The Resemblance Structure of Natural Kinds A Formal Model for Resemblance Nominalism Javier Belastegui Lazcano Supervisors: Dr. Thomas Mormann Dr. Jon P´erezLaraudogoitia A thesis presented for the degree of Doctor of Philosophy University of the Basque Country November 2020 (cc)2021 JAVIER BELASTEGUI LAZCANO(cc by-nc-sa 4.0) Declaration of Authorship I hereby declare that this dissertation, entitled \The Resemblance Structure of Natural Kinds: A Formal Model for Resemblance Nominalism" and written under the supervision of Thomas Mormann and Jon P´erezLaraudogoitia, is the result of my own original research. To the best of my knowledge and belief, it contains no material previously published or written by another person, except where it is explicitly acknowledged in the text. No part of this thesis has been submitted for a degree at any other university or institution. Javier Belastegui Lazcano 1 Para Ane, que se sabe todas mis canciones. Abstract The aim of this thesis is to better understand the ways natural kinds are re- lated to each other by species-genus relations and the ways in which the members of the kind are related to each other by resemblance relations, by making use of formal models of kinds. This is done by first analysing a Minimal Conception of Natural Kinds and then reconstructing it from the ontological assumptions of Resemblance Nominalism. The questions addressed are: 1. What is the external structure of kinds? In what ways are kinds related to each other by species-genus relations? 2. What is the internal structure of kinds? In what sense are the instances of a kind similar enough to each other? According to the Minimal Conception of Natural Kinds, kinds have two components, a set of members of the kind (the extension) and a set of natural attributes common to these objects (the intension). Several interesting features of this conception are discussed by making use of the mathematical theory of concept lattices. First, such structures provide a model for contemporary formulations of syllogistic logic. Second, kinds are ordered forming a complete lattice that follows Kant's Law of the duality between extension and intension, according to which the extension of a kind is inversely related to its intension. Finally, kinds are shown to have Aristotelian definitions in terms of genera and specific differences. Overall this results in a description of the specificity relations of kinds as an algebraic calculus. According to Resemblance Nominalism, attributes or properties are classes of similar objects. Such an approach faces Goodman's companionship and im- perfect community problems. In order to deal with these a specific nominalism, namely Aristocratic Resemblance Nominalism, is chosen. According to it, at- tributes are classes of objects resembling a given paradigm. A model for it is introduced by making use of the mathematical theory of similarity structures and of some results on the topic of quasianalysis. Two other models (the polar model and an order-theoretic model) are considered and shown to be equivalent to the previous one. The main result is that the class of lattices of kinds that a nominalist can recover uniquely by starting from these assumptions is that of complete coatomistic lattices. Several other related results are obtained, including a generalization of the similarity model that allows for paradigms with several properties and properties with several paradigms. The conclusion is that, under nominalist assumptions, the internal structure of kinds is fixed by paradigmatic objects and the external structure of kinds is that of a coatomistic lattice that satisfies the Minimal Conception of Kinds. 3 Agradecimientos My greatest intellectual debt is to my supervisor Thomas Mormann. The originality, rigour and elegance of his proposals have been a constant source of inspiration for my work. So much so that many parts of this thesis can be seen as attempts to extend some of his ideas and defy others. He has not only made invaluable comments to this thesis, but has also transmited his fascination with mathematics to me. With exemplary tolerance, trust and wisdom, he has given me freedom to explore different possibilities while preventing me from making important mistakes. He has supported me even after retiring, at more than ten thousand kilometres away from this university. For all these reasons, I am greatly indebted to him. Quiero agradecer tambi´ena Jon P´erez, que acept´oamablemente mi petici´on de incorporarse como co-director para poder finalizar este proyecto. La presente investigaci´onha sido financiada con dinero p´ublico. Por ello, agradezco al Gobierno Vasco por haberme concedido la ayuda para mi contrato predoctoral con la Universidad del Pa´ısVasco, y a ´esta´ultimapor contratarme. During the Winter of 2019, I made a brief stay at the Munich Center for Mathematical Philosophy (MCMP), with a grant provided by the institutions just mentioned. I would like to express my deepest gratitude to Hannes Leitgeb, whose work strongly influenced crucial parts of this thesis, for inviting me to join the MCMP. When I arrived at the MCMP I was warmly welcomed by many wonderful philosophers, including Matteo, Michal, Albert, Fausto, Ivano and Neil. I want to thank them for their help and for all I learnt from their discussions. Special thanks go to Bruno, whose criticism to several parts of this thesis was invaluable. I am extremely grateful to Mattias and Sophie, for reading and commenting on many parts of this thesis and for constantly transmitting to me their philo- sophical enthusiasm. I would also like to extend my gratitude to those who have made questions and objections in the talks I have given including the ones in the Jornadas de Avances, thanks to Adrian, Guglielmo and Jos´eVicente. Mi m´assincero agradecimiento a todos los miembros del n´ucleoy periferia del grupo Praxis. Mila esker, en primer lugar, a su I.P. Andoni, por su constante ayuda en el ´ambito administrativo, y en segundo lugar, a sus estudiantes y seniors, entre otros a Caterina y Nasim, Maite Arraiza, Ekai, I~nakiy Hannot. En particular, gracias a Sergio y a Ainhoa, con quien he compartido despacho durante estos cuatro a~nos,tanto por sus sabios consejos como por soportar mis divagaciones frecuentemente. Gracias a Amaia y a Maite, por su apoyo incondicional y por las interminables conversaciones y paseos. La realizaci´onde esta tesis no habr´ıasido posible sin el apoyo de mis amigos, tanto de los que pertenecen al ´ambito acad´emicocomo de los que no. Entre otros, gracias a Gorka, Lourdes, Alba, Davi y Charly. Tampoco habr´ıasido posible sin el de mi familia, el aita, la ama, mi hermana y Simone, que han cuidado de m´ıy me han proporcionado una educaci´onformidable. Gracias en especial a Julen, cuya ambici´on,creatividad y curiosidad ilimitadas han sido una fuente inagotable de motivaci´onpara m´ı. Por ´ultimo,gracias Ane. Gracias por cenar conmigo, por cantar conmigo y por hablar conmigo. Si no me he roto durante estos a~nosha sido gracias a ti. Gracias por estar ah´ı. 4 Contents 1 Introduction 10 1.1 The Problem of Natural Kinds . 11 1.2 Aim, Methodology and Results . 18 1.2.1 Aim of the Thesis . 18 1.2.2 Methodological Remarks . 20 1.2.3 Main Results . 23 1.3 Summary of the Chapters . 25 2 Theories of Natural Kinds 29 2.1 Arguments for Natural Kind Realism . 30 2.1.1 Epistemological Argument: The New Riddle of Induction 31 2.1.2 Metaphysical Argument: Universals and Sparse Properties 33 2.1.3 Semantic Argument: Reference of Natural Kind Terms . 37 2.1.4 Naturalist Argument: Scientific Classifications . 41 2.2 Scientific Essentialism . 44 2.2.1 Theoretical Roles of the Concept of Essence . 45 2.2.2 The Scientific Essentialism Programme . 48 2.2.3 Cost-Benefit Arguments . 52 2.2.4 Semantic Arguments . 53 2.2.5 Metaphysical Arguments . 58 2.3 Cluster Theories . 66 2.3.1 Homeostatic Property Clusters . 66 2.3.2 Objections to the Role of Causality . 68 2.3.3 Promiscuous Realism and Practice-Oriented Theories . 71 2.4 Conceptualism . 74 2.4.1 Conceptual Categorisation . 75 2.4.2 Conceptual Spaces . 78 2.5 Conclusion of Chapter II . 83 3 Hierarchical Structure of Natural Kinds 88 3.1 A Minimal Conception of Natural Kinds . 90 3.1.1 A Minimal Conception of Kinds and Kant's Law . 90 3.1.2 Attributes and Kinds . 93 3.2 The Hierarchical Structure of Kinds . 95 3.2.1 The Failure of the Hierarchy Condition . 95 3.2.2 Thomason's Lattice Model of Kinds . 101 3.2.3 Contemporary Syllogistic Logic . 103 3.3 The Concept Lattice Model of Kinds . 106 5 3.3.1 A Model for Natural Kinds . 106 3.3.2 Arguments for the Adequacy of the Model . 111 3.4 Aristotelian Definitions of Kinds . 118 3.4.1 Two Operations of Specific Difference . 118 3.4.2 Definitions in terms of Genera and Specific Differences . 122 3.4.3 Laws of Specific Difference . 124 3.4.4 Negations and the Hexagon of Opposition . 125 3.4.5 Differences Between this Approach and the Traditional Aristotelian Picture . 129 3.5 A Modal Picture . 131 3.6 Conclusion of Chapter III . 135 4 Resemblance Structure of Natural Attributes 138 4.1 Similarity from a Formal Point of View . 140 4.1.1 What is Similarity? . 140 4.1.2 Invariants of Categorical Similarity . 143 4.2 Objections to the Properties of Categorical Similarity . 150 4.2.1 Similarity is not Reflexive . 150 4.2.2 Similarity is not Symmetric . 151 4.2.3 Similarity is not Transitive . 160 4.3 Egalitarian Resemblance Nominalism .