrii afinicznej do euklidesowej (From affine to Euclidean geometry). Warszawa: PWN. Szmielew, W. (1983). From affine to Euclidean geometry: An axiomatic approach. Netherlands: Springer. English version of (Szmielew 1981). Tarski, A. (1929). Les fondements de la géométrie des corps (Foundations of the geometry of solids). Ksi˛ega Pamiatkowa ˛ Pierwszego Zjazdu Matematycznego (pp. 29–30). Kraków: Annales de la Societé Polonaise de Mathématique. Tarski, A. (1935). Zur Grundlegung der Booleschen Algebra. I (On the foundations of Boolean algebra). Fundamenta Mathematicæ, 24, 177–198. https://eudml.org/doc/212745. Tarski, A. (1956a). Foundations of the geometry of solids. In J. H. Woodger (Ed.), Logic, semantics, metamathematics. Papers from 1923 to 1938 (pp. 24–29). Oxford: Oxford University Press. English version of (Tarski 1929).

References

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Tarski, A. (1956b). On the foundations of Boolean algebra. In J. H. Woodger (Ed.), Logic, semantics, metamathematics. Papers from 1923 to 1938 (pp. 320–341). Oxford: Oxford University Press. English version of (Tarski 1935). Varzi, A. C. (2016). Mereology. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. http://plato.stanford.edu/archives/spr2016/entries/mereology.

Chapter 4

Theories Without the Assumption of Transitivity

4.1 Introduction We have already observed (in Sect. 1.2) that, in the literature, the transitivity of the relation is a part of is often called into question. We cited there a number of works which take different and interesting views on the matter. In this chapter, we will introduce some general approaches which will hopefully delight both defenders and opponents of the assumed transitivity of the relation. We will introduce the concept of the local transitivity of a given relation which will be such that every transitive relation is also locally transitive. We will assume only that the relation is a part of has this new property. It is evident from what we have seen over the previous three chapters that to reject transitivity complicates matters, however, because assumption (t ) has been essential up to now. We will show that without assuming the transitivity of the relation is a part of we can define the foundational concept of a mereological sum. We will formulate several proposals for the general approach to the concept being a part of a whole. By the addition of the transitivity of the relation is a part of we will obtain axiomatisations of both “existentially neutral” and “existentially involved” theories analysed in the preceding two chapters. Throughout the chapter, we will be examining certain classes of structure of the form U, , where U is a non-empty set and  is a binary relation in U which satisfies certain conditions. Obviously, the relation  is to represent the relation is a part of in the set U . Remark 4.1.1 The groundwork for this chapter is provided by the analyses undertaken in (Pietruszczak 2012, 2014). This chapter, however, departs from those analyses in a number of important ways. Firstly, in (Pietruszczak 2012) the justification of observation A appearing in the proof of Theorem 18 (Pietruszczak 2012, p. 216) is not, unfortunately, correct (in this chapter, observation Claim A appears in the proof of Theorem 4.4.11). Here, we give a corrected proof of the theorem. Furthermore, the justification of part (i) in the proof of the theorem has been significantly simplified. c Springer Nature Switzerland AG 2020  A. Pietruszczak, Foundations of the Theory of Parthood, Trends in Logic 54, https://doi.org/10.1007/978-3-030-36533-2_4

153

154

4 Theories Without the Assumption of Transitivity

Secondly, in (Pietruszczak 2012) collective sets (mereological sums) are only examined in the context of the strongest of the theories analysed therein: the theory based on the five axioms (A1)–(A5); (Pietruszczak 2012, points 8 and 9). In this strongest theory—although we are not assuming the transitivity of the relation — one can make use of only the standard concept of a mereological sum. In this chapter we will formulate the concept of a collective class (mereological sum) for a weaker theory based on axioms (A1)–(A3), (A4w ). This concept is non-standard. At the end of the chapter, we will show that, after the addition of axiom (A5), this nonstandard concept is equivalent to the standard one. As a consequence of this change we will have new ways of justifying the theorems obtained in this chapter and in (Pietruszczak 2012). Thirdly, in (Pietruszczak 2012, 2014), we were concerned exclusively with “existentially neutral” theories of parts. In this chapter, we will also examine “existentially involved” theories of parts without the assumption of transitivity (see point 4.10). Fourthly, some theorems will be given that did not appear in (Pietruszczak 2012, 2014) (inter alia, Theorem 4.4.9, Lemma 4.4.10 and Facts 4.2.4, 4.5.5 and 4.5.6). 

4.2 The First Two Axioms (Adopted Instead of Transitivity) We will not assume that the relation  is transitive. We will, however, assume that the relation  is acyclic, which seems the most natural assumption to make. We therefore accept the following (see Sect. 1.1 and Lemma A.3.2 in Appendix A): (A1) The first axiom: the relation  is acyclic. From this axiom we obtain conditions: (ac ), (as ), (irr  ) and (antis ), i.e., the relation  is also asymmetric, irreflexive and antisymmetric. Furthermore, the relation  satisfies (r ) and (antis ), i.e., it is reflexive and antisymmetric (cf. Lemma A.3.5). Besides the acyclicity of the relation  we accept that it is also locally transitive. In order to define local transitivity, we will need first to consider the concept of a path. For arbitrary x, y ∈ U , a path from x to y is an arbitrary finite sequence (u 0 , u 1 , . . . , u n ) such that: n > 0, x = u 0 , y = u n and u i  u i+1 , for i = 0, . . . , n − 1. For all x ∈ U and S ∈ 2U , a path from x to S is an arbitrary path from x to some element of the set S. With the help of the concept of a path we can also express the axiom (A1) in the following form: Fact 4.2.1 Let U = U,  satisfy axiom (A1). Then for arbitrary x, y ∈ U : if there is a path from x to y, then x = y. As a consequence, there does not exist in U any closed path, i.e., there does not exist a path from a point to that point itself.

4.2

The First Two Axioms (Adopted Instead of Transitivity)

155

Let us assume just that the relation  is locally transitive in the following sense:  ∀x,y∈U x  y =⇒ for any path (u 0 , u 1 , . . . , u n ) from x to y,   is transitive on {u 0 , u 1 , . . . , u n } .

(lt  )

(A2) The second axiom: the relation  is locally transitive. From the definition and (A2) we obtain: Lemma 4.2.2 For any path (u 0 , u 1 , . . . , u n ) and for all i, j ∈ {0, . . . , n}: if i < j then u i  u j . From the definition, the above lemma and (A1) we obtain: Fact 4.2.3 For any path (u 0 , u 1 , . . . , u n ), the set {u 0 , u 1 , . . . , u n } has n elements. Fact 4.2.4 If the structure U = U,  with unity satisfies axioms (A1) and (A2), then the relation  is transitive. Proof Assume that u 1 is the unity in U, i.e., ∀u∈U u  u 1 . Take arbitrary x, y, z ∈ U such that x  y  z. Then x  y  z  u 1 and x  u 1 , by virtue of (A1) and our assumption. Hence x  z, by virtue of Lemma 4.2.2. 

4.3 Maximally Closed Transitive Sets In this section we will assume that U = U,  satisfies axioms (A1) and (A2). We say that a path (u 0 , u 1 , . . . , u n ) is included in a subset S of the set U iff {u 0 , u 1 , . . . , u n } ⊆ S. Furthermore, we say that S is closed with respect to  iff for arbitrary x, y ∈ S, each path from x to y is included in S. We define the family CTU of subsets of the set U which are closed with respect to  in which the relation  is at the same time transitive: CTU := {S ∈ 2U : S is closed with respect to  and | S is transitive}. Alongside the operators P, I : U → 2U which we employed earlier, we will also introduce their counterparts P˘ and ˘I which are connected to the converses of the relations  and , respectively. Therefore, for an arbitrary x ∈ U we put: ˘ P(x) := {y ∈ U : x  y}, ˘I(x) := {y ∈ U : x  y}. Lemma 4.3.1 For an arbitrary x ∈ U , the sets I(x) and ˘I(x) belong to CTU . Proof Let x ∈ U . First, we will show that the relation |I(x) is transitive. Let y, z, u ∈ I(x) and y  z  u. Then, from (irr  ), (as ) and our assumption, points y, z and u are pairwise distinct and y = x = z. If u = x, i.e. y  z  x, then y  x,

156

4 Theories Without the Assumption of Transitivity

on the strength of our first assumption. If u = x, i.e. y  z  u  x, then  is transitive on the set {y, z, u, x}, since y  x and  is locally transitive. Hence y  u. Now we will show that the set I(x) is closed with respect to . Let y, z ∈ I(x) and (u 0 , u 1 , . . . , u m ) be an arbitrary path from y to z. Then, by (ac ), y = z. Since y  u 1  · · ·  u m−1  z  x, then also y = x, by virtue of (ac ), (as ) or (irr  ). Therefore either (y, u 1 , . . . , u m−1 , z, x) or (y, u 1 , . . . , u m−1 , x) is a path from y to x. Since  is locally transitive, then  is transitive on the set {y, u 1 , . . . , u m−1 , z, x}. Hence, thanks to Lemma 4.2.2, this set is included in I(x). Similarly, we can prove that for an arbitrary x ∈ U , the set ˘I(x) belongs to the  family CTU . We can now prove in a standard way the following lemma: Lemma 4.3.2 Let C be an arbitrary chain in CTU (i.e., a linearly-ordered subfam ily of the family CTU by the relation of inclusion.)1 Then C ∈ CTU . Therefore, each chain in CTU ,  has an upper bound. Let MCTU be a subfamily of the family CTU composed of all and only those sets which are maximal with respect to the relation of inclusion, i.e.: MCTU := {S ∈ CTU : ¬∃ X ∈CTU S  X }. From Lemma 4.3.2 and the Kuratowski–Zorn lemma (see Sect. B.1.2) we get: Theorem 4.3.3 Each set of CTU is included in some set of MCTU . From Lemma 4.3.1 and Theorem 4.3.3 we get: Corollary 4.3.4 (i) For each x ∈ U there are M, M  ∈ MCTU such that I(x) ⊆ M and ˘I(x) ⊆ M  . (ii) For arbitrary x, y ∈ U , if y  x then there exists an M ∈ MCTU such that x, y ∈ M. (iii) ∅ ∈ / MCTU = ∅. Moreover, from (A1) and (A2) we get: Fact 4.3.5 For each set M of MCTU , the pair M, | M  is a strictly partiallyordered set, i.e., the relation | M is transitive, irreflexive and so asymmetric. Furthermore, M is closed with respect to .

4.4 The Third Axiom 4.4.1 Definition Let U = U,  satisfy axioms (A1) and (A2). For an arbitrary subset S of the set U we put: 1 See

Appendix B.1.2.

4.4

The Third Axiom

157

˘ max (S) := {x ∈ S : ¬∃ y∈S x  y} = {x ∈ S : P(x)  S = ∅}, min (S) := {x ∈ S : ¬∃ y∈S y  x} = {x ∈ S : P(x)  S = ∅}. The elements of max (X ) (resp. min (X )) are called maximal (resp. minimal) in the set S. (A3) The third axiom: the family MCTU satisfies the following condition: for arbitrary M, M  ∈ MCTU , if M = M  then either M  M  ⊆ max (M) or M  M  ⊆ max (M  ). Remark 4.4.1 If we assume that in an irreflexive structure U = U,  the relation  is transitive, then MCTU = {U } and so U automatically satisfies axioms (A1)–(A3) (and also the stronger version of the third axiom, which is given on Sect. 4.4.4).  In the next subsection we will introduce certain auxiliary facts. In Sect. 4.4.3 we will give an equivalent version of the third axiom (see Theorem 4.4.11). Furthermore, in Sect. 4.4.4 we will give a certain stronger version (A3s ) of the third axiom. This stronger version may appear to be more intuitive than axiom (A3). Nevertheless, (A3) suffices for the proof of all the facts (from Sect. 4.4.2) that will be used in the rest of the work. Finally, in Remark 4.4.2 we will give an informal explanation of axiom (A3).

4.4.2 Some Auxiliary Facts We will examine a structure U = U,  satisfying the first three axioms in order to present some auxiliary definitions and facts. For an arbitrary M from MCTU , let us introduce the following binary relations which are counterparts of the relations ,  and  on U . For all x, y ∈ U we put: x  M y :⇐⇒ x, y ∈ M ∧ ∃z∈M (z  x ∧ z  y), x  M y :⇐⇒ x, y ∈ M ∧ x = y ∧ x  y ∧ y  x ∧ ∃z∈M (z  x ∧ z  y), x  M y :⇐⇒ x, y ∈ M ∧ ¬x  M y. By virtue of our definitions, the relations  M ,  M and  M are included in M × M and are symmetric. Furthermore, the following holds: Lemma 4.4.1 For arbitrary M from MCTU : (i) | M ⊆ | M ⊆  M ⊆ | M ,  M ⊆ | M ⊆ | M ,  M ⊆  M and | M ⊆  M . (ii) | M ⊆  M iff | M ⊆  M iff  M ⊆ | M .2 2 Observe that  M

(resp.  M ,  M ) does not have to be identical with | M (resp. | M , | M ). See, e.g., Models 4.4 and 4.5.

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4 Theories Without the Assumption of Transitivity

(iii) All elements of the set min (M) are pairs in the relation  M . (iv) ∀x,y,z∈M (z  M y ∧ z  x =⇒ y  M x). Proof (i)–(iii) Obvious. (iv) From the transitivity of the relation | M we have the transitivity of the  relation | M . Lemma 4.4.2 (i) Let x, y ∈ U and y  x. Then there is exactly one M ∈ MCTU such that x, y ∈ M. Therefore, we can put: Mxy := ( M ∈ MCTU ) x, y ∈ M. ι

(ii) Let x ∈ U and P(x) = ∅. Then there is exactly one M ∈ MCTU such that I(x) ⊆ M. Therefore, we can put: Mx := ( M ∈ MCTU ) I(x) ⊆ M. ι

(iii) For an arbitrary y ∈ P(x) we have Mxy = Mx . Proof By virtue of Corollary 4.3.4, for some M0 ∈ MCTU we have I(x) ⊆ M0 . (i) If y  x then x, y ∈ M0 . Let M ∈ MCTU and x, y ∈ M. If M = M0 , then {x, y} ⊆ max (M0 ) or {x, y} ⊆ max (M), by (A3). So, we have a contradiction. (ii) If P(x) = ∅ then for some y0 we have y0  x and y0 , x ∈ M0 . Let M ∈ MCTU and I(x) ⊆ M. Then y0 , x ∈ M. If M = M0 , then {x, y0 } ⊆ max (M0 ) or {x, y0 } ⊆ max (M), by (A3). Therefore, we obtain a contradiction. (iii) Assume for a contradiction that y ∈ P(x) and Mxy = Mx . Then {x, y} ⊆  max (Mxy ) or {x, y} ⊆ max (Mx ), by (A3). So, we obtain a contradiction. Lemma 4.4.3 Let x, y ∈ U and y  x. Then there is exactly one M ∈ MCTU such that I(x)  ˘I(y) ⊆ M. Furthermore, we have: Mxy = Mx = ( M ∈ MCTU ) ˘I(y) ⊆ M = ( M ∈ MCTU ) I(x)  ˘I(y) ⊆ M. ι

ι

Proof Let x, y ∈ U and y  x. Then x, y ∈ Mxy = Mx . We show that ˘I(y) ⊆ Mxy . Assume that y  z. Then, by virtue of Corollary 4.3.4, for some M0 ∈ MCTU we have y, z ∈ ˘I(y) ⊆ M0 . If Mxy = M0 then either y ∈ max (Mxy ) or y ∈ max (M0 ), by (A3). So, we obtain a contradiction. Therefore z ∈ M0 = Mxy . Thus, we obtain that Mxy includes both I(x) and ˘I(y). But, by Lemma 4.4.2, Mxy is the only set including I(x).  By virtue of Lemmas 4.4.2 and 4.4.3 we obtain: Corollary 4.4.4 ∀M∈MCTU ∀x∈M\min (M) M = Mx .   Corollary 4.4.5 (i) ∀M∈MCTU ∀x∈M P(x)  M = ∅ =⇒ P(x) ⊆ M . (ii) ∀M∈MCTU ∀x∈M P(x)  M =⇒ x ∈ min (M) .

4.4

The Third Axiom

159

  ˘ ˘  M = ∅ =⇒ P(x) ⊆M . (iii) ∀M∈MCTU ∀x∈M P(x)   ˘ (iv) ∀M∈MCTU ∀x∈M P(x)  M =⇒ x ∈ max (M) . Corollary 4.4.6 If M  M  ⊆ max (M) then (M  M  ) \ min (M  ) ⊆ min (M). Proof Let (a) M  M  ⊆ max (M), (b) x ∈ M, (c) x ∈ M  and (d) x ∈ / min (M  ).  Then, by (c) and (d), for some y ∈ M we have y  x. Therefore, by (a)–(c), we have  y∈ / M. So x ∈ min (M), by Corollary 4.4.5(ii).  M  Lemma 4.4.7 ∀M∈MCTU ∀x,y∈M x  y ∧ x  y =⇒ x, y ∈ min (M) . Proof Let x  M y and x  y. Then for some z ∈ U : z  x, z  y and z ∈ / M. Hence x = z = y, i.e., z  x and z  y. Therefore, x, y ∈ min (M), by Corollary 4.4.5(ii).  By virtue of Corollaries 4.3.4 and 4.4.5(i) we obtain:   x Lemma 4.4.8 ∀x,y∈U x  y =⇒ Mx = M y ∧ x M y . Proof If x  y then for some u: u  x and u  y. Hence u ∈ Mx and u ∈ M y . We will show that Mx = M y . Were it the case that Mx = M y then, thanks to (A3), either u ∈ max (Mx ) or u ∈ max (M y ), which would be a contradiction. As a result, x  x M y. The fact below says that if a structure U,  is not degenerate, then for each M ∈ MCTU the strictly partially-ordered set M, | M  is also not degenerate. Theorem 4.4.9 If a structure U,  satisfies (A1)–(A3) then Card U > 1 =⇒ ∀M∈MCTU Card M > 1. Proof Assume for a contradiction that Card U > 1 and there exists an M ∈ MCTU such that Card M = 1. Then there exists a u 0 such that M = {u 0 } and U has at least two elements u 1 and u 2 , i.e. u 1 = u 2 . Therefore, either u 0 = u 1 or u 0 = u 2 . Let us accept that the first case obtains and let us consider three subcases. ˘ 0 ). Then {u 0 }  Mu 1 ∈ CTU . Hence, by virtue Subcase 1: P(u 0 ) = ∅ = P(u of Theorem 4.3.3, there is an M  ∈ MCTU such that M  Mu 1 ⊆ M  . Therefore Mu 1 = M  = M, since M, Mu 1 ∈ MCTU . Hence we obtain a contradiction: u0 = u1. Subcase 2: P(u 0 ) = ∅. Then, by Lemma 4.4.2, there exists exactly one M  ∈ MCTU such that {u 0 } = M ⊆ I(u 0 ) ⊆ M  . Hence we have a contradiction: {u 0 } = M = I(u 0 ) = M  and P(u 0 ) = ∅. ˘ 0 ) = ∅, i.e., for some v0 we have u 0  v0 . Then, by virtue of Subcase 3: P(u Lemma 4.4.3, there exists exactly one M  ∈ MCTU such that {u 0 } = M ⊆ ˘I(u 0 ) ⊆ ˘ 0 ) = ∅. M  . Hence we have a contradiction: {u 0 } = M = ˘I(u 0 ) = M  and P(u Contradictions are derived in a similar way for the second case. 

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Lemma 4.4.10 If a structure U,  satisfies (A1)–(A3) then   ∀x,y∈U x  y ⇐⇒ ∃ M∈MCTU x  M y . Proof Left-to-right: Assume that x  y. Then, by virtue of (2.1.16), either x = y or x  y or x  y. Therefore, by virtue of Lemmas 4.4.2 and 4.4.8, there exists an M ∈ MCTU such that x  M y. Right-to-left: By logic and the inclusion  M ⊆ , for each M M ∈ MCTU . 

4.4.3 An Equivalent Version of Axiom (A3) Below is an equivalent version of our third axiom: Theorem 4.4.11 If U = U,  satisfies axioms (A1)–(A3) then (A3 ) for arbitrary M, M  ∈ MCTU , if M = M  then either M  M  ⊆ max (M)  min (M  ) or M  M  ⊆ min (M)  max (M  ). Since min (M)  max (M  ) ⊆ M  M  and max (M)  min (M  ) ⊆ M  M  , then in (A3 ) we have ‘=’ instead of ‘⊆’. Furthermore, (A3 ) entails (A3). Therefore (A3) and (A3 ) are equivalent in the class of all acyclic and locally transitive structures, i.e., those satisfying conditions (A1) and (A2). Proof Let M, M  ∈ MCTU and M = M  . Thanks to (A3): M  M  ⊆ max (M) or M  M  ⊆ max (M  ). Let us therefore examine the following three cases: (i) M  M  ⊆ max (M)  max (M  ), (ii) M  M  ⊆ max (M) and M  M   max (M  ), (iii) M  M   max (M) and M  M  ⊆ max (M  ). Ad (i) Let M  M  ⊆ max (M)  max (M  ). Thanks to M  M  ⊆ min (M) or M  M  ⊆ min (M  ). Assume for a contradiction that M  M   min (M  ) and M  M   min (M). / min (M). Hence, by virtue of CorolThen there exists an x0 such that x0 ∈ / lary 4.4.4, M = Mx0 . Furthermore, there exists a y0 ∈ M  M  such that y0 ∈ min (M  ). Hence, by virtue of Corollary 4.4.4, M  = M y0 . We therefore have Mx0 = M y0 and x0 = y0 . Since x0 , y0 ∈ max (M), then x0  y0 and y0  x0 . Furthermore, ¬x0  y0 , by virtue of Lemma 4.4.8. Hence x0  y0 , by virtue of (2.1.15). From Corollary 4.4.6, we have: x0 ∈ min (M  ) and y0 ∈ min (M). We now prove that I(x0 )  I(y0 ) ∈ CTU . Then, by virtue of Theorem 4.3.3, we get that there exists an M  ∈ MCTU such that I(x0 )  I(y0 ) ⊆ M  . Therefore, by virtue of Lemma 4.4.2, M = M  = M  and we have our contradiction. Firstly, we show that |I(x0 )I(y0 ) is transitive. Let z, u, v ∈ I(x0 )  I(y0 ), z  u and u  v. By virtue of (ac ), v = z = u = v and v  z. We show that for arbitrary

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α, β ∈ {z, u, v} the following cases are not possible: (a) α ∈ I(x0 ) and β ∈ I(y0 ); (b) β ∈ I(x0 ) and α ∈ I(y0 ). From this, we obtain that, for arbitrary α, β ∈ {z, u, v}, either or α, β ∈ I(x0 ) or α, β ∈ I(y0 ). Therefore z  v, since the sets I(x0 ) and I(y0 ) belong to CTU , by virtue of Lemma 4.3.1. Let us assume that there exist α, β ∈ {z, u, v} such that α ∈ I(x0 ) and β ∈ I(y0 ). Then α = β, α  y0 and β  y0 , because x0  y0 . If α = z and β = u (resp. β = v), then x0  z  u  y0 (resp. x0  z  u  v  y0 ). Therefore, we have a path from z to y0 and z, y0 ∈ Mx0 . Therefore, this path is included in Mx0 . Hence z  y0 . This contradicts the fact that x0  y0 . If α = u and β = v, then x0  u  v  y0 and once again we have a contradiction: x0  y0 . If α = u and β = z, then y0  z  u  x0 and we have a path from z to x0 and z, x0 ∈ M y0 . Therefore this path is included in M y0 . Hence z  x0 and once again we have a contradiction: x0  y0 . We can show in an analogous way that the remaining subcases from (a) and (b) are not possible. Secondly, we will show that I(x0 )  I(y0 ) is closed with respect to . Let z, u ∈ I(x0 )  I(y0 ) and (u 0 , u 1 , . . . , u m ) be an arbitrary path from z to u, for some m  1. Then, thanks to (ac ), z = u. If z, u ∈ I(x0 ) or z, u ∈ I(y0 ), then this path is included respectively in I(x0 ) or I(y0 ), by virtue of Lemma 4.3.1. Besides this, we will show that the remaining two possibilities are not possible: z ∈ I(x0 ) and u ∈ I(y0 ); z ∈ I(y0 ) and u ∈ I(x0 ). In the first case, x0  z  u 1  · · ·  u m−1  u  y0 . Therefore, we have a path from z to y0 and z, y0 ∈ Mx0 . Therefore this path is included in Mx0 . Hence z  y0 . This contradicts the fact that x0  y0 . We can show, in an analogous fashion, that the second case is not possible. Ad (ii) Let M  M  ⊆ max (M) and M  M   max (M  ). Hence there exists / max (M  ). Hence there exists a a y0 ∈ M  M  such that y0 ∈ max (M) and y0 ∈  y1 ∈ M such that y0  y1 . We have to prove that M  M  ⊆ min (M  ). Assume for a contradiction that M  M   min (M  ). Then there exists a x0 ∈ M  M  such that x0 ∈ max (M) / min (M  ). Therefore, there exists a x1 ∈ M  such that x1  x0 and, furand x0 ∈ thermore, x0 ∈ min (M), by virtue of our assumption and Corollary 4.4.6. Furthermore, by virtue of Lemma 4.4.2, since M = M  , x0 , y0 ∈ M  M  , / M. x1 , y1 ∈ M  , x1  x0 and y0  y1 , then x0  y0 , y0  x0 and x1 , y1 ∈ y We have M  = Mxx10 = Mx0 = M y01 = M y1 , I(x0 ) ⊆ M  , I(y1 ) ⊆ M  and ˘I(y0 ) ⊆ ˘ 0 ) ⊆ M  \ M, ˘I(y0 )  (M \ M  ) = M  (see Lemmas 4.4.3 and 4.4.2). Therefore P(y ∅ = I(x0 )  (M \ M  ) and P(x0 ) ⊆ M  \ M. We will now take a look at three auxiliary observations: Claim A There exists a path from x1 to M \ M  . Proof Assume for a contradiction that for an arbitrary u ∈ M \ M  there does not exist a path from x1 to u. We will say that a point u lies on a path (u 0 , u 1 , . . . , u n ) iff u ∈ {u 0 , u 1 , . . . , u n }. We put: P := {u ∈ U : u lies on some path from x1 to M  M  }. Since x1  x0 , then x1 ∈ P \ M. We now prove that the set M  P belongs to CTU .

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Certain further facts are worth noting of which several will be of use in what follows. Since x1 ∈ M  and M  ∈ CTU , then every path from x1 to M  M  is included in M  . Furthermore, since M  M  ⊆ max (M), then for each path from x1 to M  M  only its final point belongs to M  M  . The rest of the path (without the final point) is included in M  \ M. Firstly, we will show that M  P is closed. To this end, for arbitrary x, y ∈ M  P take an arbitrary path from x to y, i.e., an arbitrary finite sequence (u 0 , u 1 , . . . , u n ) (henceforth [x . . . y] for short) such that: n > 0, x = u 0 , y = u n and u i  u i+1 , for i = 0, . . . , n − 1. We have x = y. It is now necessary to show that the path [x . . . y] is included in M  P. To start, let us assume that x ∈ / M or y = x1 . Otherwise, we have a path (x, . . . , x1 , x0 ) included in M. And since x, x0 ∈ M and M ∈ CTU , then we have / M. Furthermore, x ∈ / P or y = x1 , because there a contradiction: x1 ∈ M and x1 ∈ does not exist a path from x1 to x1 . Below we will examine all possible cases for x and y. If x, y ∈ M then the path [x . . . y] is included in M, because M ∈ CTU . If x ∈ M and y ∈ P \ M, then y = x1 and we have a path (x1 , . . . , y, . . . , v) from x1 to M  M  . Therefore, we also have a path (x, . . . , y, . . . , v) included in M. Hence [x . . . y] is also included in M. If x = x1 and y ∈ M, then y ∈ M  M  , by virtue of our initial assumption. We therefore have a path (x1 , u 1 , . . . , u n−1 , y) from x1 to M  M  . Hence [x . . . y] is included in P. If x ∈ P \ (M  {x1 }) and y ∈ M, then we have a path (x1 , . . . , x, . . . , v) from x1 to M  M  . We therefore also have a path (x1 , u 1 , . . . , x, . . . , u n 1 , y) from x1 to M  M  , by virtue of our initial assumption. Hence [x . . . y] is included in P. If x = x1 and y ∈ P \ M, then y = x1 and we have a path (x1 , w1 , . . . , wk , y, . . . , v), where v ∈ M  M  . By changing the initial part of the path, we obtain the path (x1 , u 1 , . . . , u n−1 , y, . . . , v) from x1 to M  M  . Hence [x . . . y] is included in P. If x ∈ P \ (M  {x1 }) and y ∈ P \ (M  {x1 }), then we have two paths (x1 , . . . , x, . . . , v) and (x1 , . . . , y, . . . , w), where v, w ∈ M  M  . We therefore also have a path (x1 , . . . , x, u 1 , . . . , u n−1 , y, . . . , w) from x1 to M  M  . Therefore, the path [x . . . y] is included in P. Secondly, we will show that the relation | MP is transitive. To this end, take any x, y, z ∈ M  P such that x  y and y  z. Then z = x = y = z, by virtue of (ac ). Furthermore, observe that the following cases are not possible: (a) x ∈ M and either y ∈ P \ M or z ∈ P \ M, (b) y ∈ M and z ∈ P \ M, (c) x ∈ P \ M and y, z ∈ M. In the case where x ∈ M and z ∈ P \ M, we would have a path (x1 , . . . z, . . . , v) from x1 to M  M  . We therefore create a path (x, y, z, . . . , v) which has to be included in M. We therefore have a contradiction. Similarly, we obtain a contradiction in each subcase of (a). By contrast, were case (b) to obtain, then we would

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have a path (x1 , . . . , z, . . . , w) from x1 to M  M  . We therefore create a path (y, z, . . . w) from y to M  M  which would be included in M. Therefore, we have a contradiction: z ∈ M. In case (c), we would have a path (x1 , . . . , x, . . . , v) from x1 to M  M  . We therefore create a path (x1 , . . . , x, y, z). From our initial assumption, we have y, z ∈ M  M  . This, however, contradicts our main assumption, which says that M  M  ⊆ max (M). Thus there remain just three possible situations. In the first, where x, y, z ∈ M, we have x  z, because the relation | M is transitive. In the second, where x, y ∈ P \ M and z ∈ M, we have two paths (x1 , . . . , x, . . . , v) and (x1 , . . . , y, . . . , w) from x1 to M  M  . Since x1 ∈ M  and M  ∈ CTU , then each of the paths is included in M  ; so x, y ∈ M  . We observe that we also have a path (x1 , . . . , y, z). Therefore, z ∈ M  M  , by virtue of our initial assumption. Since the relation | M  is transitive, then x  z. In the third, where x, y, z ∈ P \ M, we have three paths (x1 , . . . , x, . . . , v), (x1 , . . . , y, . . . , w) and (x1 , . . . , z, . . . , u) from x1 to M ∩ M  . Since x1 ∈ M  and M  ∈ CTU , these paths are included in M  and x, y, z ∈ M  . Hence x  z. We have therefore demonstrated that M  P ∈ CTU . Hence, by virtue of Theorem 4.3.3, for some M  ∈ MCTU we have M  P ⊆ M  . Therefore, M = M  , because M ∈ MCTU . Hence we have a contradiction: P ⊆ M, and so  x1 ∈ M. Claim B For all v ∈ ˘I(y0 ) and u ∈ M \ M  there does not exist a path from v to u. Proof In the other case, there exists a path which either has the form (y0 , v, . . . , u), if v = y0 , or (y0 , . . . , u), if v = y0 . In both cases, the path is included in M. We therefore have a contradiction: y0  u (in the first case, we also have a contradiction: v ∈ M).  ˘ 0 ). Claim C For each u ∈ M \ M  there exists a path from u to P(y ˘ 0 ) there Proof Assume that there exists a u ∈ M \ M  such that for each v from P(y does not exist a path from u to v. Hence there does not exist a path from u to y0 . Then, thanks to Claim B and Lemma 4.3.1, ˘I(y0 )  {u} ∈ CTU . Therefore, by virtue of Theorem 4.3.3, for some M  ∈ MCTU we have : ˘I(y0 )  {u} ⊆ M  . Therefore M  = M  , by virtue of Lemma 4.4.2. Hence we obtain a contradiction: u ∈ M  .  By virtue of Claim A, for some u ∈ M \ M  there exists a path (x1 , . . . , u) from ˘ 0 ) ⊆ M  we have a path (u, . . . , v) x1 to u. By virtue of Claim C, for some v ∈ P(y from u to v. By joining these paths we obtain the path (x1 , . . . , u, . . . , v) from x1 to v. Since x1 , v ∈ M  and M  ∈ MCTU , then this path is included in M  . Hence we / M . have a contradiction: u ∈ M  and u ∈  Ad (iii) Let M  M  max (M) and M  M  ⊆ max (M  ). As in case (ii), we can prove that M  M  ⊆ min (M). Thus, MM  ⊆ max (M) min (M  ) or MM  ⊆ min (M) max (M  ). 

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4.4.4 A Stronger Version of Axiom (A3) In this subsection we will give a stronger version of the third axiom and formulate some auxiliary comments and examples. Example 4.4.1 In Model 4.1 the relation  is not transitive but it is locally transitive and acyclic. The model satisfies axioms (A1)–(A3) and axioms (A4) and (A5) that will be introduced shortly. We have: MCTU = {M1 , M2 , M3 } where M1 = {1234, 4, 123}, M2 = {4, 123, 12, 3} and M3 = {4, 12, 3, 1, 2}; M1  M2 = {4, 123} = min (M1 ) = max (M2 ); M2  M3 = {4, 12, 3} = min (M2 ) =  max (M3 ); M1  M3 = {4} = min (M1 )  max (M3 ). We observe that Model 4.1 does not satisfy the axiom below, which is evidently stronger than (A3): (A3s ) the family MCTU satisfies the following condition: for arbitrary M and M  of MCTU , if M = M  and M  M  = ∅, then either max (M) = M  M  = min (M  ) or min (M) = M  M  = max (M  ). Example 4.4.2 In addition, well-known counter-examples, which attest to the fact that the relation  is not transitive (see Sect. 1.2), satisfy (A1)–(A3) and (A3s ) in a trivial way. To see this, in these cases, the relation is tree-like. Model 4.2 illustrates this more clearly, where the relation  is locally transitive and acyclic. We have MCTU = {M1 , M2 }, where M1 = {123, 12, 23} and M2 = {12, 23, 1, 2, 3}, i.e., M1  M2 = min (M1 ) = max (M2 ). Therefore MCTU satisfies (A1)–(A5)  and (A3s ). We will illustrate this more precisely with a counter-example to the transitivity of the relation  given by Rescher (see Sect. 1.2). Model 4.1 (A1)–(A5) hold, but (A3s ) does not hold

1234 4 123 12

1

Model 4.2 The tree structure satisfies axioms (A1)–(A5) and (A3s )

3

2

1

12345 2

123

1

2

45

3

4

5

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Example 4.4.3 Let us assume that U := O  C  N  P, where O is a set of organs and C is the set of cells in the organs in O, N is the set of nuclei in the cells in the set C, and P is the set of other cells in C. We have MCTU = {M1 , M2 }, where M1 := O  C and M2 := C  N  P. Then the elements of the set K are maximal in the set M2 and minimal in the set M1 ; M1  M2 = K = min (M1 ) = max (M2 ). Therefore, the strong version of the third axiom also holds. This is evidently so in the case where we accept with Rescher that  is not transitive in U .  As we said above, the stronger version of the third axiom, (A3s ), may seem more intuitive but axiom (A3) is sufficiently strong to allow us to obtain all the facts (from Sect. 4.4.2) which have been used in the remainder of this chapter. The remark below provides some informal motivations for axioms (A3) and (A3s ). Remark 4.4.2 In essence, axioms (A3s ) and (A3) divide the universe into sets from the family MCTU which only intersect at their “extreme bounds”. Assume that M, M  ∈ MCTU , M = M  , x, y ∈ M, x, z 1 , z 2 ∈ M  and y  x  z 1  z 2 . Then, with reference to the comments in (Lyons 1977, Sect. 9.8), x is a MAXIMAL element in M (resp. MINIMAL element in M  ) with respect to  such that we can correctly say that x HAS y (resp. z 1 HAS x and z 2 HAS x). For example, the sentences below are correct: • The orchestra (z 2 ) has a violin section (z 1 ). • The orchestra (z 2 ) has a violinist (x). • The violinist (x) has a heart (y).

z1  z2 x  z2 yx

The following forms are not, however, correct: • The orchestra has the arm of the violinist. • The violin section has the violinist’s heart. We shall say that the orchestra is a “system” in which the musicians and conductor are MINIMAL elements with respect to . We can also say that it is a “closed system”, since all the musicians and the conductor are disjoint pairs (“external with respect to each other”). In addition, the subsequent axioms below are satisfied (including the fifth, which says that, for an arbitrary M ∈ MCTU all elements of the set min (M) are pairs in the relation  ; which is equivalent to the inclusion  M ⊆ ; see Lemma 4.8.1) Furthermore, each musician is also a “closed system” of his or her own body parts, in which he or she is a MAXIMAL element with respect to .  Remark 4.4.3 Using the previous comment as a point of departure, we will explain why we have accepted as an axiom the local transitivity of the relation  and not any “quasi-transitive” versions of it such as:   ∀x,y,z,u∈U x  y  z  u ∧ x  z =⇒ y  u ,   ∀x,y,z,u∈U x  y  z  u ∧ x  z =⇒ x  u ,

(a) (b)

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Model 4.3 The structure satisfies axioms (A1)–(A5) and (A3s ) hold, but does not satisfy “quasi-transitivity” conditions from Remark 4.4.3

1

abcd ab

2

a

all parts of a

cd b

all parts of b

c

all parts of c

d

all parts of d

  ∀x,y,z,u∈U x  y  z  u ∧ y  u =⇒ x  z ,   ∀x,y,z,u∈U x  y  z  u ∧ y  u =⇒ x  u .

(c) (d)

The remaining possible combinations (with respect to the acyclicity of the relation ) would be connected with the local transitivity of the relation  which we have already accepted. Let us take an orchestra composed of at least two sections including a violin section that has at least two violinists. The first two of the above conditions (a) and (b) defeat the following example: • x = left-hand thumb of violinist a, y = left hand of violinist a, z = violinist a, u = the orchestra. Conditions (c) and (d) are falsified by the following example (u as above): • x = left hand of violinist a, y = violinist a, z = the violin section. We may also formally give the structure introduced by Model 4.3, in which the family MCTU is a two-element family. Its elements should have enough parts so that axioms (A1)–(A5) are satisfied. We have distinguished in element a those parts necessary to defeat the conditions given above. 

4.5 Two Versions of the Fourth Axiom. Supplementation Principles Let U = U,  satisfy the first three axioms (A1)–(A3). In what follows, we will use the three supplementation principles we examined earlier: (SSP), (WSP) and (SP ), and for an arbitrary M of MCTU , their following “partial versions”:   ∀x,y∈M x  y =⇒ ∃z∈M (z  x ∧ z  M y) ,   ∀x,y∈M y  x =⇒ ∃z∈M (z  x ∧ z  M y) ,   ∀x,y∈M x  y =⇒ ∃z∈M (z  x ∧ z  M y) .

(SSP M ) (WSP M ) (SPM )

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Two Versions of the Fourth Axiom . . .

167

We recall that conditions (SSP), (WSP) and (SP ) characterise the concept being a part in its mereological sense, i.e., when it is transitive. Lemma 2.3.5 said that (as ), (r ) and (SSP) entail (WSP); and Lemma 2.3.8 says that (SSP) entails (SP ) and that (WSP) and (SP ) entail (SSP). Similarly, we show that “partial versions” of these lemmas also hold (i.e., when (SSP), (WSP) and (SP ) are replaced by (SSP M ), (WSP M ) and (SPM ) , respectively): Lemma 4.5.1 (i) (SSP M ), (as ) and the definitions entail (WSP M ). (ii) (SSP M ) and the definitions entail (SPM ). (iii) (WSP M ), (SPM ) and the definitions entail (SSP M ). Furthermore, from (A1)–(A3) we get: Lemma 4.5.2 Let U satisfy (A1)–(A3). Then: (i) If (SSP) holds then (SSP M ) also holds for each M of MCTU . (ii) If (SSP M ) holds for each M of MCTU , then (SP ) also holds. Proof (i) Let M ∈ MCTU , x, y ∈ M and x  y. Then either x  M y or x  M y. In the first case, we immediately obtain our thesis. In the second case, for some u ∈ M: u  x and u  y. Hence, by virtue of our assumption, x = u, i.e. u  x. Furthermore, thanks to (WSP), there exists a z such that z  x, and by virtue of Corollary 4.4.5(i), z ∈ M. Furthermore z  M y, since  ⊆  M . x (ii) Let x  y. Then, thanks to Lemma 4.4.8, Mx = M y and x M y. Furthermore, x Mx x since x  y, then thanks to (SSP ), for some z ∈ M : z  x and z M y. Hence z = x, and so z  x. We will show that z  y. To see this, assume that for some u 0 we have: u 0  z and u 0  y. Then u 0 ∈ I(y) ⊆ M y = Mx . And this entails a x  contradiction: z M y. Fact 4.5.3 There exists a structure U in which: (a) conditions (A1), (A2), (A3) and (A3s ) hold, (b) for each M of MCTU condition (SSP M ) holds, (c) conditions (WSP) and (SSP) do not hold. Thus, (WSP) and, in consequence, (SSP) do not follow from the premises: (A1), (A2), (A3s ) and (SSP M ) for each M ∈ MCTU . Proof We consider Model 4.4 with the relation  which is locally transitive and acyclic. We have MCTU = {M1 , M2 }, where M1 = {123, 12, 23} and M2 = {12, 23, 1, 2, 3}. Therefore, the family MCTU satisfies (A3), (A3s ), (SSP M1 ) and (SSP M2 ). Furthermore, we have 23  123 but there does not exist a z ∈ U such that z  123 and z  23, since 12  23. Hence (WSP) and (SSP) do not hold in U. 

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4 Theories Without the Assumption of Transitivity

Model 4.4 (A1), (A2), (A3s ) and (A4w ) hold but (WSP), (SSP), (A4) and (A5) do not

1

123

2

12

1

23

2

3

We now introduce two versions of the fourth axiom: (A4) (the stronger version): U satisfies (SSP). (A4w ) (the weaker version): (SSP M ) holds for each M ∈ MCTU . From Lemma 4.5.2(i) and Fact 4.5.3 we get: Corollary 4.5.4 In both theories based respectively on (A1)–(A3) and on (A1), (A2) and (A3s ) condition (A4) is essentially stronger than condition (A4w ). The following fact, analogous to Lemma 2.1.2(ii), also holds: Fact 4.5.5 From (A1)–(A3), (A4w ) follows the following conditions: Card U > 1 =⇒ ∀M∈MCTU ∃x,y∈M x  M y, Card U > 1 =⇒ ∃x,y∈U x  y.

(∃ M ) (∃)

Proof Ad (∃ M ) : Assume that Card U > 1 and take an arbitrary M from MCTU . Then, on the strength of Theorem 4.4.9, Card M > 1. Therefore, M has at least two elements u 1 and u 2 , i.e. u 1 = u 2 . By virtue of (antis ), u 1  u 2 or u 2  u 1 . In the first case, by virtue of (SSP M ), for some z ∈ M we have: z  u 1 and z  M u 2 . The second case is similar. Ad (∃): Assume for a contradiction that Card U > 1 and ∀x,y∈U x  y. Since MCTU = ∅, take an arbitrary M of MCTU . By virtue of (∃ M ), there are x, y ∈ M such that x  M y. By virtue of our assumption, x  y. Therefore, there exists a z such that z ∈ / M z  x and z  y. Since z ∈ / M and M ∈ MCTU , we will examine some alternative situations. There is a u such that u  z, u  x and u  y. Then, by virtue of our assumption, u  y, i.e., there is a v such that v  u and v  y. From the earlier facts and (ac ) we have v  u and v  y. By virtue of (A2), the relation  is transitive on the path (v, u, z, y). Hence we have our contradiction: u  y. There is a u such that u  z, u  x and u  y (resp. u  x and u  y). Then u ∈ M, and so the path (u, z, x) (resp. (u, z, y)) is included in M, since M is closed. Hence we have our contradiction: z ∈ M. There is a u such that u ∈ M, x  u and z  u (resp. y  u and z  u). Then, by virtue of our assumption, u  z, i.e., there is a v such that v  u and v  z. From the earlier facts and (ac ) we have v  u and v  z. By virtue of (A2), the

4.5

Two Versions of the Fourth Axiom . . .

169

relation  is transitive on the path (v, z, x, u) (resp. (v, z, y, u)). Hence we have our contradiction: z  u.  Fact 4.5.6 From (A1)–(A4) follows the following condition: Card U > 1 =⇒ ∀M∈MCTU ∃x,y∈M x  y, that is, (∃ M ) and (∃) as well. Proof Analogously as in the case of (∃ M ). We use (SSP) instead of (SSP M ).



Furthermore, we know from Lemma 2.1.2 that (∃) logically entails: Card U > 1 =⇒ ¬∃x∈U ∀u∈U x  u .

( 0)

Therefore, condition ( 0) hold in both theories based respectively on (A1)–(A3), (A4w ) and on (A1)–(A4). In Sect. 4.8 we will adopt axiom (A5), which will mean that conditions (A4) and (A4w ) will be equivalent in theories based on (A1)–(A3) and (A5). Lemma 2.4.15 says that the supplementation principle, (SSP), and our definitions entail the principle of extensionality (ext  +). Similarly, adopting (A4w ) for each M ∈ MCTU , we obtain the following “partial version” of the principle of extensionality: (ext M +) ∀x,y∈M (∅ = P(x) ∩ M ⊆ P(y) =⇒ x  y). We know that (SSP) is equivalent to the inclusion ∀x,y∈U (I(x) ⊆ O(y) ⇒ x  y). From this and (r ) we get that (ext  +) entails (SSP). Furthermore, Lemma 2.4.6 says that (r ), (t ) and our definitions entail the equivalence of conditions (SSP) and (ext  +). We can obtain an analogous result for each M ∈ MCTU , because the restriction | M is transitive. To this end, let us define an operator O M : U → 2U , by putting: O M(x) := {y ∈ U : y  M x}. With axioms (A1)–(A3), (A4w ) adopted, for an arbitrary M ∈ MCTU , the relation | M is a polarised strict partial order. Therefore, thanks to (SSP M ), Lemma 4.4.1(iv) and the inclusion I(x) ∩ M ⊆ O M(x), we obtain:   ∀x,y∈M x  y ⇐⇒ I(x) ∩ M ⊆ O M(y) ,   ∀x,y∈M x  y ⇐⇒ O M(x) ⊆ O M(y) .

(SSPM )

By appeal to Lemma 4.4.1 and Footnote 2, we may observe that the following fact obtains: Fact 4.5.7 There exists a structure U in which: (a) axioms (A1)–(A4), (A3s ) hold, (b) for some M ∈ MCTU :  M  | M ,  M  | M and | M   M .

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4 Theories Without the Assumption of Transitivity

Model 4.5 (A1)–(A4), (A3s ) and (SSP+) hold, but (A5) and (c∃∂4 ) do not hold. Furthermore, sum  SUM

1

1234 2

12

1

23

2

4

3

Proof Model 4.5 presents this structure. Clearly, the relation  is locally transitive and acyclic in this model. We have MCTU = {M1 , M2 } where M1 = {1234, 12, 23, 4} and M2 = {12, 23, 1, 2, 3, 4}. Therefore, in this structure, the family MCTU satisfies axioms (A3) and (A3s ). Furthermore, (SSP) also holds in this structure.  Finally, 12 | M1 23 but 12  M1 23.

4.6 “A Partial” Monotonicity Principle. An Equivalent Version of Axiom (A4w ) For arbitrary M ∈ MCTU and S ∈ 2U we have: 

O M[S] = {u ∈ U : ∃z∈S z  M u}.

 Therefore O M[S] is the sum of the image of the family O M [S], i.e., it is a set of M all and only those elements of U which  stand in the relation   to some  element of  the set S. We have O M[∅] = ∅, O M[S] ⊆ M and O M[S] ⊆ O(S). Let U = U,  satisfy axioms (A1)–(A3). The lemma below presents a condition which is equivalent to axiom (A4w ): Lemma 4.6.1 For each M ∈ MCTU : (i) The transitivity of the relation | M and condition (SSP M ) entail the following “partial” monotonicity principle3 :    ∀x,y∈M ∀S∈2 M S ⊆ I(y) ∧ I(x) ∩ M ⊆ O M[S] =⇒ x  y .

(mM )

(ii) (SSP M ) follows from (mM ) and the reflexivity of the relation | M . Proof (i) Let x, y ∈ M, S ⊆ M ∩ I(y) and ∀z∈M (z  x ⇒ ∃u∈S u  M z). Then ∀z∈M (z  x ⇒ ∃u∈M (u  y ∧ u  M z)). Hence, thanks to Lemma 4.4.1(iv), we get that ∀z∈M (z  x ⇒ z  M y). Hence x  y, thanks to (SSP M ).  (ii) For arbitrary x, y ∈ M of (mM ) we put S := {y}. 3 Cf.

 ). principle (Msum

4.6

“A Partial” Monotonicity Principle. An Equivalent Version of Axiom (A4w )

171

From Lemma 4.6.1 we obtain: Theorem 4.6.2 On the basis of axioms (A1)–(A3), for each M ∈ MCTU , conditions (SSP M ) and (mM ) are equivalent. Thus, in a theory based on axioms (A1)– (A3), condtion (A4w )is equivalent to the following: • (mM ) holds, for each M ∈ MCTU .

4.7 Mereological Sums in Structures Without Transitivity 4.7.1 Definitions and Basic Properties As we know, the concept of a mereological sum (collective set) is a central mereological concept. We will show that without assuming the transitivity of the relation is a part of, in structures of the form U,  which satisfy axioms (A1)–(A3) and (A4w ), we may define a relation SUM, which is the counterpart of the relation sum sum, which we examined in Chaps. 2 and 3. The relation SUM is to be a formal explication of the concept mereological sum in structures without transitivity of . We will prove that, despite only assuming the local transitivity of the relation , the relation SUM has the same properties that the relation sum has in structures with the transitive relation . We will show below that with the addition of axiom (A5), the non-standard concept of a sum is equivalent to the standard one that is defined with the help of (df sum), when we assume the transitivity of the relation . Let us assume that a structure U = U,  satisfies axioms (A1)–(A3) and (A4w ). Let us now define the following binary relation SUM in U × 2U x SUM S :⇐⇒ S ⊆ I(x) ∧    ∀M∈MCTU I(x) ⊆ M =⇒ I(x) ⊆ O M[S] .

(df SUM)

The definition above is correct thanks to Corollary 4.3.4. Observe that we can obtain counterparts of the properties (2.6.1), (2.6.3) and (2.6.7) of the relation sum: Lemma 4.7.1 For arbitrary x ∈ U and S ∈ 2U : (i) If x SUM S then S = ∅. (ii) x SUM {x} and x SUM I(x). (iii) x SUM P(x) iff P(x) = ∅. Proof Thanks to (r ), we have the inclusion I(x) ⊆ O M(x), for each M ∈ MCTU such that I(x) ⊆ M. Ad (i) If x SUM S then for some M ∈ MCTU we have I(x) ⊆ M, by virtue of Corollary 4.3.4. Therefore, ∅  = I(x) ⊆ O M[S] and, in consequence, S = ∅,  M because O [∅] = ∅. Ad (ii)  For an arbitraryM ∈ MCTU such that I(x) ⊆ M, we have {x} ⊆ I(x) ⊆ O M(x) = O M [{x}] ⊆ O M [I(x)]. Therefore, x SUM {x} and x SUM I(x).

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4 Theories Without the Assumption of Transitivity

  Ad (iii) If P(x) = ∅ then P(x) ⊆ I(x) ⊆ O M[P(x)] ⊆ O(P(x)), for an  arbitrary M ∈ MCTU such that I(x) ⊆ M. Therefore, x SUM P(x). Remark 4.7.1 (i) In the general case, the relation sum defined with the help of (df sum) is not adequate even with the adoption of axioms (A1)–(A4) and (A3s ). For example, in Model 4.5, we have 1234 sum {23, 4}. Furthermore, for axioms (A1)–(A3), (A3s ) and (A4w ) in Model 4.4, we have 123 sum {12}, 12 sum {12}, 123 sum {23} and 23 sum {23}. (ii) The properties (2.6.1), (2.6.3) and (2.6.7) of the relation sum obtain also in theories based on (A1)–(A3) and (A4w ), because in the proof of these properties we did not make use of the transitivity of the relation  . (iii) If we assume that the relation  is transitive, then MCTU = {U }. Therefore,  M = U =  and we obtain the “classical definition” of a mereological sum: i.e., we have that SUM = sum. (iv) We will obtain the identity SUM = sum, if we assume the fifth axiom (A5), as we will do in Sect. 4.8 (see Theorem 4.9.1).  The definition (df SUM) is complicated because of the possibility of elements without parts. The lemma below presents a simplified form of the equivalence defining the concept mereological sum for when we are separately examining cases relating to the possession of parts by a given element of the universe. Lemma 4.7.2 For arbitrary x ∈ U and S ∈ 2U : (i) If P(x) = ∅ then: x SUM S ⇐⇒ S = {x} ⇐⇒ x sum S. (ii) If P(x) = ∅ then: I(x) ⊆ Mx , x SUM S ⇐⇒ S ⊆ I(x) ⊆



x

OM [S].

Proof (i) Let P(x) = ∅. Then I(x) = {x} and thanks to Lemma 4.7.1(i), if x sum S or x SUM S, then ∅ = S ⊆ I(x) = {x}. Furthermore, by virtue of Lemma 4.7.1(ii), we have x sum {x} and x SUM {x}. (ii) Let P(x) = ∅. Then Mx is the only set in MCTU which includes I(x), by virtue of Lemma 4.4.2.  We therefore have the following result: Fact 4.7.3 SUM ⊆ sum. Proof Assume that x SUM S. If P(x) = ∅ then x sum S, by virtue of Lemma 4.7.2(i).  If P(x) = ∅ then from Lemma 4.7.2(ii) we have: ∅ = S ⊆ I(x) ⊆ x Mx and  I(x) ⊆ O(S)M  S. Therefore, it is also so in this case that x sum S, Mx because O [S] ⊆ O(S). 

4.7

Mereological Sums in Structures Without Transitivity

173

Moreover, we obtain: Fact 4.7.4 From (A1)–(A4), (A3s ) it does not follow that sum ⊆ SUM. Proof In Model 4.5: 1234 sum {23, 4}, but ¬1234 SUM {23, 4}.



Thanks to our definitions and Lemma 4.7.1(i) we have: Fact 4.7.5 For arbitrary x ∈ U and S ∈ 2U : x SUM S ⇐⇒ ∅ = S ⊆ I(x) ∧    ∀M∈MCTU I(x) ⊆ M ⇒ P(x) ⊆ O M[S] . Proof Left-to-right: We use Lemma 4.7.1(i) and (df SUM). Right to left: Assume that I(x) ⊆ M ∈ MCTU . Then there is a u 0 ∈ S ⊆ I(x);  so for some u ∈ S ⊆ M we have u  M x. Thus, x SUM S, by (df SUM). Thanks to the above fact and Lemma 4.7.2 we obtain: Fact 4.7.6 For arbitrary x ∈ U and S ∈ 2U , if P(x) = ∅ then x SUM S ⇐⇒ ∅ = S ⊆ I(x) ∧ P(x) ⊆



x

OM [S].

We now note that, thanks to the above facts and Lemma 4.5.1, we have: Fact 4.7.7 From (A1)–(A3) and (A4w ) it follows that for arbitrary x, y ∈ U : ∀x,y∈U (x SUM {y} ⇒ x = y).

(SSUM )

Proof Let x SUM {y} and x = y. Then y  x and y, x ∈ Mx . Furthermore, by x x virtue of (WSPM ), for some z ∈ Mx : z  x and z M y. And this gives us a contraMx  diction, because z  x and x SUM {y} entail y  z. Remark 4.7.2 (i) Condition (SSUM ) is the counterpart of condition (Ssum ) that follows from (WSP). (ii) However, for axioms (A1)–(A3), (A3s ) and (A4w ) in Model 4.4 we have 123 sum {12}, 12 sum {12}, 123 sum {23} and 23 sum {23}.  As the second result obtained from Lemma 4.6.1 we get that the relation SUM has the uniqueness properties: Fact 4.7.8 From (A1)–(A3) and (A4w ) follows:   ∀x,y∈U ∀S∈2U x SUM S ∧ y SUM S =⇒ x = y .

(USUM )

Proof Let x SUM S and y SUM S. Then, if P(x) = ∅ or P(y) = ∅, then S = {x} or S = {y}, thanks to Lemma 4.7.2. Hence x = y, by Fact 4.7.7. Let P(x) = ∅ = P(y) and we assume that x = y. Then ∅ = S ⊆ I(x) ∩ I(y) ⊆ Mx ∩ M y . We show that Mx = M y .

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4 Theories Without the Assumption of Transitivity

Assume for a contradiction that Mx = M y . Then I(x) ∩ I(y) ⊆ min (Mx ) ∩ max (M y ) or I(x) ∩ I(y) ⊆ max (Mx ) ∩ max (M y ), in virtue of (A3 ). Hence y  x and x  y. However, x  y, and so x  y, thanks to (2.1.7) and the definitions. Hence Mx = M y , by virtue of Lemma 4.4.8.  x We therefore get: x, y ∈ Mx , S ⊆ I(x), Mx ∩ I(x) ⊆ OM [S], S ⊆ I(y) and  x Mx M ∩ I(y) ⊆ O [S]. Hence x  y and y  x, by virtue of Lemma 4.6.1. There fore, thanks to (antis ), we get a contradiction: x = y.

4.7.2 Mereological Sum Versus Supremum As before, when we assumed the transitivity of the relation  and in this way also the transitivity of the relation , with the help of (df sup) we can define the binary relation sup included in U × 2U . From condition (antis ) we obtain the uniqueness of the relation sup expressed with the help of condition (Usup ). Furthermore, from (r ) and (antis ) we get (Ssup ). Lemma 4.7.9 For arbitrary x ∈ U and S ∈ 2U : (i) If P(x) = ∅ then: x sup S ⇐⇒ S = {x} ∨ (S = ∅ ∧ ∀u∈U x  u). (ii) If P(x) = ∅ then: I(x) ⊆ Mx ,

  x sup S ⇐⇒ S ⊆ I(x) ∧ ∀y∈Mx S ⊆ I(y) ⇒ x  y . Proof (i) Let P(x) = ∅. “⇒” If x sup S then S ⊆ I(x) = {x}. Therefore if S = {x} then S = ∅. “⇐” We have x sup {x} and we use (B.2.24). (ii) Let P(x) = ∅. Then Mx is the only set in MCTU which contains I(x). “⇒” By virtue of the definition. “⇐” We observe that, on the strength of our assumption, S = ∅. If S = ∅ then for any y ∈ Mx : x  y, and, in consequence, we have a contradiction: P(x) = ∅ and for any y ∈ P(x) ⊆ Mx : x  y. We will show that for / Mx an arbitrary u such that S ⊆ I(u): u ∈ Mx or x  u. Assume that S ⊆ I(u), u ∈ and x  u. Then u  x and ∅ = S ⊆ P(x) ∩ P(u); and so x  u. Hence, by virtue  of Lemmas 4.4.10, 4.4.8, we have Mx = Mu and u ∈ Mx . In Chap. 2 it was shown that if the relation  is transitive, then condition (SSP) is equivalent to the inclusion sum ⊆ sup (see Theorem 2.9.5). Below, we will generalise this result without the assumption of the transitivity of the relation . Theorem 4.7.10 On the basis of axioms (A1)–(A3), axiom (A4w ) is equivalent to the inclusion SUM ⊆ sup.

4.7

Mereological Sums in Structures Without Transitivity

175

Proof Left-to-right: Let x SUM S. We will examine two cases. Firstly, let P(x) = ∅. Then S = {x} and x sup {x}, by virtue of Lemmas 4.7.2 and 4.7.9.  x of Secondly, let P(x) = ∅. Then I(x) ⊆ Mx and S ⊆ I(x) ⊆ OM [S], by virtue x x Lemma 4.7.2(ii). Assume that y ∈ M and S ⊆ I(y). Then, since I(x) ⊆ OM [S], then x  y, on the strength of (mM ). Therefore, x sup S, by virtue of Lemma 4.7.9(ii). Right-to-left: Let M ∈ MCTU , x, y ∈ M and I(x) ∩ M ⊆ O M(y). We will prove that x  y; we therefore obtain (SSP M ). We will examine two cases. Firstly, let P(x) = ∅ or P(x)  M. Then x ∈ min (M), on the strength of our assumption and Corollary 4.4.5(ii). Hence x  y, thanks to (r ), our definitions and our assumptions. Secondly, let ∅ = P(x) ⊆ M. Then I(x) ⊆ M; therefore M = Mx . Now, on the strength of Lemma 4.7.2(ii), we can prove that x SUM S := Mx ∩ I(x) ∩  S,Mwhere x I(y). Since S ⊆ I(x), then we must show that I(x) ⊆ O [S]. Let z ∈ I(x). Then, x by virtue of our assumption, z M y, i.e., for some u ∈ Mx we have u  z and u  y. Therefore, it is also the case that u  x, thanks to the transitivity of |Mx . Hence u ∈ S and u  M z, thanks to (r ). Now, by virtue of our assumption, we have x sup S. Therefore, x  y, since S ⊆ I(y).  From (Usup ) and Theorem 4.7.10 we get: Corollary 4.7.11 From (A1)–(A3) and (A4w ) follows:   ∀x,y∈U ∀S∈2U x SUM S ∧ y sup S =⇒ x = y . Remark 4.7.3 Model 2.7 belongs to s POS+(SSP), but does not satisfy the following version of discussed earlier conditions (‡) and (‡∅ ): 

sup ⊆ SUM,



∀S∈2U ∀x∈U S = ∅ ∧ x sup S =⇒ x SUM S .

(‡SUM ) (‡SUM ) ∅

Axioms (A1)–(A3), (A3s ), (A4w ), (A4) and (A5) are evidently true in Model 2.7. Therefore, conditions (‡SUM ) and (‡SUM ) do not follow from these axioms. Notice that we will obtain SUM = sum in the theory based on (A1)–(A5) (see ). Theorem 4.9.1). So, in this theory we also get: (‡)⇔ (‡SUM ) and (‡∅ )⇔ (‡SUM ∅ The issue of conditions (‡) and (‡∅ )—connected to the problem of the “ontological involvement” of the theory—was discussed in Chap. 2. As before, also now the ) in a given theory forces the “closure” of the presence of conditions (‡SUM ) and (‡SUM ∅ structure from Model 2.7 by the sums of pairs 1 and 2; 2 and 3; 1 and 3. The model below presents the structure obtained in this way:

176

4 Theories Without the Assumption of Transitivity 123

12

1

13

2

23

3

It is the counterpart of Model 2.8, in which we took the relation  to be transitive (now, obviously, the relation is also transitive but this needs highlighting). 

4.8 The Fifth Axiom By virtue of Lemma 4.4.1, for each M ∈ MCTU the following holds: | M ⊆  M ,  M ⊆ | M , and all elements of the set min (M) are found as pairs in the relation  M . Before introducing the fifth axiom, let us observe that: Lemma 4.8.1 From (A1)–(A3) it follows that, for an arbitrary M ∈ MCTU the following conditions are equivalent: (a) (b) (c) (d)

all elements of the set min (M) are external pairs, i.e., in the relation  ,  M ⊆ | M , | M ⊆  M , | M ⊆  M .

Proof “(a) ⇒ (b)” Let x  M y (so x, y ∈ M). If ¬x | M y then x | M y. Therefore, x, y ∈ min (M), thanks to Lemma 4.4.7. Hence, thanks to (a), x  y, and in consequence x | M y as well. “(b) ⇒ (a)” From Lemma 4.4.1(iii) we obtain that all elements of the set min (M) are pairs in the relation  M , and so we use (b). “(b) ⇔ (c) ⇔ (d)” By virtue of Lemma 4.4.1(ii).  We now admit the following fifth axiom: (A5) each set M from MCTU satisfies (equivalent) conditions from Lemma 4.8.1. Example 4.8.1 Returning to Example 4.4.3, it needs to be noted that all elements of the set C (cells) are minimal in the set M1 . All cells are external with respect to each other; that is, they are in the relation . The same is true in the case of the elements of the set N ∪ P (nuclei and other parts of cells), which are minimal in the  set M2 .

4.8

The Fifth Axiom

177

From Fact 4.5.7 we get: Fact 4.8.2 There exists a structure in which: (a) axioms (A1)–(A4), (A3s ) hold, (b) axiom (A5) does not hold. Therefore axioms (A1)–(A4), (A3s ) do not imply axiom (A5).4 We will now show that axioms (A1)–(A3) and (A5) entail the equivalence of conditions (A4) and (A4w ). In order to do so, we will need the following lemma. Lemma 4.8.3 Axioms (A1)–(A3), (A4w ) and (A5) entail (WSP). Proof Let y  x. Then x, y ∈ Mx , by Lemma 4.4.2. Furthermore, x  y, thanks x x to (as ). Hence, thanks to (SSPM ), there is a z ∈ Mx such that z  x and z M y. We have z = x, i.e., z  x. Furthermore, z  y, by virtue of the fourth axiom and Lemma 4.8.1.  Therefore, by virtue of Lemmas 2.3.8(ii), 4.5.1, 4.5.2 and 4.8.3, we obtain: Theorem 4.8.4 Axioms (A1)–(A3) and (A5) imply that conditions (A4) and (A4w ) are equivalent. Furthermore, by virtue of Lemmas 4.4.1(i) and 4.8.1, we obtain: Corollary 4.8.5 Axioms (A1)–(A3) and (A5) imply that, for any M ∈ MCTU : | M =  M and | M =  M . Using this result and (SSPM ) we also have: Corollary 4.8.6 From (A1)–(A5) it follows that for an arbitrary M ∈ MCTU :   ∀x,y∈M x  y ⇐⇒ ∀z∈M (z  x ⇒ z  y) .

(SSPM )

Remark 4.8.1 If we assume that in the irreflexive structure U = U,  the relation  is transitive, then that structure automatically satisfies axioms (A1)–(A3) and (A5) (see Remark 4.4.1). By assuming the additional axiom (SSP) (i.e. (A4)), we obtain a polarised strict partial order, in which the following condition holds:   ∀x,y∈U x  y ⇐⇒ ∀z∈U (z  x ⇒ z  y) . Fact 4.8.7 Conditions (A4) and (A4w ) are independent of (A1), (A2), (A3s ) and (A5), and therefore also independent of (A1)–(A3) and (A5). Proof Model 2.1 shows that condition (SSP) is not entailed by conditions (irr  ) and (t ); i.e., the class s POS+(SSP) is a subclass of the proper class s POS. This model therefore also satisfies axioms (A1)–(A3) and (A5) (see Remarks 4.4.1 and 4.8.1), i.e., these axioms do not entail (A4). Therefore, by virtue of Theorem 4.8.4, they  also do not entail (A4w ). 4 As in the case of Fact 4.5.7, we make use here of Model 4.5. Therefore, in addition, let us add that

12 and 23 belong to min (M1 ), but 12  23, which contradicts condition (a) from Lemma 4.8.1.

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4.9 Mereological Sums for Axioms (A1)–(A5) By virtue of Lemma 4.7.2 and Corollary 4.8.5, in the theory based on axioms (A1)– (A5), we have the “classical definition” of a mereological sum: Theorem 4.9.1 Axioms (A1)–(A5) imply that: SUM = sum. So, in consequence, in the theory based on (A1)–(A5) we obtain: (‡) ⇐⇒ (‡SUM ), (‡∅ ) ⇐⇒ (‡SUM ). ∅ Proof “⊆”: By virtue of Fact 4.7.3.  “⊇”: Let x sum S, i.e., S ⊆ I(x) ⊆ O(S). Firstly, if P(x) = ∅ then we use Lemma 4.7.2(i). Secondly, if P(x) = ∅ S ⊆ I(x) ⊆ Mx . Hence ∀y∈Mx y    then Mx x x ⇒ ∃u∈S u |M y . Therefore, I(x) ⊆ O [S], by virtue of Corollary 4.8.5. The proof is, however, completed by the use of Lemma 4.7.2(ii). 

4.10 Existentially-Involved Theories The theories considered so far in this chapter have been existentially neutral. It may be said that the theory based on axioms (A1)–(A5) and the two weaker versions of it based respectively on (A1)–(A4) and (A1)–(A3), (A4w ) (or the various stronger versions of them in which axioms (A3s ) replaces axiom (A3)) correspond in a sense to the theory s POS, in which we assume the transitivity of the relation . We will now examine the axioms adopted in Chap. 3 with a view to determining those which will allow themselves to be applied without any changes and those which will require some modifications in theories in which the transitivity of the relation  is not assumed. We will only present the essential changes that have to be made (in comparison to Chap. 3). The remainder of our investigations can be carried out in a fashion analogous to that which was pursued in Chap. 3. None of the axioms adopted in Chap. 3 when added to axioms (A1)–(A5), (A3s ) generates a contradiction in the theory thereby obtained. At most, we may find that a theory becomes “trivialised” in the sense that the addition of one of the axioms entails the transitivity of the relation ; that is, that in all models we obtain MCTU = {U }. In such a case, instead of the “full version” of the axiom in question, we have to adopt its “partial version”.5 Let us add that in some cases where we use the axioms adopted in this chapter as a base, the axioms from Chap. 3 will turn out to be equivalent to their “partial versions”. 5 In

this sense, axiom (A4w ) is a “partial version” of axiom (A4).

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179

SUM 4.10.1 Axioms (‡SUM ) ∅ ) and (‡

The “existential involvement” of conditions (‡SUM ) and (‡SUM ) has been written about ∅ already in Remark 4.7.3. Let us now observe that in the theory based on (A1)–(A3) and (A4w ) they are equivalent to the following “partial versions”: ∀M∈MCTU sup  (M × 2 M ) ⊆ SUM ,   ∀M∈MCTU ∀S∈2 M ∀x∈M S = ∅ ∧ x sup S =⇒ x SUM S .

(‡SUM∂ ) (‡SUM∂ ) ∅

Firstly, it is obvious that: ) entails (‡SUM∂ ), Fact 4.10.1 (i) (‡SUM ∅ ∅ SUM SUM∂ (ii) (‡ ) entails (‡ ). Secondly, we now prove that: Fact 4.10.2 On the basis of axioms (A1)–(A3) and (A4w ): ) entails (‡SUM ), (i) (‡SUM∂ ∅ ∅ SUM∂ (ii) (‡ ) entails (‡SUM ). Proof (i) Let S = ∅ and x sup S. Let us consider the two cases. If P(x) = ∅ then S = {x}, by virtue of Lemma 4.7.9(i). Therefore, x SUM S. If P(x) = ∅ then I(x) ⊆ Mx , by virtue of Lemma 4.7.9(ii). Furthermore, x ∈ Mx and S ⊆ I(x) ⊆ Mx , ). by virtue of (df sup). Therefore, x SUM S, by virtue of (‡SUM∂ ∅ (ii) Let x sup S. If S = ∅ then x is the zero. Hence, by virtue of ( 0), it is degenerate, i.e. U = {x}. Therefore, x SUM S. If, however, S = ∅ then we repeat the considerations of the previous point.  Corollary 4.10.3 On the basis of axioms (A1)–(A3) and (A4w ): ) and (‡SUM ) are equivalent, (i) (‡SUM∂ ∅ ∅ SUM∂ (ii) (‡ ) and (‡SUM ) are equivalent.

4.10.2 Axiom (c∃) and its Various Versions This axiom, when considered in Chap. 3, had the following form:   ∀x,y∈U x  y =⇒ ∃z∈U ∀u∈U (u  z ⇔ u  x ∧ u  y) .

(c∃)

Let us compare it with the following formulas:   ∀M∈MCTU ∀x,y∈M x  M y =⇒ ∃z∈M ∀u∈U (u  z ⇔ u  x ∧ u  y) ,   ∀M∈MCTU ∀x,y∈M x  M y =⇒ ∃z∈M ∀u∈M (u  z ⇔ u  x ∧ u  y) ,

(c∃∂1 ) (c∃∂2 )

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  ∀M∈MCTU ∀x,y∈M x  y =⇒ ∃z∈M ∀u∈U (u  z ⇔ u  x ∧ u  y) ,   ∀M∈MCTU ∀x,y∈M x  y =⇒ ∃z∈M ∀u∈M (u  z ⇔ u  x ∧ u  y) .

(c∃∂3 ) (c∃∂4 )

Firstly, it is obvious that: Fact 4.10.4 (i) (c∃∂1 ) entails (c∃∂2 ). (ii) (c∃∂3 ) entails (c∃∂4 ). Secondly, by virtue of the inclusion  M ⊆  we obtain: Fact 4.10.5 (i) (c∃) entails (c∃∂1 ). (ii) (c∃∂3 ) entails (c∃∂1 ). (iii) (c∃∂4 ) entails (c∃∂2 ). Proof (i): Take an arbitrary M ∈ MCTU and x, y ∈ M such that x  M y, i.e., for some z 0 ∈ M we have z 0  x and z 0  y. Hence x  y. Therefore, by applying (c∃), we obtain a z 1 such that z 0  z 1 , since z 0  x and z 0  y. Furthermore, since z 1  z 1 , then z 1  x and z 1  y. Therefore, z 1 ∈ M, since z 0  z 1  x, z 0 , x ∈ M and M is closed. Furthermore, by virtue of our assumption, we have: ∀u∈U (u  z 1 ⇔ u  x ∧ u  y). (ii) and (iii): Obvious.  Thirdly, we we can prove: Fact 4.10.6 On the basis of axioms (A1)–(A3): conditions (c∃) and (c∃∂1 ) are equivalent. Proof “(c∃)⇒(c∃∂1 )” By Fact 4.10.5(i). “(c∃∂1 )⇒(c∃)” Take arbitrary x, y ∈ U such that x  y. Then, by virtue of Lemma 4.4.10, there is an M ∈ MCTU such that x  M y. Therefore (c∃∂1 ) gives  a z ∈ M such that: ∀u∈U (u  z ⇔ u  x ∧ u  y). Fourthly, in the light of Lemma 4.8.1, axioms (A1)–(A3) and (A5) gives the equivalence  M = . Hence we obtain: Fact 4.10.7 On the basis of axioms (A1)–(A3) and (A5): (i) (c∃), (c∃∂1 ) and (c∃∂3 ) are equivalent. (ii) (c∃∂2 ) and (c∃∂4 ) are equivalent. Now notice that: Fact 4.10.8 (i) (c∃∂4 ) does not follow from (A1), (A2), (A3s ), (A4), (c∃). (ii) (c∃∂3 ) does not follow from (A1), (A2), (A3s ), (A4), (c∃∂2 ). (iii) (c∃) does not follow from (A1), (A2), (A3s ), (A4), (A5), (c∃∂2 ). (iv) (c∃∂1 ) does not follow from (A1), (A2), (A3s ), (A4), (A5), (c∃∂2 ).

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Existentially-Involved Theories

Model 4.6 (A1)–(A5), (A3s ), (c∃∂2 ) and (c∃∂4 ) hold, but (c∃∂1 ) and (c∃) do not hold

181 1

12

23

2

1

2

3

Proof (i) Conditions (A1), (A2), (A3s ), (A4), (c∃) hold in Model 4.5. In this model, however, we have: 12  23 and 12, 23 ∈ M1 , but 12  M1 23. Hence (c∃∂4 ) does not hold. That is, for each x ∈ M1 we have x  x but the conjunction x  12 and x  23 is false. (ii) From (i) and Facts 4.10.4 and 4.10.6. (iii) and (iv) Conditions (A1), (A2), (A3s ), (A4), (A5), (c∃∂4 ) and (c∃∂2 ) hold in Model 4.6. On the other hand, (c∃∂1 ) does not hold. To see this, 12, 23 and 2 belong to M1 , a  2, but a  12. In a similar way, we show that (c∃) does not hold in Model 4.6 (we can also obtain this result from Fact 4.10.6).  Thus, by adopting axioms (A1)–(A3) we obtain various relationships between the conditions we have examined which are presented below (where ‘→’ indicates entailment):

Fact 4.10.8(i,ii) shows that even if we add conditions (A3s ) and (A4), we will not get any other entailments. Thus, the theory based on axioms (A1), (A2), (A3s ), (A4), (c∃∂3 ) is the strongest and the theory based on axioms (A1), (A2), (A3s ), (A4), (c∃∂2 ) is the weakest among theories that have axioms belonging to (A1), (A2), (A3), (A3s ), (A4), (c∃), (c∃∂1 ), (c∃∂2 ), (c∃∂3 ), (c∃∂4 ). If we adopt axioms (A1)–(A3), (A5) we obtain the relationships depicted below:

Fact 4.10.8(iii,iv) shows that even if we add conditions (A3s ) and (A4), we will not get any other entailments. Thus, three equivalent theories based on (A1), (A2), (A3s ), (A4), (A5), (c∃), on (A1), (A2), (A3s ), (A4), (A5), (c∃∂1 ), and on (A1), (A2), (A3s ), (A4), (A5), (c∃∂3 ) are the strongest among theories that have axioms belonging to (A1), (A2), (A3), (A3s ), (A4), (A5), (c∃∂1 ), (c∃∂2 ), (c∃∂3 ), (c∃∂4 ).

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In Model 4.6 it was essential that elements 12 and 23 had exactly one common part. But we obtain: Fact 4.10.9 If a structure U,  satisfies axioms (A1)–(A3), (c∃∂2 ) and the condition below   ∀x,y∈U x  y =⇒ x and y have two common par ts ,

($)

then axiom (c∃) holds in this structure. Proof Take arbitrary x, y ∈ U such that x  y. We need only consider the possible cases with respect to (2.1.16). The case x = y. Then ∀u∈U (u  x ⇔ u  x ∧ u  y). The case y  x. We show that ∀u∈U (u  y ⇔ u  x ∧ u  y). Take an arbitrary y u ∈ U such that u  y. Then u, y ∈ M y and u M y. So, by (c∃∂2 ), for some z 0 ∈ y M we have: ∀u∈M y (u  z 0 ⇔ u  x ∧ u  y). Therefore, z 0  x and z 0  y, since z 0 ∈ M y and z 0  z 0 . Moreover, y  z 0 , since y ∈ M y , y  x and y  y. So, z 0 = y, by (antis ). So, for any u ∈ U if u  y, then u  x and u  y. The case y  x: Analogously to the above. The case x  y: By ($), x and y have two common parts a1 and a2 , a1 = a2 . By x x Lemma 4.4.8, both Mx = M y and x M y; and so x M y. Hence, by (c∃∂2 ) for some z 0 ∈ Mx we have: ∀u∈Mx (u  z 0 ⇔ u  x ∧ u  y). Therefore, z 0  x and z 0  y, since z 0 ∈ Mx , z 0  z 0 and x  y. Furthermore, a1  z 0 or a2  z 0 , since a1  x, a2  x, a1  y, a2  y and a1 = a2 . In the first case, a1 , z 0 ∈ Mz0 . But also a1 , x ∈ Mx and z 0 , x ∈ Mx ; and so a1 , z 0 ∈ Mx . Hence Mz0 = Mx , since Mz0 is the only set in MCTU containing a1 and z 0 . So I(z 0 ) ⊆ Mz0 = Mx . Therefore:  ∀u∈U (u  z 0 ⇔ u  x ∧ u  y). In the second case, we reason analogously.

4.10.3 Axioms (c∃pair sup) and (c∃par SUM) These axioms have the following form:   ∀x,y∈U x Un y =⇒ ∃z∈U z sup {x, y} ,   ∀x,y∈U x Un y =⇒ ∃z∈U z SUM {x, y} .

(c∃pair sup) (c∃par SUM)

When we have axiom (A5), then this second proposition is equivalent to   ∀x,y∈U x Un y =⇒ ∃z∈U z sum {x, y} .

(c∃pair sum)

Furthermore, let us observe that by virtue of Lemmas 4.7.2 and 4.7.9 we obtain: Fact 4.10.10 (i) If z SUM {x, y}) then either z = x = y or {x, y} ⊆ Mz . (ii) If z sup {x, y} then either z = x = y or {x, y} ⊆ Mz .

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183

We will show that the conditions above are equivalent to their respective “partial versions”6 :   ∀M∈MCTU ∀x,y∈M x Un M y =⇒ ∃z∈M z sup {x, y} ,   ∀M∈MCTU ∀x,y∈M x Un M y =⇒ ∃z∈M z SUM {x, y} ,

(c∃par sup∂ ) (c∃par SUM∂ )

in which for an arbitrary M from MCTU the relation Un M is the counterpart of the relation Un and is defined by the following condition: x Un M y :⇐⇒ x, y ∈ M ∧ ∃z∈M (x  z ∧ y  z), i.e., two elements are in the relation iff they belong to M and have a common upper bound in M. Lemma 4.10.11 If a structure U,  satisfies (A1)–(A3) then   ∀x,y∈U x Un y ⇐⇒ ∃ M∈MCTU x Un M y . Proof Left-to-right: Assume that x Un y. If x = y, we take some or other M ∈ MCTU , to which belongs x. If x  y or y  x, we take M y or Mx appropriately. If, however, for some z we have x  z and y  z, then we take Mz . Right-to-left: By logic and the inclusion Un M ⊆ Un, for any M ∈ MCTU .  Fact 4.10.12 Conditions (c∃pair sup) and (c∃par sup∂ ) are equivalent in any theory having conditions (A1)–(A3) among its axioms. Proof “(c∃pair sup)⇒(c∃par sup∂ )” Take arbitrary M ∈ MCTU and x, y ∈ M such that x Un M y, i.e., for some u ∈ M we have x  u and y  u. Hence x Un y. Therefore, by applying (c∃pair sup), we get a z ∈ U such that z sup {x, y}. Hence x  z  u, since x  u and y  u. Therefore, z ∈ M, since u ∈ M and M is closed. “(c∃par sup∂ )⇒(c∃pair sup)” Take arbitrary x, y ∈ U such that x Un y. Then, by virtue of Lemma 4.10.11, there exists an M ∈ MCTU such that x Un M y. There fore, (c∃par sup∂ ) gives us a z ∈ M such that z sup {x, y}. Fact 4.10.13 Conditions (c∃par SUM) and (c∃par SUM∂ ) are equivalent in any theory having condition (A1)–(A3) and (A4w ) among its axioms. Proof “(c∃par SUM)⇒(c∃par SUM∂ )” Take arbitrary M ∈ MCTU and x, y ∈ M such that x Un M y, i.e., for some u ∈ M we have x  u and y  u. Hence x Un y. Therefore, by applying (c∃par SUM), we get a z ∈ U such that z SUM {x, y}. Hence x  z  u, since x  u and y  u, by Theorem 4.7.10. Therefore, z ∈ M, since u ∈ M and M is closed. “(c∃par SUM∂ )⇒(c∃par SUM)” Take arbitrary x, y ∈ U such that x Un y. Then, by virtue of Lemma 4.10.11, there exists an M ∈ MCTU such that x Un M y. There fore, (c∃par sup∂ ) gives us a z ∈ M such that z SUM {x, y}. 6 In

order not to complicate matters (see the point above), we will not introduce other “partial versions” of the conditions we have examined.

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4 Theories Without the Assumption of Transitivity

4.10.4 Axioms (∃pair sup) and (∃par SUM) A different situation arises for these axioms in comparison to the cases seen so far: ∀x,y∈U ∃v∈U v sup {x, y} ,

(∃pair sup)

∀x,y∈U ∃v∈U v SUM {x, y} .

(∃par SUM)

Their adoption leads to the “trivialisation” of the theory. Fact 4.10.14 Condition (t ) obtains in any theory that has (A1)–(A3) and either (∃pair sup) or (∃par SUM) among its axioms. Proof Let x  y  z. Since there is a u such that u sup {x, z} (resp. u SUM {x, z}), then {x, z} ⊆ Mu , by virtue of Lemma 4.7.9 (resp. Lemma 4.7.2). Hence, the path (x, y, z) is included in Mu , and the relation |Mu is transitive. So x  z.  Thus, (∃pair sup) and (∃par SUM) “trivialise” the family MCTU , i.e., we obtain MCTU = {U }. It is harder to show that any theory having (A1)–(A3), (A4w ) and one of the conditions below among its axioms also becomes “trivialised”: ∀M∈MCTU ∀x,y∈M ∃z∈M z sup {x, y},

(∃par sup∂1 )

∀M∈MCTU ∀x,y∈M ∃z∈M z sum {x, y}.

(∃par SUM∂1 )

Fact 4.10.15 Condition (t ) holds in any theory having (A1)–(A3), (A4w ) and either (∃par sup∂1 ) or (∃par SUM∂1 ) among its axioms. Proof Let x  y  z. Then x ∈ M y and y ∈ Mz . If M y = Mz then x  z. Let us accept, therefore, that M y = Mz . Since y ∈ M y  Mz , then either y ∈ max (M y )  / max (Mz ), then y ∈ min (Mz ) or y ∈ min (M y )  max (Mz ). Since y ∈ y z max (M )  min (M ). Furthermore, by virtue of (WSP M ), there exists a v ∈ Mz z such that v  z and v M y. By virtue of Corollary 4.4.5(i), since P(z)  M y = ∅, then P(z) ⊆ M y . Hence, v ∈ M y . Therefore, there exists a w ∈ M y such that w sup {y, v}. We have y  w  and v  w. We therefore have a contradiction: y ∈ / max (M y ). From the facts above we obtain the following result: Corollary 4.10.16 Both conditions in the two following pairs {(∃pair sup), (∃par sup∂1 )} and {(∃par SUM), (∃par SUM∂1 )} are equivalent in any theory having (A1)–(A3), (A4w ) among its axioms. Proof In the light of Facts 4.10.14 and 4.10.15, each formula from a given pair entails (t ) and together entail the second formula. 

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185

Therefore, in order to obtain some more interesting theories, we must consider the “partial versions” of conditions (∃pair sup) and (∃par SUM) below: ∀M∈MCTU ∀x,y∈M ∃v∈U v sup {x, y},

(∃par sup∂2 )

∀M∈MCTU ∀x,y∈M ∃v∈U v sum {x, y}.

(∃par SUM∂2 )

4.10.5 Le´sniewski’s Axiom Since, as we have seen, (∃pair sup) and (∃par SUM) “trivialise” the family MCTU , then Le´sniewski’s axiom will do so likewise: ∀S∈2U \{∅} ∃x∈U x SUM S .

(∃SUM)

It is obvious indeed that (∃SUM) entails (∃par SUM). It also follows from Fact 4.2.4. By putting S := U , we see that there exists an x such that x is the unity as the mereological sum of U . Hence, if the structure satisfies axioms (A1) and (A2), then the relation  is transitive. In a similar way, by virtue of Fact 4.10.15, we obtain a “trivialised” theory when to axioms (A1)–(A3), (A4w ) we add: ∀M∈MCTU ∀S∈2U \{∅} ∃x∈M x SUM S.

(∃SUM∂1 )

We must therefore consider “partial theories” based on the following condition: ∀M∈MCTU ∀S∈2 M \{∅} ∃x∈U x SUM S.

(∃SUM∂2 )

4.10.6 Weak Axioms of the Existence of a Mereological Sum For the following conditions:   ∀S∈2U \{∅} ∃u∈U S ⊆ I(u) =⇒ ∃x∈U x SUM S ,   ∀M∈MCTU ∀S∈2 M \{∅} ∃u∈M S ⊆ I(u) =⇒ ∃x∈M x SUM S .

(w1 ∃SUM) (w1 ∃SUM∂ )

we can carry out a similar analysis to that undertaken for conditions (c∃par SUM) and (∃par SUM∂1 ).

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Fact 4.10.17 Conditions (w1 ∃SUM) and (w1 ∃SUM∂ ) are equivalent in any theory having (A1)–(A3), (A4w ) among its axioms. Proof “(w1 ∃SUM)⇒(w1 ∃SUM∂ )” Take arbitrary M ∈ MCTU and S ∈ 2 M \ {∅} such that for some u ∈ M we have S ⊆ I(u). By applying (w1 ∃SUM), we obtain an x such that x SUM S. Hence S ⊆ I(x) and x  u, since x sup S, by virtue of Theorem 4.7.10. Therefore x ∈ M, since M is closed. “(w1 ∃SUM∂ )⇒(w1 ∃SUM)” Take an arbitrary S ∈ 2U \ {∅} such that there exists a u such that S ⊆ I(u). Then, if P(u) = ∅ then S = {u}. If P(u) = ∅ then S ⊆ I(u) ⊆ Mu . Therefore, in both cases, there exists an M ∈ MCTU such that S ⊆ M.  Hence (w1 ∃SUM∂ ) gives us an x ∈ M such that x sup S. Corresponding results to the above cannot be obtained for the following two conditions: ∀S∈2U \{∅} (∀y,z∈S y Un z =⇒ ∃x∈U x SUM S),

(w2 ∃SUM)

∀M∈MCTU ∀S∈2 M \{∅} (∀y,z∈S y Un z =⇒ ∃x∈M x SUM S).

(w2 ∃SUM∂ )

M

4.10.7 Axiom (SSP+) There remains just one axiom to consider from those examined in Chap. 3. It is the strong supplementation principle:    ∀x,y∈U x  y =⇒ ∃z∈U z  x ∧ z  y ∧ ∀u∈U (u  x ∧ u  y ⇒ u  z) . (SSP+) Let us recall that (SSP), i.e. (A4), follows from (SSP+), and from (A4) for each M ∈ MCTU we have (SSP M ), i.e. (A4w ). Let us compare it with its two “partial versions” below:   ∀M∈MCTU ∀x,y∈M x  y =⇒ ∃z∈M z  x ∧ z  M y ∧

(SSP+∂1 )

 ∀u∈U (u  x ∧ u  y ⇒ u  z) ,   ∀M∈MCTU ∀x,y∈M x  y =⇒ ∃z∈M z  x ∧ z  M y ∧ (SSP+∂2 )  ∀u∈M (u  x ∧ u  M y ⇒ u  z) .

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It is obvious that: Fact 4.10.18 (A4w ) follows both from (SSP+∂1 ) and from (SSP+∂2 ). Moreover, we prove: Fact 4.10.19 (i) From (A1)–(A3), (A3s ), (SSP+∂1 ), (SSP+∂2 ) does not follow (SSP), and so (SSP+) does not follow either. (ii) From (A1)–(A3), (A3s ), (SSP+) do not follow (SSP+∂2 ) and (A5). Proof (i) In Model 4.4, (A1)–(A3), (A3s ), (SSP+∂1 ) and (SSP+∂2 ) hold but (SSP) does not, and so (SSP+) does not either. (ii) In Model 4.7 (A1)–(A4), (A3s ) and (SSP+) but (A5) and (c∃∂4 ) do not. For example, 1  1234 but there does not exist an x such that x  1234, x  M1 1 and  ∀y∈M1 (y  1234 ∧ y  M1 x ⇒ y  x). Fact 4.10.20 (SSP+∂1 ) follows from (A1)–(A3), (SSP+). Proof Take arbitrary M ∈ MCTU and x, y ∈ M such that x  y. Then either x  M y or x  M y. In the first case, we immediately obtain our thesis by putting z := x. In the second case, for some u ∈ M we have u  x and u  y. Hence, by virtue of our assumption, x = u, i.e. u  x. Furthermore, thanks to (SSP+), there exists a z such that z  x, z  y and ∀v∈U (v  x ∧ v  y ⇒ v  z). Hence, by virtue of our assumption , x = z, i.e. z  x. Therefore, z ∈ M, by virtue of Corollary 4.4.5(i).  Furthermore, z  M y, since  ⊆  M . With respect to Models 4.4 and 4.7, it was essential that they did not satisfy axiom (A5). In a theory having axioms (A1)–(A5), for each M ∈ MCTU we have  M =  . We obtain the following theorem: Theorem 4.10.21 Conditions (SSP+), (SSP+∂1 ) and (SSP+∂2 ) are equivalent in any theory that has (A1)–(A3) and (A5) among its axioms. Proof “(SSP+)⇒(SSP+∂1 )” By Fact 4.10.20. “(SSP+∂1 )⇒(SSP+∂2 )” Obvious, when we consider (A5). “(SSP+∂2 )⇒(SSP+)” Assume that x  y. If x  y then put z := x. If x  y then either y  x or x  y. In both cases, we have x, y ∈ Mx . Therefore, by applying x x (SSP+∂2 ), we get a z ∈ Mx such that z  x, z M y and ∀u∈Mx (u  x ∧ u M y ⇒ u  z). However, I(x) ⊆ Mx . Therefore, by applying (A5), we obtain a corresponding condition for (SSP+). 

Model 4.7 (A1)–(A4), (A3s ) and (SSP+) hold, but (A5) and (SSP+∂2 ) do not hold

1

1234 2

1

2

3

4

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References Lyons, J. (1977). Semantics (Vol. 1). Cambridge: Cambridge University Press. https://doi.org/10. 1017/CBO9781139165693. Pietruszczak, A. (2012). Ogólna koncepcja bycia cz˛es´cia˛ cało´sci. Mereologia a nieprzechodnia relacja bycia cz˛es´cia˛ (A general concept of being a part of a whole. Mereology and the nontransitive relation of being a part.). In J. Goli´nska-Pilarek & A. Wójtowicz (Eds.), Identyczno´sc´ znaku czy znak identyczno´sci? (The identify of a sign or the sign of identity?) (pp. 157–179). Warszawa: Wydawnictwa Uniwersytetu Warszawskiego. Pietruszczak, A. (2014). A general concept of being a part of a whole. Notre Dame Journal of Formal Logic, 55(3), 359–381. https://doi.org/10.1215/00294527-2688069.

Appendix A

Logic and Set Theory

A.1

Logical Symbolism

Throughout this book we will be using standard logical symbols. So ‘¬’ is the symbol for the negation operator (“it is not the case that”) and ‘∧’, ‘∨’, ‘⇒’ and ‘⇔’ symbolise the following sentential connectives: conjunction (“and”), inclusive disjunction (“or”), material implication (“if … then”) and material equivalence (“if and only if”). For convenience, we will take it that the last two connectives dominate the first two when it comes to their connective power, thus enabling us to avoid the use of brackets. In addition, ‘∃’ is the existential quantifier (“there exists at least one … such that”) and ‘∀’ the universal quantifier (“for each … it is the case that”). The symbol ‘ :⇐⇒ ’ is used to indicate definitional equivalence and ‘:=’ to indicate definitional identity. The symbol ‘=’ itself is simply identity (“is identical to”) and ‘ =’ is the symbol of non-identity (“is not identical to”). So the formula ‘¬x = y’ we will write thus ‘x = y’. The quantifier ‘∃1 ’ means “there is exactly one … such that”. In this book, we will use the description operator ‘ ’ to build expressions of the form ( x) ϕ(x), which are individual constants whose meaning is “the unique x such that ϕ(x)’. In order to use such an expression, we must first prove that there is indeed exactly one x ϕ(x) must satisfy the following such that ϕ(x),formally: ∃1x ϕ(x), i.e., the formula  condition: ∃x ϕ(x) ∧ ∀y (ϕ(x/y) ⇒ x = y) , where ϕ(x/y) is the formula which arises from ϕ(x) by the replacement of each free occurrence of ‘x’ by ‘y’.1 The quantifiers ‘∃’, ‘∃1 ’ and ‘∀’ will bind not only variables referring to individuals but also variables referring to sets and sets which are families of sets: for example, ‘X ’, ‘Y ’, ‘Z ’, ‘S’, ‘X’, ‘Y’, ‘F ’ and ‘S’ etc. ι

ι

1 We

are obviously assuming here that the variable ‘y’ can replace every free occurrence of the variable ‘x’ in ‘ϕ(x)’. If the variable ‘y’ does not meet this condition, we choose another variable which does. © Springer Nature Switzerland AG 2020 A. Pietruszczak, Foundations of the Theory of Parthood, Trends in Logic 54, https://doi.org/10.1007/978-3-030-36533-2

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The sentential connectives above may be understood in this book just as they are in classical logic. However, when we are dealing with partial operators, we have a problem. We will introduce it by looking at the first place in which it appears, which is on Sect. 3.2. In relational structures that are models of Simons’ “minimal extensional mereology”, one may introduce a partial binary product operator, defined for all and only those pairs of individuals which overlap; that is, between which the relation . holds. We signify this relation with the symbol ‘ ’. On Sect. 3.2, the following sentence appears:   (∗) ∀x,y∈U x  y =⇒ ∃z∈U z = x y , in which ‘U ’ symbolises the universe of discourse, this being some non-empty distributive set.2 This sentence talks about the arbitrary two choices of individuals from U where the first chosen object is signified by the variable ‘x’ and the second by the variable ‘y’.3 Leaving aside trivial cases, in models of minimal extensional mereology, the relation  is not full (i.e.,  = U × U ). Therefore, we can find a model and a choice of elements x and y which do not stand in the relation  (i.e., x and y do not overlap). In this case, the expression ‘x y’ does not signify anything, and hence the sentence ‘∃v∈U v = x y’ has no truth-value, this sentence being the consequent of the conditional in (∗). However, for a given choice of x and y, the antecedent is now false (because the relational sentence ‘x  y’ has a truth-value). It will therefore suffice for us to say that any material implication with a false antecedent is true.4 Sentences lacking truth-values can only appear in theses with an implicative form and then only when the antecedent of the implication is false. We will in any case try to avoid such situations. Because of this, where possible, we will prefer a relational form of expression. We will therefore write ‘x sum S’ (“x is the sum of all elements of the set S”) and not ‘x = sum S’, because it is not the case that for all sets that there exists a sum of their elements. If the elements of a set S do not have a sum, then the sentence ‘x sum S’ is simply false (regardless of the choice of x). On the other hand, in such a case, the term-expression ‘sum S’ does not signify anything; that is, the sentence ‘x = sum S’ does not have a truth-value (regardless of x). Similarly, to express the fact that the object x is the supremum of the set S in some partial order, we will use the relational form ‘x sup S’, and not ‘x = sup S’, in which appears the name of the partial operator sup. If a set S does not have a supremum, this sentence does not have a truth-value.

A.2 Fundamental Features of Set Theory We will be concerned with a certain elementary theory in which we shall speak of distributive sets composed both of individuals and other distributive sets. The

2 In both appendices we understand the concept of a set in a distributive sense. This issue is discussed

in detail in Chap. I of (Pietruszczak 2000, 2018) and in the next section of this appendix. 3 We do not exclude the possibility of choosing the same object twice (but with different variables). 4 This is in any case the standard approach taken in mathematics when dealing with partial operators.

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fundamental concepts involved are: the concept being a distributive set and the relational concept being an element of . Set theory exists not just for the purposes of “pure mathematics” but finds application elsewhere—such as in the theory of parthood, parts, physics or more generally in formal ontology—and thus allows for the existence of objects other than distributive classes or sets.5 Such objects have no members. They are called individuals. One may say that their whole is a set or that every set is founded on the set of such individuals and empty set which means—to speak figuratively—that the “ultimate” elements out of which all sets are built are individuals and the empty set. The so-called pure theory of sets, which suffices for mathematics, involves only sets which are composed out of other sets. Here, the “ultimate” element of each set is always just the empty set.

A.2.1 Axioms of Set Theory Variables ranging over the universe of discourse are primarily the lower-case letters ‘x’, ‘y’, ‘z’, ‘u’, ‘v’, ‘w’ and ‘s’ with or without indices.6 As already mentioned, the two primitive concepts we are introducing are the concepts being a set and being an element of . That x is a set will be written symbolically thus: ‘Set x’. Furthermore, that x is an element of y (or that x belongs to y; or x is a member of y) will be written symbolically as ‘x ∈ y’ and the sentence ‘¬x ∈ y’ we will write thus: ‘x ∈ / y’. Let us observe that the sentences below are logical theorems7 :   / y) , ¬∃x Set x∧∀y (y ∈ x ⇔ Set y ∧ y ∈   ¬∃x ∀y y ∈ x ⇔ y ∈ /y .

(A.2.1) (A.2.2)

The first says that there is no set to which belong all and only those sets which are not their own elements. The second says simply that there is no object to which belong all and only those objects which are not their own elements. For (A.2.1): Assume / for a contradiction that for some x0 we have: Set x0 ∧ ∀y (y ∈ x0 ⇔ Set y ∧ y ∈ / x0 , which yields a contradiction.8 y). Hence Set x0 and: x0 ∈ x0 ⇔ Set x0 ∧ x0 ∈ For (A.2.2): Assuming for a contradiction that for some x0 we have: ∀y (y ∈ x0 ⇔ / x0 . y∈ / y), we obtain the following contradiction: x0 ∈ x0 ⇔ x0 ∈

5 The term ‘class’ is sometimes used as a synonym for ‘set’. But these terms can also have a different

meaning: a class is a set if and only if it belongs to some class. A class that is not a set (i.e., it does not belong to any class) is called a proper class. 6 Later we will introduce auxiliary variables which will enable us to formulate some short-cuts. 7 These relate to Russell’s paradox. The first is essential for establishing that there is no set of all sets. 8 Note that since x is an object, the sentence ‘x ∈ x ’ has a truth-value. 0 0 0

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The First Three Axioms Axiom of extensionality. This axiom states that:   ∀x,y Set x ∧ Set y ∧ ∀z (z ∈ x ⇔ z ∈ y) =⇒ x = y .

(Aext)

Put informally, it says that if, using the variables ‘x’ and ‘y’ to signify sets, it turns out that x and y have the same elements, then we are in fact dealing with one and the same set. Obviously, this axiom does not concern individuals, as they have no elements. Remark A.2.1 In “pure” set theory, axiom (Aext) has the following, simpler form:   ∀x,y ∀z (z ∈ x ⇔ z ∈ y) =⇒ x = y . It is thereby evident that “pure” theory is exclusively concerned with sets, which squares with what was said above about the elements of sets only being sets. Otherwise, we would obtain the following result: all individuals (entities not possessing, that is, any members), would be equal to one another and furthermore equal to the so-called empty set which has no elements (see below). Therefore, if we are to avoid a contradiction, there cannot be any individuals. In “pure” set theory, we have we have the quasi-axiom ‘∀x Set x’, which is why the predicate ‘Set’ is superfluous.

Axiom of the empty set. Let us assume that there is a set that has no element:   ∃x Set x ∧ ¬∃ y y ∈ x .

(A∅)

From this axiom and (Aext) it follows that there is exactly one such set, i.e.:    ∃x Set x ∧ ¬∃ y y ∈ x ∧ ∀z (Set z ∧ ¬∃ y y ∈ z) ⇒ z = x) .

(A.2.3)

Thus, using the uniqueness quantifier, we obtain the following:   ∃1x Set x ∧ ¬∃ y y ∈ x .

(A.2.3 )

We can therefore introduce an individual constant to designate this set, which we will call the empty set:   ∅ := ( x) Set x ∧ ¬∃ y y ∈ x . ι

We therefore have: Set ∅,

(A.2.4)

∅∈ / ∅.

(A.2.5)

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Remark A.2.2 We needn’t have adopted axiom (A∅). It would have sufficed to adopt any axiom stating that there exists some set or other (see Fact A.2.3 and Remark A.2.14). Then the other axioms would guarantee the existence of the empty set. We have chosen axiom (A∅), because it allows us to formulate the theory in a simpler way.

Notice that from (A.2.3) (or (A.2.3 )) we obtain:   ∀x Set x =⇒ x = ∅ ∨ ∃ y y ∈ x .

(A.2.6)

We therefore obtain the following:   ∀x x = ∅ ∧ ¬∃ y y ∈ x ⇐⇒ ¬ Set x ∧ ¬∃ y y ∈ x . Moreover, notice that from (A.2.1), (A.2.4) and (A.2.5) we learn only that:   / y) =⇒ (¬ Set x ∧ ∅ ∈ x . ∀x ∀y (y ∈ x ⇔ Set y ∧ y ∈

(A.2.7)

Axiom of sets. In our theory we adopt the following axiom, which, as a rule, is not adopted in theories with the predicate ‘Set’:   ∀x ∃ y y ∈ x =⇒ Set x .

(ASet)

It says that if a given object has even one element, it is a set. In other words, only sets have elements. From (A.2.4) and (ASet) we obtain the converse implication to (A.2.6). So we get:   ∀x Set x ⇐⇒ x = ∅ ∨ ∃ y y ∈ x . This sentence would not, however, provide us with a definition of the predicate ‘Set’ because, to define the empty set ∅, we had to make use of it. We can prove the following sentence, which says that there is no object to which belong all and only those sets which are not their own elements (cf. (A.2.1)):   /y . ¬∃x ∀y y ∈ x ⇐⇒ Set y ∧ y ∈

(A.2.8)

/ y). Assume for a contradiction that for some x0 we have: ∀y (y ∈ x0 ⇔ Set y ∧ y ∈ Hence, by (A.2.7), we have: ¬ Set x0 and ∅ ∈ x0 . So (ASet) yields a contradiction: Set x0 . The Lack of (ASet) as an Axiom As we have already mentioned, the axiom (ASet), as a rule, is not adopted in theories with the predicate ‘Set’. The absence of (ASet) as an axiom in a theory with the predicate ‘Set’, is typically due to the fact that set theory is most commonly constructed as “pure” set theory, where (ASet) is unnecessary as an axiom. For even if

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we had the predicate ‘Set’ (itself unnecessary), we would also have the quasi-axiom ‘∀x Set x’ (see Remark A.2.1), which would make (ASet) true. If we do not accept the sentence (ASet) as an axiom, then a question remains of whether there are other objects that have elements. In other words, different interpretations of the scope of the universe of discourse are permissible. We may admit that in the universe there are also proper classes which are not sets but have elements (see footnote 5). Obviously, in that case, all sets would be classes. In that case, however, the whole theory should be changed, including the axiom (Aext). Moreover, notice that without (ASet) we cannot prove sentence (A.2.8). We know only that if such an object were to exist, it would not be a set (see (A.2.1)). Nor would it be an individual, because the empty set ∅ would belong to it. Therefore, it would be something other than a set and an individual. It could be just a proper class. The above remarks show that the axiom (ASet) should be accepted. Auxiliary Concepts In order to develop the theory in a more comfortable way, it will be helpful to introduce some auxiliary concepts definable within the theory.9 We will say that a set x is a subset of a set y, or that a set x is included in a set y, which we shall symbolise as ‘x ⊆ y’, iff every element of x is an element of y, or symbolically:   ∀x,y x ⊆ y :⇐⇒ Set x ∧ Set y ∧ ∀z (z ∈ x ⇒ z ∈ y) . The sentence ‘¬x ⊆ y’ shall be written as ‘x  y’. Obviously, every set is contained within itself (hence the symbol ‘⊆’), the empty set is a subset of all sets and the predicate ‘⊆’ is transitive:   ∀x Set x ⇒ x ⊆ x ,   ∀x Set x ⇒ ∅ ⊆ x , ∀x ∀y ∀z (x ⊆ y ∧ y ⊆ z =⇒x ⊆ z). Furthermore, from this and (Aext) we obtain:   ∀x,y Set x ∧ Set y =⇒ (x = y ⇐⇒ x ⊆ y ∧ y ⊆ x) . We say that a set x is a proper subset of a set y, which we shall symbolise as ‘x  y’, iff x is a subset of y but distinct from it:   ∀x,y x  y :⇐⇒ x ⊆ y ∧ x = y . Obviously, we have:

9 We

have met one already: the concept of the empty set is just such an auxiliary concept.

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∀x ∀y ∀z (x  y ∧ y ⊆ z =⇒ x  z), ∀x ∀y ∀z (x ⊆ y ∧ y  z =⇒ x  z), ∀x ∀y ∀z (x  y ∧ y  z =⇒ x  z). The next auxiliary concept is the concept of being a family of sets. This applies to a set all of whose elements are sets. That x is a family of sets will be written as ‘F x’:   ∀x F x :⇐⇒ Set x ∧ ∀y (y ∈ x ⇒ Set y) . We observe that the empty set ∅ is a family of sets, i.e., we have the thesis: F ∅. Remark A.2.3 In pure set theory, the predicate ‘F’ is also superfluous. Since it contains the thesis ‘∀x Set x’, we therefore also obtain the thesis ‘∀x F x’. For since every object is a set, then if a set has an element, the element too will be a set. Despite this, some authors do employ the concept of being a family of sets in pure set theory.

Remaining Axioms and Auxiliary Concepts Axiom of sum. This axiom refers to families of sets and states that:    ∀z F z =⇒ ∃x Set x ∧ ∀y (y ∈ x ⇔ ∃u (u ∈ z ∧ y ∈ u)) .

(Asum)

It says that for any family of sets there is a set to which belong all and only those objects which belong to some sets from the family. From (Asum) and (Aext) we obtain:    ∀z F z =⇒ ∃1x Set x ∧ ∀y (y ∈ x ⇔ ∃u (u ∈ z ∧ y ∈ u)) . We can therefore introduce a (conditional) term for this set:     ∀z F z =⇒ z := ( x) Set x ∧ ∀y (y ∈ x ⇔ ∃u (u ∈ z ∧ y ∈ u)) . ι

The name of the set depends on a given family and so, ultimately, what has been defined is a certain “operation” of the set-theoretic sum of sets belonging to the family. The axiom therefore states that, for an arbitrary family z there exists its set-theoretic  sum z. Remark A.2.4 (i) Axiom (Asum) is not necessary for the empty set (family) ∅, because ∅ satisfies the postulate: Set ∅ ∧ ∀y (y ∈ ∅ ⇔ ∃u (u ∈ ∅ ∧ y ∈ u)). Thus, in this particular case:  ∅ = ∅. (ii) Just the case ofthe empty family ∅ shows that condition ‘Set x’ is needed to define the symbol ‘ ’. Otherwise, if there existed in the universe an individual i (which had no elements), then for ∅ we would have: i = ∅ and ∀y (y ∈ i ⇔ ∃u (u ∈ ∅ ∧ y ∈ u)). Therefore, ∅ and i would also be sums of the empty family ∅.

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Simplifying expressions. With the example of axiom (Asum) we see that it is worth simplifying the way formulas are expressed. Let us therefore use auxiliary variables ‘X ’, ‘Y ’, ‘Z ’ and ‘S’ for sets; and ‘F ’, ‘X’, ‘Y’ and ‘S’ for families of sets. Let us assume that for an arbitrary formula ϕ(x) with at least one free variable ‘x’, the expression ϕ(X ) arises from a formula ϕ(x) through the replacement of each free occurrence of the variable ‘x’ with the letter ‘X ’. We will use the same convention for other variables. We accept that:   ∀X ϕ(X ) is short for ∀x Set x ⇒ ϕ(x) ,   ∃ X ϕ(X ) is short for ∃x Set x ∧ ϕ(x) ,   ∃1X ϕ(X ) is short for ∃1x Set x ∧ ϕ(x) . Analogously,   ∀F ϕ(F ) is short for ∀x F x ⇒ ϕ(x) ,   ∃F ϕ(F ) is short for ∃x F x ∧ ϕ(x) ,   ∃1F ϕ(F ) is short for ∃1x F x ∧ ϕ(x) . Similar abbreviations may be made with other variables (primary and auxiliary). In this way, we find ourselves with a language that appears to have three types of variable. They do not, however, relate to three pairwise distinct universes of discourse. Variables of the first sort (‘x’, ‘y’, ‘z’ etc.) range over all objects (individuals and sets, including families). Variables of the second (‘X ’, ‘Y ’, ‘Z ’ etc.) only range over sets (including families of sets). Variables of the third type (‘F ’, ‘X’ etc.) only range over families of sets. Under this convention, (Aext), (A∅), the definitions of predicates ‘⊆’ and ‘’, (A.2.6) and some theses we have the following shortened forms:   ∀X,Y ∀z (z ∈ X ⇔ z ∈ Y ) =⇒ X = Y ,  ∃ X ¬∃ y y ∈ X,  ∀ X = ∅ ∨ ∃y y ∈ X , X   ∀X,Y X ⊆ Y ⇐⇒ ∀z (z ∈ X ⇒ z ∈ Y ) , ∀X,Y (X = Y ⇐⇒ X ⊆ Y ∧ Y ⊆ X ), ∀X X ⊆ X, ∀X ∅ ⊆ X, ∀X ∀Y ∀Z (X ⊆ Y ∧ Y ⊆ Z =⇒ X ⊆ Z), ∀X,Y X  Y ⇐⇒ X ⊆ Y ∧ X = Y , ∀X ∀Y ∀Z (X  Y ∧ Y ⊆ Z =⇒ X  Z ), ∀X ∀Y ∀Z (X ⊆ Y ∧ Y  Z =⇒ X  Z ), ∀X ∀Y ∀Z (X  Y ∧ Y  Z =⇒ X  Z ). Furthermore, for an arbitrary formula ϕ(x) with at least one free variable ‘x’, we accept that:

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  ∀x∈X ϕ(x) is short for ∀x x ∈ X ⇒ ϕ(x) ,   ∃x∈X ϕ(x) is short for ∃x x ∈ X ∧ ϕ(x) ,   ∃1x∈X ϕ(x) is short for ∃1x x ∈ X ∧ ϕ(x) . Analogously,   ∀X ∈F ϕ(X ) is short for ∀X X ∈ F ⇒ ϕ(X ) ,   ∃ X ∈F ϕ(X ) is short for ∃ X X ∈ F ∧ ϕ(X ) ,   ∃1X ∈F ϕ(X ) is short for ∃1X X ∈ F ∧ ϕ(X ) . Similar abbreviations may be made with other variables (primary and auxiliary). With these conventions and the definition of ‘F’, (Asum) has the following shortened form:   ∀F ∃ X ∀y y ∈ X ⇐⇒ ∃ Z ∈F y ∈ Z . From (Aext) therefore we have:   ∀F ∃1X ∀y y ∈ X ⇐⇒ ∃ Z ∈F y ∈ Z . Hence for an arbitrary family F we may introduce the following shortened form:  F := ( X ) ∀y (y ∈ X ⇔ ∃ Z ∈F y ∈ Z ) . ι



We can also define this set in another way:   {y : ∃ Z ∈F y ∈ Z )} := ( X ) ∀y (y ∈ X ⇔ ∃ Z ∈F y ∈ Z ) =: F . ι

So we have:

   ∀y y ∈ F ⇐⇒ ∃ S∈F y ∈ S .

(A.2.9)

In the next point, when changing variables into auxiliary variables (using our abbreviations), we will use variables with indexes (‘x  ’, ‘y  ’, ‘z  ’ etc.) in order not to “block” the use of variables without them. Non-ordered pairs and singletons. We now introduce a sentence that we may call the axiom of pairs:   ∀x,y ∃ S ∀z z ∈ S ⇐⇒ z = x ∨ z = y .

(Apair)

Remark A.2.5 Just as for axiom (A∅), the other axioms guarantee that axiom (Apair) holds (see Fact A.2.4). We have chosen axiom (Apair), because it allows us to formulate the theory in a simpler way.

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From axiom (Apair) we obtain a sentence addressing the existence of so-called singletons:   ∀x ∃ S ∀z z ∈ S ⇐⇒ z = x . From this, (Apair) and (Aext) we have:   ∀x,y ∃1S ∀z z ∈ S ⇐⇒ z = x ∨ z = y ,   ∀x ∃1S ∀z z ∈ S ⇐⇒ z = x . We may therefore designate these sets as follows:   {x, y} := ( S) ∀z z ∈ S ⇔ z = x ∨ z = y ,   {x} := ( S) ∀z z ∈ S ⇔ z = x . ι ι

The set {x, y} is called the non-ordered pair of x and y and the set {x} is called the singleton of x. We have {x} = {x, x}. Obviously, x and y are the only elements of the set {x, y} and x is the only element of the set {x}, i.e.:   ∀z z ∈ {x, y} ⇐⇒ z = x ∨ z = y ,   ∀z z ∈ {x} ⇐⇒ z = x . Since the non-exclusive alternative is commutative and idempotent, we have: {x, y} = {y, x}. We obtain:   ∀x,y,u,z {x, y} = {z, u} ⇐⇒ (x = z ∧ y = u) ∨ (x = u ∧ y = z) ,   ∀x,y {x} = {y} ⇐⇒ x = y . Notice that the following are also singletons: {{x}}, {{{x}}} etc. Non-ordered triples; finite sets. Given non-ordered pairs and singletons, we can obtain a non-ordered triple. For arbitrary objects x, y and z we have the non-ordered pair {x, y} and the singleton {z}. By virtue of (Apair), we obtain the non-ordered pair {{x, y}, {z}}, which is a family of sets. Hence, by virtue of axiom (Asum) applied to this family and with appropriate logical transformations made, we obtain:    {{x, y}, {z}} = ( S) ∀u u ∈ S ⇔ ∃ Z ∈{{x,y},{z}} u ∈ Z ,   = ( S) ∀u u ∈ S ⇔ u ∈ {x, y} ∨ u ∈ {z} ,   = ( S) ∀z u ∈ S ⇔ u = x ∨ u = y ∨ u = z , ι ι ι

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and 

{{x, y}, {z}} = {u : ∃ Z ∈{{x,y},{z}} u ∈ Z }, = {u : u ∈ {x, y} ∨ u ∈ {z}}, = {u : u = x ∨ u = y ∨ u = z}.

Hence we have:



{{x, y}, {z}} =

 {{x}, {y, z}}.

Therefore, we can put: {x, y, z} := Thus,

  {{x, y}, {z}} = {{x}, {y, z}}.

  ∀u u ∈ {x, y, z} ⇐⇒ u = x ∨ u = y ∨ u = z .

In general, we can create a finite set in an inductive fashion for arbitrary objects x1 , …, xn (n > 2): {x1 , . . . , xn } :=

 {{x1 , . . . , xn−1 }, {xn+1 }}.

whose elements are all the objects and no others, i.e.:   ∀u u ∈ {x1 , . . . , xn } ⇐⇒ u = x1 ∨ · · · ∨ u = xn . We therefore have the following general principle: the collection of a finite number of individual names in curly brackets and separated by commas creates the individual name of a set whose elements are all and only those objects whose names are used. Thus, the order of the names does not matter, and neither does it matter whether a name is used several times nor whether a given object hides behind different individual names. By way of example, consider {2, 4, 2} = {2, 2 + 2, 4} = {2, 4} = {4, 2}. We have four names here for the same two-element set. For in these four names of sets, only two objects are referred to by the names within. The numbers 2 and 4 are the only elements of the set, regardless of how its name is created. The sum of elements of sets. For arbitrary sets X , …, X n (n > 0), we obtain the finite set {X 1 , . . . , X n }, which is a family of sets. Hence, by virtue of axiom (Asum) applied to this family and with appropriate logical transformations made, we obtain:   {X, . . . , X n } = ( S) ∀z z ∈ S ⇔ ∃ Z ∈{X 1 ,...,X n } z ∈ Z ,   = ( S) ∀z z ∈ S ⇔ z ∈ X 1 ∨ · · · ∨ z ∈ X n , ι ι



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and:  {X, . . . , X n } = {z : ∃ Z ∈{X 1 ,...,X n } z ∈ Z }, = {z : z ∈ X 1 ∨ · · · ∨ z ∈ X n }. In particular case, for arbitrary sets X and Y , for the non-ordered pair {X, Y }, which is a family of sets, we obtain:   {X, Y } = ( S) ∀z z ∈ S ⇔ ∃ Z ∈{X,Y } z ∈ Z ,   = ( S) ∀z z ∈ S ⇔ z ∈ X ∨ z ∈ Y , ι



ι

and:  {X, Y } = {z : ∃ Z ∈{X,Y } z ∈ Z }, = {z : z ∈ X ∨ z ∈ Y }.  This particular case the using of the operator ‘ ’ is customarily denoted as the sum of the elements of two sets, or the sum of two sets and is expressed using the following symbol:  X  Y := {X, Y } = {z : z ∈ X ∨ z ∈ Y },   ∀z z ∈ X  Y ⇐⇒ z ∈ X ∨ z ∈ Y . Obviously, we have an “operation” which is commutative, idempotent and associative, and which has the following relationships with ‘⊆’: X Y = Y  X, X  X = {X } = X, (X  Y )  Z =  {X  Y, Z } = {X, Y, Z }, X  (Y  Z ) = {X, Y  Z } = {X, Y, Z }, (X  Y )  Z = X  (Y  Z ), X ⊆ X  Y and Y ⊆ X  Y, X  Y ⊆ Z ⇐⇒ X ⊆ Z ∧ Y ⊆ Z , X  Y ⊆ Y ⇐⇒ X ⊆ Y ⇐⇒ X  Y = Y. In general, for arbitrary sets X 1 , …, X n (n > 0) we get: X1  · · ·  Xn =

 {X 1 , . . . , X n }.

Furthermore, notice that for arbitrary objects x, y and y we have: {x, y, z} = {x, y}  {z} = {x}  {y, z} = {x}  {y}  {z}.

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In general, for arbitrary objects x1 , …, xn (n > 1), we have: {x1 , . . . , xn } = {x1 , . . . , xn−1 }  {xn } = {x1 }  · · ·  {xn }. The power set axiom. This axiom says that, for an arbitrary set, there exists a set to which belong all and all and only subsets of the first set. It may be put in the following abbreviated form:   ∀X ∃Y ∀x x ∈ Y ⇐⇒ x ⊆ X .

(Apow)

By virtue of (Aext) there exists only one such set:   ∀X ∃1Y ∀x x ∈ Y ⇐⇒ x ⊆ X .

(A.2.10)

From the definition of the predicate ‘⊆’ it follows that the set Y postulated by axiom (Apow) is a family of sets. We have:   ∀X ∀Y ∀y (∀x x ∈ Y ⇒ x ⊆ X ) ∧ y ∈ Y =⇒ Set y ,   ∀X ∀Y (∀x x ∈ Y ⇒ x ⊆ X ) =⇒ F Y . This family (set) is called the power set. Obviously, we can write (A.2.10) as:   ∀X ∃1F ∀S S ∈ F ⇐⇒ S ⊆ X . We may therefore designate this family in the following way:   2 X := ( F ) ∀S S ∈ F ⇔ S ⊆ X ,   ∀z x ∈ 2 X ⇔ z ⊆ X ,   ∀S S ∈ 2 X ⇔ S ⊆ X . ι

Remark A.2.6 The name of the family is dependent on the choice of the set X , and therefore what has been defined is a certain “operation”. The symbolic form of the name reflects the meaning of “the power set of X ”: for if X has n elements, then the

family (set) 2 X has 2n elements. Fact A.2.1 (i) The family 2∅ has exactly one element: ∅, i.e., 2∅ = {∅}. (ii) For any x, 2{x} = {∅, {x}}. (iii) For any x and y, 2{x,y} = {∅, {x}, {y}, {x, y}}. (iv) For any x1 , …, xn , 2{x1 ,...,xn } = {∅, {x1 }, . . . {xn }, {x1 , x2 }, . . . , {x1 , xn }, {x2 , x3 }, . . . , {x2 , xn }, {x1 , x2 , x3 }, . . . {x1 , . . . , xn }}. If each pair of members in {x1 , . . . , xn } is different, then 2{x1 ,...,xn } has exactly n elements.

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∅

In consequence, 22 = {∅, 2∅ } = {∅, {∅}}, 22 = 2{∅,{∅}} = {∅, {∅}, 2∅ n−1 {{∅}}, {∅, {∅}}} and for each n > 2: the family 2... elements  has exactly 2 2∅

among which are:: ∅, 2∅ , …, 2 ...  .

nב2’

(n−1)ב2’

Proof (i) The only set included in ∅ is ∅. (ii) The elements of the family 2{x} are ∅ and {x}, since ∅ ⊆ {x} and {x} ⊆ {x}. Furthermore, if a set S is an element of the family 2{x} then S ⊆ {x}, i.e., for each z: if z ∈ S then z = x. Therefore, either S = ∅ or S = {x}. (iii) The elements of the family 2{x,y} are ∅, {x}, {y} and {x, y}, since ∅ ⊆ {x}, {x} ⊆ {x, y}, {y} ⊆ {x, y} and {x, y} ⊆ {x, y}. Furthermore, if a set S is an element of the family 2{x,y} then S ⊆ {x, y}, i.e., for each z: if z ∈ S then either z = x or z = y. Therefore, either S = ∅, or S = {x}, S = {y} or S = {x, y}. (iv) Proof by induction. ∅ ∅ For 22 we put x := ∅ and use (i) and (ii). Then 22 = 2{∅} = {∅, {∅}}. 2∅ 2∅ For 22 we put x := ∅, y := {∅} and use (i)–(iii). Then 22 = 2{∅,{∅}} = {∅, {∅}, {{∅}}, {∅, {∅}}}. ∅ 2∅ ...2 , by (i)–(iv) and x := ∅, x := {∅}, …, x := 2

If n > 2, for 2... 1 2 n 

  . nב2’

(n−1)ב2’

Remark A.2.7 We have shown above that, in the set-theoretic universe of discourse, there exists an infinite number of power sets. Each of them is, however, a finite set. This first statement is not a theorem of set theory itself, however, but of its metatheory. We have not established the existence of an infinite set. The existence of such a set is established axiomatically.

Axiom of infinity. The axioms so far considered do not establish the existence of a set which has an infinite number of elements. The so-called axiom of infinity establishes precisely the existence of such a set (family):   ∃F ∅ ∈ F ∧ ∀X (X ∈ F ⇒ X  {X } ∈ F ) .

(A∞)

To any family of sets postulated by (A∞) must belong: ∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}, {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}}, …. Since all these elements are pairwise distinct, then axiom (A∞) postulates the existence of a set with an infinite number of elements. The listed elements of this set are pairwise distinct, because each of them has a different number of elements. They have, in turn, the following number of elements: 0, 1, 2, 3, 4, …. More precisely, the element in the nth place has n − 1 elements, of which each has, in turn, 0, 1, …, n − 2 elements. By induction: for n = 1 and n = 2, it is obvious. Assume that it is true that for the nth element X n . We show that then it also holds for the n + 1th element whose form is X n  {X n }. / X n , as X n has n − 1 elements and none By virtue of the inductive assumption, X n ∈

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of its elements has n − 1 elements.10 Therefore, the set X n  {X n } has n elements and each of these element has, in turn, 0, 1, …, n − 2 and n − 1 elements.11 Axiom of regularity. For convenience (and to “respect intuition”), we will also adopt the axiom of regularity. It suffices to adopt it only for families of sets:    ∀F F = ∅ =⇒ ∃ X X ∈ F ∧ ¬∃x (x ∈ X ∧ x ∈ F ) .

(Areg)

Therefore, to each non-empty family of sets belongs some set having no common element with it; i.e., is disjoint with it. From (Areg) and (Apair) the sentences below follow:   ¬∃ X,Y X ∈ Y ∧ Y ∈ X , ∀X X ∈ / X. Assume for a contradiction that for some sets X and Y : X ∈ Y and Y ∈ X . Then take the family {X, Y }, which does not satisfy (Areg), because Y ∈ X , X ∈ {X, Y }, X ∈ Y and X ∈ {X, Y }. Analogously, it can be shown than the concept being an element of is acyclic: ¬∃ X 1 ,X 2 ...,X n (X 1 ∈ X 2 ∧ · · · ∧ X n−1 ∈ X n ∧ X n ∈ X 1 ), i.e., we have a sequence of sentences falling under the scheme above, for n > 1. Finally, notice that: there is no family of sets which is composed of an infinite number of sets X 1 , X 2 , X 3 , … and no others such that: . . . ∈ X n+1 ∈ X n ∈ · · · ∈ X 2 ∈ X 1 . If such a family existed then for each n we would have X n+1 ∈ X n and X n+1 would belong to it. That is, every member of this family would have a common element with it, which contradicts (Areg). Axiom schema of replacement. Let ψ(x, y) be an arbitrary formula with at least two free variables ‘x’ and ‘y’. We will say that ψ(x, y) is univalent (or functional) iff satisfies the following condition: ∀x,y,u (ψ(x, y) ∧ ψ(x, y/u) ⇒ y = u),

10 Observe

that we are not here making use of the axiom of regularity (Areg) introduced below, from which it follows that ∀X X ∈ / X . If we were to do so, then, in the case under consideration, the theory with axiom (Areg) would be contradictory. 11 We will not be concerning ourselves with the issue of whether there is a set which has all the elements considered above and no others.

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where ψ(x, y/u) is the formula which arises from ψ(x, y) by the replacement of each free occurrence of ‘y’ by ‘u’.12 We then adopt the following axiom for an arbitrary univalent formula ψ(u, v) in which the variable ‘y  ’ does not occur as a free variable13 (in the following scheme written in a shortened form, the variables ‘X ’ and ‘Y ’ will correspond to the variables ‘x  ’ and ‘y  ’, respectively):   ∀X ∃Y ∀y y ∈ Y ⇐⇒ ∃x∈X ψ(x, y) .

(Arepψ )

Axiom (Arepψ ) therefore says that, with the help of a univalent formula ψ(x, y) satisfying our assumption about the variable ‘y  ’ (i.e. ‘Y ’), a set may be obtained from an arbitrary set. Obviously, we are dealing here not with a single axiom but with a set of axioms, for each univalent formula ψ(x, y). At the same time, the whole set falls under one schema. Remark A.2.8 We will show belong that the requirement that ‘y  ’ (i.e. ‘Y ’) does not occur as a free variable in the formula ψ(x, y) is essential. Otherwise, a contradiction / Y ’ for the results. To see this, let ψ(x, y) be ‘x = y ∧ y ∈ / y  ’; and so ‘x = y ∧ y ∈ shortened form of (Arepψ ). Then ψ(x, y) is univalent and we get for X := {∅}:   / Y ) , ∃Y ∀y y ∈ Y ⇐⇒ ∃x∈{∅} (x = y ∧ y ∈ ∀y y ∈ Y0 ⇐⇒ y ∈ {∅} ∧ y ∈ / Y0 , / Y0 , ∅ ∈ Y0 ⇐⇒ ∅ ∈ {∅} ∧ ∅ ∈ / Y0 . ∅ ∈ Y0 ⇐⇒ ∅ ∈

Thus, we obtain a contradiction. From (Aext) and (Arepψ ) for any univalent formula ψ(x, y) we obtain:   ∀X ∃1Y ∀y y ∈ Y ⇐⇒ ∃x∈X ψ(x, y) .

For an arbitrary univalent formula ψ(x, y) and an arbitrary set X , the set in question may be designated thus:   {y : ∃x∈X ψ(x, y)} := ( Y ) ∀y y ∈ Y ⇔ ∃x∈X ψ(x, y) ,   ∀y y ∈ {y : ∃x∈X ψ(x, y)} ⇐⇒ ∃x∈X ψ(x, y) . ι

We therefore see that, in this way, a set X can be used to “produce” sets whose elements are functionally assigned to the elements of X , but it may be that only to some elements of X . 12 We

are obviously assuming here that the variable ‘u’ can replace every free occurrence of the variable ‘y’ in ‘ψ(x, y)’. If the variable ‘u’ does not meet this condition, we choose another variable which does. 13 Bound occurrences of the variable can be changed into bound occurrences of another variable. We may therefore establish that the variable ‘y  ’ will not appear in ψ(x, y) at all.

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We have a number of interesting cases of an univalent formula ψ(x, y). One of them is ψ(x, y) := y = τ (x), where τ (x) is an arbitrary term with at least one variable ‘x’, but without the variable ‘y  ’ (i.e. ‘Y ’). Then the only set postulated by (Arepψ ) can also be written as {τ (x) : x ∈ X } or more simply as τ [X ]. Therefore: τ [X ] := {τ (x) : x ∈ X } := {y : ∃x∈X y = τ (x)},   ∀y y ∈ τ [X ] ⇐⇒ ∃x∈X y = τ (x) .

(A.2.11)

This expression clearly brings out how the postulated set is supposed to be the image of the set X once “transformed” by τ (see Sect. A.2.4). Other applications of the axiom schema of replacement will be introduced below. Remark A.2.9 The axiom schema of replacement is also adopted in other forms. Among them is the following (with our assumptions about the variables ‘x  ’ and ‘y  ’, i.e. ‘X ’ and ‘Y ’):   ∀x ∃1y ψ(x, y) =⇒ ∀X ∃Y ∀y y ∈ Y ⇔ ∃x (x ∈ X ∧ ψ(x, y)) . Axiom schema of subsets. It is obvious that, for an arbitrary formula χ , the formula x = y ∧ χ  is univalent. Thanks to that, from the axiom schema of replacement (Arepψ ) we obtain the so-called axiom schema of subsets, also known as the axiom schema of specification, axiom schema of separation or subset axiom scheme: Fact A.2.2 Let ϕ(y) be an arbitrary formula with at least one free variable ‘y’ in which the variable ‘y  ’ does not appear. Then we obtain the following thesis (in the following schema, written in a shortened form, the variables ‘X ’ and ‘Y ’ correspond to the variables ‘x  ’ and ‘y  ’, respectively):   ∀X ∃Y ∀y y ∈ Y ⇐⇒ y ∈ X ∧ ϕ(y) .

(Assϕ )

Proof Let ϕ(y) satisfy the assumption. We observe that the formula ψ(x, y) := x = y ∧ ϕ(y) is univalent. Furthermore, ψ(x, y) satisfies the conditions for the application of axiom (Arepψ ). Therefore we have:   ∀X ∃Y ∀y y ∈ Y ⇐⇒ ∃x∈X (x = y ∧ ϕ(y)) ,   ∀X ∃Y ∀y y ∈ Y ⇐⇒ y ∈ X ∧ ϕ(y) .

Sentence (Assϕ ) guarantees the existence of a set to which belong all and only those objects which are both elements of a set X and which satisfy the formula ϕ(y). We thereby distinguish the subsets of X . From (Assϕ ) and (Aext) we obtain:   ∀X ∃1Y ∀y y ∈ Y ⇐⇒ y ∈ X ∧ ϕ(y) . We may therefore designate this subset in the following way:

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  {y ∈ X : ϕ(y)} := ( Y ) ∀y y ∈ Y ⇔ y ∈ X ∧ ϕ(y) ,   ∀y y ∈ {y ∈ X : ϕ(y)} ⇐⇒ y ∈ X ∧ ϕ(y) . ι

Let us observe that, with the formula ψ(x, y) used in the proof of Fact A.2.2, the designation above agrees with the one which we introduced by applying axiom (Arepψ ): {y : ∃x∈X ψ(x, y)} = {y : ∃x∈X (x = y ∧ ϕ(y))} = {y : y ∈ X ∧ ϕ(y)} = {y ∈ X : ϕ(y)}. The name above denotes a set dependent on both on the set X and the formula ϕ(y). The set thus named is obviously a subset of X . We therefore see that we can “generate” only subsets of X in this way. Remark A.2.10 In Fact A.2.2 the requirement that ‘y  ’ (i.e. ‘Y ’) does not occur in free positions in ϕ(y) is essential. Without it, we could not apply this fact. On the other hand, we cannot obtain that, for an arbitrary formula ϕ(y), we have (Assϕ ) as a thesis. Otherwise, we could prove that we could obtain a contradictory theory. / Y ’ for the shortened form of For example, if we put ϕ(y) := ‘y ∈ / y  ’ (and so ‘y ∈ (Assϕ )) and X := {∅}, then we get (cf. Remark A.2.8):   / Y0 , ∀y y ∈ Y0 ⇐⇒ y ∈ {∅} ∧ y ∈ ∅ ∈ Y0 ⇐⇒ ∅ ∈ {∅} ∧ ∅ ∈ / Y0 , ∅ ∈ Y0 ⇐⇒ ∅ ∈ / Y0 . Thus, we obtain a contradiction.

Remark A.2.11 Scheme (Assϕ ) has the form adopted by Zermelo. Other equivalent forms are often encountered. One of them is as follows:    ∀X ∀y (ϕ(y) ⇒ y ∈ X ) =⇒ ∃Y ∀y y ∈ Y ⇔ ϕ(y) , where ‘y  ’ (i.e. ‘Y ’) does not occur in free positions in ϕ(y). Firstly, for an arbitrary formula ϕ(y) satisfying  our assumption and for an arbitrary set X , by (Assϕ ), for some set Y0 we have: ∀y y ∈ Y0 ⇔ y ∈ X ∧ ϕ(y) . Hence, if X satisfies the condition ∀y (ϕ(y) ⇒ y ∈ X ) then we also obtain ∀y y ∈ Y0 ⇔  ϕ(y) . So we obtain the new version for ϕ(y). Secondly, for an arbitrary formula ϕ(y) satisfying our assumption and using the new version of the formula y ∈ X ∧ ϕ(y), we get:    ∀X ∀y (y ∈ X ∧ ϕ(y) ⇒ y ∈ X ) =⇒ ∃Y ∀y y ∈ Y ⇔ y ∈ X ∧ ϕ(y) . Since the antecedent is logically true, we obtain (Assϕ ).

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Remark A.2.12 If we choose the scheme from Remark A.2.9 instead of scheme (Arepψ ), we will not get Fact A.2.2; i.e., scheme (Assϕ ) is not implied by this first

scheme. In this case we must also accept the axiom schema (Assϕ ). Remark A.2.13 In applications of set theory, we find ourselves with a universe (set) U , in which we may distinguish a subset of all and only those elements which are S’s. We therefore have to deal with the formula ‘ϕ(x) := ‘x is an S’ determining the set ESU := {x ∈ U : x is ab S}, which we may call the set of S’s (belonging to U ). This is the range or extension of the concept S in the universe U . The following holds:   ∀x x ∈ ESU ⇐⇒ x ∈ U ∧ x is an S . If the concepts S and P assign the same elements to the set U , then ESU = EPU and we say that these concepts are co-extensional. For an arbitrary x, x ∈ EUS iff x ∈ U and x is an S iff x ∈ U and x is a P iff x ∈ EPU . We are therefore using (Aext). Conversely, if ESU = EPU then the concepts S and P assign the same elements from U (now, however, we are not using (Aext) but only the logic of identity.)

The dependence of two axioms. As mentioned, two of the axioms that have been adopted so far are dependent on the others. Firstly, notice that: Fact A.2.3 The axiom of the empty set (A∅) is implied by the axiom of infinity and the axiom schema of subsets. Proof If the axiom of infinity (A∞) is written as14 :   / X ∧ ∀X ∈F ∃Y ∈F ∀y (y ∈ Y ⇔ y ∈ X ∨ y = X ) , ∃F ∃ X ∈F ∀y y ∈ then from this and (Assϕ ) we obtain (A∅). Indeed, let F∞ be the family postulated by the above version of the axiom of infinity. Furthermore, in (Assϕ ) as ϕ(y) we take ‘y = y’. Then we obtain:   ∃Y ∀y y ∈ Y ⇐⇒ y ∈ F∞ ∧ y = y . Hence, for some set Y0 we have:

/ Y0 . ∀y y ∈

Hence, we obtain the axiom of empty set: ∃ X ¬∃ y y ∈ X.

14 Using

the axioms of pair and sum we would write it as:  ∃F ∃ X ∈F ∀y y ∈ / X ∧ ∀X ∈F X  {X } ∈ F .

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Remark A.2.14 (i) In the above proof we could take ‘y ∈ / F∞ ’ instead of ‘y = y’. (ii) In the above proof, it was not necessary for us to get the family F∞ . We may adopt any axiom asserting the existence of some set X . Then we may obtain the empty set with the help of (Assϕ ) used with a set X and ϕ(y) := ‘y = y’ or ϕ(y) := ‘y ∈ / X ’. Indeed, by (Aext) and (Assϕ ), for all sets X and Y we have: {y ∈ X : y = y} = {y ∈ Y : y = y} and {y ∈ X : y ∈ / X } = {y ∈ Y : y ∈ / Y }.

Secondly we have: Fact A.2.4 The axiom of pairs (Apair) is implied by the power set axiom, the axiom of empty set and the axiom schema of replacement. In consequence, (Apair) is dependent on the power set axiom, the axiom of infinity and the axiom of empty set and the axiom schema of replacement. Proof Firstly, from the axiom schema of replacement we obtain the axiom schema of subsets (see Fact A.2.2). From the axiom schema of subsets and the axiom of infinity we obtain the axiom of empty set (A∅) (see Fact A.2.3). ∅ Secondly, by the power set axiom (Apow), we have the set 22 . We use (Arepψ ) for the univalent formula ψ(x, y) := ‘(x = ∅ ∧ y = u) ∨ (x = 2∅ ∧ y = v)’ and ∅ the set X := 22 :    ∃Y ∀y y ∈ Y ⇐⇒ ∃x∈22∅ (x = ∅ ∧ y = u) ∨ (x = 2∅ ∧ y = v) . So we have:  ∃Y ∀y y ∈Y0 ⇐⇒   ∃x (x = ∅ ∨ x = 2∅ ) ∧ ((x = ∅ ∧ y = u) ∨ (x = 2∅ ∧ y = v)) . This in turn is logically equivalent to:   ∃Y ∀y y ∈ Y ⇐⇒ y = u ∨ y = v . Hence we obtain:

  ∀u ∀v ∃Y ∀y y ∈ Y ⇐⇒ y = u ∨ y = v .

So, it suffices to change the bound variables.

Some applications of the axiom schema of subsets. Firstly, notice that with reference to (A.2.1), (A.2.2), (A.2.8) and via (Assϕ ), we obtain the following theses that say, respectively, that there does not exist a set of all sets and there does not exist a set to which everything belongs:   ¬∃ X ∀y y ∈ X ⇐⇒ Set y ,

(A.2.12)

¬∃ X ∀y y ∈ X.

(A.2.13)

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209

 For (A.2.12): Assume for a contradiction, that for some set X 0 we have: ∀y y ∈ X 0 ⇔  / y}. So, Set y . But, by (Assϕ ), we would obtain: ∃Y ∀y (y ∈ Y ⇔ y ∈ X 0 ∧ y ∈ / y}, which contradicts (A.2.1). For (A.2.13): we have: ∃Y ∀y (y ∈ Y ⇔ Set y ∧ y ∈ Assume for a contradiction that for some set X 1 we have: ∀y y ∈ X 1 . But, by (Assϕ ), we would obtain: ∃Y ∀y (y ∈ Y ⇔ y ∈ X 1 ∧ Set y}. So we have: ∃Y ∀y (y ∈ Y ⇔ Set y}, which contradicts (A.2.12). Obviously, in the light of the axiom of regularity (Areg), sentences (A.2.1) and (A.2.12) are equivalent. Secondly, we can introduce the binary operation of the product of two sets. For arbitrary sets X and S1 , …, Sn (n > 0), one of the subsets of X “produced” with the help of (Assϕ ) for ϕ(y) := ‘y ∈ S1 ∧ · · · ∧ y ∈ Sn ’ is the set: {y ∈ X : y ∈ S1 ∧ · · · ∧ y ∈ Sn }. Thus, for any set S we obtain the so-called the product of X with S: 

X  S := {y ∈ X : y ∈ S},

 ∀y x ∈ X  S ⇐⇒ y ∈ X ∧ y ∈ S . Here, the formula ϕ(y) is simply ‘y ∈ S’. We do not exclude the possibility that S = X . In this case we obtain that the “operation” ‘’ is idempotent, X  X := {y ∈ X : y ∈ X } = X. Obviously, if the sets X and S are disjoint (i.e., they have no common element), then, by virtue of (A∅) and (Aext), the set thus “produced” is the empty set: X  S = ∅ ⇐⇒ ¬∃ y (y ∈ X ∧ y ∈ S). By exchanging the variables ‘X ’ and ‘S’ we get: 

S  X := {y ∈ S : y ∈ X },

 ∀y x ∈ S  X ⇐⇒ y ∈ S ∧ y ∈ X . By virtue of (Aext) we therefore obtain that the “operation” ‘’ is commutative: X  S = S  X. Now notice that for all sets X , Y and Z we have: (X  Y )  Z = {y ∈ X  Y : y ∈ Z },  ∀y y ∈ (X  Y )  Z ⇐⇒ y ∈ X ∧ y ∈ Y ∧ y ∈ Z , X  (Y  Z ) = {y ∈ X : y ∈ Y  Z } = {y ∈ X : y ∈ Y ∧ y ∈ Z }, ∀y X  (Y  Z ) ⇐⇒ y ∈ X ∧ y ∈ Y ∧ y ∈ Z . 

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So, by (Aext), the “operation” ‘’ is associative: (X  Y )  Z = X  (Y  Z ). Obviously, the “operation” ‘’ has the following relationships with ‘⊆’: X  Y ⊆ X and X  Y ⊆ Y, X ⊆ X  Y ⇐⇒ X ⊆ Y ⇐⇒ X = X  Y. Finally, directly from our definitions and (Aext), we get that the “operations” ‘’ and ‘’ are distributive over each other so that for any set X , Y and Y the following equations hold: X  (Y  Z ) = (X  Y )  (X  Z ), X  (Y  Z ) = (X  Y )  (X  Z ). Thirdly, another set “produced” in this way is the subset of the set X which is the difference between this set and an arbitrary set S (or the relative complement of S in this set): / S},  X \ S := {y ∈ X : v ∈  /S . ∀y y ∈ X \ S ⇐⇒ y ∈ X ∧ y ∈ Here, the formula ϕ(y) is simply ‘y ∈ / S’. We do not exclude the possibility that S = X . In this case, after the application of (Aext), we have: X \ X := {y ∈ X : y ∈ / X } = ∅. Directly from our definitions and (Aext) we have: X \ S ⊆ X, X ⊆ X \ S ⇐⇒ X  S = ∅ ⇐⇒ X = X \ S. Product of a family of sets. Having (Asum) and (Assϕ ), we may introduce the concept of a product for any non- empty families of sets. This will be a generalisation of the product of a finite number of sets. LetF be an arbitrary non- empty family of sets. By virtue  (Asum), there is  a set F , where F = ∅. We use (Assϕ ) with the set F and the formula ϕ(y) := ‘∀S∈F y ∈ S’, putting: 

F := {y ∈



F : ∀S∈F y ∈ S}.

Since F = ∅, we obtain:    ∀y y ∈ F ⇐⇒ ∀S∈F y ∈ S .

(A.2.14)

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To see this, by (A.2.9), for any y we have: y∈



 F ⇐⇒ y ∈ F ∧ ∀S∈F y ∈ S ⇐⇒ ∃ S∈F y ∈ S ∧ ∀S∈F y ∈ S ⇐⇒ ∀S∈F y ∈ S

Directly from our definitions we obtain:    ∀X X ∈ F =⇒ F ⊆ X .

(A.2.15)

 We get that the “operation” ‘ ’ is a generalisation of the product of a finite number of sets, i.e., for all sets X 1 , …, X n (n > 0) we have:  {X 1 , . . . , X n } = X 1  · · ·  X n .

(A.2.16)

Axiom of choice. The axioms introduced so far either indicate the existence of a concrete set or give us a “recipe” for how to obtain one set from another. The axiom of choice is non-constructive. It says that, for each family of sets there exists a set which has exactly one common element from each non-empty set belonging to that family. However, no method is given for the construction of this set.   ∀F ∃ X ∀S∈F S = ∅ =⇒ ∃1x x ∈ X  S .

(AC)

We may also adopt the version that refers only to families of non-empty sets:   ∀F ∀S∈F S = ∅ =⇒ ∃ X ∀S∈F ∃1x x ∈ X  S .

(AC )

Obviously, both versions are equivalent, because the second follows logically from the first and we can obtain the first from the second by taking F \ {∅}. In both versions, if F := ∅, then the postulated set is simply ∅. Furthermore, in both versions of the axioms, it does not follow that the postulated set is composed only of those common elements with sets from the family F . We may, however, use axiom (Assϕ ) with the formula ϕ(y) = ‘∃ S∈F y ∈ S’ and an arbitrarily chosen set X F postulated in (AC) or (AC ) for the family F . We will then obtain the set15 : {y ∈ X F : ∃ S∈F y ∈ S}. The existence of such a set is postulated by the following versions of the axiom of choice (from which the earlier versions follow respectively):     ∀F ∃ X ∀S∈F S = ∅ =⇒ ∃1x x ∈ X  S ∧ ∀x∈X ∃ S∈F x ∈ S ,    ∀F ∀S∈F S = ∅ =⇒ ∃ X ∀S∈F ∃1x x ∈ X  S ∧ ∀x∈X ∃ S∈F x ∈ S .

(AC ) (AC )

15 In giving informal descriptions of the axiom of choice, various authors talk of just such a choice, although the form of the axiom they adopt is either (AC) or (AC ).

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Below, we will give another formulation of the axiom of choice, which will refer to the last version considered. The axiom of choice is equivalent to the Kuratowski–Zorn lemma (see Sect. B.1.2) which is used in Chap. 4.

A.2.2 Properties of the Sum and Product of a Family of Sets Sum of a family of sets. For an arbitrary family of sets F we have the set which is its sum and which satisfies the following condition: 



F,

F = {x : ∃ S∈F x ∈ S}.

It is worth noting the following equivalences: 

F ⊆ X ⇐⇒ ∀S∈F S ⊆ X ⇐⇒ F ⊆ 2 X ,

(A.2.17)

i.e., by summing the sets included in X , we obtain the set included in X . This is easily established by carrying out several equivalence-preserving transformations: 

   F ⊆ X ⇐⇒ ∀x x ∈ F ⇒ x ∈ X   ⇐⇒ ∀x ∃ S∈F x ∈ S ⇒ x ∈ X   ⇐⇒ ∀x ∀S∈F x ∈ S ⇒ x ∈ X   ⇐⇒ ∀S∈F ∀x x ∈ S ⇒ x ∈ X ⇐⇒ ∀S∈F S ⊆ X ⇐⇒ ∀S∈F S ∈ 2 X ⇐⇒ F ⊆ 2 X .

 In the general case, however, the set F needn’t belong to the family F . It may / F. even be the case that S  F  2 X but S ∈ Let us consider the special case of families of sets which are images under various transformations. Let us take an arbitrary term τ (x) which refers only to sets, i.e.: ∀y (y = τ (x) =⇒ Set y). So, by (Arepψ ), we have  the family τ [X ] of sets; see (A.2.11). Furthermore, by virtue of (Asum) the set τ [X ] exists. We obtain:    ∀y y ∈ τ [X ] ⇐⇒ ∃x∈X y ∈ τ (x) . To see this, by (A.2.9) and (A.2.11), for any y we have:

(A.2.18)

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y∈



213

  τ [X ] ⇐⇒ ∃ S∈τ [X ] y ∈ S ⇐⇒ ∃ S S ∈ τ [X ] ∧ y ∈ S   ⇐⇒ ∃ S ∃x∈X S = τ (x) ∧ y ∈ S ⇐⇒ ∃x∈X y ∈ τ (x).

We therefore adopt the following simplified expression: 

τ (x) :=



{τ (x) : x ∈ X } =



τ [X ].

(A.2.19)

x∈X

Product of a family of sets. For an arbitrary non- empty family F of sets we have the set F , which is its product and which we define as follows (see Sect. A.2.1): 

F := {y ∈



F : ∀S∈F y ∈ S}.

Hence, by (A.2.17), we obtain: ∀S∈F S ⊆ X =⇒



F = {y ∈ X : ∀S∈F y ∈ S}.

(A.2.20)

  Obviously, if ∅ = F ⊆ 2 X then F ∈ 2 X , i.e., F ⊆ X ; and so the product of a family sets included in X is also included in X . In the general case, however, the product of sets belonging to a given family needn’t belong to it. Again let us consider the special case of non-empty families of sets which are images under certain transformations. Let us take an arbitrary non- empty set X and arbitrary term τ (x) which refers only to sets. We obtain:    ∀y y ∈ τ [X ] ⇐⇒ ∀x∈X y ∈ τ (x) .

(A.2.21)

Since τ [X ] = ∅, by (A.2.11) and (A.2.14), for any y we have: y∈



  τ [X ] ⇐⇒ ∀S∈τ [X ] y ∈ S ⇐⇒ ∀S S ∈ τ [X ] ⇒ y ∈ S   ⇐⇒ ∀S ∃x∈X S = τ (x) ⇒ y ∈ S ⇐⇒ ∀x∈X y ∈ τ (x).

We therefore adopt the following simplified expression (where X = ∅): 

τ (x) :=



{τ (x) : x ∈ X }.

(A.2.22)

x∈X

Since the product of an arbitrary non- empty family of sets included in a set X is also included in X , we therefore can easily generalise a product to an arbitrary family F of sets included in X . We put (cf. (A.2.20))16 :  X

16 The

‘



x∈X

index in the expression ‘ τ (x)’.

F := {y ∈ X : ∀S∈F y ∈ S}.

 X

F ’ obviously has nothing in common with the expression

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We have here applied axiom (Assϕ ) for a set X and formula ϕ(y) := ‘∀S∈F y ∈ S’. If F = ∅ then this formula is trivially true for each y. However, we will only obtain the set X because we can distinguish it from a subset; that is:  X

∅ = X.

We also have: F = ∅ =⇒

 X

F ⊆

(A.2.23) 

F.

From this and from (A.2.20) we obtain: ∅ = F ⊆ 2 X =⇒

 X

F =



F.

We are therefore passing over “in silence” the index ‘X ’ when F = ∅. It is worth noting the following equivalence for arbitrary sets X and Y and the family F ⊆ 2 X :  F ⇐⇒ ∀S∈F Y ⊆ X  S, X X ⊆ X F ⇐⇒ ∀S∈F X ⊆ S. Y ⊆

(A.2.24)

This is easily established by carrying out several equivalence-preserving transformations: Y ⊆

 X

  F ⇐⇒ ∀y y ∈ Y ⇒ y ∈ X ∧ ∀S∈F y ∈ S   ⇐⇒ ∀y ∀S∈F y ∈ Y ⇒ y ∈ X ∧ y ∈ S   ⇐⇒ ∀S∈F ∀y y ∈ Y ⇒ y ∈ X  S ⇐⇒ ∀S∈F Y ⊆ X  S.

Generalised de Morgan’s laws. Counterparts of de Morgan’s laws hold for families of sets. Let F be a family composed of subsets of a set X . We take the univalent formula ψ(Z , S) := ‘S = X \ Z ’. Remark A.2.15 In the following we will use the axiom schema (Arepψ ) for the univalent formula ψ with other variables than ‘x’ and ‘y’. Then we will rename the variables in (Arepψ ); see footnote 13. The same applies to the formula ϕ in the

axiom schema (Assϕ ). By virtue of (Arepψ ) using with ψ(Z , S) and the family F , we have:   ∃Y ∀S S ∈ Y ⇐⇒ ∃ Z ∈F S = X \ Z . By virtue of (Aext) there exists exactly one such family, which we may write as {S : ∃ Z ∈F S = X \ Z } or {X \ Z : Z ∈ F }; cf. (A.2.11)). We may therefore take the sum and product of this family, obtaining for the sum:

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215

{X \ Z : Z ∈ F } = {x ∈ X : ∃ S (∃ Z ∈F S = X \ Z ∧ x ∈ S)} = {x ∈ X : ∃ Z ∈F x ∈ X \ Z } = {x ∈ X : ∃ Z ∈F x ∈ / Z}

and for the product: 

X {X

\ Z : Z ∈ F } = {x ∈ X : ∀S (∃ Z ∈F S = X \ Z ⇒ x ∈ S)} = {x ∈ X : ∀Z ∈F x ∈ X \ Z } = {x ∈ X : ∀Z ∈F x ∈ / Z }.

From these we obtain our generalised de Morgan’s laws for sets:   X \ F = X {X \ Z : Z ∈ F },   X \ X F = {X \ Z : Z ∈ F }.

(A.2.25) (A.2.26)

A.2.3 Cartesian Products and Relations Ordered pairs, ordered triples, ordered n-tuples. With singletons and unordered pairs in hand, we can obtain so-called ordered pairs. They are a certain sort of unordered pair. For arbitrary x, y we put: x, y := {{x}, {x, y}}. Observe that if x ∈ X and y ∈ Y , then x, y ∈ 22

X Y

. Obviously,

x, x = {{x}}. Observe that, for arbitrary x, y, z and u, we obtain: x, y = z, u ⇐⇒ x = z ∧ y = u . This, however, suffices, for the construction to be recognised as an ordered pair: x, y, z := x, y, z . From an ordered pair we can obtain an ordered triple, which is a special kind of ordered pair: x, y, z := x, y, z. Ultimately, this is just an unordered pair {{x, y}, {x, y, z}}. If x ∈ X , y ∈ Y and (X ×Y )Z . z ∈ Z , then x, y, z ∈ 22

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Once again, for arbitrary x, y, z, u, v and w, we obtain: x, y, z = u, v, w ⇐⇒ x = u ∧ y = v ∧ z = w. We can define an ordered n-tuple inductively: x1 , x2 , . . . , xn  := x1 , x2 , . . . , xn−1 , xn . Cartesian product of sets. Let X and Y be arbitrary sets. We will show that the set-theoretic axioms permit the construction from them of a set composed of all and only ordered pairs whose first member belongs to X and whose second belongs to Y . X Y To this end, we will make use of axiom (Assϕ ) for the set 22 and formula ϕ(z) := ‘∃x∈X ∃ y∈Y z = x, y’. The set thereby obtained will be designed by ‘X × Y ’ and given the name of the Cartesian product of sets X and Y . X × Y := {z ∈ 22

X Y

: ∃x∈X ∃ y∈Y z = x, y}.

Obviously ∅ × X = ∅ = X × ∅. As with when we obtained the set τ [X ] from the set X and the term τ (x) (cf. (A.2.11)), we will create a set from the sets X × Y and the term τ (x, y) with two variables ‘x’ and ‘y’ which we will designate as τ [X × Y ] or as τ [X, Y ] (the image of the set X × Y under the transformation τ ). To do this, we take a univalent formula ψ(z, u) := ∃x∈X ∃ y∈Y (z = x, y ∧ u = τ (x, y)).17 Therefore, we may apply (Arepψ ) to the the set X × Y and formula ψ(z, u) to obtain: τ [X, Y ] := τ [X × Y ] := {τ (x, y) : x ∈ X, y ∈ Y } := {u : ∃z∈X ×Y ∃x∈X ∃ y∈Y (z = x, y ∧ u = τ (x, y))}. (A.2.27) In the particular case where we take τ (x, y) := ‘x, y’, then the expression X × Y = {x, y : x ∈ X, y ∈ Y } is used. It is necessary to remember, though, that this cannot be taken as a definition of the set X × Y , because, without already having this set, it would not be possible to apply axiom (Arepψ ) to obtain (A.2.27). Just as we created the Cartesian product of two sets, so may we create the Cartesian product of X , Y and Z : X × Y × Z := (X × Y ) × Z . Since there exists a set X × Y , the justification given above (when building the product X × Y ) suffices to establish that he above definition is correct. 17 Let ψ(z, u) and ψ(z, v), i.e., for some x

1 ∈ X and y1 ∈ Y we have z = x 1 , y1  and u = τ (x 1 , y1 ), and for x2 ∈ X and y2 ∈ Y we have z = x2 , y2  and v = τ (x2 , y2 ). Then, by virtue of the property of being an ordered pair, x1 = x2 and y1 = y2 . Hence u = τ (x1 , y1 ) = τ (x2 , y2 ) = v.

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We observe that, directly from the definition, we have: (X × Y ) × Z := {u ∈ 22

(X ×Y )Z

: ∃v∈X ×Y ∃z∈Y u = v, z}

= {u ∈ 22

(X ×Y )Z

: ∃x∈X ∃ y∈Y ∃z∈Y u = x, y, z}.

Therefore X × Y × Z is the set of all ordered triples taken from X , Y and Z , since x, y, z := x, y, z. In a similar way to before, from the set X × Y × Z and term τ (x, y, z) with three variables we create a set designated by τ [X × Y × Z ] or τ [X, Y, Z ] (the image of the set X × Y × Z under the transformation τ ). To this end, we take a univalent formula ψ(u, v) := ∃x∈X ∃ y∈Y ∃z∈Z (u = x, y, z ∧ v = τ (x, y, z)). This formula is univalent. Therefore, we may apply axiom (Arepψ ) to the set X × Y × Z and this formula to obtain: τ [X, Y, Z ] := τ [X × Y × Z ] := {τ (x, y, z) : x ∈ X, y ∈ Y, z ∈ Z } := {u : ∃z∈X ×Y ×Z ∃x∈X ∃ y∈Y ∃z∈Z (z = x, y, z ∧ u = τ (x, y, z))}. Similarly, we will define inductively (for n > 1): X 1 × X 2 × · · · × X n := (X 1 × X 2 × · · · × X n−1 ) × X n . Just as above, we will show that the set is a set of all ordered n-tuples take from X 1 , …, X n . Furthermore, by applying (Arepψ ), to the term τ (x1 , . . . , xn ), we obtain: τ [X 1 , . . . , X n ] := τ [X × · · · × X n ] := {τ (x1 , . . . , xn ) : x1 ∈ X 1 , . . . ,xn ∈ X n } := {u : ∃z∈X 1 ×···×X n ∃x∈X 1 . . . ∃xn ∈X n (z = x1 , . . . , xn  ∧ u = τ (x1 , . . . , xn ))}. Relations. The set R is a relation iff there are sets X and Y such that R ⊆ X × Y . We then talk of the relation R on X × Y . Obviously, the empty set ∅ is a relation, because it is a subset of all sets, and so the empty set and the empty relation are one and the same. The set X × Y itself is also a relation, which we call a full relation on X × Y. The domain and the range of a relation R on X × Y are the following sets: dom(R) := {x ∈ X : ∃ y x, y ∈ R}. rng(R) := {y ∈ Y : ∃x x, y ∈ R}. These sets exist by virtue of (Assϕ ). Therefore R ⊆ dom(R) × rng(R). The second point of the fact below shows another way in which one may define the concept of being a relation.

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Fact A.2.5 (i) Each set S composed exclusively  of ordered pairs is a family composed exclusively of families of sets for which S is a family of set and there is  a set S. (ii) A set is a relation if  and to it. only if ordered pairs  belongexclusively  (iii) dom(R), rgn(R) ⊆ R, and so R ⊆ R× R. Proof (i) By virtue of the definition of an ordered pair and (Asum). “⇒” If R ⊆ X × Y then R is composed exclusively of ordered pairs. If Sis a   “⇐” set of some ordered pairs, then we apply (i). We will show that S ⊆ S × S.   If x, y ∈ S, i.e., {{x}, {x, y}} ∈ S, then {x}, {x, y} ∈ S, i.e., x, y ∈ S. (iii) Let R be a relation. Then, applying x ∈ dom(R) then for some y  (i) and (ii), if we have {{x}, {x, y}} ∈ R, i.e., {x} ∈ R, and x ∈ R. Similarly, if y ∈ rgn(R)  then for some x we have {{x}, {x, y}} ∈ R, i.e., {x, y} ∈ R, and y ∈ R. We do not rule out the possibility in the above considerations that the set X is itself a Cartesian product of other sets: for example, X = X 1 × · · · × X n−1 (n > 1). Then we are dealing with subsets of the set (X 1 × · · · × X n−1 ) × Y , which we call n-ary relations on X 1 × · · · × X n−1 × Y . In the case where n = 2, and so where X = X 1 , these relations are called binary relations; and where n = 3, we call them ternary relations. If X 1 = · · · = X n−1 = Y = U then we speak of an n-ary relation on U . Furthermore, if n = 2 (resp. n = 3) then we speak of binary (resp. ternary) relations on U . We remember that (Assϕ ) allows us to distinguish the subsets of a given set with the help of a formula with one free variable. We now use this axiom to distinguish the subsets of a Cartesian product X × Y aided by a formula with two free variables. In this way, we may designate relations. That is, from a formula χ (x, y) with two free variables we create a formula with one free variable by putting ϕ(z) := ∃x∈X ∃ y∈Y (z = x, y ∧ χ (x, y)). To this formula and X × Y we apply (Assϕ ) obtaining the set {z ∈ X × Y : ϕ(z)}, i.e. {z ∈ X × Y : ∃x∈X ∃ y∈Y (z = x, y ∧ χ (x, y))}. We therefore adopt the following shorthand: {x, y ∈ X × Y : χ (x, y)} := {z ∈ X × Y : ∃x∈X ∃ y∈Y (z = x, y ∧ χ (x, y))}.

(A.2.28)

Example A.2.1 The symbol ‘⊆’ is the only predicate defined with the help of the primitive predicate ‘∈’. This relation is often spoken of as the relation of inclusion. This is somewhat of a simplification. To be more precise, for an arbitrary family F of sets and the formula ‘X ⊆ Y ’ by applying the above recipe, we obtain the relation ⊆F := {X, Y  ∈ F × F : X ⊆ Y }. Also for an arbitrary set U , we obtain the relation ⊆U := {X, Y  ∈ 2U × 2U : X ⊆ Y }. The case is similarwith the primitive symbol ‘∈’. Indeed, we have the relation ∈F := {x, Y  ∈ F × F : x ∈ Y }. Also for any set U , we obtain the relation

∈U := {x, Y  ∈ U × 2U : x ∈ Y }. Example A.2.2 (Applications of set theory) Binary relations are the so-called extensions of relational concepts that refer to elements of the universe of discourse U . So,

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219

the binary relation {x, y ∈ U × U : x is a part of y} is the extension of the concept being a part of restricted to the set U . We will regard it as the relationis a part of that obtains between objects from U .

In an analogous way, we can create an n-ary relation from formulas having n free variables (n > 1). So, from a formula χ (x1 , . . . , xn ) with n free variables we create a formula with one free variable by putting ϕ(z) := ∃x1 ∈X 1 . . . ∃xn ∈X n (z = x1 , . . . , xn  ∧ χ (x1 , . . . , xn )). To this formula and to X 1 × · · · × X n we apply (Assϕ ) obtaining the set {z ∈ X 1 × · · · × X n : ϕ(z)}, i.e. {z ∈ X 1 × · · · × X n : ∃x1 ∈X 1 . . . ∃xn ∈X n (z = x1 , . . . , xn  ∧ χ (x1 , . . . , xn ))}. We may therefore adopt the following shorthand: {x1 , . . . , xn  ∈ X 1 × · · · × X n : χ (x1 , . . . , xn )} := {z ∈ X 1 × · · · × X n : ∃x1 ∈X 1 . . . ∃xn ∈X n (z = x1 , . . . , xn  ∧ χ (x1 , . . . , xn ))}. (A.2.29) We say that a relation R1 is included in a relation R2 iff R1 is included in R2 in the sense of set-containment (we then standardly write: R1 ⊆ R2 ). Furthermore, we may carry out three binary set-theoretic operations on relations just as on sets: sum , product  and difference \ . With the help of difference we define the unary operation of complementation. The complement of a relation R on X × Y is the relation (X × Y ) \ R, i.e. {x, y ∈ X × Y : x, y ∈ / R}. Obviously, for arbitrary relations R1 and R2 on X × Y we have: R1 \ R2 = R1  ((X × Y ) \ R2 ).

A.2.4 Functions, Partial Functions and Indexed Families of Sets Functions and partial functions. Let R be an arbitrary relation included in X × Y . We say that R is functional (or univalent, or right-total) iff R satisfies the following condition:   ∀x∈X ∀y,z∈Y x, y ∈ R ∧ x, z ∈ R =⇒ y = z . We say that R is left-total iff dom(R) = X , i.e. ∀x∈X ∃ y∈Y x, y ∈ R. We say that R is a function from X to Y iff R is functional and left-total, i.e. ∀x∈X ∃1y∈Y x, y ∈ R. Such a function is written thus: R : X → Y . If R is functional but not a function, then we say that R is a partial function from X to Y . It will be more convenient to use different variable-letters for functions, such as ‘F’. U Example A.2.3 (Applications of setU theory) The relation {S, x ∈ 2 × U : x sum S} is the partial function from 2 to U , because ∅ does not belong to its domain. U But is the (total) function from 2 \ {∅} to U such that for any for any S ∈ U

2 \ {∅} we have S := ( x) x sum S.

ι

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For F : X → Y for each x ∈ X we put F(x) := ( y) x, y ∈ F. The element F(x) we call the value of the function F at x. ι

Functions defined by terms. Let τ (x) be a term with the variable ‘x’. For a set X and formula ψ(y, x) := ‘y = τ (x)’, axiom (Arepψ ) designates the set τ [X ] := {τ (x) : x ∈ X }; cf. (A.2.11). We may therefore create the Cartesian product X × τ [X ]. By applying (A.2.28), we have the relation {x, y ∈ X × τ [X ] : y = τ (x)}, which is a function from X to τ [X ]. We may write it thus: X  x → τ (x) ∈ τ [X ]. Example A.2.4 With the set X as defined above, we take the family 2 X of sets and the term ‘X \ Y ’ with the one variable ‘Y ’. In this case τ [2 X ] = 2 X , since Y = X \ (X \ Y ), for each set Y of 2 X . We obtain the function {Y, Z  ∈ 2 X × 2 X : Z = X \ Y },

i.e., 2 X  Y → X \ Y ∈ 2 X . Remark A.2.16 It might seem that, to construct the function from the term τ (x) it would have sufficed to apply (A.2.28) to the formula χ (x, y) := y = τ (x) and sets X and Y , meaning that there would have been no need to appeal to axiom (Arepψ ), and in particular to the particular case of (A.2.11). However, we could have only obtained the functional relation {x, y ∈ X × Y : y = τ (x)} from (A.2.28), which could even turn out to be the empty set.

The above construction may be developed for terms with two variables. Let τ (x, y) be such a term. Axiom (Arepψ ) designates the set τ [X, Y ] := {τ (x, y) : x ∈ X, y ∈ Y }; cf. (A.2.27). We may therefore create the Cartesian product (X × Y ) × τ [X, Y ]. By applying (A.2.29), we have the relation {x, y, z ∈ X × Y × τ [X ] : z = τ (x, y)}, which is a function from X × Y to τ [X, Y ]. We may write it thus: X × Y  x, y → τ (x, y) ∈ τ [X, Y ]. Example A.2.5 The symbol ‘’ is the only so-called function symbol used in the building of terms and defined by the primitive predicate ‘∈’. This symbol is often understood as the sum-operation (function) for sets. This is somewhat of a simplification. To be more precise, for an arbitrary family of sets F satisfying the condition (this may be, for example, an arbitrary power set): if X, Y ∈ F then X  Y ∈ F , and for the term τ (X, Y ) := ‘X  Y ’, by applying the recipe above, we obtain: • τ [F , F ] = F , because X = X  X for each set X ; • the function F := {X, Y, Z  ∈ F × F × F : Z = X  Y }, otherwise written

as F : F × F  X, Y  → X  Y ∈ F . Indexed families of sets. An arbitrary function defined on some non-empty set and taking values in some family of sets is called an indexed family of sets. The set on which the function is defined is called the index set. Let F : I → F be an arbitrary indexed family of sets. Then, thanks to (Assϕ ) or (Arepψ ), there is an ordinary family of sets: {X ∈ F : ∃i∈I X = F(i)}. By applying (Arepψ ), we may express the ordinary family in the following way: {F(i) : i ∈ I }. We designate an indexed family of sets by (F(i))i∈I ’, or more simply by (X i )i∈I , where X i := F(i) for each i ∈ I .

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The sum (resp. product) of an indexed family (X i )i∈I of sets is simple the sum (resp. product) of an ordinary family {X i : i ∈ I } of sets obtained via (Asum). By applying (A.2.18) and (A.2.19) (resp. (A.2.21) and (A.2.22)), we may write these operations for indexed families in the following way: 

X i = {x : ∃i∈I x ∈ X i },

i∈I



X i = {x : ∀i∈I x ∈ X i }.

i∈I

Other versions of the axiom of choice. From (AC ) we obtain the following fact, which is often simply called the axiom of choice: Fact A.2.6For an arbitrary family F composed of non-empty sets, there is a function F : F → F such that for each Y ∈ F we have F(Y ) ∈ Y .  Proof Let F be an arbitrary family of non-empty sets. We have the set F , thanks to (Asum). By virtue of (AC ), there is a set X  such that ∀Y ∈F ∃1y y ∈ X  Y . We can therefore create the function F : F → X  F by putting F(Y ) := ( y)(y ∈ X  Y ) for each Y of F . Obviously, F(Y ) ∈ Y .

ι

The function from Fact A.2.6 “chooses” exactly one element from each set belonging to F . This is why it is called a choice function. In this version of the axiom of choice, however, what’s brought out is not that one element may be taken from each set belonging to F . It is rather this: that thanks to function F and (Arepψ ) or (Assϕ ) we have the set F[F ]. This is either an image of the family F under the transformation F, i.e. F[F ] := {x : ∃Y ∈F x = F(Y )} (cf. (A.2.11)), or a subset of  the set F , i.e. F[F ] = {x ∈ F : ∃Y ∈F x = F(Y )}. The set F[F ] satisfies the condition: ∀Y ∈F ∃1y y ∈ F[F ]  Y ∧ ∀x∈F[F ] ∃Y ∈F x ∈ Y . We have therefore obtained (AC ), and this entails other versions of the axiom of choice we have seen. Therefore we get: Fact A.2.7 On the basis of the other axioms of set-theory, the axiom of choice is equivalent to Fact A.2.6. We also have a counterpart to Fact A.2.6 dealing with indexed families of sets. Fact A.2.8 For an arbitraryindexed family (X i )i∈I composed from non-empty sets, there is a function F : I → i∈I X i such that for each i ∈ I we have F(i) ∈ X i .  Proof We have an ordinary family {X i : i ∈ I } of sets and the set i∈I X i . By virtue of (AC ), there is a setX such that ∀i∈I ∃1y y ∈ X  X i . We can therefore create the function F : I → X  i∈I X i by putting F(i) := ( y)(y ∈ X  X i ) for each i ∈ I.

ι

The function from Fact A.2.8 may also be called a choice function. Similar comments apply to those given in relation to Fact A.2.6. Therefore, the axiom of choice also follows from Fact A.2.8.

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A.3 Binary Relations A.3.1 Operations on Binary Relations Let U be an arbitrary non-empty set (a so-called universe of discourse). In general, for an arbitrary binary relation R on U , in order to shorten our formulas, we will write ‘x R y’ (resp. ‘x R y’) instead of ‘x, y ∈ R’ (resp. ‘x, y ∈ / R’). We will also write ‘x R y R z’ instead of ‘x R y ∧ y R z’. As has already been noted in the previous section, with binary relations on U , just as with subsets of the set U × U , we can perform three standard binary settheoretic operations: sum , product  and difference \ . We also have the unary operation of complementation on U defined by the condition −R := (U × U ) \ R, i.e. −R := {x, y ∈ U × U : x R y}. We obviously have R1 \ R2 = R1  −R2 , and, furthermore, all the standard theses of the so-called algebra of sets hold for the operations , , \ and −. Amongst binary relations in U we will distinguish the socalled identity relation idU . We put: idU := {x, y ∈ U × U : x = y} = {x, x : x ∈ U }. There are two further standard operations defined on binary relations on U . The first is the unary operation of taking the converse (or converse relation). The converse of a relation R is the relation R˘ := {x, y ∈ U × U : y R x}. We have R˘˘ = R. The second is the binary operation of taking the relative product, defined by the condition: R1 ◦ R2 := {x, y ∈ U × U : ∃z∈U x R1 z R2 y}. The relative product has the following properties: • • • •

idU ◦ idU = idU and R ◦ idU = R = idU ◦ R, R ◦ (R1 R2 ) = (R ◦ R1 )(R ◦ R2 ) and (R1  R2 ) ◦ R = (R1 ◦ R)  (R2 ◦ R), R ◦ (R1  idU ) = (R ◦ R1 )  R and (R1  idU ) ◦ R = (R1 ◦ R)  R, R ◦ (R1 R2 ) ⊆ (R ◦ R1 )(R ◦ R2 ) and (R1  R2 ) ◦ R ⊆ (R1 ◦ R)  (R2 ◦ R).

Furthermore, for an arbitrary natural number n > 0 and an arbitrary binary relation R on U , we define the relation R n by the following inductive conditions: R 1 = R and for arbitrary k > 0, R k+1 = R k ◦ R. To finish, let us say that for an arbitrary subset X of the set U the relation R| X is the restriction of the relation R to the set X , i.e. R| X := R  (X × X ).

A.3.2 Fundamental Properties of Binary Relations We say that a binary relation R on U is respectively reflexive, irreflexive, symmetric, asymmetric, antisymmetric, transitive and acyclic if and only if R satisfies the following conditions18 : 18 These

conditions also come in handy when the letter ‘R’ imitates a two-argument predicate; that is—having removed the restriction ‘∈ U ’ in the quantifiers—the first six conditions are written in a

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∀x∈U x R x,

(r R )

¬∃x∈U x R x,

(irr R )

∀x,y∈U (x R y =⇒ y R x),

(s R )

¬∃x,y∈U (x R y ∧ y R x),

(as R )

¬∃x,y∈U (x = y ∧ x R y ∧ y R x),

(antis R )

∀x,y,z∈U (x R y ∧ y R z =⇒ x R z),

(t R )

∀n>0 ¬∃x∈U x R x. n

(ac R )

We say also that a relation R is an equivalence relation on U iff R is reflexive, symmetric and transitive. An example of such a relation is idU . We can use some logically-equivalent versions of the above formulas. That is, R on U is respectively reflexive, irreflexive, symmetric, asymmetric, antisymmetric, transitive and acyclic if and only if R satisfies the following conditions: ∀x,y∈U (x = y =⇒ x R y),

(r R )

∀x,y∈U (x R y =⇒ x = y),

(irr R )

∀x,y∈U (x R y ⇐⇒ y R x),

(sR )

∀x,y∈U (x R y =⇒ ¬y R x),

(asR )

∀x,y∈U (x R y ∧ y R x =⇒ x = y),   ∀x,y∈U ∃z (x R z ∧ z R y) =⇒ x R y ,   ¬∃x1 ,...,xn ∈U x1 R x2 ∧ · · · ∧ xn R x1 .

(antisR ) (t R ) (acR )

In (acR ) we are dealing not with a single condition but with a set of conditions, for each n > 0. The variable ‘R’ appearing in all lemmas until the end of this section will refer to binary relations on the universe U , and these lemmas will have a general character; i.e. they will refer to arbitrary relations on U . This will not be specially highlighted so as to shorten the formulas. From the definitions above we find the following algebraic characterisation of relation-types. Lemma A.3.1 (i) R is reflexive iff idU ⊆ R. (ii) R is irreflexive iff R ⊆ −idU iff idU ⊆ −R iff idU  R = ∅. ˘ (iii) R is symmetric iff R ⊆ R˘ iff R˘ ⊆ R iff R = R. (iv) R is asymmetric iff R  R˘ = ∅ iff R ⊆ − R˘ iff R˘ ⊆ −R iff idU  R 2 = ∅. (v) R is antisymmetric iff R  R˘ ⊆ idU . (vi) R is transitive iff R ◦ R ⊆ R. first-order language and the seventh in a weak second-order language (which permits quantification over the natural numbers). Notice that we can write the seventh condition in a first-order language as a set of conditions; cf. (acR ).

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(vii) R is acyclic iff for each n > 0, idU  R n = ∅. Proof Ad (i) idU ⊆ R iff ∀x,y∈U (x = y ⇒ x R y) iff ∀x∈U x R x iff R is reflexive. Ad (ii) R ⊆ −idU iff ∀x,y∈U (x R y ⇒ x = y) iff ∀x,y∈U (x = y ⇒ ¬x R y) iff idU ⊆ −R iff ∀x∈U ¬x R y iff ¬∃x∈U x R x iff R is irreflexive. Furthermore, R ⊆ −idU iff ∀x,y∈U (x R y ⇒ x = y) iff ¬∃x, y ∈ U (x = y ∧ x R y) iff idU  R = ∅. Ad (iii) R is symmetric iff ∀x,y∈U (x R y ⇒ y R x) iff R˘ ⊆ R iff ∀x,y∈U (y R x ⇒ ˘ x R y) iff R ⊆ R˘ iff ∀x,y∈U (x R y ⇔ y R x) iff R = R. Ad (iv) R is asymmetric iff ¬∃x,y∈U (x R y ∧ y R x) iff R  R˘ = ∅. Furthermore, ¬∃x,y∈U (x R y ∧ y R x) iff ∀x,y∈U (x R y ⇒ ¬y R x) iff R ⊆ − R˘ iff ∀x,y∈U (y R x ⇒ ¬x R y) iff R˘ ⊆ −R. Finally, idU  R 2 = ∅ iff ¬∃x,y∈U (x = y ∧ x R 2 y) iff ¬∃x,y∈U (x = y ∧ ∃z∈U (x R z ∧ z R y)) iff ∀x,y∈U (x = y ⇒ ¬∃z∈U (x R z ∧ z R y)) iff ∀x∈U ¬∃z∈U (x R z ∧ z R y) iff ¬∃x,z∈U (x R z ∧ z R y) iff R is asymmetric. Ad (v) R is antisymmetric iff ¬∃x,y∈U (x = y ∧ x R y ∧ y R x) iff ∀x,y∈U (x R y ∧ y R x ⇒ x = y) iff R  R˘ ⊆ idU . Ad (vi) R is transitive iff ∀x,y,z∈U (x R z ∧ z R y ⇒ x R y) iff ∀x,y∈U (∃z (x R z ∧ z R y) ⇒ x R y) iff R ◦ R ⊆ R. Ad (vii) For each n > 0, idU  R n = ∅ iff ∀n>0 ¬∃x,y∈U (x = y ∧ x R n x) iff ∀n>0 ∀x,y∈U (x = y ⇒ ¬x R n y) iff ∀n>0 ∀x∈U ¬x R n x iff ∀n>0 ¬∃x∈U x R n x iff R is acyclic.

From Lemma A.3.1 we obtain: Lemma A.3.2 (i) R is reflexive and antisymmetric iff R  R˘ = idU . (ii) If R is reflexive and antisymmetric, then R satisfies the following condition:   ∀x,y∈U ∀u∈U (u R x ⇔ u R y) =⇒ x = y .

(∗)

(iii) If R is transitive and satisfies (∗), then R is antisymmetric. (iv) R is reflexive and transitive iff R satisfies the following condition:   ∀x,y∈U x R y ⇐⇒ ∀u∈U (u R x ⇒ u R y) . (v) (vi) (vii) (viii)

(∗∗)

R is asymmetric iff R is irreflexive and antisymmetric. If R is irreflexive and transitive, then R is acyclic. If R is acyclic, then R is asymmetric and irreflexive. If R irreflexive and transitive, then R is asymmetric and irreflexive.

Proof Ad (i)19 Let R be antisymmetric and reflexive. Then R  R˘ ⊆ idU , idU ⊆ ˘ Therefore, R  R˘ = idU . Conversely, if R  R˘ = idU then both R and idU ⊆ R. R  R˘ ⊆ idU and idU ⊆ R. Hence, R is antisymmetric and reflexive, by Lemma A.3.1(i, v). 19 All

the proofs below can also be carried out using the formulas given in the definitions and their logically equivalent versions.

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Ad (ii) Let R be reflexive and antisymmetric. Suppose that for arbitrary x, y ∈ U we have: ∀u∈U (u R x ⇔ u R y). Hence x R y and y R x, since R is reflexive. So, x = y, since R is antisymmetric. Thus, R satisfies (∗). Ad (iii) Suppose that R is transitive and satisfies (∗). Moreover, we assume that x R y and y R x. Then R satisfies ∀u∈U (u R x ⇔ u R y), since R is transitive. So, x = y, since R satisfies (∗). Thus, R is antisymmetric. Ad (iv) Let R be reflexive and transitive. For (∗∗), we assume that x R y. If u R x then u R y, since R is reflexive. So, we have the left-to-right implication in (∗∗). Now we assume: ∀u∈U (u R x ⇒ u R y). Then x R y, since R is reflexive. So, we have the converse implication in (∗∗). Let R satisfy (∗∗). Then for any x ∈ U we have: ∀y∈U (x R x ⇔ ∀u∈U (u R x ⇒ u R x)). Hence x = x. Thus, R is reflexive. Now we assume that x R y and y R z. Then, by (∗∗), we have: ∀u∈U (u R x ⇒ u R y) and ∀u∈U (u R y ⇒ u R z). Hence, ∀u∈U (u R x ⇒ u R z). So, x R z, by (∗∗). Thus, R is transitive. Ad (v) Let R be asymmetric. Then R  R˘ = ∅, by Lemma A.3.1(iv). Were it the case that idU  R = ∅ then idU  R 2 = ∅ would also be the case, i.e. R  R˘ = ∅, too. Thus, R is irreflexive, by Lemma A.3.1(ii). Furthermore, R  R˘ = ∅ ⊆ idU . Thus, R is antisymmetric, by Lemma A.3.1(v). Conversely, let R be irreflexive and antisymmetric. Then and idU  R = ∅ and R  R˘ ⊆ idU , by Lemma A.3.1(i, v). Hence R  R˘ ⊆ idU  R = ∅. Thus, R is asymmetric, by Lemma A.3.1(iv). Ad (vi) Suppose that R is transitive but not acyclic. Then for some n 0 > 0 and x ∈ U we have x R n 0 x. Therefore, for some y1 , …, yn 0 we have: x R y1 , …, yn 0 R x. Hence, x R x, since R is transitive. Hence, R is not irreflexive. Ad (vii) If R is acyclic then: ¬∃x∈U x R 2 x. Hence ¬∃x,y∈U (x R y ∧ y R x), that is, R is asymmetric. Furthermore, we use (v). Ad (viii) This is established directly by (vi) and (vii).

By using points (v) and (viii) from Lemma A.3.2 we obtain the following. Corollary A.3.3 The conditions below are logically equivalent: (a) R is irreflexive and transitive; (b) R is asymmetric and transitive.

A.3.3 “Reflexivisation”, “Irreflexivisation” and “Asymmetricisation” of Binary Relations For an arbitrary binary relation R on U we say that its “reflexivisation” is the relation R • := R  idU . The name of this relation comes from its possessing the following five properties: Lemma A.3.4 (i) R ⊆ R • and R • is reflexive. (ii) R • is the smallest reflexive relation that includes R.

226

R is reflexive iff R • ⊆ R iff R = R • . R •• = R • . R˘ • = (R • ). R \ R˘ • ⊆ R • \ R˘ • ⊆ R.

(iii) (iv) (v) (vi)

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Proof Ad (i) We have R ⊆ R  idU =: R • . Moreover, idU ⊆ R  idU =: R • ; so we use Lemma A.3.1(i). Ad (ii) By virtue of (i) we have: R • is reflexive and R ⊆ R • . Let us assume that the relation R  is reflexive and R ⊆ R  . Then also idU ⊆ R  and hence also R • := R  idU ⊆ R  . Ad (iii) R is reflexive iff idU ⊆ R iff R  idU ⊆ R iff R  idU = R iff R • = R. Ad (iv) R • := R  idU = (R  idU )  idU = R •• . Ad (v) x, y ∈ R˘ • iff x, y ∈ R˘  idU iff either x R˘ y or x = y iff either y R x or y = x iff y, x ∈ R  idU iff x, y ∈ (R • ). Ad (vi) By applying (v), we have: R \ R˘ • ⊆ R • \ R˘ • = (R  idU )  − R˘  −idU = (R  − R˘  −idU )  (idU  − R˘  −idU ) = (R  − R˘  −idU )∅⊆R.

Remembering that R˘ • = (R • ) and using Lemma A.3.1, we can formulate various connections that hold between R and R • :

Lemma A.3.5 (i) R is antisymmetric iff R • s antisymmetric. (ii) R is asymmetric iff R is irreflexive and R • is antisymmetric. (iii) R is asymmetric iff R  R˘ • = ∅. (iv) R is asymmetric iff R = R \ R˘ • = R • \ R˘ • . ˘  (idU  Proof Ad (i) Observe that R •  R˘ • = (R  idU )  ( R˘  idU ) = (R  R) • • ˘ ˘ ˘ R)  (R  idU )  idU . Hence, we have: R  R ⊆ idU iff R  R ⊆ idU . Ad (ii) Directly from (i) and Lemma A.3.2(v). ˘  (R  idU ). Hence, we Ad (iii) We have: R  R˘ • = R  ( R˘  idU ) = (R  R) ˘ obtain our theorem, because R is asymmetric iff R  R = ∅ (and then R is also irreflexive, i.e. R  idU = ∅). Ad (iv) By virtue of (iii), if R is asymmetric, then R ⊆ − R˘ • ; and so R = R \ R˘ • and R ⊆ R • \ R˘ • , since R⊆R • . The converse inclusion is given by Lemma A.3.4(vi). Conversely, we have the inclusion R ⊆ − R˘ • from the identity R = R \ R˘ • . Therefore, R is asymmetric, by virtue of (iii).

Lemma A.3.6 If R is transitive, then: (i) R ◦ R • ⊆ R and R • ◦ R ⊆ R, (ii) R • ◦ R • ⊆ R • , and so R • is transitive. Proof Assume that R is transitive. Then R ◦ R ⊆ R. Ad (i) R ◦ R • = R ◦ (R  idU ) = (R ◦ R)  (R ◦ idU ) ⊆ R  R = R. In a similar way we show that R • ◦ R ⊆ R. Ad (ii) R • ◦ R • = (R  idU ) ◦ (R  idU ) = (R ◦ R)  (R ◦ idU )  (idU ◦ R) 

(idU ◦ idU ) ⊆ R  R  R  idU = R • . Hence, R • is transitive.

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Lemma A.3.7 For any different relations R1 and R2 on U : (i) If R1 and R2 are reflexive then R1• = R2• . (ii) If R1 and R2 are irreflexive then R1• = R2• . Proof Ad (i) Suppose that R1 and R2 are reflexive and R1• = R2• . Then idU ⊆ R1 , idU ⊆ R2 and R1  idU = R2  idU . Hence, R1 = R1  idU = R2  idU = R2 . Ad (ii) Suppose that R1 and R2 are irreflexive and R1 = R2 . Then R1  idU = / ∅ = R2  idU and there are x, y ∈ U such that either both x, y ∈ R1 and x, y ∈ / R1 . In the first case, x = y; and so x, y ∈ R1• R2 , or both x, y ∈ R2 and x, y ∈ and x, y ∈ / R2• . In the second case, x, y ∈ R2• and x, y ∈ / R1• . Thus, R1• = R2• .

The operation of the “irreflexivisation” of a binary relation is analogous to the operation of “reflexivisation”. For a relation R, this relation is defined by the condition R ◦ := R \ idU = R  −idU . We obtain: Lemma A.3.8 (i) R ◦ is irreflexive and R ◦ ⊆ R. (ii) R ◦ is the greatest irreflexive relation included in R. (iii) R is irreflexive iff R ⊆ R ◦ iff R = R ◦ . (iv) R ◦◦ = R ◦ and R˘ ◦ = (R ◦ ). (v) R •◦ = R ◦ and R ◦• = R • . ˘  idU = ∅, and hence R \ R˘ ⊆ R ◦ . (vi) (R \ R)

Proof Ad (i) From Lemma A.3.1(ii). Ad (ii) By virtue of (i) we have: R ◦ is irreflexive and R ◦ ⊆ R. Assume that R  is irreflexive and that R  ⊆ R. Then R   idU = ∅ as well. If, therefore, x R  y then x R y and x = y, i.e., x R ◦ y. Therefore, R  ⊆ R ◦ . Ad (iii) R is irreflexive iff R ⊆ −idU iff R ⊆ R  −idU iff R = R  −idU . Ad (vi) There are no x, y ∈ U such that x R y ∧ ¬y R x ∧ x = y. Hence, we

have: R \ R˘ ⊆ R \ idU = R ◦ . Corollary A.3.9 (i) R is reflexive iff R = R ◦• . (ii) If R is reflexive and R ◦ is transitive, then R is transitive. (iii) R is irreflexive iff R = R •◦ . Proof Ad (i) We use Lemmas A.3.4(iii) and A.3.8(v). Ad (ii) We apply LemmaA.3.6 to R ◦ , and thereby obtain that R ◦• is transitive. However, we have R = R ◦• . Ad (iii) We use Lemma A.3.8(iii), (v).

Remembering that R˘ ◦ = (R ◦ ) and using Lemma A.3.1, we can formulate various connections that hold between R and R ◦ .

Lemma A.3.10 (i) R is antisymmetric iff R ◦ is asymmetric. (ii) R is antisymmetric iff R ◦  R˘ = ∅. ˘ (iii) R is antisymmetric iff R ◦ = R \ R.

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Proof Ad (i) Observe that R ◦  R˘ ◦ = R  R˘  −idU . Therefore R ◦  R˘ ◦ = ∅ iff R  R˘ ⊆ idU . ˘ Ad (ii) R is antisymmetric iff R  R˘ ⊆ idU iff ∅ = R  R˘  −idU = R ◦  R. ˘ Therefore, since Ad (iii) By virtue of (ii), if R is antisymmetric then R ◦ ⊆ − R. ˘ The converse inclusion is given by Lemma A.3.8(vi). R ◦ ⊆ R, we have R ◦ ⊆ R \ R.

Furthermore, if R ◦ = R \ R˘ then R ◦  R˘ = ∅, i.e. R  R˘ ⊆ idU . Lemma A.3.11 If R is antisymmetric and transitive, then: (i) R ◦ R ◦ ⊆ R ◦ and R ◦ ◦ R ⊆ R ◦ , (ii) R ◦ ◦ R ◦ ⊆ R ◦ , and so R ◦ is transitive. Proof Assume that R is antisymmetric and transitive. Then R  R˘ ⊆ idU and R ◦ R ⊆ R. Ad (i) Firstly, notice that (A): R ◦ R ◦ = R ◦ (R  −idU ) ⊆ R ◦ R ⊆ R. Secondly, notice that (B): R ◦ R ◦ ⊆ −idU . Indeed, if x, y ∈ R ◦ R ◦ then for some z 0 we have x R z 0 , z 0 R y and z 0 = y. Were it the case that x = y then x R z 0 , z 0 R x and z 0 = x, which would contradict the antisymmetry of R. So x, y ∈ R ◦ R ◦ entails x = y. From (A) and (B) we have: R ◦ R ◦ ⊆ R  −idU = R ◦ . In a similar way we show that R ◦ ◦ R ⊆ R ◦ .

Ad (ii) By (i) we have R ◦ ◦ R ◦ ⊆ R ◦ , since R ◦ ⊆ R. Corollary A.3.12 If R is asymmetric and R • is transitive, then R is transitive. Proof Suppose that R is asymmetric and R • is transitive. Then R is irreflexive and antisymmetric. Therefore, R = R ◦ and R • antisymmetric, by Lemmas A.3.8(iii) and A.3.5(i). We therefore apply Lemma A.3.11 to R • and obtain that R •◦ is transitive.

However, by Lemma A.3.9(iii), R = R •◦ , since R is irreflexive. Corollary A.3.13 If R is reflexive then the conditions below are equivalent: (a) R is antisymmetric and transitive; (b) R ◦ is irreflexive and transitive. Proof “(a) ⇒ (b)” From Lemmas A.3.8(i) and A.3.11. “(b) ⇒ (a)” From Lemmas A.3.2(viii) and A.3.10(i) and Corollary A.3.9(ii). Corollary A.3.14 The two conditions below are equivalent: (a) R is irreflexive and transitive; (b) R = R ◦ and R • is antisymmetric and transitive. Proof “(a) ⇒ (b)” From Lemmas A.3.2(viii), A.3.5(ii) and A.3.6. “(b) ⇒ (a)” From Lemma A.3.8(i) and Corollary A.3.12.

As is evident from Lemma A.3.10(i), the operation of “irreflexivisation” does not always produce an asymmetric relation. It yields such a relation when and only when the output relation is antisymmetric. Let us therefore introduce the operation of “asymmetricisation” for binary relations. For a relation R this relation is defined ˘ We obtain: by the following condition R  := R \ R˘ = R  − R.

Appendix A: Logic and Set Theory

229

Lemma A.3.15 (i) R  is asymmetric and R  ⊆ R. (ii) R is asymmetric iff R = R  . (iii) R  = R  and R˘  = (R  ).

˘  ( R˘  −R) = ∅. ˘  (R  − R) ˘ = (R  − R) Proof Ad (i) (R  − R) ˘ ˘ Ad (ii) R is asymmetric iff R  R = ∅ iff R ⊆ − R˘ iff R ⊆ R \ R˘ iff R = R \ R.    ˘  −( R˘  −R) = (R  − R) ˘  (− R ˘ Ad (iii) R = R  −(R ) = (R  − R) ˘  (R  − R˘  R) = R  .

R) = (R  − R˘  − R)



Corollary A.3.16 (i) R is antisymmetric iff R ◦ = R  . (ii) If R is reflexive, then R  ⊆ R ◦ . (iii) R is reflexive and antisymmetric iff R = R • . (iv) If R is reflexive and antisymmetric and R  is transitive, then R is also transitive. (v) R is asymmetric iff R = R • . Proof Ad (i) By virtue of Lemma A.3.10(iii). Ad (ii) If R is reflexive, then R˘ is also reflexive, and so R  − R˘ ⊆ R  −idU . Ad (iii) If R is antisymmetric, then from (i) we have R ◦ = R  . Hence R ◦• = R • , and therefore by virtue of Lemma A.3.8(v), R • = R ◦• = R • . Therefore, if R is also reflexive, then R = R • = R • . Conversely, if R = R • then R = (R  ˘  idU ⊆ − R˘  idU , and hence idU ⊆ R and R  R˘ ⊆ idU , i.e. R is reflexive − R) and antisymmetric. Ad (iv) We apply Lemma A.3.6 to R  , and obtain that R • is transitive. However, R = R • . Ad (v) From Lemma A.3.5(iv) we have: R is asymmetric iff R = R • \ R˘ • . Hence,

by using Lemma A.3.4(v), we obtain R • = R • \ (R • ) = R • \ R˘ • = R.

Lemma A.3.17 If R is transitive then: (i) R ◦ R  ⊆ R  and R  ◦ R ⊆ R  ; (ii) R  ◦ R  ⊆ R  , and so R  is transitive.20 Proof Assume that R is transitive. Then R ◦ R ⊆ R. ˘ ⊆ (R ◦ R) ⊆ R. SecAd (i) Firstly, notice that (A): R ◦ R  = R ◦ (R  − R) ˘ Indeed, if x, y ∈ R ◦ R  then for some ondly, notice that (B): R ◦ R  ⊆ − R. z 0 we have x R z 0 , z 0 R y and ¬y R z 0 . Were it the case that y R x then by transitivity we would have a contradiction: y R z 0 . From (A) and (B) we have: R ◦ R  ⊆ R  − R˘ = R  . In a similar way, we show that R  ◦ R ⊆ R  .

Ad (ii) By (i) we have R  ◦ R  ⊆ R  , since R  ⊆ R. From the facts above we obtain the following: Corollary A.3.18 For arbitrary binary relations R1 and R2 on U the conditions below are equivalent: 20 Observe

that Lemma A.3.11 follows from this lemma and Lemma A.3.16(i). We also obtain Corollary A.3.12 by applying Corollary A.3.16(v).

230

(a) (b) (c) (d)

Appendix A: Logic and Set Theory

R1 is irreflexive and transitive, and R2 = R1• ; R1 is asymmetric and transitive, and R2 = R1• ; R2 is reflexive, antisymmetric and transitive, and R1 = R2◦ ; R2 is reflexive, antisymmetric and transitive, and R1 = R2 .

Proof “(a) ⇔ (b)” We apply Corollary A.3.3. “(b) ⇒ (c)” By Lemmas A.3.4(i), A.3.5(ii) and A.3.6 along with Corollary A.3.9 (iii). “(c) ⇒ (a)” We apply Lemmas A.3.8(i) and A.3.11 along with Corollary A.3.9(i). “(c) ⇔ (d)” From Corollary A.3.16(i).

References 1. Pietruszczak, A. (2000). Metamereologia (Metamereology). Toru´n: The Nicolaus Copernicus University Press. English version (2018) Metamereology. Toru´n: The Nicolaus Copernicus University Scientific Publishing House. https://doi.org/10.12775/3961-4. 2. Pietruszczak, A. (2018). Metamereology. Toru´n: The Nicolaus Copernicus University Scientific Publishing House. English version of (Pietruszczak, 2000). https://doi.org/10.12775/3961-4.

Appendix B

Algebra

B.1 Strict Partial Orders B.1.1 Fundamental Concepts and Definitions Let U be a non-empty set and R be a binary relation on U . We say that R strictly partial orders U iff R is transitive and asymmetric. Therefore, each such relation is also acyclic, irreflexive and antisymmetric (see Lemma A.3.2). Another—logically equivalent—definition of the concept of being a strictly partially-ordering relation takes it to be transitive and irreflexive (see Corollary A.3.3). In this case—also by applying Lemma A.3.2—we find that the relation is acyclic, asymmetric and antisymmetric. If R strictly partially orders a set U , then we also say that R is a strict partial order on U , and the structure U, R we call a strictly partially-ordered set. Let s POS be the class of strictly partially-ordered sets. For ease of expression below, we will adopt the following convention: if the relation R is a strict partial order on U , then this will be designated by the symbol ‘≺’; and conversely: an arbitrary relation designated by the symbol ‘≺’ strictly partially orders a set U .21 The following sentences are therefore true22 : ¬∃x∈U x ≺ x,   ¬∃x,y∈U x ≺ y ∧ y ≺ x ,   ∀x,y,z∈U x ≺ y ∧ y ≺ z =⇒ x ≺ z ,

(irr ≺ )

∀n>0 ¬∃x∈U x ≺ x.

(ac≺ )

n

(as≺ ) (t≺ )

21 Under 22 See

this convention, the symbol ‘≺’ is a variable ranging over strict partial orders. footnote 18 in Appendix A.

© Springer Nature Switzerland AG 2020 A. Pietruszczak, Foundations of the Theory of Parthood, Trends in Logic 54, https://doi.org/10.1007/978-3-030-36533-2

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If S is a non-empty subset of U , then the relation ≺| S , i.e., the restriction of the relation ≺ to S, strictly partially orders the set S. Let  be the reflexivisation of ≺, i.e.  := ≺• . This means that  is the sum of the relations ≺ and idU , i.e.  := ≺  idU . We will express this as follows:   ∀x,y∈U x  y :⇐⇒ x ≺ y ∨ x = y .

(df )

Remark B.1.1 (i) In the light of Lemma A.3.7(ii), ≺•1 = ≺•2 iff ≺1 = ≺2 , i.e., 1 = 2 iff ≺1 = ≺2 . (ii) In the light of Lemma A.3.8(iii) and Corollary A.3.16(i), since < is antisymmetric (and irreflexive), we have ≺ = ≺◦ = ≺ . Thus, there is no need for the operations of “irreflexivisation” and “asymmetricisation” to be applied to ≺.

The general lemmas given in Sect. A.3.3 tell us that: 

∀x∈U x  x,



¬∃x,y∈U x = y ∧ x  y ∧ y  x ,   ∀x,y,z∈U x  y ∧ y  z =⇒ x  z ,   ∀x,y,z∈U x ≺ y ∧ y  z =⇒ x ≺ z ,   ∀x,y,z∈U x  y ∧ y ≺ z =⇒ x ≺ z ,   ∀x,y∈U x ≺ y ⇐⇒ x  y ∧ x = y ,   ∀x,y∈U x ≺ y ⇐⇒ x  y ∧ y  x .

(r ) (antis ) (t ) (B.1.1) (B.1.2) (B.1.3) (B.1.4)

An element x ∈ U is called an upper (lower) bound of a set S iff for each y ∈ S we have y  x (resp. y  x). We say that x ∈ U is a maximal (resp. minimal) element in a set S iff x ∈ S and there is not a y ∈ S such that x ≺ y (resp. y ≺ x). Thus, x ∈ U is a maximal (resp. minimal) element in U, ≺ iff there is not a y ∈ U such that x ≺ y (resp. y ≺ x). We say that ≺ strictly linearly (or totally) orders U iff ≺ satisfies the following condition:   (l≺ ) ∀x,y∈U x = y ∨ x ≺ y ∨ y ≺ x . We than say that the pair U, ≺ is a strictly linearly (or totally) ordered set.

B.1.2 Chains and the Kuratowski–Zorn Lemma Let U, ≺ belong to s POS and ∅ = C ⊆ U . We say that C is a chain in U, 1 ⇐⇒ ¬∃x∈U ∀u∈U x ≤ u. We can avoid using the set-theoretic operator ‘Card’, in which case the sentence above has the following equivalent form: ∃x∈U ∀u∈U x ≤ u ⇐⇒ ∃x∈U ∀u∈U x = u ⇐⇒ ∀x,u∈U x = u. Let us now observe that we could have avoided using the reflexivity of the relation ≤ in the definition of a polarised partial order26 : Fact B.3.3 (Pietruszczak 2000, 2018) Condition (r≤ ) follows logically from conditions (t≤ ) and (pol≤ ). Proof Assume for a contradiction that for some x ∈ U we have x  x. Then, by virtue of (pol⊥≤ ), for some z 0 we have: : z 0 ≤ x and z 0 ⊥ x. Were it the case that for some u 0 , u 0 ≤ z 0 then we would have from (t≤ ) the following: u 0 ≤ z 0 and u 0 ≤ x, which is impossible in virtue of the fact that z 0 ⊥ x. Therefore, ¬∃u∈U u ≤ z 0 , and hence z 0  z 0 . Now, by applying (pol⊥≤ ), once again, we have our contradiction:

∃u∈U u ≤ z 0 .

26 Obviously, if we start from U, ≺ and then take the relation , then the relation  will be reflexive

by force of its definition.

Appendix B: Algebra

243

B.3.2 Semi-polarisation in Partial Orders We know that in partially-ordered sets with zero that the relation ⊥ is empty and their polarisations boils down to their trivialization (see Fact B.3.2). Therefore, the interesting case for partially-ordered sets with zero is one where they are not themselves polarised (i.e., they are not degenerate) but their “non-zero fragments” are polarised. In this case, we need a relation other than that of ⊥ and a different concept than polarisation. Let us introduce the second auxiliary relation  of the separation of the elements in U :   x  y :⇐⇒ ∀z∈U z ≤ x ∧ z ≤ y ⇒ ∀u∈U z ≤ u .

(df )

From (B.2.11) we obtain that when P has the zero 0, then  satisfying the equivalences below for arbitrary x, y ∈ U : x  y ⇐⇒ ∀z∈U (z ≤ x ∧ z ≤ y ⇒ z = 0) ⇐⇒ ¬∃z∈U (z = 0 ∧ z ≤ x ∧ z ≤ y). Lemma B.3.4 (i) The relation  is symmetric. (ii) ⊥ ⊆ , i.e. ∀x,y∈U (x ⊥ y ⇒ x  y). (iii) ∀x,y,z∈U (z ≤ x ∧ x  y =⇒ z  y). (iv) ∀x,y∈U (x ≤ y ∧ x  y =⇒ ∀u∈U x ≤ u). (v) If P does not have a zero, then ⊥ = . (vi) If P has the zero 0 then: • ∀x∈U 0  x, • ⊥ = ∅  . Proof Ad (i) and (ii) Simple logical transformations suffice to establish them. Ad (iii) From (t≤ )and the definitions. Ad (iv) From (r≤ ) and the definitions. Ad (v) If P does not have a zero, then ¬∃z∈U ∀u∈U z ≤ u. Hence, if x  y then ¬∃z∈U (z ≤ x ∧ z ≤ y); and so x ⊥ y. Ad (vi) From (B.2.11), (B.2.12) and Lemma B.3.1(ii).

We say that P is semi-polarised (or that ≤ is the relation of semi-polarisation) iff the following condition is satisfied:    ∀x,y∈U x  y =⇒ ∃z∈U z ≤ x ∧ z  y ∧ ¬∀u∈U z ≤ u) .

(s-pol≤ )

From (B.2.11) we obtain that if P has the zero 0, then P semi-polarised iff the following condition is satisfied:    ∀x,y∈U x  y =⇒ ∃z∈U z ≤ x ∧ z  y ∧ z = 0) .

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We will show that the class of polarised partially-ordered sets is contained in the class of semi-polarised partially-ordered sets, and that in the class of partiallyordered sets without zero both concepts have the same extension. Fact B.3.5 (i) If P is polarised, then P is semi-polarised. (ii) If P does not have a zero and is semi-polarised, then P is polarised. Proof Ad (i) Assume that P is polarised and x  y. Then for some z we have z ≤ x and z ⊥ y. Hence, by virtue of Lemma B.3.1(ii), P does not have a zero. Therefore, ¬∀u∈U z ≤ u and ⊥ = , by virtue of (∃0) and Lemma B.3.4(v), respectively. Ad (ii) Assume that P does not have a zero and is semi-polarised, and that x  y. Then, by Lemma B.3.4(v), ⊥ =  and there is a z such that z ≤ x and z ⊥ y.

We will now strengthen the result given in Fact B.3.3. Fact B.3.6 Condition (r≤ ) is logically entailed by conditions (t≤ ) and (s-pol≤ ). Proof Assume for a contradiction, that for some x ∈ U we have x  x. Then, by virtue of (s-pol≤ ), for some z 0 we have: z 0 ≤ x, z 0  x and ¬∀u∈U z 0 ≤ u. Therefore, z 0 is not the zero in P. If some u is such that u ≤ z 0 , then from (t≤ ) we have: u ≤ z 0 and u ≤ x. This entails ∀v∈U u ≤ v, in virtue of z 0  x. Therefore, u is the zero in P and z 0 = u. Hence, z 0  z 0 . Now, by applying (s-pol≤ ) once more, we obtain our

contradiction: ∃u∈U (u ≤ z 0 ∧ ¬∀v∈U u ≤ v). The following facts will be of use in the remaining sections of this appendix: Lemma B.3.7 The following monotonicity principle is entailed by (t≤ ) and (s-pol≤ ): ∀S∈2U ∀x,y∈U

   ∀z∈S z ≤ y ∧ ∀u∈U (u ≤ x ∧ ∀z∈S z  u) ⇒ ∀v∈U u ≤ v  =⇒ x ≤ y .

(m≤ )

Proof Let x and y satisfy the antecedent of the implication. Take an arbitrary u ∈ U such that u ≤ x and u  y. Hence, by virtue of the first conjunct of the compound conjunction and (t≤ ), ∀z∈S z  u. Hence, by virtue of the second conjunct of the

compound conjunction, ∀v∈U u ≤ v. Hence, x ≤ y, by virtue of (s-pol≤ ). From this lemma we obtain the following directly: Corollary B.3.8 If P ∈ POS0 is semi-polarised, then ∀S∈2U ∀x,y∈U

    ∀z∈S z ≤ y ∧ ∀u∈U (u ≤ x ∧ ∀z∈S z  u) ⇒ u = 0 =⇒ x ≤ y .

The lemma below shows that the condition of semi-polarisation obtaining is equivalent to a condition connected to the relation sup≤ .27 27 This

condition says that some expansion of the relation of is a mereological sum (which we have been examining in this book) is included in the relation sup≤ . The relation sum we restrict, however, to the set (U \ {0}) × (2U \ {∅}).

Appendix B: Algebra

245

Lemma B.3.9 (i) The following sentence follows from conditions (t≤ ) and (s-pol≤ ): ∀S∈2U ∀x∈U

   ∀z∈S z ≤ x ∧ ∀u∈U (u ≤ x ∧ ∀z∈S z  u) ⇒ ∀v∈U u ≤ v  =⇒ x sup≤ S .

(∗)

(ii) Conditions (r≤ ), (t≤ ) and (∗) ental (s-pol≤ ). Proof Ad (i) Let x and S satisfy the antecedent of the implication. Take any y such x sup≤ S. that ∀z∈S z ≤ y. Then x ≤ y, by Lemma B.3.7. Therefore,  Ad (ii) Take arbitrary x and y satisfying (a): ∀u∈U u ≤ x ∧ u  y ⇒ ∀v∈U u ≤ v). We show that x ≤ y, which will give us condition (s-pol≤ ). To this end, let S0 := {z ∈ U : z ≤ x ∧ z ≤ y} and we take any u ∈ U such that y, i.e., for some w0 ∈ U we have w0 ≤ u, w0 ≤ y u ≤ x and ¬∀v∈U u ≤ v. Then u  and ¬∀v∈U w0 ≤ v. From (t≤ ) we have w0 ≤ x; and so w0 ∈ S0 . By virtue of (r≤ ), u. We show, therefore, that from (a), the conditions we have w0 ≤ w0 . Thus, w0  u ≤ x and ∀z∈S0 z  u entail ∀v∈U u ≤ v. Since we have also ∀z∈S0 z ≤ x, then x and S0 satisfy the antecedent of the assumed implication. Hence x sup≤ S0 . Since, however,

∀z∈S0 z ≤ y, we have x ≤ y. Corollary B.3.10 If P ∈ POS0, then the semi-polarisation of P is equivalent to the following condition holding:     ∀S∈2U ∀x∈U ∀z∈S z ≤ x ∧ ∀u∈U (u ≤ x ∧ ∀z∈S z  u) ⇒ u = 0 =⇒ x sup≤ S .

B.3.3 Semi-polarisation in the Class POS0 In this subsection, we will assume that P = U, ≤, 0 is an arbitrary partial order with the zero 0. We will furthermore assume that U+ := U \ {0} and ≤+ := ≤|U+ ; that is ≤+ is the restriction of the relation ≤ to the set U+ ; i.e. ≤+ := ≤  (U+ × U+ ). It is evident from Fact B.3.2 that if P is polarised, then U = {0}, i.e. that P is degenerate. Of interest, therefore, will only be those cases where P is not polarised (i.e., where it is not degenerate). If P is not polarised, then U = ∅ and we obtain a partially-ordered set P+ := U+ , ≤+ . This should be the “non-zero fragment” of the partially-ordered set P. In P+ , the relation corresponding to the relation ⊥ in P is not the restriction of ⊥ to the set U+ . For we know (see Lemma B.3.1(ii)) that since P has a zero, then ⊥ = ∅; that is, ⊥+ := ⊥|U+ = ∅ too. In P+ , the corresponding relation turns out to be the restriction of the relation  to the set U+ . We put ⊥+ := |U+ ; i.e. ⊥+ is a binary relation in U satisfying the equivalences below for arbitrary x, y ∈ U+ : x ⊥+ y :⇐⇒ ¬∃z∈U (z = 0 ∧ z ≤ x ∧ z ≤ y) ⇐⇒ ¬∃z∈U+ (z ≤+ x ∧ z ≤+ y).

(df ⊥+ )

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Appendix B: Algebra

The first of these equivalences is a re-expression (through transformation) of the restriction of the relation  to U+ . The second turns out to be a way of defining the relation ⊥ for the partially-ordered set P+ . An interesting case is where P is not polarised and P+ is. We will first show that if P+ also has a zero, then the situation described in Fact B.3.2 holds for P+ and, as a result, we obtain the following: Corollary B.3.11 If P has the zero 0 and is not polarised, and P+ is polarised and has a least element 0+ , then U+ = {0+ } and 0+ is the unity in P, i.e., 0 = 0+ = 1 and U = {0, 0+ }. Proof Let us assume that P has the zero 0 but is not polarised (and so is not degenerate either). We know from Fact B.3.2 that if P+ is polarised and has a least element 0+ , then U+ = {0+ }. Hence U = {0, 0+ }. Since 0 ≤ 0+ , it follows that 0+ is the unity in P.

To complete this section, we will present a connection between polarisation and semi-polarisation in the class POS0. Fact B.3.12 Let P belong to POS0. Then P is semi-polarised iff either P is degenerate or P+ is polarised. Proof If P is semi-polarised and non-degenerate, x, y ∈ U+ and x  y, then there is a z such that z = 0, z ≤ x and z  y. Hence z ∈ U+ and z ⊥+ y. If P is degenerate then P is polarised (Fact B.3.2) and semi-polarised (Fact B.3.5(i)). Let us therefore assume that P is non-degenerate, P+ is polarised, x, y ∈ U and x  y. Then x = 0, i.e. x ∈ U+ . If y = 0, then we take z := x, and by using (r≤ ) and Lemma B.3.4, we get: z = 0, z ≤ x and z  y. Furthermore, if

y ∈ U+ , then we use our assumption and (df ⊥+ ).

B.4 Lattices Lattices may be considered to be a certain kind of algebra; that is, as sets with the (primitive) operations of sum and product. They may also be treated as a kind of partially-ordered set, in which those operations may be defined (cf. Grätzer 1971, pp. 4–7). It will be more convenient for us, as regards their application to mereology, to take the second approach.

B.4.1 Lattices as Partially Ordered Sets A partially-ordered set U, ≤ is called a lattice iff for arbitrary x, y ∈ U the set {x, y} has a supremum and an infimum. Put formally, the pair U, ≤ is a lattice iff it belongs to POS and satisfies the following condition:

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247

∀x,y∈U ∃z,u∈U (z sup≤ {x, y} ∧ u inf≤ {x, y}).

(L)

Obviously, in virtue of conditions (B.2.17) and (B.2.19), which are true in each structure in POS, it suffices in (L) for the universal quantifier to be restricted to the case where x = y. Let L be the class of all lattices. Remark B.4.1 To define a lattice, it is not essential that the relations sup≤ and inf≤ be interdefinable (cf. (B.2.15) and (B.2.16)). Indeed, with resp to the set {x, y}, at least one of the sets {u ∈ U : u ≤ x ∧ u ≤ y} and {u ∈ U : x ≤ u ∧ y ≤ u} might not be finite, and only in such cases do we have a guarantee that suprema and infima

exist (cf. condition (L ) below.) We can prove by induction (cf. Grätzer 1971, p. 4) that a partially-ordered set U, ≤ is a lattice iff each non-empty finite subset of U has a supremum and an infimum:   ∀S∈2U 0 < Card S < ℵ0 =⇒ ∃z,u∈U (z sup≤ S ∧ u inf≤ S) .

(L )

Henceforth, let L = U, ≤ be an arbitrary lattice. In virtue of (fsup ) and (finf ), for arbitrary x, y ∈ U , the set {x, y} has only one supremum and one infimum. We can therefore define the binary operations of sum + and product · on the Cartesian product U × U , these operations taking values into U : x + y := ( z) z sup≤ {x, y},

(df +)

x · y := ( z) z inf≤ {x, y}.

(df ·)

ι

ι

The above entail the following: ∀x,y∈U x ≤ x + y, ∀x,y∈U x · y ≤ x,

∀x,y∈U y ≤ x + y,

(B.4.1)

∀x,y∈U x · y ≤ y.

(B.4.2)

Furthermore, for arbitrary x, y ∈ U : x ≤ y ⇐⇒ x + y = y, ⇐⇒ x · y = x.

(B.4.3)

From (df +) and (B.4.1) we obtain:   ∀x,y,z∈U x + y ≤ z ⇐⇒ x ≤ z ∧ y ≤ z .

(B.4.4)

Directly from (B.2.27) we have, however:   ∀x,y∈U x + y sup≤ {u ∈ X : u ≤ x ∨ u ≤ y} .

(B.4.5)

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From (df ·) and (B.4.2) we get:   ∀x,y,u∈U u ≤ x · y ⇐⇒ u ≤ x ∧ u ≤ y ,

(B.4.6)

Directly from (B.2.25) we have, however:   ∀x,y∈U x · y sup≤ {u ∈ X : u ≤ x ∧ u ≤ y} .

(B.4.7)

It is well-know that the operations + and · satisfy the following conditions (cf. Grätzer 1971; Traczyk 1970): ∀x,y∈U x + y = y + x ∀x,y,z∈U x + (y + z) = (x + y) + z

∀x,y∈U x · y = y · x ∀x,y,z∈U x · (y · z) = (x · y) · z

∀x∈U x + x = x ∀x,y∈U x + (x · y) = x

∀x∈U x · x = x ∀x,y∈U x · (x + y) = x

(B.4.8) (B.4.9) (B.4.10) (B.4.11)

Therefore, the operations + and · are commutative, associative and idempotent. We may prove by induction that for each n > 0 the following hold: ∀z,x1 ,...,xn ∈U z sup≤ {x1 , . . . , xn } ⇐⇒ z = x1 + · · · + xn ,

(B.4.12)

∀z,x1 ,...,xn ∈U z inf≤ {x1 , . . . , xn } ⇐⇒ z = x1 · · · · · xn .

(B.4.13)

Furthermore, we can show that: ∀x,y,z∈U x + (y · z) ≤ (x + y) · (x + z), ∀x,y,z∈U (x · y) + (x · z) ≤ x · (y + z),   ∀x,y,z∈U x ≤ y =⇒ x + z ≤ y + z ,   ∀x,y,z∈U x ≤ y =⇒ x · z ≤ y · z .

(B.4.14) (B.4.15) (B.4.16) (B.4.17)

From (B.4.2) we immediately obtain: Fact B.4.1 In every lattice the relation ⊥ is empty. Furthermore, the relation  in L satisfies the following condition28 :   ∀x,y∈U x  y ⇐⇒ x · y is a zero .

(B.4.18)

  Let x  y, i.e. ∀z∈U z ≤ x ∧ z ≤ y ⇒ ∀u∈U z ≤ u . Then, by virtue of (B.4.2), ∀u∈U x · y ≤ u. Therefore, L satisfies condition (∃0); that is, L has a zero and the zero is x · y. Conversely, let x · y be the zero of L, i.e., we have ∀u∈U x · y ≤ u. Take an arbitrary z such that z ≤ x and z ≤ y. Then, by virtue of (B.4.6), z ≤ x · y; by virtue of (t≤ ), however, we have ∀u∈U z ≤ u. Therefore, x  y. are not assuming in (B.4.18) that L has a zero. If L does not have a zero, then  = ∅. This concurs with Fact B.4.1 and Lemma B.3.4(v), which say that in partial orders without zero, ⊥ = . 28 We

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249

a

b

c

Model B.1 An example of lattice in which the converse implication to (B.4.19) does not hold (this is a non-distributive lattice)

Furthermore, from Lemma B.3.4(iii) and (B.4.1) we obtain:   ∀x,y,z∈U z  x + y =⇒ (z  x ∧ z  y) .

(B.4.19)

Remark B.4.2 There exist lattices in which the converse implication to (B.4.19) does not hold. An example of such a lattice is given by Model B.1, where U = {0, a, b, c, 1} and the partial order ≤ is such that 0 ≤ a, b, c ≤ 1. Then c  a, c  b and a + b = 1, but c  a + b.

B.4.2 Lattices with Unity Let us assume that a lattice L = U, ≤ has a unity 1 (we write U, ≤, 1). Then condition (B.2.7) holds and we get: ∀u∈U u + 1 = 1,

∀u∈U u · 1 = u.

(B.4.20)

Furthermore, by virtue of (B.2.8) and (B.4.3), for an arbitrary x ∈ U we have: x = 1 ⇐⇒ ∀u∈U u ≤ x, ⇐⇒ ∀u∈U u + x = x, ⇐⇒ ∀u∈U u · x = u. Let L1 be the class of all lattices with unity. It is clear that L1  POS1.

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B.4.3 Lattices with Zero Let us assume that a lattice L = U, ≤ has a zero (we write U, ≤, 0). Then, condition (B.2.10) holds and we get: ∀u∈U u + 0 = u,

∀u∈U u · 0 = 0.

(B.4.21)

Furthermore, by virtue of (B.2.11) and (B.4.3), for an arbitrary x ∈ U we have: x = 0 ⇐⇒ ∀u∈U x ≤ u, ⇐⇒ ∀u∈U x + u = u, ⇐⇒ ∀u∈U x · u = x. Let L0 be the class of all lattices with zero. It is clear that L0  POS0. From (B.4.18) and (df ) we obtain: x · y = 0 ⇐⇒ ∀z∈U (z ≤ x ∧ z ≤ y ⇒ z = 0), ⇐⇒ x  y.

(B.4.22)

We therefore also have:   ∀x,y∈U x · y = 0 ⇐⇒ ∃z∈U (z = 0 ∧ z ≤ x ∧ z ≤ y) .

(B.4.23)

Another proof of this fact: If x · y = 0, then 0 = x · y ≤ x and 0 = x · y ≤ y. Conversely, if for some z ∈ U we have z = 0, z ≤ x and z ≤ y, then 0 = z ≤ x · y, and hence x · y = 0. Let us now observe that in U, ≤, 0 we obtain the following:   ∀x,y∈U x ≤ y =⇒ ∀z∈U (z ≤ x ∧ z · y = 0 ⇒ z = 0) .

(B.4.24)

If x ≤ y, z ≤ x and z · y = 0, then from (t≤ ) we have z ≤ y, and so z = z · y = 0. In virtue of (B.4.22), the converse implication to that in (B.4.24) is equivalent to condition (s-pol≤ ), which talks of the semi-polarisation of U, ≤, 0, and so also the polarisation of the partial order U+ , ≤+ , so long as U, ≤, 0 is not degenerate, i.e. if U = {0}. Lemma B.4.2 For an arbitrary L = U, ≤, 0 the conditions below are equivalent. (a) (b) (c) (d) (e)

L is semi-polarised.    ∀x,y∈U x  y =⇒ ∃z∈U z = 0 ∧ z ≤ x ∧ z  y) .  ∀x,y∈U x  y =⇒ ∃z∈U (z = 0 ∧ z ≤ x ∧ z · y = 0) . ∀x,y∈U ∀z∈U (z ≤ x ∧ z · y = 0 ⇒ z = 0) =⇒ x ≤ y . Either U = {0} or U+ , ≤+  is polarised.

Proof “(a)⇔(b)” From the definition. “(b)⇔(c)” From (B.4.22). “(c)⇔(d)” Logical equivalence. “(d)⇔(e)” From Fact B.3.12.

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The lemma below will be of use in later parts of this appendix. Lemma B.4.3 (Pietruszczak 2000, 2018) Suppose that a lattice U, ≤, 0 satisfies the following condition: for an arbitrary S ∈ 2U there is exactly one x ∈ U such that (a) ∀z∈S z ≤ x and  (b) ∀u∈U u ≤ x ∧ ∀z∈S u · z = 0 =⇒ u = 0 .

()

Then U, ≤, 0 is semi-polarised.29 Proof Notice that, by Fact B.3.12, U, ≤, 0 is semi-polarised iff either U = {0} or U+ , ≤+  is polarised. Assume for a contradiction that U = {0} and U+ , ≤+  is not polarised. Then, by virtue of Lemma B.4.2, for some x0 , y0 ∈ U we have: (A) x0  y0 and (B) ∀u∈U (u ≤ x0 ∧ u · y0 = 0 ⇒ u = 0). Let S0 := {z : z ≤ y0 }. Since x0 · y0 ≤ y0 , then (i) x0 · y0 ∈ S0 . To begin with, let us observe that y0 satisfies conditions (a) and (b) from (*) for the set S0 . Condition (a) is in this instance a tautology. For (b): let u ≤ y0 and ∀z∈S0 u · z = 0. Then u ∈ S0 , and so u = u · u = 0. Next, we find a y1 distinct from y0 but satisfying conditions (a) and (b) from () for S0 . But this contradicts condition (). For the set S1 := S0 ∪ {x0 }, by virtue of (), there is exactly one y1 such that (ii) ∀z∈S0 z ≤ y1 , (iii) x0 ≤ y1 and (iv) ∀u (u ≤ y1 ∧ ∀z (z ∈ S0 ∨ z = x0 ⇒ u · z = 0) =⇒ u = 0). It follows from (A) and (iii) that y0 = y1 . From y1 we have that y1 satisfies condition (a) from () for S0 . Below, we will show that y1 also satisfies condition (b) from () for S0 . Take an arbitrary u such that u ≤ y1 and ∀z∈S0 u · z = 0. From this and from (i) we have (v) u · x0 · y0 = 0. Since u · x0 ≤ x0 , then u · x0 = 0, by virtue of (B) and (v). We can therefore apply (iv) to get: u = 0.

B.4.4 Bounded Lattices We say that a lattice U, ≤ is bounded (we write L = U, ≤, 0, 1) iff it is bounded, when it regarded as a partially-ordered set; that is, it is a lattice with zero and unity. Let L01 be the class of all bounded lattices. It is clear that L01  POS01. We observe the following: Fact B.4.4 Each finite lattice is bounded. Proof Let U, ≤, 0, 1 be a finite lattice. Then, by virtue of (L ), there are x and y such that x sup≤ U and y inf≤ U . Hence x = 1 and y = 0, by (B.2.23) and (B.2.24).

U, ≤, 0 is polarised, then this is equivalent to U = {0} (cf. Fact B.3.2). Then condition () is satisfied and the thesis in the lemma holds.

29 If

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B.4.5 Lattices as Algebras Lattices may be considered as algebraic structures. We say that a triple U, +, · is a lattice (as an algebraic structure) iff U is a non-empty set and + and · are binary operations on U satisfying conditions (B.4.8)–(B.4.11). It follows from the previous point and the definition of a lattice as an algebraic structure that, starting from an arbitrary lattice U, ≤, we obtain the lattice U, +, · as an algebra. Let U, +, · be a lattice (as an algebra). From (B.4.8) and (B.4.11) it follows that   ∀x,y∈U x + y = y ⇐⇒ x · y = x . We define in the lattice U, +, · (as an algebra) the binary relation ≤ with the help of condition (B.4.3). It has been proven (see, e.g., Traczyk 1970, pp. 20–21) that U, ≤ is a partially-ordered set which satisfies condition (L), where for arbitrary x, y ∈ U : x + y sup≤ {x, y} and x · y inf≤ {x, y}. Therefore, the pair U, ≤ is a lattice, regarded as a partially-ordered set. Fact B.4.5 (i) If a relational structure U, ≤ is a lattice and the binary operations + and · are defined on U with the help of (df +) and (df ·), then U, +, · is a lattice (as an algebraic structure). (ii) If an algebraic structure U, +, · is a lattice and the binary relation ≤ is defined on U with the help of (B.4.3), then U, ≤ is a lattice (as a relational structure). Let us now consider three further types of algebraic structure. The first is the algebraic structure U, +, ·, 1 such that U, +, · is a lattice (as an algebra) and 1 belongs to U and satisfies (B.4.20). It then follows from (B.4.3) that 1 satisfies condition (B.2.7); that is, U, ≤, 1 is a lattice with unity. The second is the algebraic structure U, +, ·, 0 such that U, +, · is a lattice (as an algebra) and 0 belongs to U and satisfies (B.4.21). It then follows from (B.4.3) that 0 satisfies condition (B.2.10); that is, U, ≤, 0 is a lattice with zero. The third is the algebraic structure U, +, ·, 0, 1 such that U, +, · is a lattice (as an algebra) and 0, 1 belong to U and satisfy (B.4.21) and (B.4.20), respectively. Then U, ≤, 0, 1 is a lattice with zero and unity. Fact B.4.6 (i) If U, ≤, 1 (resp. U, ≤, 0; U, ≤, 0, 1) is a lattice with unity (resp. with zero; with zero and unity) and the binary operations + and · are defined on U with the help of (df +) and (df ·), then the algebraic structure U, +, ·, 1 (resp. U, +, ·, 0; U, +, ·, 0, 1) is a lattice with unity (resp. with zero; with zero and unity). (ii) If an algebraic structure U, +, ·, 1 (resp. U, +, ·, 0; U, +, ·, 0, 1) is a lattice with unity (resp. with zero; with zero and unity) and the binary relation ≤ is defined on U with the help of (B.4.3), then U, ≤ is a lattice with unity (resp. with zero; with zero and unity).

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B.4.6 Distributive Lattices We say that a lattice U, ≤ is distributive iff inequalities (B.4.14) and (B.4.15) may be reversed; i.e.:   ∀x,y,z∈U (x + y) · (x + z) ≤ x + (y · z) ,   ∀x,y,z∈U x · (y + z) ≤ (x · y) + (x · z) .

(B.4.25) (B.4.26)

Therefore, by virtue of the antisymmetry of the relation ≤, the following conditions are satisfied in distributive lattices:   ∀x,y,z∈U x + (y · z) = (x + y) · (x + z) ,   ∀x,y,z∈U x · (y + z) = (x · y) + (x · z) .

(B.4.27) (B.4.28)

Furthermore, the following sentences are true for distributive lattices:   ∀S∈2U ∀x,y∈U x inf≤ S =⇒ y + x inf≤ {y + z : z ∈ S} ,   ∀S∈2U ∀x,y∈U x sup≤ S =⇒ y · x sup≤ {y · z : z ∈ S} .

(B.4.29) (B.4.30)

It is well-known that, in all lattices, each of six conditions (B.4.25)–(B.4.30) entails the other five. In every distributive lattice the converse implication to (B.4.19) holds. We therefore have30 :   (B.4.31) ∀x,y,z∈U z  x + y ⇐⇒ (z  x ∧ z  y) . Assume that z  x and z  y. Then, by (B.4.18) we know that the lattice has the zero 0 and that 0 = z · x = z · y. Take an arbitrary u such that u ≤ z and u ≤ x + y. Then, by virtue of (B.4.6), u ≤ z · (x + y) = (z · x) + (z · y) = 0. Hence u = 0. Therefore, z  x + y. For distributive lattices with zero we prove the following lemma which will come in useful later. Lemma B.4.7 Let U, ≤, 0 be a distributive lattice with zero. Then:   ∀S∈U ∀x,y∈U (x sup≤ S ∧ y ≤ x ∧ ∀z∈S y · z = 0) =⇒ y = 0 . Proof Take arbitrary x, y and S that satisfy the antecedent of the above implication. Then, by virtue of (B.4.3) and (B.4.30), we have, respectively, y = y · x and y · x sup≤ {y · z : z ∈ S}. We have, however, {y · z : z ∈ S} = {0}, i.e., y sup≤ {0}.

Hence, y = 0, by virtue of (sf sup ). If an algebraic structure U, +, · is a lattice—that is, if conditions (B.4.8)– (B.4.11) are satisfied—then conditions (B.4.25)–(B.4.30) are also satisfied in them. 30 The

lattice presented in Model B.1 is just such an example of a non-distributive lattice.

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Therefore, we may also consider this algebra as a distributive lattice if it satisfies one of those conditions.31 Fact B.4.8 (i) If a relational structure U, ≤ is a distributive lattice and the binary operations + and · are defined on U with the help of (df +) and (df ·), then the algebraic structure U, +, · is a distributive lattice (as an algebra). (ii) If an algebraic structure U, +, · is a distributive lattice and the binary relation ≤ is defined on U with the help of (B.4.3), then U, ≤ is a distributive lattice.

B.4.7 Complementation in Bounded Distributive Lattices Let L = U, ≤, 0, 1 be a bounded lattice. Then for arbitrary x, y ∈ U , we say that y is a complement of x iff x · y = 0 and x + y = 1. Of interest to us will be the properties of the binary relation of complementation in bounded distributive lattices. We will make use them later when discussing socalled Boolean lattices. In all the theorems to follow we will take L to be a bounded distributive lattice. Clearly, we can also consider bounded distributive lattices as algebraic structures which have the form U, +, ·, 0, 1. They have to satisfy conditions (B.4.8)–(B.4.11), (B.4.20), (B.4.21) and (B.4.27) (so also (B.4.28)). Firstly, then, let us observe that:   ∀x,y,z∈U y is a complement of x ∧ z · x = 0 =⇒ z ≤ y .

(B.4.32)

If x, y, z ∈ U satisfy the antecedent of the implication, then z = z · 1 = z · (x + y) = (z · x) + (z · y) = 0 + (z · y) = z · y. Hence z ≤ y, by (B.4.3). It follows from (B.4.32) that: Corollary B.4.9 If y is a complement of x then the two equivalent conditions below hold: (a) y is the greatest element in the set {u ∈ U : x · u = 0}, (b) x · y = 0 and y sup≤ {u ∈ U : x · u = 0}. Proof If y is a complement of x, then x · y = 0 and, by (B.4.32), for an arbitrary u ∈ U : if x · u = 0 then u ≤ y. The equivalence of conditions (i) and (ii) we have from (B.2.13).

Since there is, in a given set, at most one greatest element, we therefore obtain from Corollary B.4.9(a) the conclusion that in bounded distributed lattices each element has at most one complement, i.e.: 31 It is most often the case that just one

of the two conditions (B.4.27), (B.4.28) is adopted, because only the primitive terms of the algebra we are considering feature appear in them.

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  ∀x,y,z∈U y is a complement of x ∧ z is a complement of x =⇒ y = z . (f-com) To finish, let us prove a lemma which will come in useful later. Lemma B.4.10 (Pietruszczak 2000, 2018) If L = U, ≤ is semi-polarised then   ∀x,y∈U y is a complement of x ⇐⇒ y sup≤ {u ∈ U : x · u = 0} . Proof Let L = U, ≤, 0 be bounded distributive lattice. Notice that, by Fact B.3.12, L is semi-polarised iff either L is degenerate or U+ , ≤+  is polarised. If U = {0}, then 0 = 1, 0 is a complement of 0 and 0 sup≤ {0}. Let us therefore assume that L is non-degenerate, U+ , ≤+  is polarised and x, y belong to U . “⇒” By virtue of Corollary B.4.9(b). “⇐” Let y sup≤ {u ∈ U : x · u = 0}. Since L is distributive, then x · y sup≤ {x · u ∈ U : x · u = 0}, by virtue of (B.4.30). Therefore x · y sup≤ {0}; and so x · y = 0, by virtue of (sf sup ). Furthermore, take an arbitrary z such that z · (x + y) = 0. Since L is distributive, z · (x + y) = (z · x) + (z · y) = 0. Hence, by virtue of (B.4.1), we have (A) z · x = 0 and (B) z · y = 0. From (A) we have z ≤ y, since y sup≤ {u ∈ U : x · u = 0}. Hence z = z · y = 0, by virtue of (B). We have therefore shown that (C): ∀z (z · (x + y) = 0 ⇒ z = 0). The polarisation of U+ , ≤+  yields condition (iv) of Lemma B.4.2. Since we have (C), then by applying condition (iv) to x := 1 and y := x + y, we obtain 1 ≤ x + y. Therefore x + y = 1.

B.5 Boolean Lattices B.5.1 Definition A bounded lattice U, ≤, 0, 1 is called Boolean iff it is distributive and every element of U has a complement, i.e., the following condition holds:   ∀x∈U ∃ y∈U x · y = 0 ∧ x + y = 1 .

(B.5.1)

By (B.5.1) and (f-com), in an arbitrary Boolean lattice U, ≤, 0, 1, each element has exactly one complement, i.e.,: ∀x∈U ∃1y∈U y is a complement of x.

(B.5.2)

Therefore, we can define on U the unary operation of complementation − : U → U by putting for arbitrary x ∈ U : −x := ( y) y is a complement of x ι

= ( y) (x · y = 0 ∧ x + y = 1).

(df −)

ι

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Therefore, we immediately obtain from our definition: ∀x∈U x + −x = 1,

∀x∈U x · −x = 0.

(B.5.3)

Let BL be the class of all Boolean lattices.

B.5.2 Boolean Algebras An algebraic structure U, +, ·, −, 0, 1 is called a Boolean algebra iff U, +, ·, 0, 1 is a bounded distributive lattice (as an algebra) and − is the unary operation of complementation. Thus, in any Boolean algebra, condition (B.4.8)–(B.4.11), (B.4.20), (B.4.21), (B.4.27), (B.4.28) and (B.5.3) hold. We will show that, by starting with an arbitrary Boolean lattice U, ≤, 0, 1, we can obtain a Boolean algebra U, +, ·, −, 0, 1. As in the case of an arbitrary lattice (as an algebra), we define in the Boolean algebra U, +, ·, −, 0, 1 the binary relation ≤ with the help of condition (B.4.3). It has been proven (see Koppelberg 1989; Traczyk 1970) that this relation partially orders the set U and the relational structure U, ≤, 0, 1 is a Boolean lattice. Fact B.5.1 (i) If a relational structure U, ≤, 0, 1 is a Boolean lattice, the binary operations + and · are defined on U with the help of (df +) and (df ·), and the unary operation − is defined with the help of (df −), then the algebraic structure U, +, ·, −, 0, 1 is a Boolean algebra. (ii) If an algebraic structure U, +, ·, −, 0, 1 is a Boolean algebra and the binary relation ≤ is defined on U with the help of (B.4.3), then U, ≤, 0, 1 is a Boolean lattice.

B.5.3 Properties of Boolean Operators It is well-known that in an arbitrary Boolean lattice B = U, ≤, 0, 1, the conditions below hold (see, e.g., Koppelberg 1989): − 1 = 0,

−0 = 1

(B.5.4)

and that for arbitrary x, y ∈ U we have: − −x = x, −(x + y) = − x · −y, −(x · y) = −x + −y,

(B.5.5) (B.5.6)

x ≤ y ⇐⇒ −y ≤ −x, −x + y =1 ⇐⇒ x ≤ y ⇐⇒ x · −y = 0.

(B.5.7) (B.5.8)

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Furthermore, in BL the two (logically equivalent) conditions below hold:   ∀x,y∈U x ≤ y ⇐⇒ ∀u∈U (u ≤ x ∧ u · y = 0 ⇒ u = 0) ,   ∀x,y∈U x  y ⇐⇒ ∃u∈U (u = 0 ∧ u ≤ x ∧ u · y = 0) .

(B.5.9) (B.5.10)

The left-to-right implication in (B.5.9) is (B.4.24). Assume therefore the truth of the right-hand side of the implication in (B.5.9). We have x · −y ≤ x and x · −y · y = x · 0 = 0. Hence x · −y = 0. Therefore, x ≤ y, by virtue of (B.5.8). (The existential quantifier ‘∃’ in “⇒” from (B.5.10) may refer, for example, to z := x · −y.) From (B.5.9) and Lemma B.4.2 we obtain: Corollary B.5.2 Either B is degenerate or U+ , ≤+  is polarised. From (B.5.9) and Lemma B.4.10, for arbitrary x, y ∈ U the following holds: y = −x ⇐⇒ y sup≤ {u ∈ U : x · u = 0},

(B.5.11)

and so we also have: − x sup≤ {u ∈ U : x · u = 0}.

(B.5.12)

B.6 Grzegorczykian Lattices B.6.1 Definitions In (1955) Grzegorczyk introduced “a [elementary] theory of Boolean algebra with the zero-element but without the unity-element” (Grzegorczyk 1955, p. 92). An analysis of Grzegorczyk’s theory brings the following problems to light: • Grzegorczyk’s theory concerns—as its models—relational structures in which we have one primitive binary relation and which are special types of lattices with zero (that is, they are not algebraic structures). • In Boolean lattices (as in Boolean algebras) there is a unity and it is definable by the binary operation of complementation. In structures that are to be models of Grzegorczyk’s theory we do not in general assume the existence of a unity, and instead of the complementation operation we find the binary difference operation (or the relative complement operation).32 • Grzegorczyk should have written that his elementary theory concerns a certain class of lattices with zero (in which the existence of a unity is not assumed). For his theory concerns both lattices with and without unity. We do not indeed find there the thesis ‘¬∃x∈U ∀u∈U u ≤ x’, which would rule out the existence of a 32 In Boolean lattices (as in Boolean algebras) the difference operation can also occur because it may be simply defined by putting x − y := x · −y. We have, for an arbitrary u ∈ U : −u = 1 · −u =: 1 − u.

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unity. In addition, all complete (including all finite) models of the theory have a unity and are complete Boolean lattices (see Corollaries B.6.12, B.7.5 and B.7.6, respectively). Before presenting his theory, Grzegorczyk appealed in footnote 14 of the article to Tarski’s work (1935) on Boolean algebras. In that work Tarski was concerned with the definition of complete Boolean algebras defined with the help of a single binary relation; that is, he was considering them as relational structures of the form U, R.33 However, Tarski was not in general concerned in that work with the definition of the class of all Boolean algebras as relational structures of the form U, R. Therefore, it remains to be seen what the connection is between a Grzegorczykian lattice and a Boolean one (which need not be a complete lattice). We cannot look to Tarski’s (1935), because there is no consideration there of Boolean algebras defined as lattices of the form U, R. We will now introduce the axioms of Grzegorczyk’s theory, using the terminology he adopts in his book. We will also be talking about structures satisfying certain conditions and not about an elementary theory whose models would be those structures.34 We will call a relational structure of the form U, ≤ a Grzegorczykian lattice iff ≤ is a binary relation on U satisfying conditions B1–B7 (which will be formulated below). Conditions B1, B2, B3 and B7 overlap with conditions (r≤ ), (antis≤ ), (t≤ ) and (∃0), respectively; that is, we assume that U, ≤ is a partially-ordered set with zero where 0 := ( x) ∀u∈U x ≤ u.35 Condition B4 has the rather complicated form given below: ι

  ∀x,y∈U x  y =⇒ ∃z∈U z ≤ x ∧ z  y ∧ ∀u,v∈U (u ≤ z ∧ u ≤ y ⇒ u ≤ v)∧    ∀w∈U w ≤ x ∧ w  y ∧ ∀u,v∈U (u ≤ w ∧ u ≤ y ⇒ u ≤ v) ⇒ w ≤ z . (B4) We will simplify this expression (to something logically equivalent) by drawing on the zero 0 and the separation relation . First of all, let us (twice) move the universal quantifier binding the variable ‘v’ to the consequent of the implication: 33 We

will present Tarski’s conclusion (Tarski 1935) in point B.7.2. Tarski proved that a structure U, R with a transitive relation R is a complete Boolean lattice iff R satisfies condition (). This corresponds to condition () on Sect. B.4.4 and Tarski’s theorem itself corresponds to Theorem B.7.3 that appears later on in this appendix and which says that () is a necessary and sufficient condition for a given lattice with zero to be a Boolean lattice. 34 Both approaches are equivalent. By ignoring the restriction ‘∈ U ’ in the descriptions of quantifiers and by treating the symbol ‘≤’ as a two-argument predicate instead of the name of a binary relation on U (remember that in accordance with our convention, the expression ‘x R y’ is an abbreviation for ‘x, y ∈ R’), we obtain a statement of the elementary axioms of Grzegorczyk’s theory. 35 Conditions (r ), (antis ), (t ) and (∃0) say respectively that the relation ≤ is reflexive, antisym≤ ≤ ≤ metric and transitive, and that there is a least element in U . We therefore also use the symbol ‘≤’ (in (Grzegorczyk 1955) the symbol ‘c’ is used). Remember that in partially-ordered sets (including lattices) with zero, that zero may not be primitive but be defined by (df 0) thanks to (∃0) and (antis≤ ).

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259

  ∀x,y∈U x  y =⇒ ∃z∈U z ≤ x ∧ z  y ∧ ∀u∈U (u ≤ z ∧ u ≤ y ⇒ ∀v∈U u ≤ v) ∧  ∀w∈U w ≤ x ∧ w  y ∧   ∀u∈U (u ≤ w ∧ u ≤ y ⇒ ∀v∈U u ≤ v) ⇒ w ≤ z .

(B4 )

Now, using the separation relation  on U , defined with the help of (df ), we obtain:   ∀x,y∈U x  y =⇒ ∃z∈U z ≤ x ∧ z  y ∧ z  y ∧   ∀w∈U (w ≤ x ∧ w  y ∧ w  y) ⇒ w ≤ z .

(B4 )

And now, by using B1, i.e., (r≤ ) (see Lemma B.3.4(iv)), we obtain: Lemma B.6.1 In an arbitrary structure U, ≤ satisfying (r≤ ), for arbitrary y, z ∈ U , the following conditions are equivalent: (a) z  y ∧ z  y, (b) z  y ∧ ¬∀u∈U z ≤ u. Therefore, from (r≤ ) and (B4 ) we obtain:   ∀x,y∈U x  y =⇒ ∃z∈U z ≤ x ∧ z  y ∧ ¬∀u∈U z ≤ u ∧   ∀w∈U (w ≤ x ∧ w  y ∧ ¬∀u∈U w ≤ u) ⇒ w ≤ z .

(B.6.1)

We therefore see that inter alia condition (s-pol≤ ) is satisfied. Corollary B.6.2 An arbitrary structure U, ≤ satisfying conditions (r≤ ) and (B4) is semi-polarised. Remark B.6.1 Although we know from Fact B.3.6 that (r≤ ) is entailed by (t≤ ) and (s-pol≤ ), we cannot say that axiom B1 is dependent on the other axioms. That is, B1 was necessary in order for us to be able to derive (s-pol≤ ) from (B4).

By using the concept of the zero, we can express condition (B4 ) in another way. To this end, let us first note the following: Lemma B.6.3 In an arbitrary partially-ordered set with zero U, ≤, 0 for arbitrary y, z ∈ U , the following conditions are equivalent: (a) z  y ∧ z  y, (b) z = 0 ∧ z  y. Proof “(a) ⇒ (b)” By virtue of (B.2.10). “(b) ⇒ (a)” Were it the case that z = 0, z  y and z ≤ y then—by virtue of (r≤ )— we would have a contradiction: z = 0, z ≤ z and z ≤ y i.e., z  y.

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It follows from the above lemma that from (B4 ) we have:   ∀x,y∈U x  y =⇒ ∃z∈U z = 0 ∧ z ≤ x ∧ z  y ∧   ∀w∈U (w = 0 ∧ w ≤ x ∧ w  y) ⇒ w ≤ z .

(B.6.2)

Remark B.6.2 Condition (B4) may be restricted to the “non-zero fragment” of the structure U, ≤. To see this, let us first transform (B.6.2) into the following equivalent form:   ∀x,y∈U+ x + y =⇒ ∃z∈U+ z ≤+ x ∧ z  y ∧ (B.6.3)   ∀w∈U+ (w ≤+ x ∧ w  y) ⇒ w ≤+ z . The move from (B.6.2) to (B.6.3) involves just a restriction of the universal quantifier. The move from (B.6.3) to (B.6.2) runs as follows: if x = 0 then the antecedent of (B.6.2) is false. If x = 0 and y = 0, then we put z := x using (r≤ ) and (B.2.10). However, if x, y ∈ U+ then we use (B.6.3). By replacing relation  with ⊥+ in (B.6.3) we get:   ∀x,y∈U+ x + y =⇒ ∃z∈U+ z ≤+ x ∧ z ⊥+ y ∧   ∀w∈U+ (w ≤+ x ∧ w ⊥+ y) ⇒ w ≤+ z .

(B.6.4)

The theorem below will come in handy later: Theorem B.6.4 In an arbitrary partially-ordered set with zero U, ≤, 0 satisfying condition (B4), the following holds:   ∀S∈2U ∀x∈U x sup≤ S =⇒ ∀y∈U \{0} y ≤ x ⇒

 ∃z∈S\{0} ∃u∈U \{0} (u ≤ z ∧ u ≤ y) .

Proof Assume that (a) x sup≤ S. Let us consider various scenarios. If x = 0 then there does not exist a y such that y = 0 and y ≤ x. If U = {0}, then x = 0 and, once again, there is not a y such y = 0 and y ≤ x. If, however, S = {0} or S = ∅, then x = 0, by virtue of (sf sup ) and (B.2.24) respectively, and so again there is not a y such that y = 0 and y ≤ x. In each of these cases, the sentence holds, because the universal has an empty range of application. Let us assume therefore that U = {0}, x = 0 and ∅ = S = {0}. Furthermore, let us assume for a contradiction that there is a y0 such that (b) 0 = y0 ≤ x, and that for an arbitrary z ∈ S \ {0} there does not exist a u such that u = 0, u ≤ z and u ≤ y0 . Hence, for an arbitrary z ∈ S \ {0} we have (c): z  y0 .

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261

Now, notice that (d) y0 = x. Indeed, were it the case that y0 = x then, thanks to (a), (c) and (r≤ ), for some z 0 we would have 0 = z 0 ≤ x and z 0  x, since S  {0} = ∅. This is a contradiction, by virtue of Lemma B.3.4(iv). Therefore, x  y0 , thanks to (b), (d) and (antis≤ ). Hence, by virtue of (B.6.2), there is a u 0 such that (e) 0 = u 0 ≤ x, (f) u 0  y0 and (g) for an arbitrary w ∈ U \ {0}: if w ≤ x and w  u 0 , then w ≤ u 0 . From (a), (c) and Lemma B.3.4(vi) we obtain the following: ∀z∈S (z ≤ x ∧ z  y0 ). Hence, thanks to (g), we have ∀z∈S z ≤ u 0 . Therefore, x ≤ u 0 , thanks to (a). Hence, x = u 0 , by virtue of (e) and (antis≤ ). Therefore, by applying (b) and (f), we obtain: 0 = y0 ≤ u 0 and u 0  y0 . This is a

contradiction, by virtue of Lemma B.3.4(iv).36 For arbitrary x, y ∈ U let us define a set which will be of use in defining the difference operation: R yx := {u ∈ U : u ≤ x ∧ u  y}.

Lemma B.6.5 For arbitrary x, y ∈ U : (i) (ii) (iii) (iv)

0 ∈ R yx . R yx = {0} ⇐⇒ x ≤ y. There exists exactly one z such that z is the greatest in the set R yx . There exists exactly one z such that z ∈ R yx and z sup≤ R yx .

Proof Ad (i) By virtue of (B.2.10), 0 ≤ x; and by virtue of (antis≤ ) and our definitions, 0  y. Ad (ii) If x  y then R yx = {0}, by (B.6.2). If, however, x ≤ y, u ∈ R yx and u = 0, then u ≤ y, by (t≤ ). So, by virtue of (r≤ ), we have a contradiction: ¬u  y. Ad (iii) If x ≤ y, then we apply (ii). If, however, x  y then, by virtue of (B.6.2) and (B.2.10), the set R yx has a greatest element. In virtue of (antis≤ ) there may be only one such element. Ad (iv) From (iii) and (B.2.13).

We can therefore define the following binary difference operation on U by putting for arbitrary x, y ∈ U : x − y := ( z) z is the greatest element in R yx ι

= ( z) (z ∈ R yx ∧ z sup≤ R yx ) ι

= ( z) z sup≤

(df −)

R yx .

ι

The first equivalence we obtain from (B.2.13) and the second from Lemma B.6.5(iv) and (fsup ). Therefore—in accordance with what Grzegorczyk (1955) says—condition 36 It

follows from the sentence we have just proven that the relation sup≤ is included in a certain expansion of the relation is a mereological sum of. Lemma B.3.9 and Corollary B.3.10 address the converse inclusion; see also footnote 27.

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(B4) guarantees the existence of a difference for any arbitrary two sets in the structure.37 Let us take a look at the fundamental properties of the difference operation. Lemma B.6.6 For arbitrary x, y ∈ U : (i) (ii) (iii) (iv) (v) (vi) (vii)

x − y = 0 ⇐⇒ x ≤ y. x − x = 0, 0 − y = 0 and x − 0 = x. x − y ≤ x. ∀z∈U (z ≤ x − y ⇐⇒ z ∈ R yx ⇐⇒ z ≤ x ∧ z  y). x ≤ x − y ⇐⇒ x = x − y ⇐⇒ x  y. x − (x − y) ≤ y. ∀z∈U (z ≤ x ∧ z  x − y =⇒ z ≤ y).

Proof Ad (i) By virtue of Lemma B.6.5 and the definition of the difference operation. Ad (ii) By virtue of (i) and (r≤ ), we have x = x − 0. By virtue of Lemma B.6.5(ii) and (B.2.10), we have R 0y = {0}. Hence 0 − y = 0. In conclusion, R0x = {u ∈ U : u ≤ x}, and so x sup≤ R0x . Furthermore, from our definition, x − 0 sup≤ R0x ; and therefore x = x − 0. Ad (iii) Since x − y belongs to R yx , then x − y ≤ x. Ad (iv) Assume that z ≤ x − y. Then z ≤ x, by virtue of (iii) and (t≤ ). If u ≤ z and u ≤ y, then u ≤ x − y, by virtue of (t≤ ) and our assumptions. Therefore u = 0, since x − y  y. Hence z  y too. The converse: from the definition of the difference operation. Ad (v) In virtue of (iii) and (antis≤ ): x ≤ x − y ⇐⇒ x = x − y. Let us therefore assume that x = x − y. If x  y then x ∈ R yx . If, however, x ≤ y then x = x − y = 0. In both cases we have x  y. Conversely, assume that x  y. Then x ∈ R yx , by virtue of (r≤ ). Therefore, x ≤ x − y, since x − y sup≤ R yx . Ad (vi) Assume for a contradiction that x − (x − y)  y. Then there exists a z such that z = 0, z ≤ x − (x − y) and z  y. Hence by virtue of (iv), we have (iv), z ≤ x and z  x − y. Furthermore, z ∈ R yx . Therefore, z ≤ x − y, and so we have obtained our contradiction. Ad (vii) Let us assume that z ≤ x and z  y. Then, by virtue of (B4), there exists a u such that u = 0, u ≤ z and u  y. Therefore, by virtue of (t≤ ), u ≤ x as well. Hence u ∈ R yx . We therefore have u ≤ x − y. We have therefore obtained the following: z x − y.

We will now prove a theorem from which it will follow that condition B6, which Grzegorczyk (1955) takes as an axiom, is inessential, because it is dependent on conditions B1–B4 and B7. Theorem B.6.7 In an arbitrary partially-ordered set with zero U, ≤, 0 satisfying condition (B4), for arbitrary x, y ∈ U : 37 No operations are introduced in (Grzegorczyk 1955). All that we find after the axioms have been given are the following words: “According to B.4–B.6 there exists the difference (sum and product) for any two elements” (Grzegorczyk 1955, p. 92).

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263

(i) x − (x − y) = 0 ⇐⇒ x  y. (ii) x − (x − y) inf≤ {x, y}. Proof Ad (i) From the definition of the difference operation: : x − (x − y) = 0 iff x ≤ x − y iff x  y, by virtue of Lemma B.6.6(v). Ad (ii) By virtue of Lemma B.6.6(iii), (vi), x − (x − y) is a lower bound of the set {x, y}. We will show that it is the greatest lower bound. Let us assume for a contradiction that z ≤ x, z ≤ y and z  x − (x − y). Then, by virtue of (B4), there exists a u such that u = 0, u ≤ z and u  x − (x − y). Hence, by virtue of (t≤ ), u ≤ x y. However, by applying and u ≤ y as well. Hence, by virtue of (r≤ ), we have u  Lemma B.6.6(vii), we get u ≤ x − y. Hence, by virtue of Lemma B.6.6(iv), we have our contradiction: u  y.

Corollary B.6.8 In an arbitrary partially-ordered set with zero U, ≤, 0 satisfying condition (B4): ∀x,y∈U ∃z∈U z inf≤ {x, y},   ∀x,y∈U ∃z∈U z ≤ x ∧ z ≤ y ∧ ∀u∈U (u ≤ x ∧ u ≤ y ⇒ u ≤ z) .

(B6)

Proof The first sentence follows from Theorem B.6.7(ii). We obtain (B6) from the first sentence and (df inf≤ ) applied to the set {x, y}.

Sentence (B6) is adopted as an axiom in (Grzegorczyk 1955). Therefore, this axiom is dependent on axioms B1 and B2–B4. In the light of Theorem B.6.7(ii) and Corollary B.6.8 we may define the binary operation · of product, by putting for arbitrary x, y ∈ U : x · y := ( z) z inf≤ {x, y} = x − (x − y). ι

We can show in the usual way that the operation · satisfies condition (B.4.8)–(B.4.10), i.e., it is commutative, associative and idempotent. Condition (B.4.13) also holds. Let us now switch our attention to condition B5, which has the following form:   ∀x,y∈U ∃z∈U x ≤ z ∧ y ≤ z ∧ ∀u∈U (x ≤ u ∧ y ≤ u ⇒ z ≤ u) .

(B5)

From (B5) and (df sup≤ ) we get that, for arbitrary x and y in U , there exists a supremum for the set {x, y}. Therefore, (B5) takes the following form: ∀x,y∈U ∃z∈U z sup≤ {x, y}.

(B5 )

We can therefore define the binary operation + of sum by putting for all x, y ∈ U : x + y := ( z) z sup≤ {x, y}. ι

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We can once again show in the usual way that + satisfies conditions (B.4.8)–(B.4.10), i.e., it is commutative, associative and idempotent. Condition (B.4.12) also holds. Furthermore, with the operations + and · taken together we obtain conditions (B.4.11) and (B.4.14)–(B.4.17). Conditions (B5) and (B6) together yield condition (L). So, we obtain that each structure U, ≤ satisfying B1–B7 is a lattice, and since we also have condition B7, it is a lattice with zero. Therefore—once again, following Grzegorczyk’s own words (1955, p. 92)—conditions (B5) and (B6) guarantees the existence of a sum and product respectively for arbitrary pairs of elements. Let GL be the class of all Grzegorczykian lattices. The negation of condition (∃1) does not follow from B1–B7. This is shown by the fact that there exist Grzegorczykian lattices that satisfy (∃1): there exist finite Grzegorczykian lattices and all finite lattices are bounded (see Fact B.4.4). Furthermore, as we will show below (see Corollary B.6.12), an arbitrary finite structure is a Grzegorczykian lattice iff it is a Boolean lattice. Hence it also follows that GL  L0, because we have finite lattices which are not Boolean lattices. Furthermore, let GL1 be the class of all Grzegorczykian lattices with unity, i.e., structures form GL that satisfy condition (∃1). We will prove below that GL1 = BL (see Theorem B.6.11). It is harder to find an example of a Grzegorczykian lattice without unity. It has to be an infinite structure. Example B.6.1 Take an arbitrary infinite set S. Let U be the set of all finite subsets of S and ≤ := {x, y ∈ U × U : x ⊆ y}. Then U, ≤, ∅ is a Grzegorczykian lattice without unity. The sum, product and difference of finite sets are finite sets and thus B1–B7 are satisfied. Furthermore, only S contains all its own finite subsets but S ∈ / U.

From Theorem B.6.4, for an arbitrary G = U, ≤, 0 from GL we have:   x∨z y) . ∀x,y,z∈U \{0} z ≤ x + y =⇒ (z 

(B.6.5)

From this and from Corollary B.6.2 we get that in G also holds:   ∀x,y,z∈U z ≤ x + y ∧ z  x =⇒ z ≤ y .

(B.6.6)

If z = 0 then z ≤ y. If x = 0 then x + y = y. If y = 0 then x + y = x. Therefore, our assumption z ≤ x + y and z  x reduces to z ≤ x and z  x. This, however, yields z = 0, by virtue of (r≤ ). Let us therefore assume for a contradiction that for some x, y, z ∈ U \ {0} we have z ≤ x + y, z  x and z  y. Then, by virtue of (s-pol≤ ), there is a u such that u = 0, u ≤ z and u  y. Hence, by applying (t≤ ), we have u  x and u ≤ x + y. From the second of these facts, by virtue of (B.6.5), we obtain our contradiction: u  x or u  y. It follows from (B.6.5) that in G (as in all distributive lattices; see below Theorem B.6.9) the converse implication to that of (B.4.19) holds. We therefore have:   ∀x,y,z∈U z  x + y ⇐⇒ (z  x ∧ z  y) .

(B.6.7)

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265

If x = 0, then z  x and x + y = y, i.e., z  x + y ⇐⇒ z  y. The case is similar when y = 0. Therefore, let us assume that x = 0 = y and z  x + y. Then z = 0 and there is a u such that u = 0, u ≤ z and u ≤ x + y. Then from (B.6.5) we have z x or z  y. By applying (B.6.6), (B.6.7) and Corollary B.6.2, we obtain the following38 : Theorem B.6.9 Every Grzegorczykian lattice is distributive. Proof We will show that, for an arbitrary latticeG from GL, condition (B.4.25) holds. Assume for a contradiction that for some x, y, z ∈ U we have (x + y) · (x + z)  x + (y · z). Since G is semi-polarised (see Corollary B.6.2), for some u we have u = 0, u ≤ (x + y) · (x + z) and u  x + (y · z). Hence, by virtue of (B.4.6), we have u ≤ x + y and u ≤ x + z; whereas by virtue of (B.6.7), we have u  x and u  y · z. Hence, by virtue of (B.6.6), we have u ≤ y and u ≤ z. Therefore, by applying (B.4.6) once more, we have u ≤ y · z. However, this taken with u = 0 and u ≤ u, yields a contradiction: u  y · z.

B.6.2 Grzegorczykian Lattices Versus Boolean Lattices We will prove that the class of Grzegorczykian lattices with unity coincides with the class of Boolean lattices. Hence will obtain the result that every finite Grzegorczykian lattice is a Boolean lattice. The former fact will follow from Theorem B.6.9 and the fact below on complementation. Fact B.6.10 For an arbitrary Grzegorczykian lattice with unity U, ≤, 0, 1, each element of U has a complement, i.e.:   ∀x∈U ∃ y∈U x · y = 0 ∧ x + y = 1 . Proof It suffices to show that, for any x ∈ U : x · (1 − x) = 0 and x + (1 − x) = 1. By virtue of (B.4.22), we have: x · (1 − x) = 0 iff x  1 − x. Assume that x  1 − x. Then there is a u such that u = 0, u ≤ x and u ≤ 1 − x. From this last result, by virtue of Lemma B.6.6(iv), we have u  x. However, this together with u = 0 and u ≤ u, yields a contradiction: u  x. Therefore, x · (1 − x) = 0. Assume for a contradiction that 1  x + (1 − x). Then, by virtue of (s-pol≤ ), there is a u such that u = 0, u ≤ 1 and u  x + (1 − x). Then, by virtue of (B.6.7), we have u  x and u  1 − x. From this last result, by virtue of Lemma B.6.6(vii), we have u ≤ x. This together with u = 0 and u ≤ u, yields a contradiction: u  x.

Theorem B.6.11 GL1 = BL, i.e., a given structure is a Grzegorczykian lattice with unity iff it is a Boolean lattice. 38 Although

order.

(B.6.7) holds in all distributive lattices, we had to prove the theorems in a different

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Proof “⇒” If a given lattice belongs to GL1, then it is bound, distributive (see Theorem B.6.9) and each element has a complement (see Fact B.6.10). It is therefore a Boolean lattice. “⇐” Assume that L = U, ≤, 0, 1 belongs to BL. We know that L is bounded: i.e., that conditions (∃0), (∃1) and (L) hold. Therefore, conditions B1–B3 and B5–B7 are satisfied. We define in this lattice the binary operation · of product and the unary complement operation −. By (B.4.2), (B.5.8), (B.5.3), for all x, y ∈ U we have: x · −y ≤ x, x  y ⇐⇒ x · −y = 0, ∀u∈U (u ≤ x · −y ∧ u ≤ y ⇒ u = 0). Hence, we obtain conditions (B.5.9) and (B.5.10). Now assume that w = 0, w ≤ x and ∀u∈U (u ≤ w ∧ u ≤ y ⇒ u = 0). Then w · y = 0, thanks to (B.4.22). Hence w ≤ −y, thanks to (B.5.5) and (B.5.8). By virtue of (B.4.8), (B.4.10) and (B.4.17), we have w = w · w ≤ x · w = w · x ≤ −y · x = x · −y. Thus, z := x · −y satisfies (B4).

Corollary B.6.12 Every finite Grzegorczykian lattice is a Boolean lattice. Proof In the light of Fact B.4.4, every finite lattice is bounded and so has a unity. Therefore, by virtue of the left-to-right direction of Theorem B.6.11, each finite Grzegorczykian lattice is a Boolean lattice.

In the light of Fact B.6.10 and its proof, Theorem B.6.11 and the general principles of definition for the complement operation in Boolean lattices (see Sect. B.5.2), we see that, in an arbitrary Grzegorczykian lattice L = U, ≤, 0, 1 with unity, we can define the operation of complementation by putting for an arbitrary x ∈ U : −x := 1 − x = ( y) (x · y = 0 ∧ x + y = 1). ι

In the other direction, by drawing on the right-to-left direction of the proof of Theorem B.6.11, we see that in an arbitrary Boolean lattice we can define the binary operation of difference by putting for arbitrary x, y ∈ U : x − y := x · −y. The identity GL1 = BL shows that there is a great difference between Grzegorczykian lattices with unity and Grzegorczykian lattices without unity. Remark B.6.3 Let U, ≤, 0 be a non-degenerate Grzegorczykian lattice without unity. It is not possible to find an element 1 outside of the set U such that the partiallyordered set U, ≤, 0, 1 would be a Grzegorczykian lattice with unity in which U ∗ := U  {1} and ≤∗ := ≤ (U ∗ × {1}); that is, in which ≤∗ would be a binary relation on U that would be an expansion of relation ≤ such that 1 would be the unity in U ∗ , ≤∗ .

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In other words, the relation ≤∗ would have to satisfy the following condition (for arbitrary x, y ∈ U ∗ )39 : x ≤∗ y :⇐⇒ x ≤ y ∨ y = 1. This occurs for the following reason. For arbitrary x, y ∈ U we have x + y ∈ U . Therefore, for an arbitrary x ∈ U \ {0} we do not find a y ∈ U ∗ that would be the complement of x. Indeed, for y ∈ U we have x + y = 1, whereas for y = 1 we have x · y = x = 0. We remember that were U ∗ , ≤∗ , 0, 1 a Grzegorczykian lattice with

unity, then it would also be a Boolean lattice as a consequence.40

B.7 Completeness of Structures B.7.1 Complete Partial Orders—Complete Lattices We say that a partially-ordered set P = U, ≤ is complete iff each subset of the universe U has a supremum in U : ∀S∈2U ∃x∈U x sup≤ S.

(complsup )

By virtue of (B.2.15) and (B.2.16), condition (complsup ) is equivalent to the following: (complinf ) ∀S∈2U ∃x∈U x inf≤ S. Therefore P is complete iff condition (complinf ) is satisfied. Fact B.7.1 For an arbitrary partially-ordered set P the following conditions are equivalent: (a) P is complete, (b) P is a complete lattice. the lattice U, ≤ is degenerate, i.e. if U = {0}, then we obtain in the manner described above the two-element Boolean lattice U ∗ , ≤∗ , in which U ∗ = {0, 1 and ≤∗ is a reflexive relation on U ∗ such that 0 ≤∗ 1 and 1 ∗ 0. Let us also observe that the non-degenerate lattice U, ≤, 0 cannot arise from some Grzegorczykian lattice with unity U1 , ≤1 , 0, 1 (i.e., a Boolean lattice), by “discarding” that unity (i.e., when U = U1 \ {1} and ≤ is the restriction of the relation ≤1 to U ). Indeed, we have 0 = 1 and for some x ∈ U it the case that 0 = x = 1, −x = 0 and x + −x = 1. Therefore, by “discarding” the unity, the structure is no longer a lattice. 40 This is well illustrated by the structure given in Example B.6.1, in which U is the set of all subsets of the infinite set S and ≤ is the relation of inclusion. For arbitrary finite sets x and y, their set-theoretic sum x + y is also a finite set, i.e., one that is belongs to U . Therefore, no matter how we add element 1 which is to be greater than all the elements of U , the identities x · y = ∅ and x + y = 1 would not be satisfied, for x ∈ U \ {∅} and y ∈ U  {1} (because under the assumptions we made x · 1 = x = ∅ and 1 = x + y ∈ U , for y = 1). 39 If

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Proof P is complete then it satisfies both conditions (complsup ) and (complinf ) which entail condition (L). On the other hand, every lattice is a partially-ordered set.

Fact B.7.2 Every finite lattice is complete. Proof Let L = U, ≤ be a finite lattice. Since L satisfies condition (L ) and the set U is non-empty and finite, there exists an x such that x inf≤ U . By virtue of (B.2.24), we have x sup≤ ∅. This and condition (L ) show that L also satisfies condition

(complsup ). Remark B.7.1 The concept of completeness can also be applied in the context of an algebraic structure on U that has the operation + : U × U → U (resp. · : U × U → U ), so long as the relation ≤ defined by the condition ‘∀x,y∈U (x ≤ y ⇔ x + y = y)’ (resp. ‘∀x,y∈U (x ≤ y ⇔ x · y = x)’) partially orders U . We then say that such a structure U, +, . . . (resp. U, ·, . . .) is complete iff the partially-ordered set U, ≤ is complete. The concept of completeness can therefore be applied to lattices (as relational structures) and to Boolean algebras.

The following sentences hold, in the light of condition (complsup ) : ∃x∈U x sup≤ U, ∃x∈U x sup≤ ∅. We therefore know, by (B.2.23) and (B.2.24), that in an arbitrary complete partiallyordered set P there exist a unity and a zero. It is customary to designate them by ‘1’ and ‘0’. We therefore have: 1 = ( x) x sup≤ U, 0 = ( x) x sup≤ ∅. ι ι

P is therefore bounded. By using (B.2.23) and (B.2.24) once more, we have: 1 = ( x) x inf≤ ∅, 0 = ( x) x inf≤ U. ι ι

We could have equally used condition (complinf ) to obtain: ∃x∈U x inf≤ ∅, ∃x∈U x inf≤ U, and then by using them along with (B.2.23) and (B.2.24) we would have obtained the four identities given above them. In conclusion, we can say that an arbitrary complete partially-ordered set is a complete bounded lattice. In the lattice L = U, ≤, in virtue of conditions (complsup ),

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(fsup ), (complinf ) and (finf ), in addition to the operations +, · : U × U → U , we have two functions on the set 2U which take values in the set U : the operations of supremum sup and infimum: inf: sup(S) := ( x) x sup≤ S,

(df sup)

inf(S) := ( x) x inf≤ S.

(df inf)

ι

ι

We will show that by using these operations in a complete lattice, conditions (B.4.14) and (B.4.15) can be generalised, taking on the following forms: ∀x∈U ∀S∈2U sup{x · z : z ∈ S} ≤ x · sup S, ∀x∈U ∀S∈2U x + inf S ≤ inf{x + z : z ∈ S}.

(B.7.1) (B.7.2)

For (B.7.1): from the definition of a supremum, ∀z∈S z ≤ sup S. Hence, by virtue of (B.4.17), ∀z∈S x · z ≤ x · sup S. Therefore, x · sup S is an upper bound of the set {x · z : z ∈ S}. From this, (df sup≤ ) and (df sup) we obtain (B.7.1). By using (B.4.16), (df inf≤ ) and (df inf), we prove (B.7.2). In an analogous way. Furthermore, by using the operations sup and inf in a complete lattice we can write distributivity conditions (B.4.29) and (B.4.30) as follows: ∀x∈U ∀S∈2U x · sup(S) = sup{x · z : z ∈ S},

(B.7.3)

∀x∈U ∀S∈2U x + inf(S) = inf{x + z : z ∈ S}.

(B.7.4)

It is well-known that in every complete lattice, each of the four conditions (B.4.27), (B.4.28), (B.7.3) and (B.7.4) entails the other three.

B.7.2 Complete Boolean Lattices (Boolean Algebras). Tarski’s Theorem A Boolean lattice U, ≤, 0, 1 is complete iff it is complete as a partially-ordered set. Let CBL be the class of all complete Boolean lattices. Similarly, the Boolean algebra U, +, ·, −, 0, 1 is complete iff the partially-ordered set U, ≤ that we obtain from it is complete, where x ≤ y ⇐⇒ x + y = y, dla x, y ∈ U , for all x, y ∈ U (cf. Remark B.7.1). Since finite Boolean lattices exist and each finite lattice is complete, CBL = ∅. Furthermore, we have BL  CBL, because there are known examples of incomplete Boolean lattices. Example B.7.1 Let FC(IN) be a family of all finite and all co-finite (i.e., finitely complemented) subsets of the set of natural numbers IN. The family FC(IN) is a

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field of sets41 and so FC(IN), ⊆, ∅, IN is a Boolean lattice.42 It is an incomplete lattice because (e.g.) the family E of all finite sets composed of even numbers does not  have a supremum in the lattice FC(IN),  ⊆, ∅, IN. pf : {{n} : n ∈ IN} ⊆ FC(IN), E is the set of all even numbers but E ∈ / FC(IN), and only this set could be the

supremum of the family E in the lattice FC(IN), ⊆, ∅, IN.43 In his (1935), Tarski examined the equivalence of various sets of axioms of complete Boolean lattices (complete Boolean algebras). We will introduce the axiom he obtained later in this section, in Theorem B.7.4. The result of Tarski is related to Theorem B.7.3, which for gives a necessary and sufficient condition lattices with zero to be complete Boolean lattices. In the proof of Theorem B.7.3 we will not have to use Theorem B.7.4 at all. We need only those facts already established in this appendix. Theorem B.7.3 (Pietruszczak 2000, 2018) Let L = U, ≤, 0 be a lattice with zero. Then L is a complete Boolean lattice iff relation ≤ satisfies condition () from Sect. B.4.4. Proof “⇒” Take an arbitrary S ∈ 2U and put x := sup S. Then, by virtue of Lemma B.4.7, x satisfies, for the set S, conditions (a) and (b) from () respectively. To finish, we will show that x is the only element in U satisfying, for the set S, conditions (a) and (b) from (). Assume that y ∈ U also satisfies, for the set S, conditions (a) and (b) from (). Since L is semi-polarised, then from Corollary B.3.8 we obtain the following: x ≤ y and y ≤ x.44 Hence x = y. “⇐” Let L = U, ≤, 0 be a lattice with zero in which condition () holds. By virtue of Lemma B.4.3, L is semi-polarised. We next prove that L is a complete lattice. Take an arbitrary S ∈ 2U . By virtue of our assumption, there exists (for S) exactly one x0 ∈ U satisfying conditions (a) and (b) from (). Take an arbitrary y ∈ U that is the supremum for the set S; and so ∀z∈S z ≤ y. From this and from (b) for x0 and S, by virtue of Corollary B.3.8, we have x0 ≤ y. This together with (a) for x0 and S shows that x0 sup≤ S. We therefore have x0 = sup(S). Furthermore, it follows from the above that sup S is the only element in U which satisfies (for S) conditions (a) and (b) from (). We therefore have:   ∀u∈U u ≤ sup S ∧ ∀z∈S u · z = 0 =⇒ u = 0 .

(b )

It follows from the arbitrary choice of S that L is complete. Therefore, L is also bounded: 1 = sup(U ). call a family F of subsets of a set S a field of sets iff S ∈ F and for arbitrary X, Y ∈ F : X  Y ∈ F and S \ X ∈ F . Therefore, ∅ ∈ F as well for arbitrary X, Y ∈ F : X  Y ∈ F . 42 In saying that the family of sets F is a lattice, we are taking the relation ≤ := {X, Y  ∈ F × F : X ⊆ Y } as a partial order (as in Example B.6.1). We will write this lattice as F , ⊆, . . . for short. 43 If F is a field of subsets of a set S such that, for each z ∈ S, {z} ∈ F , then for any Y ⊆ F and  X ∈ F : X sup⊆ Y iff X = Y (see Frankiewicz and Zbierski 1992, p. 11). 44 For lattices with zero, the sentence from Corollary B.3.8 reduces to:     ∀S∈2U ∀x∈U ∀z∈S z ≤ y ∧ ∀u∈U (u ≤ x ∧ ∀z∈S z · u = 0) ⇒ u = 0 =⇒ x ≤ y . 41 We

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We now prove distributivity condition (B.7.3). Let x ∈ U and let S ∈ 2U . By virtue of (B.7.1), we have sup{x · z : z ∈ S} ≤ x · sup S. It therefore suffices to show the converse inequality. We put x S := sup{x · z : z ∈ S}. It follows directly from the definition of a supremum that (A) ∀u∈S x · u ≤ x S . Assume for a contradiction that x · sup S  x S . Hence, by virtue of (iv), there exists a v such that (B) v = 0, (C) v ≤ x · sup S and (D) v · x S = 0. We obtain from (A), by virtue of (B.4.17), (i) ∀u∈S v · x · u ≤ v · x S . From this and from (D), we obtain (ii) ∀u∈S v · x · u = 0. From (C) it follows that v = v · x · sup S. Hence x · v = x · v · x · sup S = x · v · sup S. So we have (iii) x · v ≤ sup S. Now, by applying (b ), (ii), (iii) and (C) we obtain 0 = x · v = v. This contradicts (B). To finish, since L is a complete lattice, for an arbitrary x ∈ U the set {z ∈ U : x · z = 0} has a supremum which is, by Lemma B.4.10, the complement of x.

The result of Tarski from (1935), which concerns the equivalence of different sets of axioms for complete Boolean algebras, may be introduced in the form of the following theorem (employing the terminology we have been using in this appendix): Theorem B.7.4 (Tarski 1935, Theorems 1 and 2) Let U be a an arbitrary nonempty set and R be arbitrary transitive relation on U . Then, for R to be reflexive and antisymmetric, and for the partially-ordered set U, R to be a complete Boolean lattice, it is both necessary and sufficient that R satisfies the following condition: For an arbitrary S ∈ 2U there exists exactly one x ∈ U such that (a) ∀z∈S z R x and  (b) ∀u∈U u R x ∧ ∀z∈S ∀w∈U (w R u ∧ w R z ⇒ ∀v∈U w R v) =⇒ ∀v∈U u R v . () Condition () has a somewhat intricate form. This is a consequence of the fact that it has been formulated using only the relation R (whose transitivity was earlier assumed). Were we to take the lattice U, ≤, 0 with zero in place of U, R, then condition () would reduce to condition () from Theorem B.7.3.45 To see this, first observe that, with condition () thus modified, point (a) from condition () is simply point (a) from condition (). Next, observe that the formulas ‘∀v∈U w ≤ v’ and ‘∀v∈U u ≤ v’ just define the zero 0 in the lattice U, ≤ and therefore say the same thing as the formulas ‘w = 0’ and ‘u = 0’, respectively. Thanks to this, the formula: • ∀w∈U (w ≤ u ∧ w ≤ z ⇒ ∀v∈U w ≤ v) takes on the form: • ∀w∈U (w ≤ u ∧ w ≤ z ⇒ w = 0). This may be simplified to B.7.4 says that the pair U, R in which relation R is transitive and in which condition () holds, is a bonded lattice. 45 Theorem

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• u·z =0 thanks to condition (B.4.22), which holds in all lattices with zero. We therefore obtain:   • ∀u∈U u ≤ x ∧ ∀z∈S u · z = 0 =⇒ u = 0 , that is, point (b) of () for lattices with zero reduces to point (b) of (). It can been see from the above that Theorem B.7.3 follows from Theorem B.7.4.

B.7.3 Complete Grzegorczykian Lattices Versus Complete Boolean Lattices We know from point B.7.1 that each complete Grzegorczykian lattice is bounded, i.e., that it has a unity. Let CGL be the class of all complete Grzegorczykian lattices. We then have CGL ⊆ GL1 = BL, by virtue of Theorem B.6.11. Hence, since CGL = CGL1, we also have CGL = CGL1 = CBL. Corollary B.7.5 CGL = CBL, i.e., a given structure is a complete Grzegorczykian lattice iff it is a complete Boolean lattice. Furthermore, from Corollary B.6.12 and Fakt B.7.2 we obtain the following: Corollary B.7.6 Every finite Grzegorczyk lattice is a complete Boolean lattice.

B.8 Atoms, Atomicness and Atomisticness B.8.1 General Definition of an Atom Let P = U, ≤ be a partially-ordered set. We now give the definition of the concept of being an atom which will be independent of whether P has or does not have a zero. An P is an arbitrary element such that it is not the zero and such that there is no element smaller than it which is not the zero. Let us therefore introduce the following predicate:   ∀x∈U x is an atom :⇐⇒ ¬x is a zero ∧ ¬∃u∈U (u  x ∧ ¬u is a zero) . (df atom) Let At be the set of all atoms in P. Obviously, this may the empty set (regardless of whether P has or does not have a zero). The definition is easily seen to be logically equivalent to both of the following:   ∀x∈U x ∈ At ⇐⇒ x is not a zero ∧ ∀u∈U (u  x ⇒ u is a zero) ,   ∀x∈U x ∈ At ⇔ x is not a zero ∧ ∀u∈U (u ≤ x ⇒ u is a zero ∨ x = u) .

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B.8.2 Atoms in Partially-Ordered Sets with Zero Let P = U, ≤, 0 be a partially-ordered set with zero. Then the definition of the concept being an atom can be given the following shorter form:  ∀x∈U x ∈ At ⇐⇒ x = 0 ∧ ¬∃u∈U (u  x ∧ u = 0) ⇐⇒ x = 0 ∧ ∀u∈U (u  x ⇒ u = 0)



(B.8.1)

⇐⇒ x = 0 ∧ ∀u∈U (u ≤ x ⇒ u = 0 ∨ u = x) . If P is a non-degenerate structure, i.e., U = {0} and 0 = 1, then P does not have any atoms.

B.8.3 Atoms in Non-degenerate Partially Ordered Sets Without Zero Let P = U, ≤ be a partially-ordered set without zero. We then obtain: At = {x ∈ U : ¬∃u∈U u  x} =: min≤ (U ), This is because the definition of the concept being an atom may be shortened in the following way:   ∀x∈U x ∈ At ⇐⇒ ¬∃u∈U u  x ⇐⇒ ∀u∈U (u ≤ x ⇒ u = x) .

B.8.4 Atomistic and Atomic Partial Orders We say that a partially-ordered set P = U, ≤ is atomistic iff an arbitrary non-zero element from it is a supremum of the set Atx := {a ∈ At : a ≤ x}. Fact B.8.1 The two conditions below are equivalent: (a) P is atomistic,   (b) ∀x,y∈U ∀a∈At (a ≤ x ⇒ a ≤ y) =⇒ x ≤ y . Proof For “(a) ⇒ (b)” Take arbitrary x, y ∈ U . If x is a zero then (b) holds. Let us suppose, therefore, that x is not a zero and assume the antecedent of the implication in (b), i.e., ∀a∈At (a ≤ x ⇒ a ≤ y). Since x sup≤ Atx , then ∀a∈Atx a ≤ x; and so ∀a∈Atx a ≤ y. Therefore, x ≤ y, by the definition of a supremum. For “(b) ⇒ (a)” Take an arbitrary x ∈ U such that x is not a zero. It is an upper bound of the set Atx . If y is also an upper bound of the set Atx , then the antecedent

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of the implication in (b) is satisfied. Therefore x ≤ y. Hence x sup≤ Atx , by the definition of a supremum.

From this and (t≤ ) we obtain: Corollary B.8.2 If P is atomistic then   ∀x,y∈U x ≤ y ⇐⇒ ∀a∈At (a ≤ x ⇒ a ≤ y) . We say that P is atomic iff every non-zero element in P is either an atom or some atom is smaller than it: i.e., ∀x∈U (x is a zero ∨ x ∈ At ∨ ∃a∈At a  x), that is, ∀x∈U (x is not a zero ⇒ ∃a∈At a ≤ x). Fact B.8.3 If P is atomistic then it is atomic. Proof Take an arbitrary non-zero element x from U . If x sup≤ Atx then Atx = ∅, by virtue of (B.2.24).

Fact B.8.4 If P is atomic and polarised, then it is atomistic. Proof Let x be an arbitrary non-zero element from U . Then x is an upper bound of the set Atx . Let us suppose that y is also an upper bound of the set At x but that x  y.  Then, by virtue of (pol≤ ), there exists a z such that z ≤ x and ¬∃u∈U (u ≤ z ∧ u ≤ y) . Since P is atomic, there exists an a ∈ At such that a ≤ z. Hence, a ∈ Atx , by virtue of (t≤ ). We therefore also have a ≤ y, which is to say that we have reached a contradiction.

Therefore, x ≤ y, and this gives us x sup≤ Atx . In the case where P is not atomic, we may equally say that it is non-atomic. We say that P is atomless iff there are no atoms in its universe. If P is atomless, then it is also non-atomic but the converse need not follow. Finally, we obtain the following in an obvious way: Fact B.8.5 If P is atomless, then either it is degenerate or it has an infinite universe.

B.9 Quasi-orders Let R be a binary relation on a set U . We say that R is a quasi-order on U iff R is reflexive and transitive. We then call the structure U, R a quasi-ordered set. Let QOS be the class of all quasi-ordered sets. Clearly, each partial order on U is also a quasi-order on U . There exist, however, proper quasi-orders on U , that is, quasi-orders which are not antisymmetric. Hence POS  QOS. By virtue of Lemma A.3.10(i) and Corollary A.3.16(i, ii), respectively, if R is not antisymmetric, then R ◦ is not asymmetric and R   R ◦ , i.e., R \ R˘  R \ idU . It will be useful to adopt the following convention henceforth: if a relation R is a quasi-order on U , then this will be denoted by the symbol ‘’; and, conversely, an

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arbitrary relation denoted by the symbol ‘’ quasi-orders U .46 Under this convention, the following two sentences are true:

∀x,y,z∈U



∀x∈U x  x,

(r )

 x  y ∧ y  z =⇒ x  z .

(t )

By virtue of Lemma A.3.2(ii), they are in turn both equivalent to the following condition:   ∀x,y∈U x  y ⇐⇒ ∀z∈U (z  x ⇒ z  y) . Let < be the “asymmetricisation” of the relation , i.e. < :=  . This means that < is the set-theoretic difference of the relations  and , i.e. < :=  \ . We may express this with the following sentence:



  ∀x,y∈U x < y ⇐⇒ x  y ∧ y  x .

(df