FORMAL ONTOLOGY AND CONCEPTUAL REALISM
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FORMAL ONTOLOGY AND CONCEPTUAL REALISM
SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Editor-in-Chief:
VINCENT F. HENDRICKS, Roskilde University, Roskilde, Denmark JOHN SYMONS, University of Texas at El Paso, U.S.A.
Honorary Editor:
JAAKKO HINTIKKA, Boston University, U.S.A.
Editors: DIRK VAN DALEN, University of Utrecht, The Netherlands THEO A.F. KUIPERS, University of Groningen, The Netherlands TEDDY SEIDENFELD, Carnegie Mellon University, U.S.A. PATRICK SUPPES, Stanford University, California, U.S.A. ´ JAN WOLENSKI, Jagiellonian University, Kraków, Poland
VOLUME 339
FORMAL ONTOLOGY AND CONCEPTUAL REALISM
by
Nino B. Cocchiarella Indiana University, Bloomington, IN, U.S.A.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4020-6203-2 (HB) ISBN 978-1-4020-6204-9 (e-book)
Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com
Printed on acid-free paper
All Rights Reserved © 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.
Dedicated to the village of Fragneto L’Abate, Provincia di Benevento, and to my family and friends in Italy
Contents Preface
xi
Introduction
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Formal Ontology
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1 Formal Ontology and Conceptual Realism 1.1 Formal Ontology as a Characteristica Universalis 1.2 Radical Empiricism and the Logical Construction 1.3 Commonsense Versus Scientific Understanding . 1.4 The Nexus of Predication . . . . . . . . . . . . . 1.5 Univocal Versus Multiple Senses of Being . . . . 1.6 Predication and Preeminent Being . . . . . . . . 1.7 Categorial Analysis and Transcendental Logic . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 The Completeness Problem . . . . . . . . . . . . 1.9 Set-Theoretic Semantics . . . . . . . . . . . . . . 1.10 Conceptual Realism . . . . . . . . . . . . . . . . 1.11 Summary and Concluding Remarks . . . . . . . . 2 Time, Being, and Existence 2.1 Possibilism versus Actualism . . . . . 2.2 Logics of Actual and Possible Objects 2.3 Set-theoretic Semantics . . . . . . . . 2.4 Axioms in Possibilist Logic . . . . . . 2.5 A First-order Actualist Logic . . . . . 2.6 Tense Logic . . . . . . . . . . . . . . . 2.7 Temporal Modes of Being . . . . . . . 2.8 Past and Future Objects . . . . . . . . 2.9 Modality Within Tense Logic . . . . . 2.10 Causal Tenses in Relativity Theory . . 2.11 Summary and Concluding Remarks . . 2.12 Appendix . . . . . . . . . . . . . . . . vii
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viii 3 Logical Necessity and Logical Atomism 3.1 The Ontology of Logical Atomism . . . . . . . . . 3.2 The Primary Semantics of Logical Necessity . . . . 3.3 The Modal Thesis of Anti-Essentialism . . . . . . . 3.4 An Incompleteness Theorem . . . . . . . . . . . . . 3.5 The Semantics of Metaphysical Necessity . . . . . 3.6 Metaphysical Versus Natural Necessity . . . . . . . 3.6.1 The Concordance Model of a Multiverse . . 3.6.2 The Multiverse of the Many-Worlds Model 3.7 Summary and Concluding Remarks . . . . . . . . .
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4 Formal Theories of Predication 4.1 Logical Realism . . . . . . . . . . . . . . . 4.2 Nominalism . . . . . . . . . . . . . . . . . 4.3 Constructive Conceptualism . . . . . . . . 4.4 Ramification and Holistic Conceptualism . 4.5 The Logic of Nominalized Predicates . . . 4.6 Summary and Concluding Remarks . . . .
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5 Formal Theories of Predication Part II 5.1 Homogeneous Stratification . . . . . . . 5.2 Frege’s Logic Reconstructed . . . . . . . 5.3 Conceptual Intensional Realism . . . . . 5.4 Hyperintensionality . . . . . . . . . . . . 5.5 Summary and Concluding Remarks . . .
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6 Intensional Possible Worlds 6.1 Actualism Versus Possibilism Redux 6.2 Intensional Possible Worlds . . . . . 6.3 Summary and Concluding Remarks . 6.4 Appendix 1 . . . . . . . . . . . . . .
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II
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Conceptual Realism
7 The 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11
Nexus of Predication Predication in Natural Realism . . . . Conceptualism . . . . . . . . . . . . . Referential Concepts . . . . . . . . . . Singular Reference and Proper Names Definite Descriptions . . . . . . . . . . Nominalization as Deactivation . . . . The Content of Referential Concepts . The Two Levels of Analysis . . . . . . Ontology of the Natural Numbers . . . Ontology of Fictional Objects . . . . . Summary and Concluding Remarks . .
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CONTENTS
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8 Medieval Logic and Conceptual Realism 8.1 Terminist Logic and Mental Language . . 8.2 Ockham’s Early Theory of Ficta . . . . . 8.3 Ockham’s Later Theory of Concepts . . . 8.4 Personal Supposition and Reference . . . 8.5 The Identity Theory of the Copula . . . . 8.6 Ascending and Descending . . . . . . . . . 8.7 How Confused is Merely Confused . . . . 8.8 Summary and Concluding Remarks . . . .
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169 169 174 176 178 180 183 191 193
Geach Against General Reference Geach’s Negation Argument . . . . . . . . . . Disjunction and Conjunction Arguments . . . Active Versus Deactivated Concepts . . . . . Deactivation and Geach’s Arguments . . . . . Geach’s Arguments Against Complex Names Relative Pronouns as Referential Expressions Summary and Concluding Remarks . . . . . .
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10 Le´ sniewski’s Ontology 10.1 Le´sniewski’s Logic of Names . . . . . . . . . . . 10.2 The Simple Logic of Names . . . . . . . . . . . 10.3 Consistency and Decidability . . . . . . . . . . 10.4 A Reduction of Le´sniewski’s System . . . . . . 10.5 Pragmatic Uses of Proper and Common Names 10.6 Classes as Many as the Extensions of Names . 10.7 Summary and Concluding Remarks . . . . . . .
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11 Plurals and the Logic of Classes as Many 11.1 The Logic of Classes as Many . . . . . . . 11.2 Extensional Identity . . . . . . . . . . . . 11.3 The Universal Class . . . . . . . . . . . . 11.4 Intersection, Union, and Complementation 11.5 Le´sniewskian Theses Revisited . . . . . . 11.6 Groups and the Semantics of Plurals . . . 11.7 Plural Reference and Predication . . . . . 11.8 Cardinal Numbers and Plural Quantifiers 11.9 Summary and Concluding Remarks . . . . 11.10Appendix 1: A Set-Theoretic Semantics . 11.11Appendix 2: Bell’s System M . . . . . . .
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9 On 9.1 9.2 9.3 9.4 9.5 9.6 9.7
Logic of Natural Kinds Conceptual Natural Realism . . . . . The Problem with Moderate Realism Modal Moderate Realism . . . . . . Aristotelian Essentialism . . . . . . .
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General Versus Individual Essences Summary and Concluding Remarks Appendix 1 . . . . . . . . . . . . . Appendix 2 . . . . . . . . . . . . .
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Afterword on Truth-Makers
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Bibliography
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Index
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Preface This book is based on a series of lectures given in Rome in 2004 at the Lateran Pontifical University under the auspices of the Science, Theology, and the Ontological Quest (STOQ) project, which was supported by the Pontifical Council for Culture with the support of the John Templeton Foundation. The project was aimed at developing a dialogue between science, philosophy and theology at a time when many theoretical, ethical and cultural challenges are being raised by new developments in science. The director of the project is Professor Gianfranco Basti of the Pontifical Lateran University, whom I want to thank for inviting me to give the lectures and for his many helpful comments. I also want to thank Professor Michele Malatesta and Professor Philip Larrey and Mr. Ciro De Florio for their comments and participation in the lectures. I also want to thank Professor Greg Landini for comments and corrections of my discussion of hyperintensionality in chapter five.
xi
Introduction The history of philosophy is replete with different metaphysical schemes of the ontological structure of the world. These schemes have generally been described in informal, intuitive terms, and the arguments for and against them, including their consistency and adequacy as explanatory frameworks, have generally been given in even more informal terms. The goal of formal ontology is to correct for these deficiencies. By formally reconstructing an intuitive, informal ontological scheme as a formal ontology we can better determine the consistency and adequacy of that scheme; and then by comparing different reconstructed schemes with one another as formal ontologies we can better evaluate the arguments for and against them, and come to a decision as to which system it is best to adopt. This book is divided into two parts. The first part is on formal ontology and how different informal ontological systems can be formally developed and compared with one another. An abstract set-theoretic framework, which we call comparative formal ontology, can be used for this purpose without assuming that set theory is itself a superseding ontological system. The second part of this book is on the formal construction and defense of a particular ontological scheme called conceptual realism. Conceptual realism is to be preferred to alternative formal ontologies for the reasons briefly described below, and for others as well that are given in more detail in various parts of the book. Conceptual realism, in other words, is put forward here as the best ontological system to adopt.
1. Formal Ontology Formal ontology, as we explain in chapter one, is a discipline in which the formal methods of mathematical logic are combined with the intuitive, philosophical analyses and principles of ontology, where by ontology we mean the study and analysis of being qua being, including in particular the different categories of being and how those categories are connected with the nexus of predication in language, thought and reality. The purpose of formal ontology is to bring together the clarity, precision and methodology of logical analyses on the one hand with the philosophical significance of ontological analyses on the other. The phrase ‘formal ontology’ was first used by Edmund Husserl in his Formal and Transcendental Logic (1929). For Husserl a formal ontology is supposed to be a system of logic taken as a universal theory of science, and in particular as xiii
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the justifying discipline for science. Instead of actually constructing a formal ontology, however, Husserl turned to the study of the categorial structures of what he took to be a transcendental subjectivity for the justification of such a logic. That is not the kind of study that is presented here. The phrase ‘formal ontology’ was again used in 1969 at a memorial conference for Bertrand Russell at Indiana University.1 Later, in 1972, a conference on formal ontology was held at the University of Victoria in Victoria, BC, Canada.2 Today the subject of formal ontology has become more widely discussed, especially in Europe, by many philosophers, logicians, and cognitive and computer scientists working on programs for knowledge representation. In March of 1993, for example, an international conference was held in Padova, Italy, on the role of formal ontology in information technology.3 Since then there have been at least three other international conferences on formal ontology in information systems.4 The idea of a formal ontology, even if not the phrase itself, goes back to Gottfried Leibniz’s notion of a characteristica universalis, the general features of which are explained in chapter one. Leibniz was a mathematician and he did make preliminary attempts at constructing a system of logic that would function as a characteristica universalis. Unfortunately, logic had still not progressed beyond Aristotle’s theory of the syllogism, and we have only fragments and the general idea of what such a system should achieve. In addition, because his logic was strictly algebraic, Leibniz did not deal with the central feature of a formal ontology, namely how the different ontological categories are connected with the nexus of predication. This is also true of many of the systems constructed in analytic metaphysics, where a logical analysis of different aspects of language, thought and reality are represented. These systems are really fragments or component parts of implicit, or as yet unspecified, formal ontologies with which they are compatible. The first full system of logic that was designed to function as a characteristica universalis and that also provided a formal theory of predication was Gottlob Frege’s system in his 1893, Der Grundgesetze der Arithmetik.5 This system was subject to Russell’s paradox, but, as we explain in Part I, Frege’s logic can be reconstructed and shown to be equiconsistent with the theory of simple types. Russell’s own attempt in his 1903 Principles of Mathematics was also subject to his paradox, but, Russell’s implicit system at that time can also be consistently reconstructed. Both Frege’s and Russell’s early ontology were 1 See
Cocchiarella 1974. The papers for that conference were published in Nakhnikian 1974. Victoria Conference on Formal Ontology was held in October of 1972. The papers for the conference were later published in the Israeli journal, Philosophia, vol, 4, no.1, 1974. 3 The papers from that conference, including my paper, “Knowledge Representation in Conceptual Realism,” were published later in a special issue of the International Journal of Human-Computer Studies, vol. 43 (5/6), 1995. For more general information on formal ontology see the web site http://www.formalontology.it. 4 See, e.g., Guarino 1998, Welty and Smith 2001, and Varzi and Vieu 2004. 5 Frege was adamant that his logic was “not a mere calculus ratiocinator, but a lingua characteristica in the Leibnizian sense” (Frege 1972, p. 90). 2 The
2. TIME, BEING AND EXISTENCE
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based on a modern form of Platonism, which in this book we call logical realism. Frege’s ontology was extensionalist, however, whereas Russell’s was intensionalist, and as a result their respective accounts of the nexus of predication were quite different. Nevertheless, as reconstructed here, both Frege’s and Russell’s early systems of logic can be taken as variants of a logical realist type of formal ontology. Russell’s later theory of ramified types was also originally described by Russell as a version of Platonism, but in fact it is more suitable to a constructive form of conceptual realism.6 In part 1 of this book we describe a system of ramified second-order logic with nominalized predicates as a formal ontology for constructive conceptual realism, which is a subsystem of the larger framework of conceptual realism.
2. Time, Being and Existence A criterion of adequacy for any formal ontology is that it should provide a logically perspicuous representation of our commonsense understanding of the world, and not just of our scientific understanding. A central feature of our commonsense understanding is how we are conceptually oriented in time with respect to the past, the present and the future, and the question arises as to how we can best represent this orientation. It is inappropriate to represent it in terms of a tenseless idiom of moments (or intervals of time) of a coordinate system, as is commonly done in scientific theories; for that amounts to replacing our commonsense understanding with a scientific view. A more appropriate representation is one that respects the form and content of our commonsense speech and mental acts about the past, the present and the future. Formally, this can best be done in terms of a logic of tense operators, or in what is now called tense logic, which we discuss in chapter two. Now the most natural formal ontology for tense logic is conceptualism, or a formal ontology such as conceptual realism that contains conceptualism as a component. This is because what tense operators represent in conceptualism are certain cognitive schemata regarding our orientation in time, cognitive schemata that are in fact fundamental to both the form and content of our conceptual activity. Thought and communication are inextricably temporal phenomena, and it is the cognitive schemata underlying our use of tense that structures that phenomena temporally in terms of the past, the present and the future. A second criterion of adequacy for a formal ontology is that it must explain and provide an ontological ground for the distinction between being and existence, or, if it rejects that distinction why it does so. Put simply, the problem is: Can there be things that do not exist? Or is being the same as existence? Different formal ontologies will answer these questions in different ways. The logic of time in conceptual realism provides the clearest ontological ground for such a distinction in terms of the tense-logical distinction between past, present and future objects, i.e., the distinction between things that did exist, do exist, or will exist, or what in the proposed book we call realia, as 6 See
Cocchiarella 1991 for a defense and explanation of this claim.
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opposed to existentia, which is restricted to the things that exist at the time we speak or think, i.e., the time we take to be the present of our commonsense framework. The present, in other words, unlike the tenseless medium of our scientific theories, is indexical, and refers at any moment of time to that moment itself. We explain in chapter two how the categories of time, being and existence can be represented in tense logic.
3. Ontology and Modality Another criterion of adequacy for a formal ontology is that it must explain the ontological grounds, or nature, of modality, i.e., of such modal notions as necessity and possibility. A set-theoretic semantics for modal logic may be useful for showing consistency or completeness, but it does not of itself provide an ontological ground for modality. Chapter two of the proposed book deals not just with the fundamental categories of time, being and existence, but with modality as well. Some of the earliest views of necessity and possibility are grounded in such a framework. As we explain in chapter two, even the temporal and modal distinctions of the special theory of relativity theory can be best understood within the framework of conceptual realism. There is more to the distinction between being and existence than that between past, present and future objects, of course, and there is also more to modality than what can be grounded in time and the special theory of relativity. The abstract intensional objects of conceptual realism, for example, do not exist as concrete objects, but, as we explain later, they also are not Platonic forms. They do not exist in an independent Platonic realm, in other words, but rather have a mode of being dependant upon the evolution of culture and consciousness. Similarly, the natural or causal modalities of the logic of natural kinds and natural properties and relations that we describe below for conceptual realism go beyond the modalities that can be grounded on the logic of tenses, including the causal tenses of the special theory of relativity. But both of these developments, i.e., the abstract being of intensional objects and the causal modalities of the logic of natural kinds, are extensions of the distinctions made in chapter two and not rejections of those distinctions. Indeed, the logic of actual and possible objects described in chapter two is retained, or only slightly modified, throughout the remainder of the book. Finally, it should be noted that all too often it has been assumed that the only modalities appropriate for a formal ontology are the logical modalities, e.g., logical necessity and possibility. In chapter three we explain how logical necessity and possibility can be understood as modalities, but only within the ontological framework of logical atomism. It is only in an ontology of simple objects and simple properties and relations as the bases of logically independent atomic states of affairs that logical necessity and possibility can be made sense of as modalities, as opposed to semantic properties of sentences. Logical atomism, of course, is not an adequate formal ontology for either our commonsense or our scientific framework, and we are not putting it forward as such here. Rather,
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the point is that it serves to explain why logical necessity and possibility cannot be ontologically grounded on a formal ontology of complex objects and complex states of affairs that are not logically reducible to simple objects and logically independent atomic states.
4. Formal Theories of Predication Part I of this book, as already noted, is on formal ontology, and especially comparative formal ontology. Comparative formal ontology is the preferable, if not also the proper, domain of many issues and disputes in metaphysics, epistemology, and the methodology of the deductive sciences. For just as the construction of a particular formal ontology lends clarity and precision to our informal categorial analyses and serves as a guide to our intuitions, so too comparative formal ontology can be developed so as to provide clear and precise criteria for constructing and comparing different formal ontologies so that ultimately we can make a rational decision about which such system we should ourselves adopt. Different formal ontologies are primarily based on different formal theories of predication, which in turn are based on different theories of universals, the three most important being nominalism, conceptualism, and realism. A basic feature of a formal ontology, in other words, is a formal theory of predication based on a theory of universals. A key aspect of such a theory is how the categories of being, especially the category of objects and the category of universals, are related to one another, and how the unity of the nexus of predication is explained in terms of those categories. Such a categorial analysis indicates another basic feature of a formal ontology, namely, how it represents the categorial structure of the world, and in particular whether it can represent the categorial structure of our commonsense understanding of the world as well that of our scientific theories, without the two being in conflict. A formal ontology is not just a formal axiomatic development, in other words, but rather it is a system in which ontological categories are represented by logical categories, and ontological analyses by logical analyses. We have already constructed and compared a number of such systems in Cocchiarella 1986 and 1987, and in fact a detailed consistent reconstruction and analysis of nominalism and Gottlob Frege’s and Bertrand Russell’s forms of logical realism have already been given in those previous books. For that reason we do not go into great detail of their views in the present book, but some coverage of that reconstruction is necessary in order explain the core part of conceptual realism, which has a number features in common with those systems, and which is the framework we will develop and defend in part II of this book. That reconstruction is given in chapters four and five. Nominalism, which denies that there are any universals, whether real or conceptual, is logically the weakest formal ontology. From a logical point of view what is interesting about nominalism is the kind of constraint it imposes on a theory of predication. That constraint, however, has a more interesting counterpart in constructive conceptualism and represents an important stage of
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cognitive development, a stage that is an essential part of conceptualism and the more general framework of conceptual realism. The more general framework allows the constraint to be acknowledged and yet also transcended, whereas the similar constraint in nominalism leaves no room for such transcendence. We discuss and explain the nature of this constraint and why it can be transcended in constructive conceptualism but not in nominalism in chapter four. This difference between nominalism and conceptualism is one of the reasons why conceptualism, as a formal ontology, is to be preferred over nominalism. There is another reason as well, namely, that predication in language, which is the only form of predication acknowledged in nominalism, depends upon our cognitive capacity for language, including in particular our rule-following cognitive capacities underlying the use of referential and predicable expressions. A cognitive theory of predication is needed to explain predication in language, in other words, and that is precisely what the form of conceptualism we defend here is designed to do. In fact, the referential and predicable concepts of our form of conceptualism are none other than the rule-following cognitive capacities underlying the use of referential and predicable expressions, and the unity of the nexus of predication in conceptualism is what underlies and accounts for the unity of predication in language. Conceptualism is to be preferred over nominalism because, unlike the latter, which is based on an unexplained account of predication in language, it is framed in terms of a theory of predication, i.e., a theory of predication about the cognitive structure of our speech and mental acts, and therefore a theory of thought that underlies and explains predication in language.
5. Conceptual Realism The purpose of comparative formal ontology, we have noted, is to provide clear and precise criteria by which to judge the adequacy of a particular proposed system of formal ontology. Conceptual realism, the system we think is best and have adopted, contains, in addition to a conceptualist theory of predication, an intensional realism that is based on a logic of nominalized predicates, and a natural realism that is based on a logic of natural kinds. Unlike the a priori approach of the transcendental method, which claims to be independent of the laws of nature and our evolutionary history, i.e., of our status as biological beings with a culture and history that shapes our language and much of our thought, conceptual realism is framed within the context of a naturalistic epistemology and a naturalistic approach to the relation between language and thought, thought and reality, and our scientific knowledge of the world. This naturalistic approach is one of the reasons why conceptual realism is to be preferred to a logical realism as a modern form of Platonism as well as to a transcendental idealism such a Husserl’s. The following are some of the features of conceptual realism that we will cover in this book. The realism part of conceptual realism, we have said, contains both a natural realism and an intensional realism, each of which can be developed as separate
5. CONCEPTUAL REALISM
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subsystems, and both of which are not only consistent in themselves but compatible with each other within the larger framework. We call these two subsystems conceptual natural realism and conceptual intensional realism. Conceptual natural realism represents a modal form of moderate realism, which, by being extended to include a logic of natural kinds, can be developed into a modern form of Aristotelian essentialism. Conceptual intensional realism, on the other hand, represents a modern counterpart of Platonism based on the intensional contents of our referential and predicable concepts. These two subsystems capture the more important ontological features of both logical realism and natural realism as theories of universals while also explaining our epistemic access to the abstract intensional objects of logical realism on the one hand and the natural kinds and natural properties and relations of natural realism on the other. The two subsystems are compatible in the larger framework of conceptual realism, as we have said, because each is constructed on the basis of a different logical aspect of that framework. Conceptual intensional realism, for example, is based on a logical analysis of nominalized predicates and propositional forms as abstract singular terms, i.e., a logical analysis of the abstract nouns and nominal phrases that we use in describing the intensional contents of our speech and mental acts. The intensional objects that are denoted by these abstract singular terms serve the same purposes in conceptual intensional realism that abstract objects serve in logical realism as a modern form of Platonism. The difference is that, unlike Platonic Forms, the intensional objects of conceptual realism do not exist independently of mind and the natural world, the way they do in logical realism, but are products of the evolution of culture and language, and especially of the institutionalized linguistic practice of nominalization. In this way our epistemic grasp of abstract intensional objects is explained in terms of the concepts that underlie our rule-following cognitive capacities in the use of language. The natural kinds and natural properties and relations of conceptual natural realism, on the other hand, are not intensional objects; and in fact they are not objects at all but are rather unsaturated causal structures that are complementary to the structures of natural kinds of things. Unlike conceptual intensional realism, which is based on the logic of nominalized predicates, and hence is directed upon an “object”-ification of predicable concepts, conceptual natural realism is directed upon the structure of reality and depends upon empirical assumptions as to whether or not there are natural properties or relations corresponding to particular predicable concepts, and similarly whether or not there are natural kinds corresponding to particular sortal common-name concepts. The difference between Plato’s and Aristotle’s ontologies has been one of the basic issues of debate in the history of philosophy. The way both forms of realism are contained within the general framework of conceptual realism shows how a modern form of Aristotelian essentialism is compatible with an intensional logic that is a counterpart to a modern form of Platonism. This is another reason why conceptual realism is the formal ontology that is best to adopt. There is an importance difference between conceptual intensional realism and logical realism, however, if one extends and compares a metaphysical necessity
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INTRODUCTION
and possibility in the latter with a conceptual necessity and possibility to the former. The metaphysical modalities allow for a much stronger system in terms of what in chapter six are called intensional possible worlds, which can be expressed either as world propositions or as world properties. The assumption that there are such intensional worlds is justifiable in modal logical realism but, as we explain in chapter six, not at all in conceptual intensional realism.
6. Predication in Conceptual Realism Conceptual realism, we have noted, is based upon a cognitive theory of predication, and referential and predicable concepts are in fact the rule-following cognitive capacities that underlie our use of referential and predicable expressions. As explained in chapter seven, the nexus of predication in conceptual realism is the result of jointly exercising a referential and predicable concept as complementary cognitive structures, and as such it is what accounts for both predication in language and the unity of thought. An important feature of this theory is that it gives a unified account of both general and singular reference, a feature it has in common with the terminist logic of Ockham and other medieval philosophers. The theory also provides an account of complex predicate expressions that contain abstract noun phrases, such as infinitives and gerunds, and also complex predicate expressions with quantifier phrases occurring as direct-object expressions of transitive verbs. Conceptually, as we explain in chapter seven, the content of such a quantifier phrase and the referential concept it stands for is “object”-ified through a doubly reflexive abstraction that by deactivation and “nominalization” of the quantifier phrase first generates a predicable concept and then the intensional content of that predicable concept. All direct objects of speech and thought are intensionalized in this way so that a parallel analysis is given for both ‘Sofia finds a unicorn’ and for ‘Sofia seeks a unicorn’. And yet, relations, such as F inds, that are extensional in their second argument positions can still be distinguished from those that are not, such as Seeks, by appropriate meaning postulates. The same doubly reflexive abstraction explains the three different types of expressions that represent the natural number concepts, namely first, as numerical quantifier phrases, such as ‘three dogs’, ‘two cats, ‘five chairs’, etc., then, second, as the cardinal number predicates ‘has n instances’, or ‘has n members’, and the third as the numerals ‘1’, ‘2’, ‘3’, etc., i.e., as objectual terms that purport to name the natural numbers as abstract objects. Finally, the deactivation of referential expressions that is a part of this cognitive theory of predication is also involved in fictional discourse and in stories in general. The objects of fiction, on this account, are none other than the intensional objects that deactivated referential expressions denote as abstract objectual terms. This account of the ontology of fictional objects explains their “incompleteness” as well as their status as intensional content.
7. EXPLAINING MEDIEVAL SUPPOSITION THEORY
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7. Explaining Medieval Supposition Theory and Defending General Reference The unified account of both general and singular reference that is part of our cognitive theory of predication is a feature, we have noted, that it has in common with medieval terminist logic. In chapter eight we explain how the medieval notion of supposition, though different from reference, nevertheless can be explained in terms of our cognitive theory. We also explain in that chapter how the medieval doctrine of the identity of the copula for categorical propositions can be understood in terms of our theory of reference. These historically problematic notions, in other words, can be explained and accounted for in conceptual realism, which is another reason why we think it is the best formal ontology to adopt. It is well-known, of course, that Peter Geach, in his book Reference and Generality, has argued against the possibility of a coherent account of general reference, including in particular the medieval theory of supposition. In chapter nine, we give a detailed discussion of Geach’s arguments along with a refutation of those arguments. A key feature of our refutation is the notion of the deactivation of referential concepts, a notion that is basic to our analysis of the direct objects of intensional verbs. We also give an account in that chapter of co-reference, as, e.g., in so-called donkey sentences, in terms of a variable-binding ‘that’-operator.
8. The Logic of Names in Le´ sniewski’s Ontology Versus in Conceptual Realism The category of names in conceptual realism’s theory of reference contains both proper and common names (common count nouns). It is through this broader notion of a name that general as well as singular reference is part of this theory. A single category of names, both proper and common, is also a feature of Le´sniewski’s ontology, which has been called a logic of names. In chapter ten we formally describe both Le´sniewski’s ontology, or logic of names, and the simple logic of names that is part of our cognitive theory of reference. We then show how Le´sniewski’s ontology is reducible to the simple logic of names of our cognitive theory. Names, whether proper or common, occur as parts of quantifier phrases in the simple logic of names of our cognitive theory. In the broader theory of reference of conceptual realism, however, names, whether proper or common, can also be “nominalized”, i.e., transformed into objectual terms that can occur as arguments of predicates. The result is a logic of “classes as many,” where by a class as many we mean essentially the notion that was originally described by Russell in his 1903 Principles of Mathematics. Nominalized common names, in this logic, provide both a semantic and an ontological ground for a logic of
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INTRODUCTION
plurals within the more general framework of conceptual realism. We describe such a logic in chapter eleven. Semantically, such a logic is needed to account for irreducible forms of plural reference and predication. The well-known Geach sentence ‘Some critics admire only each other’, for example, which semantically says that there is a group of critics who admire only other members of the group, cannot be analyzed in first-order logic alone; and it would be both semantically and ontologically misleading to analyze it in terms of set theory. Unlike a set, a group in the sense intended here is a plurality of individuals and not an abstract object. A logic of plurals is needed not just as a semantical framework for plural reference and predication in natural language and our commonsense framework, but, and perhaps more importantly, also for an ontological account of the properties of groups of objects in our scientific theories. The temperature and pressure of a volume of gas, for example, are really properties of the group of atoms or molecules in that volume rather than properties of the individual atoms or molecules that make it up. The visual, auditory, and other sensory properties of different modules of the brain are properties of the groups of neurons that make up those modules rather than of the individual neurons in the group. Similarly, the dispersion and redistribution of different populations of species of plants and animals are statistical properties of the groups of plants and animals and not of the individuals in those groups. Groups, which are classes as many of two or more objects, are plural objects, and as such they are values of the objectual variables in this ontology.
9. Conceptual Realism and Aristotelian Essentialism Conceptualism and natural realism have a clear affinity for each other, even though they do not have the same overall logical structure. Conceptualism, for example, presupposes some form of natural realism as the causal ground of our capacity for language and thought, and natural realism presupposes conceptualism as a framework by which it can be articulated as a formal ontology. Historically, in fact, the two ontologies have often been confused with one another, so that sometimes it was said that a universal “exists” in a double way, one being in the mind and the other in things in the world. Abelard, for example, held that a universal “exists” first as a common likeness in things, and then as a concept that exists in the human intellect through the mind’s power to abstract from our perception of things by attending to the likeness in them. Aristotle is sometimes also said to have held such a view. Aquinas, however, was clear about the distinction and maintained that a concept and a natural property or natural kind are not really the same universal, and that in fact they do not even have the same mode of being. The distinction is one of the issues discussed in chapter twelve. Natural realism is usually described as a moderate realism, where universals
10. CRITERIA OF ADEQUACY
xxiii
exist only in re, i.e., in things. As a scientifically acceptable ontology, however, this is much too restrictive a view. As we explain in chapter twelve, many of the natural properties and relations that we now know to characterize atoms and compounds as physical complexes did not have any instances at all at the time of the Big Bang, i.e., when the universe began. Yet, these natural properties and relations must be acknowledged in any natural realism that is adequate for science. What we propose instead in conceptual realism is a modal moderate realism, where the modality in question is ontologically grounded in nature and its causal matrix. A logic of natural kinds can also be developed within this framework, moreover, so that the result is a reconstruction of Aristotelian essentialism. Natural kinds are general essences in this ontology, and not individual essences, which are usually found in modal versions of logical realism. The question of whether there can be individual essences in a modern form of Aristotelian essentialism is briefly considered in chapter twelve.
10. Criteria of Adequacy Criteria of adequacy for a formal ontology that we have indicated so far can be summarized as follows. • A formal ontology must provide a logically perspicuous representation of our commonsense understanding of the world as well as our scientific understanding. • A formal ontology must explain the distinction between being and existence, i.e., give an ontological grounding of that distinction, or if it denies the distinction then explain why it does so and why the result is an adequate ontological framework. • A formal ontology should provide an ontological, and not just a settheoretical, account of modality. • A formal ontology must explain the nature of predication in thought as well as in language and indicate what theory of universals is part of that explanation. It is because conceptual realism fulfills these criteria of adequacy, as well as others indicated throughout this book, that it is the best formal ontology to adopt.
Part I
Formal Ontology
1
Chapter 1
Formal Ontology and Conceptual Realism Formal ontology is a discipline in which the formal methods of mathematical logic are combined with the intuitive, philosophical analyses and principles of ontology.1 In this way formal ontology brings together the clarity, precision and methodology of logical analysis on the one hand with the philosophical significance of ontological analysis on the other. Father I.M. Bochenski has said of ontology, for example, that it is a “sort of prolegomenon to logic” in that whereas ontology is an intuitive, informal inquiry into the categorial aspects of reality in general, “logic is the systematic formal, axiomatic elaboration of this material predigested by ontology.”2 Ontology, which is the study of being qua being (Aristotle, Meta. 1031a), has been a principal part of metaphysics since ancient times. Metaphysics itself has usually been divided into ontology and cosmology, where • ontology = the study of being as such, and • cosmology = the study of the physical universe at large; i.e., space, time, nature and causality. Implicit in this division is a distinction between methodologies. The methodology of cosmology, for example, is based on the analysis of such categories as space, time, matter, and causality, where the goal is to discover by observation and experiment the laws connecting those categories and their constituents with one another, including in particular the natural kinds of things (beings) in nature. The methodology of ontology, on the other hand, is based on the analysis of ontological categories, i.e., categories of being, where the goal is to discover the laws connecting these categories and the entities in them with one another. 1 This chapter is an extended version of my essay, “Formal Ontology,” in Burkhardt and Smith 1991. 2 Bochenski 1974.
3
4
CHAPTER 1. FORMAL ONTOLOGY AND CONCEPTUAL REALISM
The particular sciences that are part of cosmology and that are concerned with particular natural kinds of beings may be prior to ontology in the order of discovery—and even in the order of conception. But as an analysis of ontological categories, ontology is a science that contains the ontological forms, if not the specific content, of the ideas and principles of the different natural sciences, and in that sense it is a science that is prior to all of the others. Similarly, logic contains the logical forms, but not the specific content, of the different scientific theories that make up the content of our knowledge of nature, and in that sense logic is a science prior to all others. Thus, when the logico-grammatical forms and principles of logic are formulated with the idea of representing the different categories of being and the laws connecting them, i.e., when ontological and logical categories are combined in a unified framework, then the result, which is what we mean by formal ontology, is a comprehensive deductive science that is prior to all others in both logical and ontological structure. In addition, by proving the consistency of the logical framework we can therefore also show that the intuitive ontological framework is consistent as well.
1.1
Formal Ontology as a Characteristica Universalis
A system of logic can be constructed under two quite different aspects. On the one hand, it can be developed as a formal calculus and studied independently of whatever content it might be used to represent. Such a formal system in that case is only a calculus ratiocinator. On the other hand, a system of logic can be constructed somewhat along the lines of what Leibniz called a characteristica universalis. Such a system, according to Leibniz, was to serve three main purposes. The first was that of an international auxiliary language that would enable the people of different countries to speak and communicate with one another. Apparently, because Latin was no longer a “living” language and new trade routes were opening up to lands with many different local languages, the possibility of such an international auxiliary language was widely considered and discussed in the 17th and 18th centuries.3 There were in fact a number of proposals and partial constructions of such a language during that period, but none of them succeeded in being used by more than a handful of people. It was only towards the end of the 19th century when Esperanto was constructed that such a language came to be used by as many as eight million people. At present, however, the question of whether even Esperanto will succeed in fulfilling that purpose seems very much in doubt. Ido is another such language, which was constructed in 1907 by a committee of linguists, but it has not been used since about 1930.4 In any case, notwithstanding its visionary goal, the idea of an international auxiliary language is not the purpose of a formal ontology. 3 Cf. 4 Cf.
Cohen 1954 and Knowlson 1975. Van Themaat 1962.
1.1. FORMAL ONTOLOGY AS A CHARACTERISTICA UNIVERSALIS 5 The second and third purposes Leibniz set for his characteristica universalis are what distinguish it from its precursors and give his program its formal or logistic methodology. The second purpose is that the universal character is to be based upon an ars combinatoria, i.e., an ideography or system of symbolization, that would enable it to provide a logical analysis of all of the actual and possible concepts that might arise in science. Such an ars combinatoria would contain both a theory of logical form, i.e., a theory of all the possible forms that a meaningful expression might have in such a language, and a theory of definitional forms, i.e., a theory of the operations whereby one could construct new concepts on the basis of already given concepts. The third purpose was that the universal character must contain a calculus ratiocinator , and in particular a complete system of deduction and valid argument forms, by which, through a study of the consequences, or implications, of what was already known, it could serve as an instrument of knowledge. These two purposes are central to the notion of a formal ontology. With a universal character that could serve these purposes, Leibniz thought that a unified encyclopedia of science could be developed about the world, and that, by its means, the universal character would then also amount to a characteristica realis, i.e., a representational system that would enable us to see into the inner nature of things and guide our reasoning about reality like an Ariadne’s thread.5 In other words, in Leibniz’s program for a characteristica universalis we have an attempt to encompass the relationships between language and reality, language and thought, and language and knowledge, especially as represented in terms of scientific theories. In two fundamental parts of the program, namely, the construction of an ars combinatoria and a calculus ratiocinator, we also have two critical components that are necessary for a formal ontology. The idea of a characteristica realis, i.e., a unified encyclopedia of science, is also important for a formal ontology. That is because a formal ontology, as a logistic system, must be structurally rich enough so that in principle every scientific theory can be formulated within it so that the result would be a system of metaphysics containing both an ontology and a cosmology. Of course, this will be possible only by adding to the general framework of a formal ontology appropriate nonlogical constants, axioms, and meaning postulates that represent the basic concepts and principles of a given science. In addition, though this is not required from a strictly scientific point of view, a formal ontology should be sufficiently structured so that with the addition of suitable nonlogical constants and meaning postulates a logical analysis of every meaningful declarative sentence of natural language can be given within it. That is, a formal ontology should be able to contain a semantics for natural language that captures the ontology of our commonsense framework. In that case such a logistic system can be taken not only as a characteristica realis, but also as a lingua philosophica. Of course, prior to the introduction of such constants and postulates, whether for science or natural language, a formal ontology is essentially just a shell containing the logico-ontological categorial forms and principles of science and of 5 Cp.
Cohen 1954, p. 50.
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CHAPTER 1. FORMAL ONTOLOGY AND CONCEPTUAL REALISM
our commonsense understanding of the world. There is still one important component of a formal ontology that Leibniz did not include or discuss as part of a characteristica universalis. This is the component that deals explicitly with the analysis of ontological categories and how those categories are connected with the nexus of predication in language, thought and reality. It is this component that distinguishes formal ontology from analytic metaphysics in the sense of formally constructed systems that represent one or more aspects of reality. Analytic metaphysics is a part of formal ontology, to be sure, but it is not itself a formal ontology unless it deals explicitly with the formal analysis ontological categories and their connection with the nexus of predication in language, thought and reality.
1.2
Radical Empiricism and the Logical Construction of the World
Both Gottlob Frege and Bertrand Russell viewed the logical systems they constructed as a framework for a universal characteristica, and each had a particular analysis of the nexus of predication, which we will discuss in chapters three, four, and five. Russell, for example, described his theory of types as “a logically perfect language”, by which he meant a language that would show at a glance the logical structure of the facts that are described by its means.6 According to Russell, such a language “would be one in which everything that we might wish to say in the way of propositions that are intelligible to us, could be said, and in which, further, structure would always be made explicit”.7 All that needs to be added to the theory of logical types to be such a language, Russell maintained, is a vocabulary of (nonlogical) descriptive constants that correspond to the meaningful words and phrases of natural science and ordinary language. The constants of pure mathematics, unlike those of the natural sciences, do not need to be added to the framework because they, according to Russell, are all definable in purely logical terms within the framework itself. Knowledge of pure mathematics is explainable, in other words, as logical knowledge—a view known as logicism—and, in particular, as knowledge of the propositions that are provable in the theory of logical types independently of any vocabulary of descriptive constants. Despite his logicism regarding mathematics, Russell was a radical empiricist as far as our knowledge of physical or concrete existence was concerned. All our empirical knowledge of the world, he maintained, must be reducible to our knowledge of what is given in experience, by which he meant that it must be constructible within the framework of the theory of logical types from the lowest level of objects, which he assumed to be events corresponding to our sensory experience. It was by means of logical constructions within this framework that Russell proposed to bridge the gulf between the world of physical science and 6 Cp.
“The Philosophy of Logical Atomism” (1918) in Russell 1956. 1959, p. 165.
7 Russell
1.2. RADICAL EMPIRICISM AND THE LOGICAL CONSTRUCTION
7
the world of sense, and he was guided in this regard by “the maxim which inspires all scientific philosophizing, namely ‘Occam’s razor’: Entities are not to be multiplied without necessity. In other words, in dealing with any subjectmatter, find out what entities are undeniably involved, and state everything in terms of these entities”.8 Thus, because sense-data are the entities that are “undeniably involved” in all of our empirical knowledge according to Russell, “the only justification possible” for such knowledge “must be one which exhibits matter as a logical construction from sense-data”.9 Though Russell gave a number of examples of how such a construction might be given, it was Rudolf Carnap who, using nothing more than the framework of the simple theory of logical types (and a certain amount of empirical science, such as gestalt psychology, about the structure of experience), gave the most detailed analysis indicating how we might reconstruct all our knowledge of the world in terms of what is given in experience.10 This meant in particular that all of the concepts of science could be analyzed and reduced to certain basic concepts that apply to the content of what is given in experience. One of the important patterns for such an analysis is known today as definition by abstraction, whereby, relative to a given equivalence relation (i.e., a relation that is reflexive, symmetric and transitive), certain concepts are identified with (or represented by) the equivalence classes that are generated by that equivalence relation. This pattern was used by Frege and Russell in the analysis of the natural numbers (as based on the equivalence relation of equinumerosity, or oneone correspondence), and was then used again in the analysis of the negative and positive integers, the rational numbers, the real numbers, and even the imaginary numbers. Carnap, who took definition by abstraction as indicative of the “proper analysis” of a concept, generalized the pattern into a form that he called “quasi analysis” (but which, he acknowledged, really amounted to a form of synthesis), which could be based on relations of partial similarity instead of full similarity, i.e., on relations that amount to something less than an equivalence relation. The concepts definable in terms of a quasi-analysis specify in what respect things (especially items of experience) that stand to one another in a relation of partial similarity agree, i.e., the respect in which they are in part, but not fully, similar.11 In this way Carnap was able to define the various sense modalities (as classes of qualities that intuitively belong to the same sense modality, where concepts for sense qualities are definable in terms of a partial similarity between elementary experiences), including in particular the visual sense and color concepts (as determined by the three-dimensional ordering relation of the color solid). Using the various sense modalities, Carnap went on to construct the fourdimensional space-time world of perceptual objects, which, with all its various sense qualities, “has only provisional validity”, and which, for that reason, “must give way to the strictly unambiguous but completely quality-free world 8 Russell
1914, p.112. p. 106. 10 Cp. Carnap 1967. 11 Cp. Carnap, op. cit., sections 71-74. 9 Ibid.,
8
CHAPTER 1. FORMAL ONTOLOGY AND CONCEPTUAL REALISM
of physics”.12 We will not go into the details of Carnap’s “constructional definitions” here, but we should note that all that Carnap meant by a logical analysis by means of such definitions was translatability into his constructional language—i.e., into an applied form of the simple theory of logical types as based on a primitive descriptive constant for a certain relation of partial similarity between elementary experiences. Such a translation need not, and in general did not, preserve synonymy, nor did it in any sense amount to an ontological reduction of ordinary physical objects to the sensory objects of experience. What it did preserve, according to Carnap, was a material equivalence (i.e., an equivalence of truth-value) between the sentences of ordinary language, or of a scientific theory, and the sentences of the constructional language.13 Carnap’s project was not a formal ontology in the sense intended here, unless we view it as representing the radical empiricist doctrine that there is no more to reality than what we can construct in terms of sense data, or what Carnap called elementary experiences (Elementarelebnisse). One extreme form of this position is ontological solipsism, which is not at all the same as the methodological solipsism that Carnap took it to be. In addition, the one important component that is missing is an analysis of the nexus of predication and an account of our commonsense understanding of the world.
1.3
Commonsense Versus Scientific Understanding
Our commonsense understanding of the world is sometimes said to be in conflict with our scientific understanding, which, on this view, is taken as providing the only proper criteria for truth. It is also claimed that the construction of a logistic system as the basis of a unified encyclopedia of science can represent only our scientific understanding, because by its very nature such a system can operate only with concepts and principles that have sharp and exact boundaries, such as the concepts and principles we strive to formulate in our scientific theories. The same cannot be said, according to this so-called “eliminativist view,” of the concepts and principles of our commonsense understanding. That is, the concepts underlying our use of natural language do not have sharp boundaries, and do not require the kind of precision of thought that is the goal of scientific knowledge, which alone can provide an adequate criterion of truth. Many of the words and phrases of natural language by which we express our commonsense understanding, for example, are vague or ambiguous, and as such are unsuitable for the kind of logical representation involved in our methodology. Gottlob Frege expressed this view in comparing the difference between his logical system and ordinary language with that between a microscope and the human eye. Even though the eye is superior to the microscope, Frege observed, “because of the 12 Ibid.,
p. 207. same claim is made in Goodman 1951 for an alternative constructional language based on nominalist principles that are opposed to the predicate quantifiers of type theory. 13 The
1.3. COMMONSENSE VERSUS SCIENTIFIC UNDERSTANDING
9
range of its possible uses and the versatility with which it can adapt to the most diverse circumstances,” nevertheless, “as soon as scientific goals demand great sharpness of resolution, the eye proves to be insufficient”.14 That only our scientific understanding can provide an adequate criterion of truth about the natural world does not mean that our commonsense understanding gives a false picture of the world, or a picture that, for the purposes of knowledge, ought to be eliminated.15 No doubt, many of our commonsense beliefs and concepts about the natural world have been revised and corrected over the millennia, and probably many will be revised or corrected in times to come. The concept water, for example, has been replaced by the concept H2 O in the scientific context of the atomic theory of matter, where the concept H2 O is systematically related to the concept H for hydrogen and the concept O for oxygen.16 This does not mean that the concept water is somehow misleading and that the role it plays in our commonsense framework is to be eliminated. Indeed, not only has the concept continued to be functionally useful in everyday contexts, but it also continues to serve in scientific contexts as well. It is not just our commonsense concepts that are important for an understanding of the world, however, but also how we structure our thought in our commonsense framework as well. How we reason and argue in this framework are preconditions of scientific knowledge and theorizing. Scientific understanding depends, in other words, both conceptually and pragmatically upon our commonsense understanding, including the way the world is categorially structured, and the way we reason in terms of that structure. In this regard, the representation of our scientific knowledge involves more than the representation of a large number of facts or beliefs about the objects in a given domain of scientific inquiry, regardless of whether those facts or beliefs are in conflict with what is believed by common sense. In particular, it involves the criteria for valid reasoning that we bring to bear on our commonsense arguments, and the way those arguments are structured in terms of the categorial structure of our commonsense understanding. It is precisely the formal representation of the categorial structure of our commonsense framework, as well as the criteria for valid reasoning within that framework that is one of the goals of formal ontology and a criterion of adequacy. The arguments that we find in natural language and in terms of which we articulate our reasoning can be evaluated as valid or invalid only with respect to a logical theory, and in particular one that provides an adequate formal representation of the basic categories of natural language and the commonsense framework expressed in its use. The adequacy of such a theory is judged on the basis of how well it agrees with our commonsense intuition of which arguments are valid and which invalid. We are not claiming here that the ontology of our commonsense framework, based as it is on perceptible objects and their qualitative features, is also fun14 Frege
1879, p, 6. e.g., “Philosophy and the Scientific Image of man”, in Sellars 1963, for a account of the eliminativist view. 16 Cf. Fodor 1993, p.86. 15 See,
10 CHAPTER 1. FORMAL ONTOLOGY AND CONCEPTUAL REALISM damental to science. Certainly, our commonsense framework is prior in the order of conception, but it is not necessarily prior in the order of being. This distinction is especially important in regard to the mind-body problem and the nature of consciousness where the eliminativist view against our commonsense understanding has been argued most forcefully. But some caution is necessary here because the mind-body problem really divides into at least two different sub-problems: (a) the study of the relations between physiological states and certain states of consciousness, and (b) the study of the emergence of consciousness, meaning, and the self and its relation to its body. consciousness, is a problem that is studied by experts in neuropsychology and other neurosciences, and as such it is a proper part of a characteristica realis. The second problem, i.e., the problem of the emergence of consciousness, meaning and the self can be solved, on the other hand, only by taking natural language, intentionality, and our commonsense framework into account, which means the inclusion within formal ontology of an intensional logic that can be used to represent our commonsense understanding and the contents of our beliefs and theories, including the fables and stories that are part of our culture and of our individual mental spaces. Such an intensional logic will provide an account of the ontology of fictional objects in terms of the contents of our concepts, and it will contain a logic of our various cognitive modalities, including a logic of knowledge and belief.
1.4
The Nexus of Predication
Leibniz’s own ideography for his characteristica universalis was algebraic and, as we have noted, did not deal with the central feature of either a conceptual or ontological theory of logical form—namely, the nexus of predication. How predication is represented in a formal ontology depends on different theories of universals, where by a universal we mean that which can be predicated of things (Aristotle, De Int. 17a39). Traditionally, there have been three main theories of universals: nominalism, conceptualism, and realism. The difference between these three types of theories depends on what each takes to be predicable of things. In this regard, we will distinguish between: • predication in language (nominalism), • predication in thought (conceptualism), and • predication in reality (realism). All three types of theories agree that there is predication in language, and in particular that predicates can be predicated of things in the sense of being true or false of them. Nominalism goes further in maintaining that only predicates (or really prdicate tokens) can be predicated of things, that is, that there are no universals other than the predicate expressions of some language or other:
1.4. THE NEXUS OF PREDICATION
11
Nominalism: only predicates are true or false of things; there are no universals that predicates stand for. Conceptualism opposes nominalism and maintains that predicates can be true or false of things only because they stand for concepts, where predicable concepts are the cognitive capacities—intelligible universals—that underlie predication in thought and our rule-following abilities in the use of the predicate expressions of natural language. Conceptualism: predication in thought underlies predication in language; predicable concepts are rule-following cognitive capacities regarding the use of predicate expressions. Realism also opposes nominalism in maintaining that there are real universals, namely, properties and relations, that are the basis of predication in reality. Realism: there are real properties and relations that are the basis of predication in reality. There are two distinct types of realism that should be distinguished; namely, various forms of logical realism as modern forms of Platonism, and various forms of natural realism, with at least one being a modern form of Aristotle’s theory of natural kinds. Realism logical realism natural realism with natural kinds without natural kinds ↓ (Aristotelian essentialism) Both forms of realism are compatible with conceptualism, but natural realism, especially Aristotle’s theory of natural kinds, is closely connected with the kind of conceptualism we will describe in later chapters. That is because, natural realism as a formal ontology presupposes some form of conceptualism in order even to be articulated; and the kind of conceptualism that we will later defend depends in turn on some form of natural realism as its causal basis. How conceptualism is compatible with logical realism, and how natural realism and a certain modern form of conceptualism are intimately connected are issues we will take up later in our discussion of what we call conceptual realism. Corresponding to these different theories of universals, there are different formal ontologies containing different formal theories of predication, each representing some variant of one of these alternatives. That means that there will be different comprehensive systems of formal ontology. Each formal ontology, of course, will view itself internally as the final arbiter of all logical and ontological distinctions. But the study of different possible formal ontologies, their consistency, adequacy, and relative strength with respect to one another, and, similarly, the study of the alternative subtheories that might be realized in the
12 CHAPTER 1. FORMAL ONTOLOGY AND CONCEPTUAL REALISM different branches of a comprehensive formal ontology, may together be called comparative formal ontology. A comprehensive system of formal ontology will in general have different branches or subsystems within which different ontological tasks can be carried out. One such branch, for example, might be a theory of parts and wholes, which would include a relation of foundation regarding how some parts are founded or dependent upon other parts or wholes.17 There might also be a theory of extensive and intensive magnitudes, i.e., a measurement theory, and a theory of continuants and of the existence of the latter in space and time.18
1.5
Univocal Versus Multiple Senses of Being
One important distinction between different systems of formal ontology is whether being is taken as univocal or as having different senses. It will have different senses when different types or categories of expressions are understood as representing different categories of being, in which case there will also be different types of variables bound by quantifiers having the entities of those different categories as their values. Such is the case in both conceptualism and some variants of realism. Where being is univocal, on the other hand, i.e., where there is just one ontological category of being (being simpliciter ), only one type of quantifiable variable will have semantic significance. This does not mean that there are no different “kinds”, or sorts, of being, but only that in such a framework being is a genus, and that the different kinds of being all fall within that genus. In a formal ontology for nominalism, for example, there will be no ontological category corresponding to any grammatical category other than that of objectual terms (logical subjects), and in particular there will be no ontological category or mode of being corresponding to the grammatical category of predicate expressions. Only objectual variables, i.e., the category of variables having objectual terms as their substituends, will have semantic-ontological significance in such a formal ontology. Predicate variables, and quantifiers binding such, if admitted at all, must then be given only a substitutional and not a semantic interpretation, which means that certain constraints must be imposed on the logic of the predicate quantifiers in such a formal ontology. Most nominalists in fact eschew even such a substitutional interpretation of predicate quantifiers and describe their ontology only in terms of first-order logic where there is but one type of boundable variable, i.e., where, as in W.V.O. Quine’s phrase: to be = to be the value of a bound objectual variable. It should be noted, however, that, unlike traditional nominalists, some contemporary nominalists (e.g., Nelson Goodman), take abstract objects (e.g., qualia) as well as concrete objects to fall under their supposedly univocal sense 17 Cf.
18 Cf.
Husserl 1900, Volume 2, Investigation III, and Barry Smith 1982. Brentano, 1933.
1.5. UNIVOCAL VERSUS MULTIPLE SENSES OF BEING
13
of being.19 This means that although there is but one ontological category of being in such an ontology, there may still be different “kinds” of being. That is, in such a system being is a genus, which is not at all the same as being multivalent. Nominalism: being is univocal; i.e., being is a genus. Being is also univocal in some forms of realism (regarding universals). This would appear to be the case, for example, in the ultra-realism of certain early scholastic philosophers for whom the realm of being is the realm only of universals (as in the teachings of John Scottus Eriugena and Remigius of Auxerre). It is certainly univocal in the case of certain contemporary forms of logical realism, where properties, relations, concrete objects, and perhaps states of affairs as well, are different kinds, as opposed to, categories of being. A formal ontology for such realists is developed today much as it is in nominalism, namely, as an axiomatic first-order logic with primitive predicates standing for certain basic ontological notions. Indeed, except perhaps for the distinction between an intensional and an extensional logic, there is little to distinguish realists who take being to be univocal from such nominalists as Goodman who include abstract objects as values of their objectual variables and who describe such objects axiomatically (e.g., in terms of a mereological relation of overlap, or of part-towhole).20 This is particularly true of those realists who, in effect, replace the extensional membership relation of an axiomatic set theory by an intensional relation of exemplification, and, dropping the axiom of extensionality, call the result a theory of properties.21 Logical realism: being is univocal (i.e., being is a genus) if predication is based on a relation of: 1. membership (set theory), or 2. exemplification (in first-order logic), or 3. part-to-whole (mereology). Formal ontology, in other words, for both the nominalist and that kind of realist who takes being to be univocal and who has abstract as well as concrete objects as values of their object variables, i.e., for whom being is a genus, is really no different from an applied theory of first-order logic. That is, it is no different from a first-order logic to which primitive “nonlogical” (descriptive?) constants and axioms are added and taken as describing certain basic ontological notions. In such a framework, it would seem, the dividing line between the logical and the nonlogical, or between pure formal ontology and its applications, has become somewhat blurred, if not entirely arbitrary. 19 Cf.
Goodman 1956, p. 17. also Goodman and Quine 1947. 21 Cf., Bealer 1982. 20 See
14 CHAPTER 1. FORMAL ONTOLOGY AND CONCEPTUAL REALISM
1.6
Predication and Preeminent Being
Beginning with Aristotle, the standard assumption in the history of ontology has been that being is not a genus, i.e., that there are different senses of being, and that the principal method of ontology is categorial analysis. This raises the problem of how the different categories of being fit together, and of whether one of the senses or category of being is preeminent and the others somehow dependent on that sense or category of being. The different categorial analyses that have been proposed as a resolution of this problem have all turned in one way or another on a theory of predication, i.e., on how the different categories fit together in the nexus of predication, and they have differed from one another primarily on whether the analysis of the fundamental forms of predication is to be directed upon the structure of reality or the structure of thought. In formal ontology, the resolution of this problem involves the construction of a formal theory of predication. Aristotle’s categorial analysis, for example, is directed upon the structure of the natural world and not upon the structure of thought, and the preeminent mode of being is that of concrete individual things, or primary substances. Aristotle’s realism regarding species, genera, and universals is a form of natural realism, it should be emphasized, and not of logical realism. Also, unlike logical realism, Aristotle’s realism is a moderate realism, though, as we indicate below, a modal moderate realism is better suited to a modern form of Aristotelian essentialism. • Moderate realism = the ontological thesis that universals exist only in rebus, i.e., in things in the world. • Modal moderate realism = the ontological thesis that universals exist only in things that, as a matter of a natural or causal possibility, could exist in nature, even if in fact no such things actually do exist in nature. Predication is explained in Aristotle’s realism in terms of two ontological configurations that together characterize the essence-accident distinction of Aristotelian essentialism. These are the so-called essential predicative nexus between an individual and the species or genera, i.e., the natural kinds, to which it belongs, and the accidental, or nonessential, predicative nexus between an individual and the universals that inhere in it. A formal theory of predication constructed as an Aristotelian formal ontology must respect this distinction between essential and accidental predication, and it must do so in terms of an adequate representation of the two ontological configurations underlying predication in an Aristotelian ontology. Aristotle’s moderate natural realism has two types of predication: 1. Predication of species, genera (natural kinds). 2. Predication of properties and relations. As a formal ontology, Aristotelian essentialism must contain a logic of natural kinds. In addition, as a form of moderate realism it must impose the constraint that every natural kind, property or relation is instantiated, because every natural kind, property or relation exists only in rebus. This constraint leads to
1.6. PREDICATION AND PREEMINENT BEING
15
Aristotle’s problem of the fixity of species, according to which members of a species cannot come to be except from earlier members of that species, and that therefore there can be no evolution of new species. The fixity of species: Members of a species cannot come to be except from earlier members of that species. Therefore, there can be no evolution of new species. This problem can be resolved, however, in a modified Aristotelian formal ontology of modal natural realism, where the modal category of natural necessity and possibility is part of the framework of the formal ontology. On this modified account, instead of requiring that every natural property or relation actually be instantiated at any given time, we require only that such an instantiation be within the realm of natural possibility, a possibility that might arise in time and changing circumstances and not just in other possible worlds. Such a formal ontology, needless to say, will contain a modal logic for natural necessity and possibility, as well as a logic of natural kinds that is to be described in terms of that modal logic.22 Natural necessity, we will later argue, is a causal modality based on natural kinds and the laws of nature, and as such it is not the same as logical necessity. As modalities, logical necessity and possibility, we will later argue, can be made sense of only in an ontology of logical atomism, an ontology in which there are no causal relations and no natural necessity as a causal modality. Plato’s ontology is also directed upon the structure of reality, but the preeminent mode of being in this framework is not that of concrete or sensible objects but of the Ideas, or Forms. This leads to the problem of how and in what sense concrete objects participate in Ideas, and also to the problem of how and in what sense Ideas are “things” or abstract objects separate from the concrete objects that participate in them. A Platonist theory of predication in contemporary formal ontology is the basis of logical realism, where it is assumed that a property or relation exists corresponding to each well-formed predicate expression (or open formula) of logical grammar, regardless of whether or not it is even logically possible that such a property or relation have an instance. When applied as a foundation for mathematics (as was Plato’s own original intent), logical realism is also called ontological logicism. The best-known form of logical realism today is Bertrand Russell’s theory of logical types, which Russell developed as a way to avoid his famous paradox of predication (upon which his paradox of membership is based), a paradox not unrelated to Plato’s problem of the separate reality of Ideas. Whether and to what extent Russell’s theory of logical types can satisfactorily resolve either of Plato’s problems and be the basis of an adequate realist formal ontology is an issue that belongs to what we have called comparative formal ontology. 22 Cf.
Cocchiarella 1976 and 1996.
16 CHAPTER 1. FORMAL ONTOLOGY AND CONCEPTUAL REALISM
1.7
Categorial Analysis and Transcendental Logic
Kant’s categorial analysis, unlike Aristotle’s, is directed upon the structure of thought and experience rather than upon the structure of reality. The categories function on this account to articulate the logical forms of judgments and not as the general causes or grounds of concrete being. There is no preeminent mode of being identified in this analysis, accordingly, other than that of the transcendental subject, whose synthetic unity of apperception is what unifies the categories that are the bases of the different possible judgments that can be made. What categories there are and how they fit together to determine the concept of an object in general is determined through a “transcendental deduction” from Kant’s table of judgments, i.e., from the different possible forms that judgments might have according to Kant. It is for this reason that the logic determined by this kind of categorial analysis is called transcendental logic. The transcendental logic of Husserl in his later work is perhaps one of the best-known versions of this type of approach to formal ontology. According to Husserl, logic, as formal ontology, is a universal theory of science, and as such it is the justifying discipline for science. But even logic itself must be justified, Husserl insists, and it is that justification that is the task of transcendental logic. This means that the grounds of the categorial structures that determine the logical forms of pure logic are to be found in a transcendental subjectivity, and it is to a transcendental critique of such grounds that Husserl turns in his later philosophical work. On the basis of such a critique, for example, Husserl gives subjective versions of the laws and rules of logic, such as the law of contradiction, the principle of excluded middle, and the rules of modus ponens and modus tollens, claiming that it is only in such subjective versions that there can be found the a priori structures of the evidence for the objective versions of those laws and rules.23 Husserl also claims on the basis of such grounds that every judgment can be decided24 , and that a “multiplicity,” such as the system of natural numbers, is to be “defined, not by just any formal axiom system, but by a ‘complete’ one”.25 That is, according to Husserl: the axiom-system formally defining such a multiplicity is distinguished by the circumstance that any proposition ... that can be constructed, in accordance with the grammar of pure logic, out of the concepts ... occurring in that system is either true—that is to say: an analytic (purely deducible) consequence of the axioms— or ‘false’—that is to say: an analytic contradiction—; tertium non datur.26 Unfortunately, while such claims for transcendental logic are admirable ideals, they are nevertheless in conflict with certain well-known results of 23 Husserl
1929, §§75-8. §§79-80. 25 Ibid., §31, p. 96. 26 Ibid.. Cf. also Husserl 1913 §72, pp. 187f. 24 Ibid.,
1.8. THE COMPLETENESS PROBLEM
17
mathematical logic, such as Kurt G¨odel’s first incompleteness theorem.
1.8
The Completeness Problem
The transcendental approach to categorial analysis, as this last observation indicates, raises the important problem of the completeness of formal ontology. It does this, moreover, not in just one but in at least two ways: first, as the problem of the completeness of the categories; and, second, as the problem of the completeness of the laws of consequence regarding the logical forms generated by those categories. Two problems of the completeness of formal ontology: 1. the completeness of the categories; and 2. the completeness of the deductive laws with respect to those categories. For Aristotle, for whom the categories are the most general “causes” or grounds of concrete being, and for whom categorial analysis is directed upon the structure of reality, the categories and their systematization must be discovered by an inductive abstraction and reflection on the structure of reality as it is revealed in the development of scientific knowledge, and therefore the question of the completeness of the categories and of their systematization can never be settled as a matter of a priori knowledge. This is true of natural realism in general. Natural realism: the categories of nature and their laws are not knowable a priori. For Kant and the transcendental approach, however, the categories and the principles that flow from them have an a priori validity that is grounded in the understanding and pure reason respectively—or, as on Husserl’s approach, in a transcendental phenomenology—and the question of the “unconditioned completeness” of both is said to be not only practical but also necessary. The difficulty with this position for Kant is that neither the system of categories nor the laws of logic described in terms of those categories can be viewed as providing an adequate system of formal ontology as we have described it above. Kant’s description of logic, for example, restricts it to the valid forms of the syllogism, which can in no sense account for the complexity of many intuitively valid arguments of natural language, not to mention the complexity of proofs in mathematics. Husserl, unlike Kant, does not himself attempt to settle the matter of a complete system of categories, nor therefore of a complete system of the laws of logic or formal ontology; but he does maintain that such completeness is not only possible but necessary, and that the results achieved regarding the categories and their systematization must ultimately be grounded on the a priori structures of the evidence of a transcendental subjectivity. Transcendental logic: the categories and their laws are knowable a priori. The transcendental approach in general, in other words, or at least the a priori nature of its methodology as originally described, leaves no room for
18 CHAPTER 1. FORMAL ONTOLOGY AND CONCEPTUAL REALISM inductive methods or new developments in either logic or categorial analysis, especially in the way both are affected by new results in scientific theory (e.g., the logic of quantum mechanics27 and the way that logic relates to the logic of macrophysical objects) or in theoretical linguistics, (e.g., universal grammar and the way that grammar is related to the pure logical grammar of a formal ontology), or even in cognitive science (e.g., artificial intelligence and the way that the computational theory of mind is related to the categorial and deductive structure of logic). Some categorial analyses not knowable a priori: 1. The logic of quantum mechanics and how that logic relates to the logic of macrophysical objects. 2. Theoretical linguistics: is there a universal grammar underlying all natural languages? And, if so, how is that grammar related to the pure logical grammar of a formal ontology? 3. Cognitive science and artificial intelligence: are there categories and laws of thought that can be represented in formal ontology? And, if so, how are these categories and laws related to the categories of nature? And can they be simulated (duplicated?) in artificial intelligence? Despite the difficulties with the problem of completeness of the a priori methodology of the transcendental approach, it does not follow that we must give up the view that an analysis of the forms of predication is to be directed primarily upon the structure of thought. There are alternatives other than the transcendental idealism of either Kant or Husserl that such a view might adopt. Jean Piaget’s genetic epistemology with its “functional” (as opposed to absolute) a priori is such an alternative, for example, and so is Konrad Lorenz’s biological Kantianism with its evolutionarily determined (and therefore nontranscendental) a priori.28 Some non-transcendental approaches: 1. Jean Piaget’s genetic epistemology (a non-absolute “functional” a priori). 2. Konrad Lorenz’s biological Kantianism (an evolutionarily determined a priori). Any version of a naturalized epistemology, in other words, where an a posteriori element would be allowed a role in the construction of a formal ontology, might serve as such an alternative; and in fact such a naturalized epistemology is presupposed by conceptual realism, which we will describe in more detail later. The comparison of these alternatives, and a study of their adequacy (as well as of the adequacy of a more complete and perhaps modified account of transcendental apriority) as epistemological grounds for a categorial analysis that is directed upon the structure of thought, are issues that properly belong to comparative formal ontology. The transcendental approach claims to be independent of our status as biologically, culturally, and historically determined 27 Cf.
28 Cf.
Putnam 1969 and Dummett 1976. Piaget 1972 and Lorenz 1962.
1.9. SET-THEORETIC SEMANTICS
19
beings, and therefore independent of the laws of nature and our evolutionary history.
1.9
Set-Theoretic Semantics
The problem of the completeness of a formal ontology brings up a methodological issue that is important to note here. This is the issue of how different research programs can be carried out in restricted branches or subdomains of a formal ontology without first deciding whether or not the categorial analysis of that formal ontology is to be directed upon the structure of thought or the structure of reality. We do not always have to decide in advance whether or not there must (or even ever can) be a final completeness to the categories or of the laws of logic before undertaking such a research program. In particular, we can try to establish restricted or relative notions of completeness for special areas of a formal ontology, and we can then compare and evaluate those results in the context of comparative formal ontology. The construction of abstract formal systems and model-theoretic semantics within set theory will be especially useful in carrying out and comparing such research programs. In other words, set theory is an ideal framework within which to carry out comparative analyses of different formal systems proposed either as a formal ontology or a subsystem of such. Set theory is not itself a formal ontology, it should be noted, in that it does not contain a theory of predication. We must be cautious in our use of set theory, however, and especially in how we apply such well-known mathematical results as Kurt G¨odel’s incompleteness theorems. G¨ odel’s first incompleteness theorem, for example, does not show, as is commonly claimed, that every second-order predicate logic must be incomplete, where by second-order predicate logic we mean an extension of first-order predicate logic in which quantifiers are allowed to reach into the positions that predicates occupy as well as of the subject or argument positions of those predicates. Rather, what G¨ odel’s theorem shows is that second-order predicate logic is incomplete with respect to its so-called standard set-theoretic semantics. In particular, we must not confuse membership in a set with predication of a concept, property, or relation. Nor should we wrongly identify the logical concept of a class, i.e., the concept of a class as the extension of a concept, property or relation, with the mathematical concept of a set, i.e., a set in the sense of the iterative concept, which is based on Georg Cantor’s power-set theorem that the set of all subsets of any given set always has a greater cardinality than that set. Cantor’s theorem, for example, while essential to the iterative concept of set, will in fact fail in certain special cases of the logical concept of a class—such as, e.g., the universal class, which is the extension of the concept of self-identity. For this reason we should note that 1. a representation of concepts by sets in a set-theoretical semantics will not always result in the same logical structure as a representation of those concepts by the classes that are their extensions, and
20 CHAPTER 1. FORMAL ONTOLOGY AND CONCEPTUAL REALISM 2. an incompleteness theorem based on the one kind of structure need not imply an incompleteness theorem based on the other.29 We should distinguish accordingly: • (a) The logical notion of a class (as one or as many) as the extension of (and therefore having its being in) a concept. • (b) The mathematical iterative notion of a set (which has its being in its members). A set-theoretical semantics for a formal theory of predication must not be confused, in other words, with a semantics for that theory based on its own forms of predication taken primitively. For the latter is based on the very forms of predication that it is designed to interpret, and it is in that sense an internal semantics for that theory, while the set-theoretical semantics, being based on the membership relation of a framework not internal to the theory itself, is an external semantics for that theory. This means that in constructing a settheoretical semantics for a formal theory of predication we must be cautious not to confuse and literally identify the internal content or mode of significance of the forms of predication of that theory with the external model-theoretic content of the membership relation, or, as in the case of a set-theoretic possible-worlds semantics, with the external content of any function (e.g., on models as settheoretic representatives of possible worlds) defined in terms of the semantically external membership relation. If we do not confuse predication with membership in this way, then we will be able to see why the incompleteness of secondorder predicate logic with respect to its standard set-theoretical semantics need not automatically apply to any formal ontology designed to include secondorder logic as part of its formal theory of predication. The careful separation and clarification of these issues is a topic that belongs to the methodology of comparative formal ontology. Distinguish: 1. Predication in a formal theory of predication corresponding to a given theory of universals. 2. Membership in a set based on the iterative concept of set. G¨ odel’s first incompleteness theorem does show that any formal ontology that includes arithmetic as part of its pure formal content must be deductively incomplete; that is, not every well-formed sentence of the pure logical grammar of such a formal ontology will be such that either it or its negation is provable in that formal ontology. This does impose a limitation on what can be deductively achieved in such a formal ontology, and it requires a modification, if not a complete revision, of any categorial analysis, such as Husserl’s, where the ideal of deductive completeness even for an “infinite multiplicity” such as the system of natural numbers is taken as an essential part of that analysis. 29 See Cocchiarella 1988 and 1992 for a discussion and example of a framework in which Cantor’s theorem fails.
1.10. CONCEPTUAL REALISM
21
The deductive incompleteness of an ontology that contains arithmetic is not the same as the incompleteness of the categorial structure of that ontology, in other words, and in particular it does not show that the formal theory of predication that is part of that structure is incomplete. What must be resolved in a formal ontology that is to contain arithmetic as part of its pure formal content is the problem of how the possible completeness of its internal content as a formal theory of predication is to be distinguished from its deductive incompleteness, and how within that pure formal content we are to characterize the content of arithmetic (and perhaps, more generally, all of classical mathematics as well).30 Finally, in regard to G¨ odel’s second incompleteness theorem what must also be resolved for such a formal ontology is the question of how, and with what sort of significance or content, we are to prove its consistency, since such a proof is not available within that formal ontology itself. Again, these are issues that are to be investigated not so much in a particular formal ontology as in comparative formal ontology.
1.10
Conceptual Realism
Comparative formal ontology, as our remarks have indicated throughout, is the proper domain of many issues and disputes in metaphysics, epistemology, and the methodology of the deductive sciences. Just as the construction of a particular formal ontology lends clarity and precision to our informal categorial analyses and serves as a guide to our intuitions, so too comparative formal ontology can be developed so as to provide clear and precise criteria by which to judge the adequacy of a particular system of formal ontology and by which we might be guided in our comparison and evaluation of different proposals for such systems. It is only by constructing and comparing different formal ontologies that we can make a rational decision about which such system we should ourselves ultimately adopt. Since 1969 I have constructed and compared a number of such systems and have come to the conclusion that the framework of conceptual realism is the formal ontology that we should adopt. Unlike the a priori approach of the transcendental method, which claims to be independent of the laws of nature and our evolutionary history, i.e., of our status as biological beings with a culture and history that shapes our language and much of our thought, conceptual realism is framed within the context of a naturalistic epistemology and a naturalistic approach to the relation between language and thought, thought and reality, and our scientific knowledge of the world. The following are some of the features of conceptual realism that we will cover in this book. As a conceptualist theory about the mental acts that underlie reference and predication in language and thought, the categorial analyses of conceptual re30 We should keep in mind in this context the distinction between a logical analysis of the concept of a natural number on the one hand, which may be part of a pure formal theory of predication, and the axioms, such as infinity, that must be added to the logical background in order to account for the standard laws of arithmetic.
22 CHAPTER 1. FORMAL ONTOLOGY AND CONCEPTUAL REALISM alism are primarily directed upon the structure of thought. But what guides us in these analyses is the structure of natural language as a representational system, and in particular as a representational system that is categorially structured and logically oriented. Our methodology, in other words, is based on a linguistic and logical analysis of our speech and mental acts, and not, e.g., on a phenomenological reduction of those acts. The realism part of conceptual realism, as we will see, contains both a natural realism and an intensional realism, each of which can be developed as separate subsystems, one containing a modern form of Aristotelian essentialism, and the other containing a modern counterpart of Platonism based on the intensional contents of our speech and mental acts. We call these two subsystems conceptual natural realism and conceptual intensional realism. The realism of conceptual realism contains two subsystems: 1. a conceptual natural realism (as a modern form of Aristotelian essentialism), and 2. a conceptual intensional realism (as a modern counterpart of Platonism). In addition to the categorial analyses that are directed upon our speech and mental acts, conceptual natural realism also contains a categorial analysis that is directed upon the structure of reality, and in particular an analysis in which natural properties and relations are taken as corresponding to some, but not all, of our predicable concepts, and natural kinds are taken as corresponding to some, but not all, of our sortal common-name concepts. Natural kinds are not properties in this framework. The category of natural kinds is the realist analogue of a category of common-name concepts and not of predicable concepts. Common-name concepts are a fundamental part of conceptual realism’s theory of reference, just as predicable concepts are a fundamental part of conceptual realism’s theory of predication. Proper as well as common names are part of this theory of reference, and together both are described in a separate logic of names as another subsystem of conceptual realism. As we will explain later, S. Le´sniewski’s ontology, which has also been described as a logic of names, is reducible to our conceptualist logic of names.31 Conceptual intensional realism, as we have developed it, is a logic of nominalized predicates and propositional forms as abstract singular terms, i.e., a logic of the abstract nouns and nominal phrases that we use in describing the intensional contents of our speech and mental acts. The intensional objects that are denoted by these abstract singular terms serve the same purposes in conceptual intensional realism that abstract objects serve in logical realism as a modern form of Platonism. The difference is that, unlike Platonic Forms, the intensional objects of conceptual realism do not exist independently of mind and the natural world, the way they do in logical realism, but are products of the evolution of culture and language, and especially of the institutionalized linguistic practice of nominalization. 31 See Cocchiarella 2001 for a proof of this reduction. For a description of Le´ sniewski’s ontology, see Slupecki, 1955.
1.11. SUMMARY AND CONCLUDING REMARKS
23
The way both forms of realism are contained within the general framework of conceptual realism shows how a modern form of Aristotelian essentialism is compatible with an intensional logic that is a counterpart to a modern form of Platonism.
1.11
Summary and Concluding Remarks
• Metaphysics consists of the separate disciplines of ontology and cosmology, each with their respective methodologies. • Formal ontology connects logical categories—especially the categories involved in predication—with ontological categories. • The goal of a formal ontology is the construction of a lingua philosophica, or characteristica universalis, as explicated in terms of an ars combinatoria and a calculus ratiocinator as part of a formal theory of predication. • A formal ontology should serve as the framework of a characteristica realis, and hence as the basis of a formal approach to science and cosmology. It should also serve as a framework for our commonsense understanding of the world. • The central feature of a formal ontology is how it represents the nexus of predication, which depends on what theory of universals it assumes. • The three main theories of universals are nominalism, conceptualism, and (logical or natural) realism. • The analysis of the fundamental forms of predication of a formal ontology may be directed upon the structure of reality or upon the structure of thought. • Natural realism, and in particular Aristotle’s ontology, is directed upon the structure of the natural world, and the preeminent mode of being is that of concrete individual things, or primary substances. There are two major forms of natural realism, moderate realism and modal moderate realism. • Aristotle’s moderate natural realism has two types of predication: predication of species and genera (natural kinds), and predication of properties and relations. • Kant’s and Husserl’s categorial analyses, unlike Aristotle’s, are directed upon the structure of thought and experience rather than upon the structure of reality. The categories function on this account to articulate the logical forms of judgments and not as the general causes or grounds of concrete being. • Husserl’s formal ontology is based on a transcendental logic in which the laws and rules of logic are justified in terms of subjective analyses of presumed a priori structures that provide the evidence for the objective versions of those laws and rules. • There are two problems regarding the completeness of a formal ontology: first, the problem of the completeness of the categories of an ontology, and second, the problem of the completeness of the deductive laws that are based on those categories. • Set theory provides only an external semantics for a formal ontology, unless that ontology is set theory itself, which has no nexus of predication, and hence
24 CHAPTER 1. FORMAL ONTOLOGY AND CONCEPTUAL REALISM strictly speaking is not a formal ontology. An incompleteness theorem for a formal ontology based a set-theoretic semantics need not show that the ontology is incomplete with respect to an internal semantics. In particular, sometimes general models are a better repesentation of a formal ontology’s internal semantics than are so-called “standard” models. • Conceptual realism is a formal ontology framed within the context of a naturalistic epistemology and a naturalistic approach to the relations between language, thought, and reality as based on our scientific knowledge of the world. • Conceptual realism is based on a conceptualist account of the speech and mental acts that underlie reference and predication. It is directed in that regard primarily upon the structure of thought. But, because its methodology is based on a linguistic and logical analysis of our speech and mental acts, it is not committed to a phenomenological reduction of those acts. Nor does it preclude such a reduction. • Conceptual realism contains both a natural realism and an intensional realism, each of which can be developed as separate subsystems that are compatible within the larger framework, one containing a modern form of Aristotelian essentialism, and the other containing a modern counterpart of Platonism based on the intensional contents of our speech and mental acts.
Chapter 2
Time, Being, and Existence One criterion of adequacy for a formal ontology, we have said, is that it should provide a logically perspicuous representation of our commonsense understanding of the world, and not just of our scientific understanding.1 Now a central feature of our commonsense understanding is how we are conceptually oriented in time with respect to the past, the present and the future, and the question arises as to how we can best represent this orientation. It is inappropriate to represent it in terms of a tenseless idiom of moments or intervals of time of a coordinate system, as is commonly done in scientific theories; for that amounts to replacing our commonsense understanding with a scientific view. A more appropriate representation is one that respects the form and content of our commonsense speech and mental acts about the past, the present and the future. Formally, this can best be done in terms of a logic of tense operators, or in what is now called tense logic.2 The most natural formal ontology for tense logic is conceptual realism. That is because what tense operators represent in conceptual realism are certain cognitive schemata regarding our orientation in time and that are fundamental to both the form and content of our conceptual activity. Thought and communication are inextricably temporal phenomena, and it is the cognitive schemata underlying our use of tense that structures that phenomena temporally in terms of the past, the present and the future. A second criterion of adequacy for a formal ontology is that it must explain and provide an ontological ground for the distinction between being and existence, or, if it rejects that distinction why it does so. Put simply, the problem is: Can there be things that do not exist? Or is being the same as existence? 1 This chapter is an extension and deveopment of my article “Quantification, Time, and Necessity,” in Lambert 1991. 2 For an excellent book on tense logic and related philosophical issues, see Prior 1967. Other texts are Gabbay 1976 and Benthem 1983.
25
26
CHAPTER 2. TIME, BEING, AND EXISTENCE
Now the logic of time in conceptual realism provides the clearest ontological ground for such a distinction in terms of the tense-logical distinction between past, present and future objects, i.e., the distinction between things that did exist, do exist, or will exist, or what in the proposed book I call realia, as opposed to existentia, which is restricted to the things that exist at the time we speak or think, i.e., the time we take to be the present of our commonsense framework. The present, in other words, unlike the tenseless medium of our scientific theories, is indexical, and refers at any moment of time to that moment itself. Another criterion of adequacy for a formal ontology is that it must explain the ontological grounds, or nature, of modality, i.e., of such modal notions as necessity and possibility, as opposed to merely giving a set-theoretic semantics for modal logic. In this chapter we will deal not just with the fundamental categories of time, being and existence, but with modality as well. We explain below how some of the earliest views of necessity and possibility are grounded in such a framework as tense logic. As we explain below, even the temporal and modal distinctions of the special theory of relativity theory can be understood best within the framework of conceptual realism. In what follows we will develop separate logics for both possibilism and actualism, and then we will extend these logics to both possibilist and actualist versions of quantified tense logic, where by possible objects we mean only realia, i.e., the things that did, do, or will exist. These logics will serve not only as essential component parts of our larger framework of conceptual realism as a formal ontology, but also as paradigmatic examples of how different parts or aspects of a formal ontology can be developed independently of constructing a comprehensive system all at once. They also illustrate how the model-theoretic methodology of set theory can be used to guide our intuitions in axiomatically developing the formal systems we construct as part of a formal ontology.
2.1
Possibilism versus Actualism
The two main parts of metaphysics, we have noted, consists of ontology and cosmology, where ontology is the study of being, and cosmology is the study of the physical universe, i.e., the world of natural objects and the space-time manifold in which they exist. If existence is the mode of being of the natural objects of the space-time manifold—i.e., of “actual” objects—then the question arises as to whether or not being is the same as existence, and how this difference, or sameness, is to be represented in formal ontology. We will call the two positions one can take on this issue possibilism and actualism, respectively. In possibilism, there are objects that do not now exist but could exist in the physical universe, and hence being is not the same as existence. In actualism being is the same as existence. Possibilism: There are objects (i.e., objects that have being or) that possibly exist but that do not in fact exist. Therefore: Existence = Being.
2.1. POSSIBILISM VERSUS ACTUALISM
27
Actualism: Everything that is (has being) exists. Therefore: Existence = Being. In formal ontology, possibilism is developed as a logic of actual and possible objects. Whatever exists in such a logic has being, but it is not necessary that whatever has being exists; that is, there can be things that do not exist. Much depends, of course, on what is meant here by ‘can’. Does it depend, for example, on the merely possible existence of objects that never in fact exist in the spacetime manifold? Or is there a weaker, less committal sense of modality by which we can say that there can exist objects that do not now exist? The answer is there are a number of such senses, all explainable in terms of time or the space-time manifold. We can explain the difference between being and existence, first of all, in terms of the notion of a local time (Eigenzeit) of a world-line of space-time. Within the framework of a possibilist tense logic, for example, being encompasses past, present, and future objects with respect to such a local time, while existence encompasses only those objects that presently exist.3 No doctrine of merely possible existence is needed in such a framework to explain the distinction between existence and being. We can interpret modality, in other words, so that it can be true to say that some things do not exist, namely past and future things that do not now exist. In fact, there are potentially infinitely many different modal logics that can be interpreted within the framework of tense logic. In this respect, tense logic provides a paradigmatic framework within which possibilism can be given a logically perspicuous representation as a formal ontology. Tense logic also provides a paradigmatic framework for actualism as well. Instead of possible objects, actualism assumes that there can be vacuous proper names, i.e., proper names that name nothing. Some names, for example, may have named something in the past, but now name nothing because those things no longer exist; and hence the statement that some things do not exist can be true in a semantic, metalinguistic sense, i.e., as a statement about the denotations, or lack of denotations, of proper names. What is needed, according to actualism, is not that we should distinguish the concept of existence from the concept of being, but only that we should modify the way that the concept of existence (being) is represented in standard first-order predicate logic with identity. On this view, a first-order logic of existence should allow for the possibility that some of our singular terms might fail to denote an existent object, which, according to actualism, is only to say that those singular terms denote nothing, rather than that what they denote are objects (beings) that do not exist. Such a logic for actualism amounts to what today is called a logic free of existential presuppositions, or simply free logic.4 The logic of actualism = free logic, i.e. logic free of existential presuppositions regarding the denotations of singular terms. 3 For some philosophers, e.g., Arthur Prior, being encompasses only past and present objects, apparently because, unlike the past and the present, the future is as yet undetermined. See Prior 1967, chapter viii. 4 See Lambert 1991 for a collection of papers on free logic and its philosophical applications.
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28
In what follows we shall first formulate a logic of actual and possible objects in which existence and being are assumed to be distinct concepts that are represented by different universal quantifiers ∀e and ∀, respectively, with the existential quantifiers ∃e and ∃ defined in terms of ∀e and ∀ and negation in the usual way. The free logic of actual objects, where existence is not distinguished from being—but also where it is not assumed that all singular terms denote—is then described as a certain subsystem of the logic of actual and possible objects. Of course, it is only from the perspective of possibilism that the logic of actual objects is to be viewed as a proper subsystem of the logic of being, because, according to possibilism, the logic of being includes the logic of actual objects as well. From the perspective of actualism, the logic of actual objects is all there is to the logic of being. Although both the free logic of actualism and the logic of possibilism have their most natural applications in tense and modal logic, we will first formulate these logics without presupposing any such larger encompassing framework. We will then describe a framework for tense logic where we distinguish an application of the logic of actual and possible objects from an application of the free logic of actual objects simpliciter. After that, we will explain how different modal logics can be interpreted in terms of tense logic, and how an application of the logic of actual and possible objects in modal logic can be distinguished from an application of the free logic of actual objects simpliciter.5 Tense logic, as these developments will indicate, is a paradigmatic framework in which to formally represent the differences between actualism and possibilism.
2.2
Logics of Actual and Possible Objects
We will initially consider only the first-order logic of actual and possible objects. Later, after we have considered different formal theories of predication, we will extend the logic into a fuller account of being and existence. We turn first to the syntax of the logic. As logical constants, we have the following: 1. The negation sign: ¬ 2. The (material) conditional sign: → 3. The conjunction sign: ∧ 4. The disjunction sign: ∨ 5. The biconditional sign: ↔ 5 For an account of the kinds of qualifications that are required in the statement of the laws involving the interplay of quantifiers, tenses, and modal operators, or what are called de re modalities, see Appendix 1 of this chapter. The tense-logical frameworks for which these laws are stated provide logically perspicuous representations of the differences between actualism and possibilism, including a restricted version of temporal possibilism where determinate being includes only what did or does exist, leaving the future as an indeterminate realm of nonbeing.
2.2. LOGICS OF ACTUAL AND POSSIBLE OBJECTS
29
6. The identity sign: = 7. The possibilist universal and existential quantifiers: ∀, ∃ 8. The actualist universal and existential quantifiers: ∀e , ∃e When stating axioms, we will assume that ¬, →, =, ∀, and ∀e are the only primitive logical constants, and that the others are defined in terms of these in the usual way. We do this only for the convenience of not having to deal with too many axioms when proving metatheorems. We take a formal language L to be a set of objectual constants and predicates of arbitrary (finite) degrees. We define a formal language in this way because we want every formal language to have the same logical grammar and therefore differ from other formal languages only in the objectual and predicate constants in that language. Objectual constants are the symbolic counterparts of proper names in this sort of logic, and n-place predicate constants are the symbolic counterparts of n-ary relation expressions, with one-place predicate constants the counterparts of monadic predicates. Whether or not predicate constants stand for concepts or properties and intensional relations depends on what formal theory of predication is assumed in the larger framework, i.e., the sort of framework that we will turn to later in our discussion of formal theories of predication. Also, whether the use of objectual constants is the best way to represent proper names and singular reference is a matter we will turn to later as well. For now, we note only that this is the standard, modern way to represent proper names. • Objectual constants6 : symbolic counterparts of proper names. • n-place predicate constants: symbolic counterparts of n-place predicate expressions of natural language, for some natural number n. • A formal language L: a set of objectual and predicate constants. The objectual terms, or for brevity, terms, of a formal language L are the objectual variables and the objectual constants in that language. Atomic formulas of L are the identity formulas of L, i.e., formulas of the form a = b, or the result of concatenating an n-place predicate constant of L with n many singular terms of L. • The (objectual) terms of L =df {a : a is either an objectual constant in L or an objectual variable}. • The atomic formulas of L =df {a = b : a, b are terms of L}∪{F (a1 , ..., an ) : for some natural number n, F is an n-place predicate constant in L and a1 , ..., an are terms of L}. 6 We use the phrase ‘objectual constant’ and ‘objectual variable’ instead of the more usual ‘individual constant’ and ‘individual variable’ so as to accommodate our account in a later chapter of plural expressions that name plural objects, i.e., objects that are not individuals in the sense of single entities.
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We use two quantifiers—though only one style of objectual variable—one for quantification over possible objects, or possibilia, and the other for quantification over actual objects. The formulas of a language L are those objects that belong to every set K containing the atomic formulas of L and such that ¬ϕ, (ϕ → ψ), (∀xϕ), (∀e xϕ) ∈ K whenever ϕ, ψ ∈ K and x is an objectual variable. As indicated, we use Greek letters as variables for expressions of the syntactical metalanguage (set theory). • χ is a (first-order) formula of L if, and only if, for all sets K, if (1) every atomic formulas of L is in K and (2) for all ϕ, ψ ∈ K and objectual variables x, ¬ϕ, (ϕ → ψ), (∀xϕ), and (∀e xϕ) ∈ K, then χ ∈ K. Note: The induction principle for formulas follows from this definition. Induction Principle: If L is a (formal) language, then if: (1) every atomic formula of L ∈ K, and (2) for all ϕ, ψ ∈ K and all objectual variables x, ¬ϕ, (ϕ → ψ), (∀xϕ), and (∀e xϕ) ∈ K, then every formula of L ∈ K.
2.3
Set-theoretic Semantics
A set-theoretic semantics for the logic of actual and possible objects is only a mathematical tool. It does not explain the difference between being and existence but merely models it mathematically. In this respect it guides our intuitions about how validity and logical consequence are to be determined in the logic. A model for a formal language L is characterized in terms a universe U of actual objects, a nonempty domain of discourse of possible objects D containing the universe of actual objects, and an assignment R of extensions drawn from the domain of discourse to the objectual and predicate constants in L. The extension of an objectual constant is what is denoted by that constant, and the extension of an n-place predicate constant F is the set of n-tuples of objects in the domain that F is understood to be true of in the model. Definition: A is a model for a formal language L if, and only if, for some U, D, R, (1) A = U, D, R, (2) D is a nonempty set, (3) U ⊆ D, and (4) R is a function on L such that for each objectual constant in L, R(a) ∈ D, and for each natural number n and each n-place predicate constant F in L, R(F ) ⊆ D n , i.e., R(F ) is a set of n-tuples drawn from D. The assumption that the domain of discourse of possible objects is not empty can be dropped, and later we will have reasons to do just that; but, insofar as
2.4. AXIOMS IN POSSIBILIST LOGIC
31
the domain is restricted to concrete objects—i.e., insofar as it does not include any abstract objects—then, as applied to time, the assumption says only that some concrete object exists at some time or other, which seems appropriate; and as applied to possible worlds of concreta, it says only that some concrete object exist in some world or other, which again seems appropriate. The notions of satisfaction and truth of a formula of a language L in a model for L are defined in the usual Tarski manner, except that the satisfaction clause for the actual quantifier applies only to the universe of the model in question, whereas the satisfaction clause for the possible quantifier covers the entire domain of discourse, i.e., the set of possibilia of the model. We will not go into those details here. A formula ϕ is said to be logically true if for some language L of which ϕ is a formula, ϕ is true in every model suited to L. Definition: ϕ is logically true if, and only if, for some language L, ϕ is a formula of L and ϕ is true in every model A suited to L.
2.4
Axioms in Possibilist Logic
We turn now to an axiomatization of the logic of actual and possible objects as our first-order description of possibilism. We note as well that a first-order logic of actualism is properly contained in this version of possibilism. Where ϕ, ψ, χ are formulas, x, y are variables, and a, b are (objectual) terms (variables or constants), we take all instances of the following schemas to be axioms of the logic of actual and possible objects. (A1)
ϕ → (ψ → ϕ)
(A2)
[ϕ → (ψ → χ)] → [(ϕ → ψ) → (ϕ → χ)]
(A3)
(¬ϕ → ¬ψ) → (ψ → ϕ)
(A4)
(∀x)[ϕ → ψ] → [(∀x)ϕ → (∀x)ψ]
(A5)
(∀e x)[ϕ → ψ] → [(∀e x)ϕ → (∀e x)ψ]
(A6)
ϕ → (∀x)ϕ,
(A7)
(∀x)ϕ → (∀ x)ϕ
where x is not free in ϕ e
(A8)
(∃x)(a = x),
(A9)
(∀e x)(∃e y)(x = y)
(A10)
where x is not a where x, y are distinct variables
a = b → (ϕ → ψ), where ϕ, ψ are atomic formulas and ψ is obtained from ϕ by replacing an occurrence of b by a
As inference rules we assume only modus ponens and the rule of universal generalization The turnstile, , is read as ‘is a theorem of the logic of actual and possible objects’. MP:
If ϕ → ψ and ϕ, then ψ.
CHAPTER 2. TIME, BEING, AND EXISTENCE
32 UG:
If ϕ, then (∀x)ϕ.
By axiom (A7) and (UG), we also have as a derived rule: UGe :
If ϕ, then (∀e x)ϕ.
Axioms (1)–(A3), as is well-known, suffice to validate all tautologous formulas, which we will assume hereafter. Note that where a, b, c are terms (variables or constants), the transitivity of identity law, a = b → (b = c → a = c), is an immediate consequence of axiom (A10). The reflexive law of self-identity, a = a, is a consequence of axioms (A10), (UG), (A4), (A6) (A8) and tautologous transformations.7 The symmetry of identity law for identity then follows from (A10), transitivity and reflexivity.8 Finally, note that by a simple induction on formulas, Leibniz’s law, (LL), a = b → (ϕ ↔ ψ), (LL) where ψ is obtained from ϕ by replacing one or more free occurrences of a by free occurrences of b, is provable.9 In other words, the full logic of identity, which we will assume hereafter, is contained in this system. Note that the principle of universal instantiation, (∀x)ϕ → ϕ(a/x),
(UI)
where a can be properly substituted for x in ϕ, is not one of our axiom schemas.10 This is convenient because when extending the logic by adding tense and modal operators we generally have to revise this principle when taken as an axiom so as to cover the cases when x has de re occurrences in ϕ, i.e., when x occurs within 7 That is, where a and y are distinct terms, then by (A10), a = y → (a = y → a = a), and hence, a = y → a = a, and therefore a = a → a = y. Thus, by (UG) and (A4), (∀y)(a = a) → (∀y)(a = y),. But by (A6), a = a → (∀y)(a = a), and hence a = a → (∀y)(a = y). That is, by tautology, ¬(∀y)(a = y) → a = a. But, by (A8) and the definition of ∃, ¬(∀y)(a = y), from which it follows that a = a. 8 THat is, by (A10), a = b → (a = a → b = a). But a = a, so therefore by tautology and modus ponens, a = b → b = a. 9 The case for atomic formulas is a consequence of (A10), the symmetry of identity law, and tautologous transformations. The other cases follow by the inductive hypothesis, tautologous transformations, and finally (A4) and (A5). 10 The notion of proper substitution of a term for a variable that is needed for (UI) is also not involved in any of the axioms; and even the notion of bondage and freedom in (A6) can be replaced by the notion of an occurrence simpliciter. This is very convenient because these notions are complex and difficult for students to grasp at first. Also, it is convenient to avoid these notions when using G¨ odel’s arithmetization technique, because they add so much complexity to that technique. A yet further, and perhaps even more important reason is noted above.
2.4. AXIOMS IN POSSIBILIST LOGIC
33
the scope of a tense or modal operator—or any other intensional operator. The restrictions needed in each case can be determined by seeing what is needed in the proof by induction on Leibniz’s law, (LL), from which this principle follows. That is, by (LL), a = x → [ϕ → ϕ(a/x)], where ϕ(a/x) is the result of replacing all free occurrences of x by a (which we assume to be distinct from x). Then, by tautologous transformations, (UG), (A4) and (A6), we have (∃x)(a = x) → [(∀x)ϕ → ϕ(a/x)], from which, by axiom (A8) and modus ponens, (UI) follows. The law for existential generalization , ϕ(a/x) → (∃x)ϕ, (EG) is of course the converse of (UI) and therefore provable on its basis. Axiom (A8) is the important axiom here. One should not think that it is redundant because it is provable by (EG) from the law of self-identity. Of course, a = a → (∃x)(a = x) is an an instance of (EG); but (EG) is provable from (UI), and, as noted, in the proof of (UI) we need (A8), i.e., (∃x)(a = x). So it would be circular reasoning to try to prove the latter in terms of (EG). What axiom (A8) says in effect is that every objectual term denotes a possible object (as a value of the bound objectual variables), even if that object does not exist (as a value of the variables bound by the actualist quantifier, ∀e ). That is not a problem if the logic does not introduce complex objectual terms, i.e., objectual terms that might contain formulas that describe impossible situations (such as ‘the round square’), or if the logic is not extended to a situation where certain complex objectual terms must fail to denote on pain otherwise of resulting in a contradiction. (This is what in fact happens, as we will see in the next chapter when we extend the logic to second-order predicate logic with nominalized predicates as abstract objectual terms.) It is this latter situation that we will later be concerned with, in which case we will then have to replace axiom (A8) by (∀x)(∃y)(x = y), which is the possibilist counterpart of the actualist axiom (A9). Finally, we note that If we restrict ourselves to formulas in which the actualist quantifier does not occur, then axioms (A1)-(A4), (A6), (A8), and (A10) yield all and only the standard logical truths of first-order predicate logic.11 The standard logical truths, moreover, are none other than the logical truths as defined above when formulas are restricted to those in which the actualist quantifier does not occur. What these results show, in other words, is that the 11 The completeness of the system as described above is due to D. Kalish and R. Montague 1965, their result being obtained by a modification of an original formulation by A. Tarski.
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34
logic of possible objects is none other than standard first-order predicate logic with identity. Hereafter we will assume all the known results about this logic. The main result of this section is that a formula is a theorem of the logic of actual and possible objects if, and only if, it is logically true. Metatheorem: For all formulas ϕ, ϕ if, and only if, ϕ is logically true.
2.5
A First-order Actualist Logic
Essentially the same proof that we described above for (UI) applies to the principle of universal instantiation for the actualist quantifier, except that now the antecedent condition in one step of that proof is not an axiom. That is, by Leibniz’s law, (UG), (A5), the counterpart of (A6) for ∀e (which is derivable from (A6) and (A7)), we have (∃e y)(a = y) → [(∀e x)ϕ → ϕ(a/x)],
(UIe )
which is the actualist version of universal instantiation (where a and y are distinct terms). We cannot assume the antecedent here unless we are given as a separate premise that the objectual term a denotes an actual, existent, object (as a value of the variable bound by ∃e ). Note that in addition to the quantifier concept of existence represented by ∀e (and its dual ∃e ), the predicable concept of existence can be defined in this first-order logic as follows (where a is distinct from the variable y): E!(a) =df (∃e y)(a = x). Now by an E -formula, let us understand a formula in which the possible quantifier, ∀, does not occur. Definition: ϕ is an E-formula =df ϕ is a formula in which the possibilist quantifier, ∀, does not occur. Note that if we restrict ourselves to E-formulas, then neither (A6e )
ϕ → (∀e x)ϕ,
(A8e )
a = a,
where x is not free in ϕ,
nor where a is a objectual term
are provable. That is, the proofs of these valid formulas depend on several possibilist axioms, namely, (A6), (A7) and (A8). That means that if we want to consider only actualism, and not the full logic of possible and actual objects, then we need to take both of these schemas as axioms of the logic of actualism. Indeed, it can be shown that axioms (Al)-(A3), (A5), (A9), (A10), together with these schemas, (A6e ) and (A8e ), yield all and only those logical truths that are E-formulas.12 12 See
Cocchiarella 1966.
2.6. TENSE LOGIC
35
Metatheorem: All logical truths that are E-formulas are derivable from axiom schemas (Al)-(A3), (A5), (A9), (A10), (A6e ) and (A8e ) with modus ponens and (UGe ) as the only inference rules. Thus, whereas 1. the standard formulas that are logically true constitute the logic of possible objects simpliciter, 2. the E-formulas that are logically true constitute the logic of actual objects simpliciter. 3. All of the formulas together—i.e., the standard formulas, the E-formulas, and the formulas that contain both the possible and the actual quantifiers— that are logically true constitute the logic of actual and possible objects, which can be shown to be complete for the semantics described above.
2.6
Tense Logic
As already indicated, one of the most natural applications of the logic of actual and possible objects is in tense logic, where existence applies only to the things that presently exist, and possible objects are none other than past, present, or future objects, i.e., objects that either did exist, do exist, or will exist. The most natural formal ontology for tense logic is conceptual realism. This is because as forms of conceptual activity, thought and communication are inextricably temporal phenomena, and to ignore this fact in the construction of a formal ontology is to court possible confusion of the Platonic with the conceptual view of intensionality. Propositions on the conceptualist view, for example, are not abstract entities existing in a platonic realm independently of all conceptual activity. Rather, according to conceptual realism, they are conceptual constructs corresponding to a projection on the level of objects of the truth-conditions of our temporally located assertions. However, on our present level of analysis, where propositional attitudes are not being considered, their status as constructs can be temporarily ignored. What is also a construction, but which, should not be ignored in a conceptualist framework, are certain cognitive schemata characterizing our conceptual orientation in time and implicit in the form and content of our assertions as mental acts. These schemata, whether explicitly recognized as such or not, are usually represented or modelled in terms of a tenseless idiom (such as our settheoretic metalanguage) in which reference can be made to moments or intervals of time (as objects of a special type). Of course, for most scientific purposes such a representation is quite in order. But to represent them only in this way in a context where our concern is with a perspicuous representation of the form of our assertions as speech or mental acts might well mislead us into thinking that the schemata in question are not essential to the form and content of an
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36
assertion after all—the way they are not essential to the form and content of a proposition on the Platonic view. Now it is important to note that even though the cognitive schemata in question can be modelled in terms of a tenseless idiom of moments or intervals of time, as in fact they are in our set-theoretic metalanguage, they are really themselves the conceptually prior conditions that lead to the construction of our referential concepts for moments or intervals of time. In other words, in terms of conceptual priority, these cognitive schemata are implicitly presupposed by the same tenseless idiom in which they are set-theoretically modelled. In this regard, despite the use of moments or intervals of time in the semantic clauses of the metalanguage, there is no need for conceptualism to assume that moments or intervals of time are independently existing objects as opposed to being merely constructions out of the different events that actually occur in nature, constructions that can be given within a tense-logically based language. Now because the temporal schemata implicit in our assertions enable us to orientate ourselves in time in terms of the distinction between the past, the present, and the future, a more appropriate or perspicuous representation of these schemata is one based upon a system of quantified tense logic in which tenses are represented by tense operators. As applied in thought and communication, what these operators correspond to is our ability to refer to what was the case, what is the case, and what will be the case—and to do so, moreover, without having first to construct referential concepts for moments or intervals of time. In the simplest case we have operators only for the past and the future. P it was the case that ... F it will be the case that ... We do not need an operator for the simple present tense, ‘it is the case that’, because it is already represented in the simple indicative mood of our predicates. With negation applied both before and after a tense operator, we can shorten the long reading of ¬P¬ , namely, ‘it was not the case that it was the case that it was not the case’ to simply ‘it was always the case’. A similar shorter reading applies to ¬F¬ as well. In other words,. we also have the following readings: ¬P¬ it always was the case that ... ¬F¬ it always will be the case that ... Tensed formulas are defined inductively as follows. Definition: ϕ is a tensed formula of a language L if, and only if, ϕ is in every set K such that (1) every atomic formula of L is in K, and (2) whenever ϕ, ψ ∈ K and x is a variable, then ¬ϕ, (ϕ → ψ), Pϕ, F ϕ, (∀x)ϕ, (∀e x)ϕ ∈ K. We will avoid going into all of the details here of a set-theoretical semantics for tense logic.13 Briefly, the idea is that we consider the earlier-than relation of a local time (Eigenzeit ) of a world-line in space-time. Though it is natural to 13 For
such details see Cochiarella 1966 or 1974.
2.6. TENSE LOGIC
37
assume that this relation is a serial ordering, i.e., that it is transitive, asymmetric and connected, we initially impose no other constraints at all upon it, and then consider validity with respect to the different kinds of structures that it can have, e.g., that it is discrete, dense, or continuous, has a beginning and end, or neither, etc. What is important is that we distinguish the objects that exist at each moment of a local time from the objects that exist at any of the other moments of that local time. In this respect a local time determines a history of the world from its unique point of point . Formally, this can stated as follows. Definition: If L is a (formal) language and A is a model (as defined earlier) for a language L, then (1) UA = the universe (of existing objects) of A, and (2) DA = the set of possibilia of A.
Definition: If L is a language and R is a serial relation, then B is an Rhistory with respect to L if there are a nonempty index set I included in the field of R and an I-termed sequence A of models suited to L such that (i) B = R, Ai∈I ; (ii) I is identical with the field of R if I has more than one element; and (iii) ∪j∈I UAj ⊆ DAi , for all i ∈ I; and (iv) DAi = DAj , for all i, j ∈ I. Where R, Ai∈I is such a history, we take the members of the set I to be the moments of the local time with respect to which the history is determined and R to be the earlier-than relation ordering those moments. The structure of R is the temporal structure of that history. It may, for example, have a beginning, or an end, both, or neither, and it may be discrete, dense, or continuous, and so on. Condition (iii) stipulates that whatever is actual at one time or another in a history is a possible object of that history. Condition (iv) states the requirement that whatever is a possible object at one moment of a history is a possible object at any other moment of that history. A complete description of the world relative to the language L and a moment i of a history R, Ai∈I is given by the model Ai that is associated with i in that history. Except where tense operators are involved, satisfaction and truth in a history R, Ai∈I at a given moment i of the history is understood as satisfaction and truth in the model Ai . The satisfaction clauses for the tense operators have the obvious references to the models associated with the moments before and after the moment i. Validity in a history is defined as truth at all times in that history. If R is a relation, then ϕ is said to be R-valid if ϕ is a tensed formula of some language L such that for each R-history B with respect to L, ϕ is valid in B. A tensed schematic formula ϕ is understood to characterize a class K of relations if for each relation R, ϕ is R-valid if, and only if, R ∈ K.14 14 See
Cocchiarella 1966 and 1974 for the details of this semantics.
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38
Definition: If ϕ is a tensed formula of a language L and B =R, Ai∈I is an history, then: (1) ϕ is valid in B if, and only if, for all i ∈ I, ϕ is true at i in B; (2) ϕ is R-valid if, and only if, for each R-history B , ϕ is valid in B ; and (3) ϕ characterizes a class K of relations if, and only if, for each relation R, ϕ is R-valid if, and only if, R ∈ K. Special schematic formulas can be shown to characterize various classes of relations. For example, the tensed formulas PPϕ → Pϕ F Fϕ → F ϕ characterize the class of transitive relations, whereas the converse schemas Pϕ → PPϕ Fϕ → FF ϕ characterize the class of dense relations, i.e., where between any two moments of time there is always another moment of time.15 The most natural assumption, we have said, is that the earlier-than relation of a local time (Eigenzeit ) of a world-line is a serial ordering, i.e., that the relation is transitive, asymmetric and connected. This is not just a matter of “logical purity,” as Arthur Prior has suggested, but of how we conceive the structure of the earlier-than relation of a local time.16 The “logical purity” assumed in this regard is an abstraction from all features of the structure of the earlier-than relation of a local time other than its seriality. That is, the structure is assumed to be invariant for all serial orderings regardless whether time has a beginning, end, or neither, and whether time is discrete, dense and continuous or not. Prior suggested, however, that “if we really want to be safe, it’s odd to begin by insisting on linearity [i.e., seriality], and it might be better ... to confine one’s ‘basic’ laws to those which put no special assumptions on the earlier-than relation at all,” including, e.g., the condition of transitivity as well as that of connectedness.17 This kind of purity abstracts beyond the fact that it is the structure of the earlier-than relation of a local time and characterizes instead the structure of a binary relation simpliciter. That much abstraction might be an appropriate characterization of the accessibility relation between possible worlds in modal logic, but it abstracts too much insofar as it is the structure of the earlier-than relation of a local time that is in question. 15 The
formulas ¬F ¬ϕ → Fϕ ¬P¬ϕ → Pϕ
characterize time as having no end, or beginning, respectively. 16 See Prior 1967, p. 51, for a discussion of this issue. 17 Ibid.
2.6. TENSE LOGIC
39
Time, as well as causality, is a cosmological category, which, for some, might indicate that “tense logic is not really logic but physics, or that it has a good deal of physics ‘built into it’,” as Prior has noted.18 That in fact may be true to some extent, because insofar as such cosmological categories as time and causality can be said to have a logic of their own they are to that extent also part of formal ontology. In any case, it is clear that we are not dealing with the structure of time if the earlier-than relation is not assumed to be transitive. The issue of the connectedness of the earlier-than relation of a local time might be thought to be another matter, however, especially in the context of Einstein’s theory of special relativity. But, as we will argue in section 11 below, the claim that Einstein’s theory of special relativity shows that connectedness does not apply is based on a confusion of: 1. the causal signal relation, which is not connected, between the different momentary states of different world lines with 2. the earlier-than relation line, which is connected.
of a local time (Eigenzeit) of a given world
In any case, in regard to the characterization of logical truth as extended to all tensed formulas, we restrict our considerations—in deference to this fundamental feature of (local) time—to serial histories, i.e., histories whose temporal ordering is a series. Definition: ϕ is tense-logically true if for some language L of which ϕ is a tensed formula, ϕ is valid in every serial history suited to L. Metatheorem: ϕ is tense-logically true if, and only if, for every serial ordering R, ϕ is R-valid. Given modus ponens and universal generalization for ∀, and the following as inference rules (i) if t ϕ, then t ¬P¬ϕ (ii) if t ϕ, then t ¬F¬ϕ then these rules together with all instances of (A1)-(A10) of the logic of actual and possible objects (applied now to tensed formulas) and all instances of the following axiom schemas as well yield all and only the tense-logical truths: (A11)
¬P¬(ϕ → ψ) → (Pϕ → Pψ)
(A12)
¬F¬(ϕ → ψ) → (F ϕ → F ψ)
(A13)
ϕ → ¬P¬Fϕ
(A14)
ϕ → ¬F¬Pϕ
(A15)
PPϕ → Pϕ
(A16)
F Fϕ → F ϕ
18 Ibid.
CHAPTER 2. TIME, BEING, AND EXISTENCE
40 (A17)
Pϕ ∧ Pψ → P(ϕ ∧ ψ) ∨ P(ϕ ∧ Pψ) ∨ P(ψ ∧ Pϕ)
(A18)
F ϕ ∧ Fψ → F (ϕ ∧ ψ) ∨ F(ϕ ∧ Fψ) ∨ F(ψ ∧ Fϕ)
(A19)
x = y → ¬P¬(x=y)∧¬F¬(x=y)
where x, y are variables
Metatheorem: For all tensed formulas ϕ, t ϕ if, and only if, ϕ is tenselogically true.19 A completeness theorem for actualist tensed logic is also forthcoming if we restrict ourselves to tensed E-formulas, i.e., those formulas in which the possible quantifier does not occur. Restricted to tensed E-formulas, the axioms for actualism are as follows: Tensed Actualist axioms: (A1)-(A3), (A5), (A6e ), (A8e ), (A9), and (A10)(A19). We will also need the inference rules listed above—but with universal generalization for ∀e instead of ∀—as well as the following (somewhat complex) rules: (iii) If t ¬P¬(ϕ1 → ¬P¬(ϕ1 → ... → ¬P¬(ϕn−1 → ¬P¬ϕn )...)), and x is not free in ϕ1 , ..., ϕn−1 , then t ¬P¬(ϕ1 → ¬P¬(ϕ1 → ... → ¬P¬(ϕn−1 → ¬P¬(∀e x)ϕn )...)). (iv) If t ¬F¬(ϕ1 → ¬F¬(ϕ1 → ... → ¬F¬(ϕn−1 → ¬F¬ϕn )...)), and x is not free in ϕ1 , ..., ϕn−1 , then t ¬F¬(ϕ1 → ¬F¬(ϕ1 → ... → ¬F¬(ϕn−1 → ¬F¬(∀e x)ϕn )...)). The above axioms and rules yield all and only those tense-logical truths that are tensed E-formulas.20
2.7
Temporal Modes of Being
In assuming that being and existence are not the same concept, possibilism does not also assume that whatever is (i.e., whatever has being) either did exist, does exist, or will exist, a thesis we shall call temporal possibilism. We will call the objects of temporal possibilism realia. Formally, this thesis is stated as follows: Temporal Possibilism: (∀x)[PE!(x) ∨ E!(x) ∨ FE!(x)]. Realia: What did, does, or will exist. If we add this formula as a new axiom, then to render it tense-logically true we need only require that the condition stated in clause (iii) of the definition of an R-history be an identity rather than just an inclusion. 19 See
20 See
Cocchiarella 1966 for a proof of this metatheorem. Cocchiarella 1966.
2.7. TEMPORAL MODES OF BEING
41
It should be noted that Aristotle seems to have held such a view in that he thought that whatever is possible is realizable in time, which, for Aristotle, has no beginning or end. A more restrictive view than temporal possibilism—but one that still falls short of actualism—is that only the past and the present are metaphysically determinate, and for that reason only objects that either do exist or did exist have being. The future, being indeterminate metaphysically as well as epistemically has no being. Being, on this account, covers only past or present existence. Future objects have no being but only come into being in the present when they exist, and then continue to have being in the past. We can characterize this position by first defining quantification over past objects, and then quantification over past and present objects, as follows (where ∃p and ∃pp are defined in the usual way as the duals of ∀p and ∀pp , respectively): (∀p x)ϕ =df (∀x)[PE!(x) → ϕ] (∀pp x)ϕ =df (∀x)[PE!(x) ∨ E!(x) → ϕ] The metaphysical thesis that being comprises only what either did exist or does exist can now be expressed as follows: (∀x)(∃pp y)(x = y). Alternatively, instead of having the concept of being in such a framework represented by the possibilist quantifier ∀, we can take it to be represented directly by ∀pp as a primitive quantifier together with ∀e for the concept of existence. A sound and complete axiom set for this system is then given by (A1)-(A3), (A5), (A9), (A10), together with the schemas: (∀pp x)(ϕ → ψ) → [(∀pp x)ϕ → (∀pp x)ψ], ϕ → (∀pp x)ϕ, (∀pp x)ϕ → (∀e x)ϕ, (∀pp x)(∃pp y)(x = y), (∀pp x)[(∃pp y)(x = y) ∧ ¬F ¬(∃pp y)(x (∀pp x)¬P¬ϕ → ¬P¬(∀pp x)ϕ, ¬F¬(∀pp x)ϕ → (∀pp x)¬F¬ϕ, a = a,
where x is not free in ϕ, where x, y are distinct variables, = y)],
where a is an arbitrary term.
The inference rules for this system are modus ponens, universal generalization for ∀pp , rules (i), (ii) as described earlier, and the counterpart of rule (iv) using ∀pp in place of ∀e .21 21 The
counterpart of rule (iii) with ∀pp in place of ∀e is provable, and it does not need to be taken as a primitive rule in this system.
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2.8
Past and Future Objects
Actualists claim that quantificational reference to either past or future objects is possible only indirectly (de dicto)—i.e., through the occurrence of an actualist quantifier within the scope of a tense operator.22 This is true of some of our references to past objects, as for example in an assertion of, Someone did exist who was a King of France. In this case, the apparent reference to a past object can be accounted for as follows: P(∃e x)King-of -F rance(x), where the reference to a past object is not direct but indirect, i.e., within the scope of a past-tense operator. Here, by a direct quantificational reference to a past object that no longer exists (or a future one that has yet to exist) we mean one in which the quantifier is outside the scope of a tense operator. Note: What is apparently is not possible on this account about a direct quantificational reference to past objects that no longer exist is our present inability to actually confront and apply the relevant identity criteria to objects that do not now exist. A present ability to identify past or future objects, however, is not the same as the ability to actually confront and identify those objects in the present; that is, our existential inability to do the latter is not the same as, and should not be confused with, what is only presumed to be our inability to directly refer to past or future objects. Indeed, the fact is that we can and do make direct reference to realia, and to past and future objects in particular, and that we do so not only in ordinary discourse but also, and especially, in most if not all of our scientific theories. The real problem is not that we cannot directly refer to past and future objects, but rather how it is that conceptually we come to do so. One explanation of how this comes to be can be seen in the analysis of the following English sentences: 1. There did exist someone who is an ancestor of everyone now existing. 2. There will exist someone who will have everyone now existing as an ancestor. Assuming, for simplicity, that we are quantifying only over persons, it is clear that (1) and (2) cannot be represented by: 3. P(∃e x)(∀y)Ancestor-of (x, y) 4. F (∃e x)(∀y)Ancestor-of (y, x). 22 Cf.
Prior, 1967, Chapter 8.
2.8. PAST AND FUTURE OBJECTS
43
What (3) and (4) represent are the different sentences: 5. There did exist someone who was an ancestor of everyone then existing. 6. There will exist someone who will have everyone then existing as an ancestor. Of course, in temporal possibilism, referential concepts are available that enable us to refer directly to past and future objects. Thus, for quantification over past objects we have the quantifier ∃p and for quantification over future objects we have ∃f as the future-counterpart of ∃p . Using these quantifiers, the obvious representation of (1) and (2) is: 7. (∃p x)(∀y)Ancestor-of (x, y) 8. (∃f x)(∀y)F Ancestor-of (y, x). We should note here that the relational ancestor concept is such that: x is an ancestor of y only at those times when either y exists and x did exist, though x need not still exist at the time in question, or when x has continued to exist even though y has ceased to exist. When y no longer exists as well as x, we say that x was an ancestor of y; and where y has yet to exist, we say that x will be an ancestor of y. Now although these last analyses are not available in actualist tense logic, nevertheless semantical equivalences for them are available once we allow us the use of the now-operator , N . N:
It is now the case that ...
The now-operator is unlike the simple present tense in that it always brings us back to the present even when it occurs within the scope of either the pastor future-tense operators. Thus, although the indirect references to past and future objects in (3) and (4) fail to provide adequate representations of (1) and (2), the same indirect references followed by the now-operator succeed in capturing the direct references given in (7) and (8): 9. P(∃x)N (∀y)Ancestor-of (x, y) 10. F (∃x)N (∀y)FAncestor-of (y, x). In other words, at least relative to any present-tense context, we can in general account for direct reference to past and future objects, and hence to all of the objects of temporal possibilism, as follows: (∀p x)ϕ ↔ ¬P¬(∀e x)N ϕ (∀f x)ϕ ↔ ¬F¬(∀e x)N ϕ
44
CHAPTER 2. TIME, BEING, AND EXISTENCE (∀x)ϕ ↔ (∀p x)ϕ ∧ ϕ ∧ (∀f x)ϕ.
These equivalences, it should be noted, cannot be used other than in a present tense context; that is, the above use of the now-operator would be inappropriate when the equivalences are stated within the scope of a past- or future-tense operator, because in that case the direct reference to past or future objects would be from a point of time other than the present. Formally, what is needed in such a case is the introduction of a so-called “backwards-looking” operator, such as the then-operator, which can be correlated with occurrences of past or future tense operators within whose scope they lie and that semantically evaluate the formulas to which they are themselves applied in terms of the past or future times already referred to by the tense operators they are correlated with23 . Backwards-looking operators, in other words, enable us to conceptually return to a past or future time already referred to in a given context in the same way that the now-operator enables us to return to the present. In that regard, their role in the cognitive schemata characterizing our conceptual orientation in time and implicit in each of our assertions is essentially a projection of the role of the now-operator. We will not formulate the semantics of these backwards-looking operators here. But we do want to note that by means of such operators we can account for the development of referential concepts by which we can refer directly to past or future objects. Such an account is already implicit in the fact that such direct references to past or future objects can be made with respect to the present alone. This shows that whereas the reference is direct at least in effect, nevertheless the application of any identity criteria associated with such reference will itself be indirect, and in particular, not such as to require a present confrontation, even if only in principle, with a past or future object.
2.9
Modality Within Tense Logic
It is significant that the first modal concepts to be discussed and analyzed in the history of philosophy are concepts based on the distinction between the past, the present, and the future, that is, concepts that can be analyzed in terms of the temporal modalities that are represented by the standard tense operators. The Megaric logician Diodorus, for example, is reported as having argued that the possible is that which either is or will be the case, and that the necessary is that which is and always will be the case.24 Formally, the Diodorean modalities can be defined as follows: ♦f ϕ =df (ϕ ∨ Fϕ) f ϕ =df ϕ ∧ ¬F ¬ϕ ∴ f ϕ ↔ ¬♦f ¬ϕ 23 Cf.
24 See
Vlach,1973 and Saarinen 1976. Prior 1967, chapter 2, for a discussion of Diodorus’s argument.
2.9. MODALITY WITHIN TENSE LOGIC
45
Aristotle, on the other hand, included the past as part of what is possible; that is, for Aristotle the possible is that which either was, is, or will be the case in what he assumed to be the infinity of time, and therefore the necessary is what is always the case25 : ♦t ϕ =df Pϕ ∨ ϕ ∨ Fϕ t ϕ =df ¬P¬ϕ ∧ ϕ ∧ ¬F ¬ϕ ∴ t ϕ ↔ ¬♦t ¬ϕ Both Aristotle and Diodorus assumed that time is real and not ideal. In other words, the Diodorean and Aristotelian temporal modalities are understood to be real modalities based on the nature of time. In fact they provide a paradigm by which we might understand what is meant by a real, as opposed to a merely formal, modality such as logical necessity. These temporally-based modalities contain an explanatory, concrete interpretation of what is called the accessibility relation between possible worlds in modal logic, except that worlds are now construed as momentary states of the universe as described by the models associated with the moments of a local time. That is, where possible worlds are momentary descriptive states (models) of the universe with respect to the local time (Eigenzeit ) of a given world-line, then the relation of accessibility between worlds is ontologically grounded in terms of the earlier-than relation of that local time. The Aristotelian modalities are stronger than the Diodorean, of course, and in fact they provide a complete semantics for the quantified modal logic known as S5. Definition: If L is a language, then ϕ is an S5t -formula of L if, and only if, ϕ belongs to every set K containing the atomic formulas of L and such that ¬ϕ, t ϕ, ♦t ϕ, (ϕ → ψ), (∀x)ϕ, (∀e x)ϕ ∈ K whenever ϕ, ψ ∈ K and x is a variable. Definition: ϕ is S5-valid if, and only if, ϕ is an S5t -formula that is tenselogically true. We obtain the system we call S5t if to the axioms (A1)-(A10) of the logic of actual and possible objects we add all instances of schemas of the following forms: (S5t -1) (S5t -2) (S5t -3)
t ϕ → ϕ t (ϕ → ψ) → (t ϕ → t ψ) ♦t ϕ → t ♦t ϕ
(S5t -4)
(x = y) → t (x = y),
25 See
where x, y are variables.
Hintikka, 1973, Chapters V and IX. Aristotle may have intended his notion of possible to apply to individuals as well, a position that is validated in the quantified modal logic described in this section.
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As inference rules we take in addition to modus ponens and universal generalization for ∀ the following: (MG)
if S5t ϕ, then S5t t ϕ .
Note that by (S5t -4), (A8) and (MG), (∃y)t (x = y) is provable, and from this it can be shown that the Carnap-Barcan formula, (∀x)t ϕ ↔ t (∀x)ϕ is also provable. A completeness theorem for this version of quantified S5 modal logic can be proved in the usual way. Metatheorem: For each S5t -formula ϕ, S5t ϕ if, and only if, ϕ is S5-valid. For an actualist S5 modal logic, we need only restrict the S5-formulas to those that are tensed E-formulas—i.e., tensed formulas in which the possibilist quantifier does not occur—and use only the logic of actual objects as described earlier together with the axiom schemas (S5t -1)-(S5t -4) and one new inference rule added to those of S5t . That is, where S5te is that subsystem of S5t that is the result of replacing (Al)-(A10) of the logic of actual and possible objects by the axioms for the logic of actual objects simpliciter and adding to the inference rules of S5t the following: If S5te t (ϕ1 → t (ϕ2 → ... → t (ϕn−1 → t ϕn )...)), and x is not free in ϕ1 , ..., ϕn−1 , then S5te t (ϕ1 → t (ϕ2 → ... → t (ϕn−1 → t (∀e x)ϕn )...)). A completeness theorem can be shown for the actualist modal logic S5te . Metatheorem: For each S5t -formula ϕ that is also an E-formula, S5te ϕ if, and only if, ϕ is S5-valid. The Diodorean modalities, we have noted, are weaker than the Aristotelian modalities, and the corresponding quantified modal logic is not S5 but the weaker system known as S4.3. Definition: If L is a language, then ϕ is an S4.3t -formula of L if, and only if, ϕ belongs to every set K containing the atomic formulas of L and such that ¬ϕ, ♦f ϕ, (ϕ → ψ), (∀x)ϕ, (∀e x)ϕ ∈ K whenever ϕ, ψ ∈ K and x is a variable.
2.9. MODALITY WITHIN TENSE LOGIC
47
Definition: ϕ is S4.3-valid if, and only if, ϕ is an S4.3t -formula that is tenselogically true. We obtain the system we call S4.3t if we add to the axioms (A1)-(A10) of the logic of actual and possible objects all instances of schemas of the following forms: (S4.3t -1) (S4.3t -2) (S4.3t -3) (S4.3t -4) (S4.3t -5) (S4.3t -6)
f ϕ → ϕ f (ϕ → ψ) → (f ϕ → f ψ) f ϕ → f f ϕ ♦f ϕ ∧ ♦f ψ → ♦f (ϕ ∧ ψ) ∨ ♦f (ϕ ∧ ♦f ψ) ∨ ♦f (ψ ∧ ♦f ϕ) ♦f (x = y) → f (x = y) (∀x)f ϕ → f (∀x)ϕ
As inference rules for S4.3t we have the same inference as those for S5t , except except for having f where t occurs in those rules. It can be shown that for each S4.3t -formula ϕ, ϕ is a theorem of S4.3t if, and only if, ϕ is S4.3-valid, which is our completeness theorem for S4.3t . Metatheorem: For each S4.3t -formula ϕ, S4.3t ϕ if, and only if, ϕ is S4.3valid. For an actualist S4.3t modal logic we need first to restrict the tensed S4.3t formulas to tensed E-formulas. Then, to obtain the subsystem S4.3te of S4.3t when the latter is restricted to E-formulas, we must first delete the axiom schema (S4.3t -6), which is not an E-formula, and replace (A1)-(A10) of the logic of actual and possible objects by the axioms of the logic of actual objects simpliciter. We then adopt the same modal inference rules as already described for S5te , except for using f instead of t in those rules. Then, it can be shown that for each S4.3-formula ϕ that is also an E-formula, ϕ is a theorem of S4.3te if, and only if, ϕ is S4.3-valid, which is our completeness theorem for S4.3t -formulas when the latter are restricted to E-formulas. Metatheorem: For each S4.3t -formula ϕ that is also an E-formula, S4.3te ϕ if, and only if, ϕ is S4.3-valid. Infinitely many other modal logics can be generated in ways similar to the above by various combination of tenses—e.g., merely iterating new occurrences of F in the definition of the Diodorean modalities will lead to new modalities. In addition to these temporal notions of modality, the semantics for yet another can be given corresponding roughly to the idea that a formula is conditionally necessary (in a given history at a given moment of that history) because of the way the past has been. The semantics for this notion also yields a completeness theorem for an S5 type modal structure, and it may be used for a partial or full explication of the notions of causal modality and counterfactuals.
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2.10
Causal Tenses in Relativity Theory
One of the defects of Aristotle’s notion of necessity as a paradigm of a temporallybased modality is its exclusion of certain situations that are possible in special relativity theory as a result of the finite limiting velocity of causal influences, such as a light signal moving from one point of space-time to another. For example, relative to the present of a given local time T0 , a state of affairs can come to have been the case, according to special relativity, without its ever actually being the case.26 That is, where F Pϕ represents ϕ’s coming (in the future) to have been the case (in the past of that future), and ¬♦t ϕ represents ϕ’s never actually being the case, the situation envisaged in special relativity might be thought to be represented by: F Pϕ ∧ ¬♦t ϕ.
(Rel)
This conjunction is incompatible with the connectedness assumption of the local time T0 in question; for on the basis of the assumption of connectedness, F Pϕ → Pϕ ∨ ϕ ∨ Fϕ is tense-logically true, and therefore, by definition of ♦t , F Pϕ → ♦t ϕ is also tense-logically true. That is, F Pϕ, the first conjunct of (Rel), implies ♦t ϕ, which contradicts the second conjunct of (Rel), ¬♦t ϕ. The connectedness assumption cannot be given up, moreover, without violating the notion of a local time or of a world-line as an inertial reference frame upon which that local time is based. The notion of a local time is a fundamental construct not only of conceptualism and our common-sense framework but of natural science as well, as in the assumption of an Eigenzeit in relativity theory. In conceptualism the connectedness of a local time is part of the notion of the self as a center of conceptual activity, and in fact it is one of the principles upon which the tense-logical cognitive schemata characterizing our conceptual orientation in time are constructed. This is not to say that in the development of the concept of a self as a center of conceptual activity we do not ever come to conceive of the ordering of events from perspectives other than our own. Indeed, by a process that Jean Piaget calls decentering, children at the stage of concrete operational thought (7–11 years) develop the ability to conceive of projections from their own positions to that of others in their environment; and subsequently, by means of that ability, they are able to form operational concepts of space and time whose systematic coordination results essentially in the structure of projective geometry.27 Spatial considerations aside, however, and with respect to time alone, the cognitive schemata implicit in the ability to conceive of such projections can be 26 Cf. 27 Cf.
Putnam, 1967. Piaget 1972.
2.10. CAUSAL TENSES IN RELATIVITY THEORY
49
represented in part by means of tense operators corresponding to those already representing the past and the future as viewed from one’s own local time. That is, because the projections in question are to be based on actual causal connections between the momentary descriptive states of inertial frames, or world-lines, we can represent the cognitive schemata implicit in such projections by what we will here call causal-tense operators. Pc Fc
: it causally was the case that ... : it causally will be the case that ...
Semantically, the causal-tense operators go beyond the standard tenses by requiring us to consider not just a single local time but a causally-connected system of local times. The causal connections are between the momentary states of the different inertial reference frames upon which such local times are based; and, given the finite limiting velocity of light, these causal connections can be represented by a signal relation between the moments of the local times themselves—so long as we assume that the sets of moments of different local times are disjoint. (This assumption is harmless if we think of a moment of a local time as an ordered pair one constituent of which is the inertial frame upon which that local time is based.) The only constraint that should be imposed on such a signal relation is that it be a strict partial ordering, i.e., transitive and asymmetric.28 Thus, by the causal past, as represented by Pc , we mean not just the past with respect to the here-now of our own local time, but also the past with respect to any momentary state of any other world-line that can send a signal to our here-now; and by the causal future, as represented by Fc , we mean not just the future with respect to our here-now, but the future of any momentary state of any world-line to which we can send a signal from here-now. The geometric structure at a given momentary state of a world-line of a causally connected system is that of a Minkowski light-cone. That is, at each momentary state X of a world-line there is both a prior light cone (the causal past) consisting of all the momentary states (or space-time points) of world-lines that can send a signal to X and a posterior light cone (causal future) of all the momentary states (or space-time points) of world-lines that can receive a signal from X. Momentary states are then said to be simultaneous if no signal relation can be sent from one to the other. The causal past (prior light-cone) of the here-now momentary state X of a world-line = the momentary states of world-lines that can send a signal to X. 28 See Cocchiarella 1984, section 15, for the details of this semantics. The signal relation, incidentally, provides yet another example of a concrete interpretation of an accessibility relation between possible worlds, reconstrued now as the momentary states of the universe at different space-time points.
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The causal future (posterior light-cone) of the here-now momentary state X of a world line = the momentary states of world-lines to which a signal can be sent from X. A momentary state X of a world-line W1 is simultaneous with a momentary state Y of a world-line W2 if, and only if, no signal can be sent from X to Y , nor from Y to X. Now because the signal relation has a finite limiting velocity, simultaneity will not be a transitive relation. As a result any one of a number of momentary states of one world-line can be simultaneous with the same momentary state of another world-line. This is what leads to the type of situation described by Putnam: Let Oscar be a person whose whole world-line is outside of the light-cone of me-now. Let me-future be a future ‘stage’ of me such that Oscar is in the lower half of the light cone of me-future [i.e., the prior cone of me-future]. Then, when that future becomes the present, it will be true to say that Oscar existed, although it will never have had such a truth value to say in the present tense ‘Oscar exists now’. Things could come to have been, without its ever having been true that they are!29 What all this indicates is that the possibility according to special relativity theory of a state of affairs coming to have been the case without its ever actually being the case is a possibility that should be represented in terms of the causal tense-operators Fc ϕ and Pc ϕ—i.e., in terms of the causal past and causal future—and not in terms of the simple past- and future-tense operators P and F , i.e., the past and future according to the ordering of events within a single local time. Note that because there is a causal connection from the earlier to the later momentary states of the same local time, the signal relation is assumed to contain as a proper part the connected temporal ordering of the moments of each of the local times in such a causally connected system.30 The following, in other words, are valid theses of such a causally connected system: Pϕ → Pc ϕ F ϕ → Fc ϕ But note also that, because the signal relation has a finite limiting velocity, the converses of these theses will not also be valid in such a system. Were we to reject the assumption of relativity theory that there is a finite limit to causal influences, namely, the speed of light—as was implicit in classical physics and is still implicit in our commonsense framework where simultaneity is assumed to be 29 Putnam
1967, p. 204. Carnap, 1958, Sections 49–50, for such an analysis of the notion of a causally connected system of local times. 30 See
2.10. CAUSAL TENSES IN RELATIVITY THEORY
51
absolute across space-time—then we would validate the converses of the above theses, in which case the causal tense operators would be completely redundant, which explains why they have no counterparts in natural language, which, prior to the special theory of relativity, allowed for unlimited causal influences. A related point is that unlike the cognitive schemata of the standard tense operators whose semantics is based on a single local time, those represented by the causal tense-operators are not such as must be present in one form or another in every speech or mental act. They are derived schemata, in other words, constructed on the basis of those decentering abilities whereby we are able to conceive of the ordering of events from a perspective other than our own. The importance and real significance of these derived schemata was unappreciated until the advent of special relativity. One important consequence of the divergence of the causal from the standard tense operators is the invalidity of F c Pc ϕ → Pc ϕ ∨ ϕ ∨ F c ϕ and therefore the consistency of Fc Pc ϕ ∧ ¬♦t ϕ. Unlike its earlier counterpart in terms of the standard tenses, this last formula is the appropriate representation of the possibility in special relativity of a state of affairs coming (in the causal future) to have been the case (in the causal past) without its ever actually being the case (in a given local time). Indeed, not only can this formula be true at some moment of a local time of a causally connected system, but so can the following formula31 : [Pc ♦t ϕ ∨ Fc ♦t ϕ] ∧ ¬♦t ϕ. Quantification over realia, which now includes things that exist in space-time with respect to any local time and not just with respect to a given local time, also finds justification in special relativity. For just as some states of affairs can come to have been the case in the causal past of the causal future without their actually ever being the case, so too there can be things that do not exist in the past, present or future of our own local time, but which nevertheless might exist in a causally connected local time at a moment that is simultaneous with our present. In this regard, reference to such objects as real even if not presently existing would seem hardly controversial—or at least not at that stage of conceptual development where our decentering abilities enable us to construct referential concepts that respect other points of view causally connected with our own. Realia encompass all the objects of temporal possibilism and possibly more as well. 31 This formula would be true at a given moment t of a local time X if in either the prior cone or posterior cone of that moment there is a space-time point t of a world-line Y such that ϕ is always true in Y , even though ϕ is never true in X. Putnam’s Oscar example indicates how this is possible in relativity theory.
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Finally, it should be noted that there is also a causal counterpart to Diodorus’s notion of possibility as what either is or will be the case, namely, possibility as what either is or causally will be the case: ♦cf ϕ =df ϕ ∨ Fc ϕ. Instead of the modal logic S4.3, this causal Diodorean notion of possibility results in the modal logic S4. Moreover, if we also assume, as is usual in special relativity, that the causal futures of any two moments t, t of two local times of a causally connected system eventually intersect, i.e., that there is a moment w of a local time such that both t and t can send a signal to w, then the thesis Fc ¬Fc ¬ϕ → ¬Fc ¬Fc ϕ will be validated, and the causal Diodorean notion of possibility will then result in the modal system S4.2,32 i.e., the system S4 plus the thesis ♦f c f c ϕ → f c ♦f c ϕ. Many other modal concepts can also be characterized in terms of a causally connected system of local times, including, e.g., the notion of something being necessary because of the way the past has been. What is distinctive about them all is the unproblematic sense in which they can be taken as material or metaphysical modalities. Tense logic is not the only framework in which both the logic of actual and possible objects and the logic of actual objects simpliciter have natural applications and in which the differences between possibilism and actualism can be made perspicuous. There is also, for example, the logic of belief and knowledge and the differences between the possible quantifier and the actual quantifier binding variables occurring free within the scope of operators for propositional attitudes. Still, even these other frameworks must presuppose some account of the logic of tenses, in which case the differences between possibilism and actualism within tense logic becomes paradigmatic. Indeed, as we have indicated, this is certainly the case for the differences between possibilism and actualism in modal logic, since some of the very first modal concepts ever to be discussed in the history of philosophy have been modal concepts that can be analyzed in the framework of tense logic.
2.11
Summary and Concluding Remarks
• A formal ontology must provide an ontological ground for the distinction between being and existence, or, if the distinction is rejected, an adequate account of why it is rejected. This is a criterion of adequacy for any formal ontology. 32 Cf.
Prior, 1967, p. 203.
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53
• A first-order logic of actual and possible objects is a component part of conceptual realism. A fuller ontological account of the distinction between actualism and possibilism is given in chapter six, section one. • Our commonsense understanding includes a representation of how we are conceptually oriented in time with respect to the past, the present and the future. • Tense logic provides a formal representation that respects the form and content of our commonsense speech and mental acts about the past, the present and the future. • Because tense operators represent cognitive schemata regarding our orientation in time, conceptual realism is the most natural formal ontology for a tense-logical representation of our commonsense understanding of time. • Conceptual realism provides the clearest ontological ground for the distinction between being and existence in terms of the tense-logical distinction between past, present and future objects, i.e., the distinction between things that did exist, do exist, or will exist. • Another criterion of adequacy for a formal ontology is that it must explain the ontological grounds, or nature, of modality, i.e., of such modal notions as necessity and possibility, as opposed to merely giving a set-theoretic semantics for modal logic. • Some of the earliest ontological views of modality, going back as far as Aristotle and Diodorus, are temporal notions. Different notions of necessity and possibility can be grounded in the tense-logical part of conceptual realism. • The analysis of necessity and possibility within tense logic provides a clear and unproblematic paradigm by which to understand the notion of a possible world and the accessibility relation between possible worlds. • Different temporal modes of being can be represented within the tenselogical framework of conceptual realism. • Special relativity is not incompatible with the connectedness of a local time (Eigenzeit ) as represented by the axioms of standard tense logic. • The cognitive schemata implicit in our ability to conceive of projections to other reference frames, and hence to other local times, can be represented in terms of causal tense operators based on a signal relation between different space-time points. • The causal past and the causal future that are represented by the causal tense operators are not the same as the simple past and the simple future. It is with respect to the causal tense operators that the connectedness thesis fails to be valid, not with respect to the standard tense operators.
2.12
Appendix
Once the logic of actual and possible objects is extended by introduction of tense and modal operators, there are certain complications that arise in the application of Leibniz’s law and the law of universal instantiation (and its dual, existential generalization) in tense and modal contexts. We describe some of
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these features here as well as certain laws regarding the commutation of tense and modal operators with quantifier phrases. In describing some of the theorem schemas involving quantifiers, tenses and modal operators in these different logics, we shall use the following notation: (t) t : is a theorem of tense logic (of local time) with quantification over actual and possible objects. (te ) te : is a theorem of tense logic (of local time) with quantification over just actual objects. : is a theorem of tense logic (of local time) with quantification over (tp ) tp just past and present objects. e
p
Note that what is provable in (t ) is provable in (t ), and what is provable in (tp ) is provable in (t). That is, {ϕ : te ϕ} ⊆ {ϕ : tp ϕ} ⊆ {ϕ : t ϕ} We may use te , accordingly, to state what is provable in all three systems, p and tp for what is provable in (t ) and (t). We will also use the following counterparts of notions already defined: (∀f x)ϕ =df (∀x)[FE!(x) → ϕ] ♦p ϕ =df (ϕ ∨ Pϕ) p ϕ =df ϕ ∧ ¬P¬ϕ
Leibniz’s Law: Assume that a, b are objectual constants, ϕ is a tensed formula, and ψ is obtained from ϕ by replacing one or more occurrences of a by occurrences of b. Then, we have the following theses about Leibniz’s law: (1) te t (a = b) → (ϕ ↔ ψ) (2) te (x = y) → (ϕ ↔ ψ),
where x, y are variables.
(3) te (a = b) → (ϕ ↔ ψ), tense operator.
if a does not occur in ϕ within the scope of a
(4) te p (a = b) → (ϕ ↔ ψ), a future tense operator.
if a does not occur in ϕ within the scope of
(5) te f (a = b) → (ϕ ↔ ψ), a past tense operator.
if a does not occur in ϕ within the scope of
2.12. APPENDIX
55
Identity and Non-identity: Although identity and non-identity as expressed in terms of objectual variables is always necessary, that is, te (x = y) → t (x = y) te (x = y) → t (x = y), The same theses are not true for objectual constants—unless they are “rigid designators,” i.e., denote the same object at all times, which is symbolized as (∃x)t (a = x). Without assuming that, the relevant qualifications are as follows: (6) t (∃x)t (a = x) ∧ (∃y)t (b = y) → [a = b ↔ t (a = b)]∧ [a = b ↔ t (a = b)] te (∃e x)t (a = x) ∧ (∃e y)t (b = y) → [a = b ↔ t (a = b)]∧ [a = b ↔ t (a = b)] tp (∃pp x)t (a = x) ∧ (∃pp y)t (b = y) → [a = b ↔ t (a = b)]∧ [a = b ↔ t (a = b)] Similar theorems hold when t is uniformly replaced throughout (6) by f and p , respectively.
Universal Instantiation: The law of universal instantiation does not hold in general in these logics without qualification. The different qualifications are as follows, where x and y are variables, a is a term distinct from y, and a is free for x in ϕ: (7) t (∃y)t (a = y) → [(∀x)ϕ → ϕ(a/y)] te (∃e y)t (a = y) → [(∀e x)ϕ → ϕ(a/y)] tp (∃pp y)t (a = y) → [(∀pp x)ϕ → ϕ(a/y)] (8) If either a is a variable or x does not occur in ϕ within the scope of a past or future tense operator, then: t (∀x)ϕ → ϕ(a/y) te (∃e y)(a = y) → [(∀e x)ϕ → ϕ(a/y)] tp (∃pp y)(a = y) → [(∀pp x)ϕ → ϕ(a/y)]
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56
(9) If x does not occur in ϕ within the scope of a future tense operator, then: tp (∃y)p (a = y) → [(∀x)ϕ → ϕ(a/y)] te (∃e y)p (a = y) → [(∀e x)ϕ → ϕ(a/y)] tp (∃pp y)p (a = y) → [(∀pp x)ϕ → ϕ(a/y)] (10) If x does not occur in ϕ within the scope of a past tense operator, then: t (∃y)f (a = y) → [(∀x)ϕ → ϕ(a/y)] te (∃e y)f (a = y) → [(∀e x)ϕ → ϕ(a/y)] tp (∃pp y)f (a = y) → [(∀pp x)ϕ → ϕ(a/y)]
Laws of Commutation: The possible quantifier ∃ commutes with both the past and future tense operators and therefore with ♦f , ♦p , and ♦t as well. Dually, ∀ commutes with ¬P¬ and ¬F¬ and therefore with f , p , and t as well: (11) t P(∃x)ϕ ↔ (∃x)Pϕ t F (∃x)ϕ ↔ (∃x)Fϕ
t ¬P¬(∀x)ϕ ↔ (∀x)¬P¬ϕ t ¬F¬(∀x)ϕ ↔ (∀x)¬F¬ϕ
t ♦f (∃x)ϕ ↔ (∃x)♦f ϕ
t f (∀x)ϕ ↔ (∀x)f ϕ
t ♦p (∃x)ϕ ↔ (∃x)♦p ϕ
t p (∀x)ϕ ↔ (∀x)p ϕ
t ♦t (∃x)ϕ ↔ (∃x)♦t ϕ
t t (∀x)ϕ ↔ (∀x)t ϕ
The actual quantifier ∃e does not commute with the past or future tense operators except under special conditions, and even then different conditions are required for each direction—unless it is assumed that nothing ever comes to exist or ceases to exist, in symbols: t (∀e x)t E!(x)
Nothing ever comes to exist or ceases to exist,
in which case ∃e commutes with ♦f , ♦p , and ♦t (and therefore, by duality, ∀e commutes with ¬P¬, ¬F¬, f , p , and t ). (12) te (∀e x)¬P¬E!(x) → [(∃e x)Pϕ → P(∃e x)ϕ] te ¬P¬(∀e x)¬F¬E!(x) → [P(∃e x)ϕ → (∃e x)Pϕ] te (∀e x)¬F¬E!(x) → [(∃e x)F ϕ → F (∃e x)ϕ]
2.12. APPENDIX
57
te ¬F¬(∀e x)¬P¬E!(x) → [F (∃e x)ϕ → (∃e x)Fϕ] te t (∀e x)t E!(x) → [(∃e x)Pϕ ↔ (∃e x)Pϕ] te t (∀e x)t E!(x) → [(∃e x)F ϕ ↔ F (∃e x)ϕ] te t (∀e x)t E!(x) → [(∃e x)♦f ϕ ↔ ♦f (∃e x)ϕ] te t (∀e x)t E!(x) → [(∃e x)♦p ϕ ↔ ♦p (∃e x)ϕ] te t (∀e x)t E!(x) → [(∃e x)♦t ϕ ↔ ♦t (∃e x)ϕ] Assumptions weaker than the condition that nothing ever comes into or goes out of existence—such as that everything presently existing always has existed and always will exist, or that everything now existing will never cease to exist, or that everything now existing always has existed yield commutations in only one direction. (∀e x)p E!(x)
Everything presently existing always has existed
(∀e x)f E!(x)
Everything presently existing always will exist.
(∀e x)t E!(x) ist.
Everything presently existing always has and always will ex-
te (∀e x)t E!(x) → [t (∀e x)ϕ → (∀e x)t ϕ] te (∀e x)f E!(x) → [f (∀e x)ϕ → (∀e x)f ϕ] te (∀e x)p E!(x) → [p (∀e x)ϕ → (∀e x)p ϕ] The quantifier ∃pp commutes with the past and future tense operators in only one direction, each the converse to the other, and therefore it commutes with ♦p and ♦f in only one direction as well. Similarly, ∀pp commutes with ¬P¬ and ¬F¬, and therefore with p and f in only one direction: (13) tp P(∃pp x)ϕ → (∃pp x)Pϕ
tp (∀pp x)¬P¬ϕ → ¬P¬(∀pp x)ϕ
tp (∃pp x)F ϕ → F (∃pp x)ϕ
tp ¬F¬(∀pp x)ϕ → (∀pp x)¬F¬ϕ
tp (∃pp x)♦f ϕ → ♦f (∃pp x)ϕ
tp f (∀pp x)ϕ → (∀pp x)f ϕ
tp ♦p (∃pp x)ϕ → (∃pp x)♦p ϕ
tp (∀pp x)p ϕ → p (∀pp x)ϕ
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∀pp commutes with p in both directions if every past and present object always was a past or present object: tp (∀pp x)p [E!(x) ∨ PE!(x)] → [p (∀pp x)ϕ ↔ (∀pp x)p ϕ] Strong conditions are needed in order to commute ∀pp with t , and in fact only a very strong condition suffices for commutation in both directions: tp (∀pp x)t [E!(x) ∨ PE!(x)] → [t (∀pp x)ϕ → (∀pp x)t ϕ] tp t (∀pp x)t [E!(x) ∨ PE!(x)] → [t (∀pp x)ϕ → (∀pp x)t ϕ].
Chapter 3
Logical Necessity and Logical Atomism As indicated in the previous chapter, the analysis of necessity and possibility within tense logic provides a clear and unproblematic paradigm by which to understand the notion of a possible world and the accessibility relation between possible worlds. It also shows the unproblematic nature of quantifying into modal contexts so long as possible objects are restricted to realia. The situation is more problematic, however, once we turn to the different notions of logical necessity and possibility, or a metaphysical necessity and possibility that goes beyond causal modalities and the space-time continuum. The problem with logical and metaphysical necessity and possibility, according to its critics, is quantification into modal contexts, which is traditionally known as de re modality. By allowing such quantification, these critics argue, we become committed to essentialism, and perhaps a bloated universe of logically possible objects as well. The essentialism is avoidable, it is claimed, but only by turning to a Platonic realm of individual concepts whose existence is no less dubious or problematic than logically possible objects. Moreover, basing one’s semantics on individual concepts would in effect render all identity statements containing only proper names either necessarily true or necessarily false—i.e., there would then be no contingent identity statements containing only proper names.1 These claims are not true independently of what formal ontology we adopt. The claim that identity statements containing only proper names would be either necessarily true or necessarily false does not depend, for example, on a commitment to individual concepts, but on whether or not proper names are “rigid designators,” i.e., on whether a proper name is assumed to denote the same object in every possible world, or at any time, at which that object exists. The commitment to essentialism that these critics have in mind, moreover, is not Aristotelian essentialism, even though Quine, who has been the most 1 Cf.
Quine 1943, 1947, 1953.
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60
vocal of the critics, explicitly condemns it as such.2 The examples Quine gives of essentialism, such as, the number nine is necessarily, i.e., essentially, greater than seven, but nine is only contingently and not essentially the number of planets, and everything is necessarily self-identical, is not based on an ontology of natural kinds and have nothing to do with Aristotelian essentialism. Apparently, what Quine and the other critics of modal logic have in mind is either a logical essentialism based on logical modalities or a metaphysical essentialismbased on some form of realism other than natural realism and perhaps even some sort of logical modalities broadly understood. Now there is an ontology in which all of these objections to modal logic, including especially those directed against de re modalities, can be shown to be completely false. This is the ontology of logical atomism, where there are no individual concepts and no possibilia other than the simple objects that exist in the actual world. In addition, it is probably only in logical atomism that logical necessity and possibility find their clearest, and perhaps their only, adequate explication. And yet, not only is logical essentialism false in logical atomism, but it is refutable as well. In other words, with respect to the logical modalities, a modal thesis of anti-essentialism is valid in logical atomism, and one consequence of this is that all de re logical modalities are reducible to de dicto logical modalities, and hence that there is no problem of de re logical modalities. Logical atomism: (1) There are no individual concepts and no possibilia, i.e., existence = being. (2) Logical essentialism is refuted because the modal thesis of antiessentialism is logically true in this framework. (3) All de re logical modalities are reducible to de dicto logical modalities. These results do not mean that logical atomism provides the kind of formal ontology we should adopt, and in fact there are good reasons why just the opposite is the case. Nevertheless, logical atomism does provide the paradigm framework by which to understand logical necessity and possibility, and it shows why a logical essentialism based on this kind of necessity—as opposed, e.g., to a natural necessity—not only does not follow but is actually refuted. Now opposed to logical atomism, but on a par with it in its referential interpretation of quantifiers and proper names, is Saul Kripke’s semantics for what he calls metaphysical necessity.3 There are no individual concepts in Kripke’s semantics, and yet proper names are “rigid designators,” which means that there can be no contingent identity statements containing only proper names. But 2 Cf. 3 Cf.
Quine 1953, p. 173–174. Kripke, 1971, p. 150.
3.1. THE ONTOLOGY OF LOGICAL ATOMISM
61
unlike what is meant by the clear and primary meaning of the phrase “all possible worlds” in logical atomism, and with respect to which logical necessity is interpreted, Kripke’s semantics allows for a “cut-down” on the totality of possible worlds, so that the notion of “all possible worlds,” and hence necessity, has a secondary meaning in his semantics.4 Such a secondary meaning or “cut-down” on the notion of “all possible worlds,” and therefore on necessity, amounts to an initial, but incomplete, step toward something like Aristotelian essentialism. The problem with this initial step, and why it is incomplete, is its failure to provide any ontological content—as opposed to a merely formal, set-theoretic structure—to what is meant by necessity and possibility. This problem of the secondary meaning of necessity, or “cut-down,” on the notion of “all possible worlds,” is only compounded, moreover, by adding to this semantics a relation of accessibility between possible worlds. What must then be explained, in other words, is not just the philosophical significance of the “cut-down” on the notion of “all possible worlds,” but also the ontological content of the accessibility relation between possible worlds. The real problem of quantified modal logic for an ontology other than logical atomism, in other words, is to give an ontological account of the “cut-down” on the notion of “all possible worlds” and of the accessibility relation between possible worlds. The question is: can this really be done other than in tense logic or Aristotelian essentialism, both of which are contained in conceptual realism?
3.1
The Ontology of Logical Atomism
Reality, according to logical atomism, consists of the existence and nonexistence of atomic states of affairs, where the existence of a state of affairs is “a positive fact” and its nonexistence “a negative fact”.5 The actual world, in other words, consists of all that is the case, namely the totality of facts, whether positive or negative.6 Every other possible world consists of the same atomic states of affairs that make up reality, except that what are positive facts in one world can be negative facts in another, with every possible combination of atomic states of affairs being realized in some possible world or other.7 The totality of possible worlds, in other words, is completely determined by all the combinations of the existence or nonexistence of the atomic states of affairs that make up reality. 4 A secondary semantics for necessity stands to the primary semantics in essentially the same way that nonstandard models for second-order logic stand to standard models. See Cocchiarella 1975. 5 Wittgenstein 1961, 2.06. For a fuller discussion of logical atomism as a formal ontology, see chapters 6 and 7 of Cocchiarella 1987. 6 Wittgenstein 1961, 1. It is an issue as to whether the Tractatus allowed for negative facts. There are negative facts in Russell’s version of logical atomism. 7 If we did not include negative facts, then a world that contains none of the states of affairs that “exist” in the actual world—i.e., that would contain as positive facts all of the negative facts of the actual world—would be an empty world, a world devoid of all facts, and hence of all objects as well, and therefore not a possible world at all.
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The direction of this determination is important. Atomic states of affairs do not have being (the-case-or-not-the-case) because they exist (are the case) in some possible worlds; rather, possible worlds are possible because they are resolvable into the atomic states of affairs that make up reality. The determination of the totality of possible worlds in terms of the atomic states of affairs that make up reality is what makes logical atomism a paradigm framework for the semantics of logical necessity. Every atomic state of affairs is a configuration of objects, and therefore because every state of affairs is a positive or negative fact in each possible world, each possible world consists of the same totality of objects as every other possible world. There is no distinction, accordingly, between the existence and being of objects. In the ontological grammar of logical atomism, in other words, there is no distinction between the possibilist quantifiers ∀ and ∃ and the actualist quantifiers ∀e and ∃e , and for that reason we will not include the latter in the formal ontology of logical atomism. Ordinary proper names of natural language are not “logically proper names”in the framework of logical atomism. The things they name, if they name anything at all, are not the simple objects that are the constituents of atomic states of affairs. The names of ordinary language have a sense (Sinn), moreover, in so far as they are introduced into discourse with identity criteria, usually provided by a sortal common noun with which they are associated.8 The logically proper names of logical atomism have no sense other than what they denote. In other words, in logical atomism, “a name means (bedeutet ) an object. The object is its meaning (Bedeutung).”9 Different identity criteria have no bearing on the simple objects of logical atomism, and (pseudo) identity propositions, strictly speaking, have no sense (Sinn), i.e., they do not represent an atomic state of affairs. Semantically, what this comes to is that logically proper names, or objectual constants, are rigid designators; that is, their introduction into formal languages requires that the formula (∃x)(a = x) be logically true in the primary semantics for each objectual constant a. Kripke also claims that proper names are rigid designators, but his proper names are those of ordinary language, and his necessity is metaphysical and not logical necessity. Nevertheless, in agreement with logical atomism the function of a proper name, according to Kripke, is simply to refer, and not to describe the object named10 ; and this applies even when we fix the reference of a proper 8 Cf. Geach, 1980, p. 63f. Individual constants cannot be vacuous in logical atomism, moreover, which means that the free logic of the quantifiers ∀e and ∃e is to be excluded, as opposed to the logic of the possibilist quantifiers ∀ and ∃, which, as already noted is standard predicate logic. 9 Wittgenstein 1961, 3.203. 10 Kripke, 1971, p. 140.
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63
name by means of a definite description—for the relation between a proper name and a description used to fix the reference of the name is not that of synonymy.11 Because logically proper names are rigid designators, there are no contingent identity statements involving only proper names. Accordingly, where a, b are objectual constants, (a = b) → (a = b) (a = b) → (a = b) are both understood to be logically true in logical atomism, and in Kripke’s framework of metaphysical necessity as well. The fact that there can be no contingent identities or non-identities in logical atomism is reflected, moreover, in the logical truth of both of the formulas (∀x)(∀y)[(x = y) → (x = y)], (∀x)(∀y)[(x = y) → (x = y)] in the semantics of logical atomism.12 But then even in the framework of Kripke’s metaphysical necessity (where quantifiers also refer directly to objects), an object cannot but be the object that it is, nor can one object be identical with another—a metaphysical fact which is reflected in the above formulas being valid in that framework as well. Another observation about the ontology of logical atomism is that the number of objects in the world is part of its logical scaffolding.13 That is, for each positive integer n, it is either logically necessary or impossible that there are exactly n objects in the world; and if the number of objects is infinite, then, for each positive integer n, it is logically necessary that there are at least n objects in the world.14 This is true in logical atomism because every possible world consists of the same totality of objects. One important consequence of the fact that every possible world (of a given logical space) consists of the same totality of objects is the logical truth of the Carnap-Barcan formula (and its converse) (∀x)ϕ ↔ (∀x)ϕ. Carnap, it should be noted, was the first to actually give a semantic argument justifying the logical truth of this principle.15 The idea, in effect, is that any universally quantified sentence (∀x)ϕ, no matter whether ϕ contains occurrences of modal operators or not, “is to be interpreted as a joint assertion for all values of the variable.”16 11 Ibid.,
pp. 156f. Carnap 1946 for the first clear recognition of the validity of these noncontingent identity theses. 13 This observation was first made by Ramsey in his adoption of logical atomism. Cf. Ramsey 1960. 14 Cf. Cocchiarella, 1975, Section 5. 15 Cf. Carnap, 1946, p.37 and 1947, Section 40. Unlike Carnap, Barcan assumed the formula as an axiom, and gave no explanation of why it should be accepted. 16 Carnap 1947, p. 37. 12 See
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3.2
The Primary Semantics of Logical Necessity
Let us now turn to what we take to be the primary semantics of logical necessity.17 Our terminology will proceed as a natural extension of the syntax and semantics of standard first-order logic with identity. A formal language consists then just of predicate constants (of various finite degree) and objectual constants. As primitive logical constants we take: →
the material conditional sign,
¬
the negation sign,
∀
the universal quantifier,
=
the identity sign, and
the logical necessity sign.
The conjunction, disjunction, biconditional, existential quantifier and possibility signs—∧, ∨, ↔, ∃ and ♦, respectively—are understood to be defined in the usual way as metalinguistic abbreviatory devices. The formulas of a language are defined as in the logic of actual and possible objects, except that now the quantifier ∀e for existent objects is excluded and the logical necessity operator is included. Because all possible objects are actual objects in logical atomism, we can restrict the notion of an model suited for a formal language L as follows. Definition: If L is a language, then a model A is an L-model if, and only if, for some nonempty set D and function R on L, A = D, R, and for each objectual constant a ∈ L, R(a) ∈ D and for each positive integer n and each n-place predicate F n ∈ L, R(F n ) ⊆ Dn , i.e., R(F n ) is a set of n-tuples of members of D. Note: In the context of logical atomism, a model D, R for a language L represents a possible world of a logical space based upon D as the universe of objects of that space and L as the predicates characterizing the atomic states of affairs of that space. The possible worlds of this logical space are the L-models having D as their domain and that assign to each objectual constant a in L the same denotation that R assigns to a, because a is a “rigid designator”. Definition: If A = D, R is a model for a language L representing the actual world, then the logical space determined by A = the totality of possible worlds based on A, in symbols W lds(A), is defined as follows: W lds(A) = {D,R : D,R is an L-model and for all objectual constants a in L, R (a) = R(a)}. 17 One or another version of this primary semantics for logical necessity, it should be noted, occurs in Carnap, 1946; Kanger, 1957; Beth, 1960 and Montague, 1960. Only Carnap, however, was clear about the association of his semantics with logical atomism.
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Definition: A is an assignment (of values to variables) in a domain D if, and only if, A is a function with the set of objectual variables as domain and such that for each variable x, A(x) ∈ D. Definition: If A is an assignment in a domain D, x is an objectual variable and d ∈ D, then A(d/x) is an assignment in D that is exactly like A except for its assigning d to x. Definition: If L is a language, A = D, R is an L-model, A is an assignment in D, and a is a variable or an individual constant in L, then (the denotation of a in A): A(a) if a is a variable denA,A = . R(a) if a is an objectual constant The satisfaction in A of a formula ϕ of L by an assignment A in D, in symbols A, A |= ϕ, is recursively defined as follows: 1. A, A |= (a = b) iff denA,A (a) = denA,A (b); 2. A, A |= F n (denA,A (a1 , ..., an ) iff denA,A (a1 ), ..., denA,A (an ) ∈ R(F n ); 3. A, A |= ¬ϕ iff A, A ϕ; 4. A, A |= (ϕ → ψ) iff either A, A ϕ or A, A |= ψ; 5. A, A |= (∀x)ϕ iff for all d ∈ D, A, A(d/x) |= ϕ; and 6. A, A |= ϕ iff for all B ∈ W lds(A), B, A ϕ. The truth of a formula in a model (indexed by a language suitable to that formula) is as usual the satisfaction of the formula by every assignment in the universe of the model. Logical truth is then truth in every model (indexed by any appropriate language). Definition: If L is a language, ϕ is a standard formula of L, A = D, R, and A is an L-model, then ϕ is true in A if, and only if, for each assignment A in D, A, A |= ϕ. Definition: ϕ is logically true if, and only if, for some language L, ϕ is a formula of L, and ϕ is true in every L-model. These definitions are the natural extensions of the same semantical concepts as defined for the modal free formulas of standard first-order predicate logic with identity. Note that every model, because it specifies both a domain and a language, determines both a unique logical space and a possible world of that space. In this regard, the clause for the necessity operator in the above definition of satisfaction is the natural extension of the standard definition of satisfaction and interprets the necessity operator as ranging over all the possible worlds (models) of the logical space to which the given one belongs.
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3.3
The Modal Thesis of Anti-Essentialism
Now it may be objected that logical atomism is an inappropriate framework upon which to base a system of quantified modal logic; for if any framework is a paradigm of anti-essentialism, it is logical atomism with its ontologically simple objects and independent atomic states of affairs. The objection is void and begs the question, because it assumes that any system of quantified modal logic is committed to essentialism insofar as it allows quantifiers to reach into modal contexts, i.e., insofar as it allows de re modalities. Now essentialism means different things in different ontologies. For Aristotle and Aquinas the doctrine of essentialism is a form of natural realism, only in addition to natural properties and relations there are also natural kinds hierarchically ordered in terms of species and genera. On this doctrine, if an object belongs to a natural kind, then it necessarily belongs to that natural kind, i.e., it belongs to that kind in every possible world in which it exists. But a natural kind of object will also have natural properties that are not essential to it, i.e., properties that it has in some but not in all possible worlds in which it exists. This doctrine is similar to Quine’s characterization of what he calls “Aristotelian essentialism,” namely, “the doctrine that some of the attributes of a thing (quite independently of the language in which that thing is referred to, if at all) may be essential to the thing and others accidental.”18 But Quine’s characterization fails to distinguish natural kinds from attributes in general, and in fact, as we will see in our later development of the logic of natural kinds, natural kinds are not “attributes” at all, at least not in the sense of the natural properties that an object might have. Instead, natural kinds are certain types of causal structures that have a complementary relationship with natural properties and relations, and as such they are general essences as opposed to the individual essences of, e.g.,Alvin Plantinga’s ontology.19 Now, according to Quine “what Aristotelian essentialism says” is that you can have open sentences, e.g., ϕx and ψx such that ϕ is necessary to x but ψ is not necessary to x even though ψ is true of x, which formally can be symbolized as follows:20 (∃x)[ϕx ∧ ¬ψx ∧ ψx]. As an example of this so-called “Aristotelian essentialism”, Quine gives the following, (∃x)[(x > 5) ∧ there are just x planets ∧ ¬there are just x planets]. Note that here (x > 5) represents the “attribute” of being greater than 5, a condition that in no sense can be understood as a natural kind in Aristotle’s natural realism. According to Quine, “something yet stronger can be shown”, namely 18 Cp,
e.g., Quine 1966, pp.173f. Plantinga 1974, chapter 5, §2. Also, see our discussion of the difference between general and individual essence in chapter ??, §5. 20 Ibid. 19 See
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67
(∀x)[(x = x) ∧ (∀x)(x = x ∧ P ) ∧ (∀x)¬(x = x ∧ P ). Here, again, (x = x), the condition of being self-identical, does not represent a natural kind, nor even a natural property in Aristotle natural realism. Selfidentity, unlike belonging to a natural kind, is a condition that is necessary to all objects and in no way marks off what is essential to some objects as opposed to others. If we ignore Aristotelian essentialism and the issue of natural kinds, then Quine’s characterization can be slightly reformulated as the more interesting doctrine that there are some conditions (predicates, properties, or concepts) that are necessarily true of some objects, but, unlike self-identity, not of others. It is this more interesting version of essentialism that is actually invalidated in logical atomism where the opposite thesis of anti-essentialism is logically true. This anti-essentialist thesis was first formulated and validated by Rudolf Carnap in his state-description semantics, which, in the case of an infinite domain, is equivalent to the above semantics for logical atomism.21 The general idea of the modal thesis of anti-essentialism is that if a predicate expression or open formula ϕ in which no objectual constants occur can be true of some objects in a given universe (satisfying a given identity-difference condition with respect to the variables free in ϕ), then ϕ can be true of any objects in that universe (satisfying the same identity-difference conditions). In other words, no condition is essential to some objects that is not essential to all, which is as it should be if necessity means logical necessity.22 The restriction to identity-difference conditions mentioned above can be dropped, it should be noted, if nested quantifiers are interpreted exclusively and not, as we have done, inclusively where, e.g., it is allowed that the value of y in (∀x)(∃y)ϕ(x, y) can be the same as the value of x, as for example in (∀x)(∃y)(x = y).23 Now our point is that when nested quantifiers are interpreted exclusively, then identity and difference formulas are superfluous—which is especially appropriate in logical atomism where an identity formula does not represent an atomic state of affairs.24 Retaining the inclusive interpretation and identity as primitive, however, an identity-difference condition is defined as follows. Definition: If x1 , ..., xn are distinct objectual variables, then an identity-difference condition for x1 , ..., xn is a conjunction of one each but not both of the formulas (xi = xj ) or (xi = xj ), for all i, j such that 1 ≤ i < j ≤ n. 21 See Carnap, 1946, T10-3.c, p.56, for the first formulation of this thesis ever to be given, and also Parsons, 1969 for a much later formulation. Note, however, that whereas in Carnap’s semantics the thesis is valid, in Parson’s semantics the thesis is merely consistent. 22 If objectual constants do occur in a formula, they can be replaced uniformly by distinct new objectual variables not already occurring in the formula. 23 See Hintikka, 1956 for a development of the exclusive interpretation. 24 Cf. Wittgenstein, 1961, and Cocchiarella, 1987, chapter V1.
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Note: Because there are only a finite number of nonequivalent identity-difference conditions for x1 , ..., xn , we can assume an ordering, ID1 (x1 , ..., xn ),..., IDj (x1 , ..., xn ), is given of all of these nonequivalent conditions. The modal thesis of anti-essentialism may now be stated for all formulas ϕ in which no objectual constants occur as follows: for all positive integers j such that 1 ≤ j ≤ n, every formula of the form, (∃x1 )...(∃xn )[IDj(x1 , ..., xn ) ∧ ϕ] → (∀x1 )...(∀xn )[IDj(x1 , ..., xn ) → ϕ] is to be logically true, where x1 , ..., xn are all the distinct objectual variables occurring free in ϕ. We can also phrase this thesis in terms of its equivalent contrapositive form: (∃x1 )...(∃xn )[IDj(x1 , ..., xn ) ∧ ♦ϕ] → (∀x1 )...(∀xn )[IDj(x1 , ..., xn ) → ♦ϕ] Note: Where n = 0, the above formula is understood to be just (ϕ → ϕ); and where n = 1, it is understood to be just (∃x)ϕ → (∀x)ϕ, or equivalently (∃x)♦ϕ → (∀x)♦ϕ. The first of these last formulas state that if something is essentially ϕ, then everything essentially ϕ. The validation of the thesis in our present semantics is easily seen to be a consequence of the following lemma (whose proof is by a simple induction on the formulas of L). That is, given that some objects satisfy ϕ in some L-model, then, by the following lemma, any permutation of those objects with any others in a domain D of the same size will also satisfy ϕ in some other L-model with that domain D, i.e., there will be an isomorphism between the two L-models.
LEMMA: If L is a language, A, B are L-models, and h is an isomorphism of A with B, then for all formulas ϕ of L and all assignments A in the universe of A, A, A ϕ if, and only if, B, A/h |= ϕ.25 As already noted, one of the consequences of the modal thesis of antiessentialism is the reduction of all de re formulas to de dicto formulas. Such a consequence indicates the correctness of our association of the present semantics with logical atomism. 25 We understand h to be an isomorphism of A with B if (1) h is a 1-1 mapping of the domain of A onto the domain of B, (2) for each individual constant a ∈ L, denB,A/h (a) = h(denA,A (a)), and (3)for all positive intergers n and n-place predicate constants F ∈ L, the extension F is assigned in B is {h(d1) , ..., h(dn ) : for d1 , ...dn in the extension that F is assigned in A}. Also, we take the relative product A/h to be that assignement in the domain of B such that for all variables x, A/h(x) = h(A(x)). In the case of an atomic formula in the inductive argument for this lemma, we have A, A |= F (a1 , ..., an ) iff denA,A (a1 ), ..., denA,A (an ) ∈ R(F ) iff h(denA,A (a1 )), ..., h(denA,A (an )) ∈ R (F ), where R, R are the assignments to predicate constants in A and B, respectively; and therefore A, A |= F (a1 , ..., an ) iff B, A/h |= F (a1 , ..., an ). The remaining cases follow in each case by the inductive hypothesis.
3.4. AN INCOMPLETENESS THEOREM
69
Note: A de re formula ϕ is one in which some objectual variable has a free occurrence in a subformula of ϕ of the form ψ, and hence a variable that can be bound by a quantifier applied to ϕ. A de dicto formula is a formula that is not de re. De Re Elimination Theorem: For each de re formula ϕ, there is a de dicto formula ψ such that (ϕ ↔ ψ) is logically true.26
3.4
An Incompleteness Theorem
Another result of the semantics for logical atomism is its essential incompleteness with respect to any language containing at least one relational predicate. This result depends on the conditional possibility of there being infinitely many objects, and a relational predicate is needed in order to state such an infinitary condition. In other words, an essential incompleteness theorem results if there are relational states of affairs in logical atomism. If there are only properties, i.e., monadic states of affairs, then the formal ontology is not only complete but decidable as well. The above semantics yields both a completeness theorem and a decision procedure for logical truth, in other words, for any language containing only monadic predicates. The incompleteness theorem for languages containing a relational predicate is easily seen to follow from the following lemma and the well-known fact that the modal-free nonlogical truths of a first-order language containing at least one relational predicate is not recursively enumerable.27
LEMMA: If ψ is a sentence that is satisfiable, but only in an infinite model, and ϕ is a modal-free and identity-free sentence and ϕ, ψ contain no objectual constants, then (ψ → ¬ϕ) is logically true iff ϕ is not logically true.28 26 A proof of this theorem can be found in McKay, 1975. Briefly, where x , ..., x are all the n 1 distinct individual variables occurring free in ϕ and ID1 (x1 , ..., xn ), ..., Dk (x1 , ..., xn ) are all the nonequivalent identity-difference conditions for x1 , ..., xn , then the equivalence in question can be shown if ψ is obtained from ϕ by replacing each subwff χ of ϕ by:
27 Cf.
[ID1 (x1 , ...xn ) ∧ ∀x1 ...∀xn (ID1 (x1 , ..., xn )
→
χ)] ∨ ...
∨[IDk (x1 , ..., xn ) ∧ ∀x1 ...∀xn (IDk (x1, ..., xn)
→
χ)].
Cocchiarella, 1975. Assume the antecedent and, for the left-to-right direction that (ψ → ¬ϕ) is logically true. We note that if ϕ were logically true, then it would be true in every L-model for any language L of which ϕ is a formula; but then ϕ would be true in an infinite L-model A in which ψ is satisfiable, in which case, by assumption, ¬ϕ would be true A as well; but that is impossible because ϕ would then be false in some L-model when by assumption ϕ is logically true, and therefore true in every L-model. For the right-to-left direction, suppose 28 Proof.
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THEOREM: If L is a language containing at least one relational predicate, then the set of formulas of L that are logically true is not recursively enumerable. Proof. Because the modal and identity free formulas of L that are not logically true are not recursively enumerable, it follows by the above lemma that the logically true formulas of L of the form (ψ → ¬ϕ) are also not recursively enumerable, and hence that the set of formulas of L that are logically true is not recursively enumerable. This last result does not affect the association we have made of the primary semantics with logical atomism. Indeed, given the L¨ owenheim-Skolem theorem, what the above lemma shows is that there is a complete concurrence between logical necessity as an internal condition of modal free propositions, or of their corresponding states of affairs, and logical truth as a semantical condition of the modal free sentences expressing those propositions, or representing their corresponding states of affairs. And that of course is as it should be if the operator for logical necessity is to have only formal and no material content. Finally, it should be noted that the above incompleteness theorem explains why Carnap was not able to prove the completeness of the system of quantified modal logic formulated in Carnap 1946. For on the assumption that the number of objects in the universe is denumerably infinite, Carnap’s state description semantics is essentially that of the primary semantics restricted to denumerably infinite models; and, of course, precisely because the models are denumerably infinite, the above incompleteness theorem applies to Carnap’s formulation as well. Thus, the reason why Carnap was unable to carry though his proof of completeness is finally answered.
3.5
The Semantics of Metaphysical Necessity
Like the situation in standard second-order logic, the incompleteness of the primary semantics can be avoided by allowing the quantification over possible worlds in the interpretation of necessity to refer not to all of the possible worlds (models) of a given logical space but only to those in a given non-empty set of such. ϕ is not logically true. Let A be an arbitrary L-model for any language L of which ϕ and ψ are formulas. It suffices to show that (ψ → ¬ϕ) is true in A. If ψ is not satisfiable in A, then (ψ → ¬ϕ) is vacuously true in A. Suppose then that ψ is satisfiable in A. Then, by hypothesis, D, the domain of A, is infinite. Now because ϕ is modal and identity free and not logically true, then, by L¨ owenheim-Skolem theorem, ϕ must be false in some L-model B having D as its domain, and hence, because no objectual constants occur in ϕ or ψ, B ∈ W lds(A). But then, by the semantic clause for , ¬ϕ is true in A, and therefore so is (ψ → ¬ϕ).
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71
That is, by allowing a “cut-down” on the notion of “all possible worlds” in the interpretation of necessity, we can obtain a completeness theorem instead of an incompleteness theorem. Of course, the world in question must be in the “cut-down” as part of the definition of satisfaction. Accordingly, where L is a language and D is a non-empty set, we understand a model structure based on D and L to be a pair A,K, where A ∈ K, K is a set of L-models all having D as their domain of discourse and all of the objectual constants in L are assigned the same denotation in each L-model in K. Definition: If L is a language and D is a nonempty set, then A,K is a model structure based on D and L if, and only if, A ∈ K and K is a set of L-models all having D as their domain of discourse and all agreeing on the assignment of members of D to the objectual constants in L, i.e., the assignment of a members of D to objectual constants in L is the same for each member of K. The satisfaction of a formula ϕ of L in such a model structure by an assignment A in D, in symbols A, K), A |= ϕ, is recursively defined exactly as in the primary semantics except for clause (6), which is now defined as follows: 6. A, K, A |= ϕ iff for all B ∈ K, B, K, A |= ϕ. Instead of logical truth, a formula is understood to be universally valid if it is satisfied by every assignment in every model structure based on a language to which the formula belongs. Definition: ϕ is universally valid if, and only if, for every language L, every nonempty domain D, every model structure A, K based on D and L, and every assignment A in A, if ϕ is a formula of L, then A, K, A |= ϕ. Where QS5 is standard first-order logic with identity supplemented with the axioms of S5 propositional modal logic, a completeness theorem for the secondary semantics of logical necessity was proved by Kripke in 1959. Completeness Theorem: A set Γ of formulas is consistent in QS5 if, and only if, all the members of Γ are simultaneously satisfiable in a model structure; and (therefore) a formula ϕ is a theorem of QS5 if, and only if, ϕ is universally valid. Despite the above completeness theorem, the secondary semantics has too high a price to pay as far as logical atomism is concerned. Unlike the situation in the primary semantics, the secondary semantics does not validate the modal thesis of anti-essentialism—i.e., it is false that every instance of the thesis is universally valid.
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Example 1 As an example of a false instance of the thesis, consider a model structure A, K, where K = {A, B}, D = {d1 , d2 }, d1 = d2 , and F is a monadic predicate that is assigned only d1 in B and nothing in A. Then (∃x)♦F (x) is true in A, K, whereas (∀x)♦F (x) is false in A, K, which invalidates the modal thesis of anti-essentialism (in the case of a monadic formula). The reason why the modal thesis of anti-essentialism can be invalidated in the secondary semantics is because necessity no longer represents an invariance through all the possible worlds of a given logical space but only through those in a nonempty set of such worlds. In this way, necessity is no longer a purely formal concept having no material content the way it is in logical atomism. Instead, necessity is now allowed to represent an internal condition of states of affairs— i.e., a condition that has material and not merely formal content—for what is invariant through all of the members of such a nonempty set need not be invariant though all the possible worlds (models) of the logical space to which those in the set belong. One example of how such material content affects the implicit metaphysical background can be found in monadic modal predicate logic. First let us note a well-known fact about modal-free monadic predicate logic. Note: Modal-free monadic predicate logic is decidable and no modal-free monadic formula can be true in an infinite model unless it is true in a finite model as well. Therefore, any substitution instance of a modal-free monadic formula for a relational predicate in an axiom of infinity—i.e., a formula that is true only in an infinite domain of discourse—is not only false but logically false; and hence, contrary to certain metaphysical views there can be no modal-free analysis or reduction of all relational predicates (or open formulas with two or more free variables) in terms only of monadic predicates, i.e., in terms only of modal-free monadic formulas. Now the same result also holds for quantified monadic modal logic with respect to the primary semantics, where “all possible worlds” means all possible worlds. Theorem: Modal monadic predicate logic is also decidable with respect to the primary semantics of logical atomism; and therefore no monadic formula, modal-free or otherwise, can be true in an infinite model unless it is also true in a finite model.29 That is, there can be no reduction of all relational predicates or open formulas in terms only of monadic formulas, modal free or otherwise. 29 Cf. Cocchiarella 1975. This is proved by interpreting as a string of universal quantifiers on the predicates occurring within the scope of , and thereby translating modal formulas into modal-free formulas of second-order monadic predicate logic, which is known to be decidable. It is also shown in Cocchiarella 1975 that if second-order monadic predicate logic is given a secondary, “cut-down” semantics, then, unlike its primary semantics, the resulting logic is no longer decidable.
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73
The following formula, e.g., is true in some models based on an infinite domain but false in all models based on a finite domain. This formula, in other words, can be taken as an axiom of infinity. (∀x)¬R(x, x) ∧ (∀x)(∃y)R(x, y) ∧ (∀x)(∀y)(∀z)[R(x, y) ∧ R(y, z) → R(x, z)] Consider substituting the open formula (with two free variables) ♦[F (x)∧G(y)], with two monadic predicate constants F and G, for the two-place predicate R in this formula. The first conjunct then yields the following substitution instance. (∀x)¬♦[F (x) ∧ G(x)], which is equivalent to ¬♦(∃x)[F (x) ∧ G(x)]. But the formula ♦(∃x)[F (x) ∧ G(x)] is logically true in the primary semantics of logical atomism, and hence its negation is logically false, which shows that that the conjunction that is an instance of the above infinity formula is logically false. In other words, where L is any language having the monadic predicates F and G as members, then given any nonempty domain D there will be an Lmodel A in which [F (x) ∧ G(x)] is satisfied by some member of D, and hence (∃x)[F (x) ∧ G(x)] will be true in A, which means that ♦(∃x)[F (x) ∧ G(x)] will be true in any world (L-model) in the logical space determined by A, i.e., true in any world in W lds(A), and hence that ♦(∃x)[F (x) ∧ G(x)] is logically true with respect to the primary semantics of logical atomism. Thus, substituting ♦[F (x) ∧ G(y)] for R in the above formula representing an axiom of infinity results in a logically false sentence in the primary semantics. With respect to the secondary semantics, however, the situation is quite different, because all we need do is exclude all of the L-models in the logical space based on L and D in which (∃x)[F (x) ∧ G(x)] is true. The “cut-down” or resulting model structure B, K will be such that ¬♦(∃x)[F (x) ∧ G(x)] is true in all of the models in K. Equivalently, the formula (∀x)[F (x) → ¬G(x)], which clearly represents a necessary, i.e., an internal, relation between being an F and not-being a G, will be true in B, K with respect to the secondary semantics. Instead of modal monadic predicate logic being decidable the way it is in the primary semantics, modal monadic predicate logic is undecidable in the secondary semantics, as Kripke has shown. Moreover, on the basis of that semantics a modal analysis of relational predicates in terms of monadic predicates can in general be given. 30 Indeed, substituting ♦[F (x) ∧ G(y)] for the binary predicate R in the above infinity axiom results in a modal monadic sentence 30 Cf.
Kripke 1962.
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that is true in some model structure based on an infinite universe and false in all model structures based on a finite domain.31 Somehow, in other words, by means of a “cut-down” on the notion of “all possible worlds”, relational content has been incorporated into the semantics for necessity, and thereby of possibility as well. In this respect, the secondary semantics is not the semantics of a merely formal, or logical, necessity but of a necessity having material, nonlogical content as well.
3.6
Metaphysical Versus Natural Necessity
Kripke does not describe the necessity of his semantics as a formal, or logical, necessity, but as a metaphysical necessity. He has also argued that not every necessary proposition is a priori, and that not every a posteriori proposition is contingent.32 It is because Kripke is concerned with a metaphysical necessity and not a logical, or formal, necessity that not every necessary proposition needs to be a priori, nor every a posteriori proposition contingent. But such a position cannot be taken as a refutation of the claim in logical atomism that every logically necessary proposition is a priori and that every a posteriori proposition is logically contingent. These are two different metaphysical frameworks, each with its own notion of necessity and thereby of contingency as well. The ontology of logical atomism’s framework is very clear and precise, moreover, whereas it is not at all clear just what ontological framework Kripke has in mind with his notion of “metaphysical necessity”. Now we can extend the notion of a model structure B, K based on a domain D so that instead of having D represent the same universe of existing objects in all of the worlds in K, D would represent only the same domain of possible objects of the structure B, K, i.e., the union, or sum, of all of the objects that exist in some world or other in K. We would then distinguish a universe of existing objects for each world A ∈ K (i.e., the objects that exist in A and not necessarily in the other worlds in K) from the possible objects made up of the objects that exist in some world or other in K, just as we did in the semantics for the logic of actual and possible objects in tense logic where instead of worlds we had moments of time. We would then reintroduce the actualist quantifiers ∀e and ∃e and the free logic of actualism to represent the restricted quantification over existing objects. We could either retain the full logic of actual and possible objects in that case, or we could restrict ourselves to just the actualist modal logic, depending on whether we want to represent modal possibilism or modal actualism.33 31 As shown in Cocchiarella 1975, a parallel result holds for second-order monadic predicate logic. That is, although second-order monadic predicate logic is decidable with respect to its standard set-theoretic semantics, it is not decidable with respect to a secondary semantics that allows for a “cut-down” on the notion of all values of the monadic predicate variables. 32 Cf. Kripke 1971, p. 150. 33 See Cocchiarella 1984, section 6.
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We can also deepen the material content of the secondary semantics by adding a relation R of accessibility between possible worlds, i.e., between the models in K, where B, K is a model structure, so that necessity in a world A ∈ K is an invariance condition with respect to the worlds in K that are accessible from A.34 If R is an equivalence relation, then the modal logic S5, whether as actualist or possibilist, characterizes the class of model structures that result. Other quantified modal logics characterize classes of model structures in which the accessibility relation between models is weaker than an equivalence relation. The modal logic S4, for example, characterizes the class of model structures in which the accessibility relation is transitive and reflexive. The question remains, however, as to just what ontology is represented by these kinds of set-theoretic semantics, and in particular what notion of necessity other than logical necessity is in question. Calling the result a metaphysical modality is not an adequate answer. We need a philosophical, and in particular an ontological, account of what principles determine the “cut-down” on possible worlds (models), and how the accessibility relation between worlds is to be explained in terms of such principles. A set-theoretic structure with respect to which a completeness theorem can be proved is not itself such an ontological or philosophical account. In the ontology of conceptual realism, for example, the “cut-down” on possible worlds can be accounted for either in terms of time, as in tense logic, or in terms of the network of natural laws that are part of nature’s causal matrix. Not all logically possible states of affairs will be realized in time, so that time itself provides a metaphysical ground for such a “cut-down”. That is why the Aristotelian and Diodorean modalities definable in terms of a local time are unproblematic. The earlier-than relation of a local time, or the signal relation of a causally connected system of local times also provide unproblematic metaphysical bases for different accessibility relations, as well as ontological grounds for different modal logics. Similarly, an ontological ground for such a “cut-down” on all logically possible worlds can be given in terms of the set L of laws of nature. Thus, for example, where K is the set of possible worlds of a multiverse35 in which all of the laws in L are true, and B ∈ K, the model structure B, K would characterize an invariance condition based on the laws in L. Different sets of laws would then determine different model structures.36 But because each model structure B, K would be determined by the same set of laws, then all of the worlds in K would have the same laws of nature, and hence each would be accessible from every other member of K. The modal logic that results would then be S5. 34 Ibid.,
section 7. Kaku 2005 for account of the multiverse, or megaverse, of parallel worlds as described in modern cosmology, and especially as based on string theory and the inflationary universe. 36 According to the Astronomer Royal of Great Britain, Sir Martin Rees, “what’s conventionally called ‘the universe’ could be just one member of an ensemble. Countless other ways may exist in which the laws are different. The universe in which we’ve emerged belongs to the unusual subset that permits complexity and consciousness to develop.” (Quoted in Kaku 2005. p. 15.) 35 See
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3.6.1
The Concordance Model of a Multiverse
One example of a multiverse is what is known as the concordance model of cosmology. The universe, as is now well-known, is expanding, and because of that expansion the most distant objects we can see now with the most powerful telescopes are about 40 billion lightyears (4 × 1026 meters) away.37 A sphere of this radius, and hence our universe as well, is called a Hubble volume; and, according to some cosmologists, beyond the Hubble horizon there are other universes with the same laws of nature as ours, but with possibly different initial conditions. Space is assumed to be infinite on this model and because the distribution of matter is relatively uniform and represented by an ergodic random field it is also assumed that “there are infinitely many other regions the size of our observable universe, where every possible cosmic history is played out.”38 On this view the curvature of space is flat, which excludes a curved, bounded, and hence finite, space-time such as Einstein once favored. Of course the universe could be hyperspherical, and hence finite after all, but because it is so large it appears flat and Euclidean to us, just as a small part of the Earth’s surface appears flat to us. Nevertheless, on this model, it is assumed both that space is infinite and that there are infinitely many physical objects in the multiverse; that is, even though there are only finitely many objects in each region, the total number of objects is infinite because the number of regions is infinite. This multiverse is essentially just the cosmos, however, because if the cosmic expansion were to decelerate then it would be physically possible to travel to regions beyond the Hubble horizon. On the other hand, if the acceleration of the universe’s expansion continues indefinitely, as most cosmologists today assume, then the rate of expansion will exceed the speed of light and the different regions will amount essentially to different possible worlds, all with the same laws of nature as ours. The objects in these regions can then be considered as physically possible objects, and quantifying over them would be different from quantifying over the “actual” objects of our region. The logic of a formal ontology representing this situation would then be an S5 modal logic with actualist and possibilist quantifiers.
3.6.2
The Multiverse of the Many-Worlds Model
Another example of a multiverse are the parallel worlds in the omnium of the many-worlds interpretation of quantum mechanics. In quantum mechanics (QM) each particle in the universe is associated with a probability wave that specifies the different probabilities of where that particle might be located anywhere in the universe at each moment. Whether a particle is the same as its wave function, or whether the wave function is merely a mathematical construct 37 See Tegmark [2003]. Although the universe is only 13.7 billion years old and the light that is now reaching us from the most distant stars took that many years to reach us, those stars, because of the expansion of the universe, are now more than 13.7 billion lightyears away. They are in fact now 40 billion lightyears away. 38 Ibid.
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that describes the particle’s motion is one of the issues that distinguishes different versions of QM. In standard quantum mechanics, when a measurement is made and a particle is observed at a given location, then the probability of finding it at that location becomes 100 percent while the probability of finding it at any other location at that time drops to zero. This is what is meant in saying that the wave function “collapses”. The many-worlds interpretation, (MWI), denies that a particle’s wave function ever collapses. Instead of a collapse of the wave function, what happens according to the MWI is that every potential outcome described in the particle’s probability function is realized in a separate parallel world, so that anything that could happen in the sense of being physically possible according to QM in fact does happen in some parallel world.39 All of the worlds accessible in this way from a given world when a measurement is made at a given moment have the same past up to that moment, but, except for the laws of nature, they differ thereafter in some way. An infinite number of parallel worlds populated by copies of ourselves is assumed in this way, where all of the worlds “co-exist” in a quantum superposition.40 Although the objects in those worlds are not “real” in the same sense in which the objects of our universe are real, nevertheless, they have an ontological status as objects of the multiverse, or what following Roger Penrose might preferably be called the omnium.41 This type of situation is represented in a formal ontology in terms of an S4 modal logic in which necessity and possibility are based on what is physically possible in QM.42 In other words, we distinguish between quantifying over the real objects of our universe from quantifying over the objects in other possible (parallel) worlds. There would then seem to be infinitely many possible objects, i.e., objects that exist in some possible world of the multiverse, even though in our universe there are only finitely many objects.43 It is by these kinds of interpretations, or models of cosmology, that we can give content to what is meant by a “cut-down” on the notion of all possible worlds, and thereby on the notions of necessity and possibility. The real problem of quantified modal logic for an ontology other than logical atomism is to give an ontological account of the “cutdown” on the notion of “all possible worlds” and of the accessibility relation between possible worlds. 39 In Everett’s original version of the axioms of MWI no account was given of how the branching into different parallel worlds takes place. Later proposals by Graham and DeWitt introduce the complicated notion of a measuring device that results in observations (by humans or automata) upon which the splitting into parallel worlds is based. See De Witt & Graham [1973]. 40 Penrose [2004], p. 784. 41 Penrose [2004], p.784. 42 The modal logic is S4 because the accessibility relation between possible worlds is a partial ordering determined by the wavefunctions that split each universe into its related parallel universes. The result in effect is a branched-tree model of the universe something like the semantics for S4 in terms of the causal-tense operators described in chapter 2, §10. 43 For a fuller account of MWI see De Witt & Graham [1973].
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We have already indicated how such an account can be given for conceptual realism in terms of time and also, as described above, in terms of a cosmological model of the multiverse. In a later chapter we will base a modal logic of a natural or causal necessity on the laws of nature or the causal mechanisms of a hierarchically structured universe of the natural kinds that make up the causal nexus of the world. The question is: can this really be done other than in the modalities constructible within tense logic on the basis of time or the causal or natural modalities of a cosmological model of the multiverse?
3.7
Summary and Concluding Remarks
• Logical atomism provides a paradigmatic example of a formal ontology in which being is the same as existence. • Logical atomism is also the paradigm of a formal ontology in which a strictly formal interpretation of logical necessity and possibility can be given. This is because it is only in an ontology of simple objects and simple properties and relations as the bases of logically independent atomic states of affairs that an absolute totality of possible worlds is uniquely determined; and it is only with respect to this totality, as opposed to arbitrary sets of possible worlds, that logical necessity and possibility can be made sense of as modalities. • The modal thesis of anti-essentialism is logically true in the formal ontology of logical atomism. • All de re logical modalities are reducible to de dicto logical modalities in the formal ontology of logical atomism. • Every logically necessary proposition is logically true and therefore a priori in logical atomism, and every a posteriori proposition is logically contingent. • If there are only properties, and hence only monadic states of affairs, then the formal ontology of logical atomism is both complete and decidable. • But if there is at least one relation and infinitely many objects, then logical atomism is essentially incomplete. • A completeness theorem is forthcoming but only by allowing for arbitrary “cut-downs” on the totality of possible worlds. • Allowing for arbitrary “cut-downs” on the totality of possible worlds introduces material content into the modalities of a formal ontology, which is antithetical to logical atomism, and results in something other than logical necessity and possibility. • The modal thesis of anti-essentialism is consistent, but not universally valid, in a semantics based on a “cut-down” to arbitrary non-empty sets of possible worlds. • A “cut-down” on the set of possible worlds can be accounted for in terms of time, as in tense logic, where an ontological account can also be given of the accessibility relation beween worlds as momentary states of a local time. • A “cut-down” on the set of possible worlds can also be accounted for in terms of the network of natural laws that determine nature’s causal matrix.
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79
But in that case we are no longer dealing with logical necessity and possibility but a causal necessity and possibility; and of course not every proposition that is causally necessary is a priori, nor is every a posteriori proposition causally contingent. • There are alternative theories of a multiverse of possible worlds in which a “cut-down” on the totality of possible worlds can be ontologically grounded: one, e.g., is the concordance model, which validates the laws of S5 modal logic, and another is the many-world interpretation of quantum mechanics, which validates the laws of S4 modal logic.
Chapter 4
Formal Theories of Predication A formal ontology, we have said, is based on a theory of predication and not on set theory as a theory of membership. Set theory, of course, might be used as a model-theoretic guide in the construction of a theory of predication; but such a guide can be misleading if we confuse membership with predication. Unlike set theory, a theory of predication depends on what theory of universals it is designed to represent, where, by a universal we mean that which can be predicated of things. A universal is not just an abstract entity such as a set, in other words, but something that has a predicative nature, which sets do not have. Our methodology in representing a theory of predication is to reconstruct it as a second-order predicate logic that includes the salient features of that theory. Now by a second-order predicate logic we mean an extension of firstorder predicate logic in which quantifiers are allowed to reach into the positions that predicates occupy in formulas (sentence forms), as well as into the subject or argument positions of those predicates.1 This means that just as the quantifiers of first-order logic are indexed by object variables, which are said to be bound by those quantifiers, so too the quantifiers of second-order logic are indexed by predicate variables, which are said to be bound by those quantifiers. In this respect we need only add n-place predicate variables, for each natural number n, to our syntax for first-order logic. We will use for this purpose the capital letters F n , Gn , H n , with or without numerical subscripts, as n-place predicate variables; but we will generally drop the superscript when the context makes clear the degree of the predicate variable. Sometimes, for relational predicates, i.e., where n > 1, we will use R and S as relational predicate variables as well. We still understand a formal language L to be a set of object and predicate constants. The atomic formulas of such a language L are defined in the same way 1 Here, by first-order predicate logic we mean the logic of possible objects described in chapter 2.
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as in first-order logic, except that now the predicate of an atomic formula might be a predicate variable instead of a predicate constant. Thus, the (second-order) formulas of a language L are understood to be defined as follows. Definition: χ is a (second-order) formula of a language L if, and only if, for all sets K, if (1) every atomic formula of L is in K, and (2) for all ϕ, ψ ∈ K, all object variables x, and, for each natural number n, all n-place predicate variables F n , ¬ϕ, (ϕ → ψ), (∀x)ϕ, and (∀F n )ϕ ∈ K . It is formally convenient, incidentally, to take propositional variables to be 0place predicate variables, and hence to allow for atomic formulas of the form F 0 . In other words, a propositional variable is a predicate variable that takes zero many terms as arguments to result on an atomic formula. We will generally use the capital letters P , Q, with or without numerical subscripts as propositional variables. A principle of induction over second-order formulas follows of course just as it did in first-order logic. By way of axioms, we add the following to the ten we already gave for standard first-order logic. These axioms are the distribution law for predicate quantifiers, and the law of vacuous quantification: (A11)
(∀F n )[ϕ → ψ] → [(∀x)ϕ → (∀x)ψ]
(A12)
ϕ → (∀F n )ϕ,
where F n is not free in ϕ
We also add the inference rule of universal generalization for predicate quantifiers: UG2 :
4.1
If ϕ, then (∀F n )ϕ.
Logical Realism
The axioms described so far apply to nominalism and conceptualism, as well as to logical realism. What distinguishes logical realism is an axiom schema that we call an impredicative comprehension principle, (CP). This principle is stated as follows: (∃F n )(∀x1 )...(∀xn )[F (x1 , ..., xn ) ↔ ϕ], (CP) where ϕ is a (second-order) formula in which F n does not occur free, and x1 , ..., xn are pairwise distinct object variables occurring free in ϕ. What the comprehension principle does in logical realism is posit the existence of a universal corresponding to every second-order formula ϕ. This is so even when ϕ is a contradictory formula, e.g., ¬[G(x) → G(x)]. This is because, contrary to what has sometimes been held in the history of philosophy, there are properties and relations, that on logical grounds alone cannot have any instances. Such properties and relations cannot be excluded without seriously affecting the logic of logical realism. In particular, because we cannot effectively decide even when a first-order formula is contradictory, the logic would then not be recursively axiomatizable.
4.1. LOGICAL REALISM
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The comprehension principle, (CP), is said to be impredicative because it posits properties and relations in terms of totalities to which they belong. Formally, this is a consequence of the fact that even formulas with bound predicate quantifiers are allowed to specify, i.e., “comprehend”, properties and relations. Note that by a simple inductive argument on the structure of the formula ϕ, a second-order analogue of Leibniz’s law is provable independently of (CP): (∀x1 )...(∀xn )[F (x1 , ..., xn ) ↔ ϕ] → (ψ ↔ ψ[ϕ/F (x1 , ..., xn )]), where ψ[ϕ/F (x1 , ..., xn )] is the result of properly substituting ϕ for the free occurrences of F in ψ with respect to the object variables x1 , ..., xn .2 From this, by UG2 , axioms (A11), (A12) and elementary transformations, what follows as a theorem schema is the law of universal instantiation of formulas for predicate variables: (∀F )ψ → ψ[ϕ/F (x1 , ..., xn )], (UI2 ) where ϕ can be properly substituted for F in ψ. The contrapositive of (UI2 ) is of course the second-order law for existential generalization, ψ[ϕ/F (x1 , ..., xn )] → (∃F )ψ.
(EG2 )
The second-order predicate logic described so far, where the comprehension principle (CP) is valid for any formula ϕ, is sometimes called “standard” second-order logic. The use of “standard” in this context should not be confused with the notion of a “standard” set-theoretic semantics for second-order logic, i.e., a semantics based on confusing predication with membership in a set D, where all sets of n-tuples drawn from the power-set of Dn are taken as the values of the n-place predicate variables.3 Second-order predicate logic, it is well-known, is essentially incomplete with respect to this so-called “standard set-theoretic semantics”. But, as we have already noted (in chapter one), whether or not that incompleteness applies to the theory of universals that is the basis of a second-order predicate logic is another matter altogether. Now it is noteworthy that had we assumed (UI2 ) as an axiom schema instead of (CP), then, by (EG2 ), the comprehension principle (CP) would be derivable as a theorem schema instead. This raises the question of whether or not there are any reasons to prefer (CP) over (UI2 ), or (UI2 ) over (CP). 2 If x , ...x are pairwise distinct object variables, then ψ[ϕ/F (x , ..., x )] is just ψ unless n n 1 1 the following two conditions are satisfied: (1) no free occurrence of F n in ψ occurs within a subformula of ψ of the form (∀a)χ, where a is a predicate or object variable other than x1 , ...xn that occurs free in ϕ; and (2) for all terms a1 , ..., an , if F (a1 , ..., an ) occurs in ψ in such a way that the occurrence of F n in question is a free occurrence, then for each i such that 1 ≤ i ≤ n, if ai is a variable, then there is no subformula of ϕ of the form (∀ai )χ in which the variable xi has a free occurrence. If these two conditions are satisfied, then ψ[ϕ/F (x1 , ..., xn )] is the result of replacing, for all terms a1 , ..., an , each occurrence of F (a1 , ..., an ) in ψ at which F n is free by an occurrence of ϕ(a1 /x1 , ..., an /xn ). When conditions (1) and (2) hold, we say that ϕ can be properly substituted for F n in ψ. 3 By D n we understand the set of n-tuples drawn from D. In the so-called “standard” set-theoretic semantics for second-order logic, the power-set of D n is taken as the range of values assigned to n-place predicate variables. The set theory involved here is assumed to be based on the iterative concept of set and Cantor’s power-set theorem.
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One reason to prefer (UI2 ) over (CP) is that whereas (CP) is existential in form (UI2 ) is universal. Gottlob Frege, who was the first to formulate a version of second-order logic, argued that the laws of logic should be universal in form, which is why he had a version of (UI2 ) as one of his basic laws. Bertrand Russell assumed a version of (EG2 ), the contrapositve of (UI2 ), as a “primitive proposition” of his higher-order logic. This principle, according to Russell, “gives the only method of proving ‘existence theorems’”4 , which suggests that Russell did not think that “existence theorems” of the form (CP) were provable in his logic. It is elegant, perhaps, to have the basic laws of logic all be universal in form, but there is something to be said for putting our existential posits up front, and that is precisely what (CP) does—just as the related axiom (A8), (∃x)(a = x), for first-order logic puts our our existential presuppositions up front for objectual terms. The latter, i.e., axiom (A8), may in fact be rejected, as we will see, once predicates are allowed to be transformed into objectual terms as counterparts of abstract nouns. In other words, in a somewhat larger ontological context that we will consider later, where complex objectual terms are allowed, certain of these complex objectual terms will lead to a contradiction if axiom (A8), which is the first-order counterpart of (CP), is not modified along the lines of “free logic”. Can (CP) also be rejected in that larger ontological context? No, at least not in logical realism. In the ontology of natural realism, however, the assumption that a natural property or relation corresponds to any given predicate or formula is at best an empirical hypothesis. That is, just as whether or not an objectual term denotes an object in free logic is not a question that can be settled by logical considerations alone, so too in natural realism the question whether or not a given predicate or open formula stands for a natural property or relation cannot be settled by logic alone. The comprehension principle (CP), in other words, is not a valid thesis in natural realism. We will forego giving a fuller analysis of natural realism at this point, however, until a later lecture when we consider conceptual natural realism. The status of the comprehension principle, (CP), as these remarks indicate, is an important part of the question of what metaphysical theory of universals is being assumed as the basis of our logic as a formal theory of predication.
4.2
Nominalism
In nominalism, the basic thesis is that there are no universals, and that there is only predication in language. This suggests that the comprehension principle (CP) must be false in nominalism, which is why the formal theory of predication that is commonly associated with nominalism is standard first order predicate logic with identity. In fact, however, the situation is a bit more complicated than that. 4 Russell
& Whitehead 1910, p. 131.
4.2. NOMINALISM
85
It is true that according to nominalism first-order predicate logic gives a logically perspicuous representation of the predicative nature of the predicate expressions of language. According to nominalism it is the logico-grammatical roles that predicates have in the logical forms of first-order predicate logic that explains their predicative nature.5 Nominalism: The logico-grammatical roles that predicate expressions have in the logical forms of first-order predicate logic explains their predicative nature. Predicate constants are of course assigned the paradigmatic roles in this explanation, but this does not mean that predicate constants are the only predicative expressions that must be accounted for in nominalism. In particular, any open first-order formula of a formal language L, relative to the free object variables occurring in that formula, can be used to define a new predicate constant of L.6 Such an open formula would constitute the definiens of a possible definition for a predicate constant not already in that language. Accordingly, an open formula must be understood as implicitly representing a predicate expression of that formal language. Potentially, of course there are infinitely many such predicate constants that might be introduced into a formal language in this way, and some account must be given in nominalism of their predicative role. Question: how can nominalism, as a theory of predication, represent the predicative role of open first-order formulas. Now an account is forthcoming by extending standard first order predicate logic to a second order predicate logic in which predicate quantifiers are interpreted substitutionally. That is, we can account for all of the nominalistically acceptable predicative expressions of an applied first-order language without actually introducing new predicate constants by simply turning to a second order predicate logic in which predicate quantifiers are interpreted substitutionally and where predicate variables have only first-order formulas as their substituents.7 There are constraints that such an interpretation imposes, of course, and in fact, as we have shown elsewhere, those constraints are precisely those imposed on the comprehension principle in standard “predicative” second-order logic.8 Here, it should be noted, the use of the word ‘predicative’ is based on Bertrand Russell’s terminology in Principia Mathematica, where the restriction in question was a part of his theory of ramified types. Apparently, because of 5 Strictly speaking, nominalism deals primarily with predicate tokens, written or spoken, and not with predicates per se. The latter, however can be construed as classes as many of predicates of similar tokens, where classes as many are as described in chapter 11. We will ignore that further complication of nominalism here. 6 By an open formula we mean a formula in which some variables have a free occurrence. 7 Strictly speaking predicate variables will have as subtituends any formula in which no bound predicate variable occurs, which means that free predicate variables are allowed to occur in such substituends. 8 See Cocchiarella 1980.
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the liar and other semantical paradoxes, Russell, despite his being a logical realist at the time, thought that only the so-called “predicative” formulas should be taken as representing a property or relation.9 The restriction, simply put, is that no formula containing bound predicate variables is to be allowed in the comprehension principle. The comprehension principle (CP), in other words, is to be restricted as follows: (∃F n )(∀x1 )...(∀xn )[F (x1 , ..., xn ) ↔ ϕ],
(CP!)
where ϕ is a formula in which (1) no predicate variable has a bound occurrence, (2) F n does not occur free in ϕ, and (3) x1 , ..., xn are pairwise distinct object variables occurring free in ϕ.10 Under a substitutional interpretational the appearance of an existential posit regarding the existence of a universal in the quantifier prefix (∃F n ) is just that, an appearance and nothing more. In an applied formal language, this principle involves no ontological commitments under such an interpretation beyond those one is already committed to in the use of the first-order formulas of that language. That is, by interpreting predicate quantifiers substitutionally, (CP!) will not commit us ontologically to anything we are not already committed to in our use of first-order formulas, and, as we have said, it is the logico-grammatical role of predicate expressions in first-order logic that is the basis of nominalism’s theory of predication. Note that because the second-order analogue of Leibniz’s law is provable independently of (CP), it then is provable in our nominalistic logic just as it was in the logic for logical realism. But then, just as the universal instantiation law, (UI2 ), is provable in logical realism, we have a restricted version also provable in nominalism. That is, if no predicate variable has a bound occurrence in ϕ, then, by (CP!), the following is provable in nominalism, (∀F )ψ → ψ[ϕ/F ],
(UI!2 )
where ϕ can be properly substituted for F in ψ. From (UI!2 ), we can then derive the restricted version of existential generalization for predicates. What these observations indicate is that a comprehension principle can be used to make explicit what is definable in a given applied language, as well as to indicate, as in logical realism, what our existential posits are regarding universals. 9 Russell’s logical realism is most pronounced in his 1903 Principles of Mathematics. His later 1910 view in Principia Mathematica might more appropriately be described as a form of conceptual Platonism, even though Russell never explicitly described himself this way. From 1914 on, especially in his logical atomist phase, Russell is best described as a natural realist. See Cocchiarella 1991 for a description of Russell’s higher-order logic as a form of conceptual Platonism. 10 Russell used the exclamation mark as a way to indicate which formulas of his type theory were “predicative”. We use it here in the metalanguage as a way to indicate the relevant restriction on the comprehension principle.
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Thus, where L is a formal language, P n is an n-place predicate constant not in L, and ψ is a first-order formula in which x1 , ..., xn are all of the pairwise distinct object variables occurring free, then (∀x1 )...(∀xn )[P (x1 , ..., xn ) ↔ ψ] is said to be a possible definition in L of P . Now it is just such a possible definition that is posited in (CP!2 ). In other words, as the definiens of such a possible definition, the first-order formula ψ is implicitly understood to be a complex predicate of the language L. Of course, if the above were an explicit definition in L, then, by (EG!2 ), the relevant instance of (CP!) follows as provable in L. The kind of definitions that are excluded in nominalism but allowed in logical realism are the so-called “impredicative” definitions; that is, those that in realist terms represent properties and relations that seem to presuppose a totality to which they belong. The definition of a least upper bound in real number theory is such a definition, for example, because, by definition, a least upper bound of a set of real numbers, is one of the upper bounds in that set.11 The exclusion of impredicative definitions is sometimes called the Poicar´e-Russell vicious-circle principle, because Henri and Bertrand Russell were the first to recognize and characterize such a principle.12
4.3
Constructive Conceptualism
The notion of an “impredicative” definition is important in conceptualism as well as in nominalism, and it is basic to an important stage of concept-formation. Conceptualism, as we have noted, differs from nominalism in that it assumes that there are universals, namely concepts, that are the semantic grounds for the correct application of predicate expressions. Of course, conceptualism also differs from realism in that concepts are not assumed to exist independently of the human capacity for thought and concept-formation. Conceptualism is a sociobiologically based theory of the human capacity for thought and concept-formation, and, more to the point, systematic conceptformation. Concepts themselves are types of cognitive capacities, and it is their exercise as such that underlies the speech and mental acts that constitutes our thoughts and communications with one another. But thought and communication exist only as coordinated activities that are systematically related to one another through the logical operations of thought; and it is with respect to the idealized closure of these operations that concept-formation is said to be systematic. It is only as a result of this closure, moreover, that the unity of thought as a field of internal cognitive activity is possible. 11 Similarly, the definition of the set of natural numbers as the intersection of all sets to which 0 belongs and that are closed under the successor operator is also impredicative because it is defined in terms of a totality (proper class) to which it belongs. 12 Cf. Poincar´ e 1906 and Russell 1906.
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The coordination and closure of concepts does not occur all at once in the development of human thought, of course, nor is the structure of the closure the same at all stages in that development. In fact, the human child proceeds through stages of cognitive development that are of increasing structural complexity, corresponding in part to the increasing complexity of the child’s developing brain. These stages, as Jean Piaget has noted, emerge as states of cognitive equilibrium with respect to certain regulatory processes that are constitutive of systematic concept-formation.13 Different stages proceed as transformations from one state of cognitive equilibrium to another of increased structural complexity, where the need for such transformations arises from the child’s interaction with his environment and the tacit realization of the inadequacy of the earlier stages to understand certain aspects of the world of his experience. The later stages are states of “increasing re-equilibration” of the intellect, in other words, so that the result is an improved representation of the world.14 Now there is an important stage of cognitive equilibrium of logical operations that immediately precedes the construction of so-called “impredicative” concepts, which usually does not occur until post-adolescence. We refer to the logic of this stage as constructive conceptualism. The later, succeeding more mature stage at which “impredicative” concept-formation is realized is called holistic conceptualism, though we will generally refer to it later simply as conceptualism. It is in constructive conceptualism that “impredicative” definitions are excluded, and this exclusion occurs in the form that the comprehension principle takes in constructive conceptualism, which can be formally described as follows: (∀G1 )...(∀Gk )(∃F )(∀x1 )...(∀xn )[F (x1 , ..., xn ) ↔ ϕ],
(CCP!)
where (1) ϕ is a pure second-order formula, i.e., one in which no nonlogical constants occur, (2) F is an n-place predicate variable such that neither it nor the identity sign occur in ϕ, (3) ϕ is “predicative” in nominalism’s purely grammatical sense, i.e., no predicate variable has a bound occurrence in ϕ, (4) G1 , ..., Gk are all of the distinct predicate variables occurring (free) in ϕ, and (5) x1 , ..., xn are pairwise distinct object variables. Now we should note that by the rule for universal generalization of quantifiers, (UG), every instance of the conceptualist principle (CCP!) is derivable from the nominalist principle (CP!). But not every instance of the nominalist principle (CP!) is an instance of the conceptualist principle (CCP!). In an applied formal language L, the formula ϕ in instances of (CP!) will in general 13 Cp.
Piaget 1977. p. 13 and §6 of chapter 1,
14 Ibid.,
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be a first-order formula of L, and it may contain the identity sign and any predicate constant of L. Instances of (CCP!), on the other hand, contain neither the identity sign nor any predicate constants of L. Predicate constants are excluded from instances of (CCP!) because, unlike the situation in nominalism, a predicate constant (or first-order formula in terms of which such a constant might be defined) might not stand for a “predicative” concept, i.e., it might not stand for a value of the bound predicate variables. This is because the logic of predicate quantifiers in constructive conceptualism is like the logic of first-order quantifiers in free logic in that the logic is free of existential presuppositions regarding predicate constants, which means that a predicate constant must stand for a “predicative” concept in order to be a substituend of the bound predicate variables of the logic. The predicate quantifiers in nominalism, on the other hand, function like the objectual quantifiers of standard first-order logic; and that is because, as the paradigms of predication in nominalism, predicate constants do not differ from one another in their predicative role, which is why, under a substitutional interpretation, all predicate constants are substituends of the bound predicate variables. Consider, for example, a language L containing ‘∈’ as a primitive two-place predicate constant, and suppose we formulate a theory of membership in L with the following as a second-order axiom: (∀F )(∃y)(∀x)[x ∈ y ↔ F (x)].
(C)
Now, in nominalism, where predicate quantifiers are interpreted substitutionally, this axiom seems quite plausible as a thesis, stipulating in effect that every predicate expression has an extension. But as plausible a thesis as that might be, it leads directly to Russell’s paradox. For, by the nominalist comprehension principle, (CP!), (∃F )(∀x)[F (x) ↔ x ∈ / x], (D) is provable under such an interpretation; and, because no predicate quantifier occurs in x ∈ / x, then, by (UI!2 ), x ∈ / x represents a predicate expression that can be properly substituted for F in a universal instantiation of (C). In constructive conceptualism, on the other hand, (D) is not an instance of the conceptualist principle, (CCP!), and all that follows by Russell’s argument from (C) is the fact that ‘∈’ cannot stand for a “predicative” (relational) concept. That is, instead of the contradiction that results when predicate quantifiers are interpreted substitutionally, (C), when taken as an axiom of a theory of membership in constructive conceptualism, leads only to the result that the membership predicate does not stand for a “predicative” (relational) concept: ¬(∃R)(∀x)(∀y)[R(x, y) ↔ x ∈ y]. In other words, as a plausible thesis to the effect that every “predicative” concept has an extension, (C) is consistent, not inconsistent, in constructive conceptualism.
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On the nominalist strategy, the notion of a “predicative” context is purely grammatical in terms of logical syntax; that is, an open formula is “predicative” in nominalism just in case it contains no bound predicate variables. In constructive conceptualism, the notion of a “predicative” context is semantical, which means that in addition to being “predicative” in nominalism’s purely grammatical sense, it must also stand for a “predicative” concept. It is for this reason that the second-order logic of constructive conceptualism must be free of existential presuppositions regarding predicate constants, which is why the binary predicate, ‘∈’, in a theory of membership having (C) as an axiom, cannot stand for a value of the bound predicate variables. In general, how we determine which, if any, of the primitive predicate constants of an applied language and theory stand for a “predicative” concept depends on the domain of discourse of that language and theory and how that domain is to be conceptually represented. In particular, those primitive predicates that are to be taken as standing for a “predicative” concept will be stipulated as doing so in terms of the “meaning postulates” of that theory, whereas those that are not will usually occur in axioms that determine that fact. Identity and its role in a logical theory marks another important difference between nominalism and constructive conceptualism. In nominalism, identity is definable in any applied language with finitely many predicate constants. This is because such a definition can be given in terms of a formula representing indiscernibility with respect to those predicate constants. Suppose, for example L is a language with two just two predicate constants, a one-place predicate constant P , and a two-place predicate constant R. Then, identity can be defined in theories formulated in terms of L as follows: a = b ↔ [P (a) ↔ P (b)] ∧ [R(a, a) ↔ R(b, a)] ∧ [R(a, b) ↔ R(b, b)]∧ [R(a, a) ↔ R(a, b)] ∧ [R(b, a) ↔ R(b, b)] In other words, in any given application based on finitely many predicate constants, which we may assume to be the standard situation, identity, in nominalism, is reducible to a first-order formula, which is why the identity sign is allowed to occur in instances of (CP!) under its nominalistic, substitutional interpretation. Such a definition will not suffice in constructive conceptualism, on the other hand, because the first-order formula in question, even were it to stand for a “predicative” concept, cannot justify the substitutivity of identicals in nonpredicative contexts. The identity sign is not eliminable, or otherwise reducible, in constructive conceptualism, in other words, because, on the basis of Leibniz’s law, identity must allow for full substitutivity even in nonpredicative contexts. Thus, whereas, x = y ↔ (∀F )[F (x) ↔ F (y)], is provable in nominalism’s second-order logic, as based on its substitutional interpretation, the right-to-left direction of this same formula is not provable in the logic of constructive conceptualism. Finally, note that although “impredicative” definitions are not allowed in nominalism, they are not precluded in the logic of constructive conceptualism.
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The difference is determined by the role free predicate variables have in each of these frameworks. In nominalism, free predicate variables must be construed as dummy schema letters, which in an applied language stand for arbitrary first-order formulas of that theory. This means that the substitution rule, if ψ, then ψ[ϕ/G(x1 , ..., xn )], is valid on the substitutional interpretation only when ϕ is “predicative” in nominalism’s purely grammatical sense, i.e., only when no predicate variable has a bound occurrence in ϕ. Indeed, the rule must be restricted in this way because, otherwise, by taking ψ to be the following instance of (CP!), (∃F )(∀x1 )...(∀xn )[F (x1 , ..., xn ) ↔ G(x1 , ..., xn )], we would be able to derive the full, unrestricted impredicative comprehension principle, (CP), by substituting ϕ for G, and, thereby, transcend the substitutional interpretation of predicate quantifiers. In the predicate logic of constructive conceptualism, on the other hand, the above substitution rule is valid for all formulas, regardless whether or not they contain any bound predicate variables.15 But, unlike the situation in nominalism, the validity of such a rule does not lead to the unrestricted impredicative comprehension principle. In particular, the above instance of (CP!) is not also an instance of (CCP!). This means that the notion of a possible (explicit) definition of a predicate constant is broader in constructive conceptualism than it is in nominalism, where only first-order formulas are allowed as definiens. Nevertheless, on the basis of the rule of substitution for all formulas, it can be shown that definitions in constructive conceptualism whose definiens contain a bound predicate variable will still be noncreative and will still allow for the eliminability of defined predicate constants.16 Thus, even though constructive conceptualism validates only a “predicative” comprehension principle, i.e., a comprehension principle encompassing laws of compositionality that are in accordance with the vicious circle principle, it nevertheless allows for impredicative definitions of predicate constants, that is, of predicate constants that do not stand for values of the bound predicate variables and that cannot therefore be existentially generalized upon.
4.4
Ramification and Holistic Conceptualism
The difference between nominalism and constructive conceptualism, we have said, is similar to that between standard first-order logic and first-order logic free of existential presuppositions regarding objectual terms. The freedom from such presuppositions for predicates in constructive conceptualism indicates how 15 A similar substitution rule for singular terms is also valid in free logic incidentally, i.e., singular terms can be validly substituted for free object variables even though they cannot be validly substituted for bound object variables. 16 See Cocchiarella 1986a, chap. 2, sec. 3, for a proof of this claim.
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concept-formation is essentially an open process, and that part of that process is a certain cognitive tension, or disequilibrium, between the predicates and formulas that stand for concepts at a given stage of concept-formation and those that do not. This disequilibrium in concept-formation is the real driving force of what is known as “ramified” second-order logic, though, strictly speaking, ramified second-order logic is usually associated with nominalism and not with conceptualism.17 We can close the “gap” between predicates that stand for “predicative” concepts at a given stage of concept-formation and those that do not by introducing new predicate quantifiers in addition to the original ones. These predicate quantifiers will still be within the confines of a restricted, constructive comprehension principle. That is, a new comprehension principle would be added that allowed formulas containing predicate variables bound by the original predicate quantifiers, but not also formulas containing predicate variables bound by the new predicate quantifiers. This will close the “gap” between those formulas that stand for “predicative” concepts at the initial stage and those that do not, because the latter now stand for “predicative” concepts at the new, second stage. Of course in proceeding in this way we open up a new “gap” between the formulas that stand for “predicative” concepts at the new stage of conceptformation and those that do not. But then we can go on to close this new “gap” by introducing predicate quantifiers that are new to this stage, along with a similar comprehension principle. This process that continues on in this way is what is known as “ramification”. Formally, the process can be described in terms of a potentially infinite sequence of predicate quantifiers ∀1 , ∃1 , ..., ∀j , ∃j , ... (for each positive integer j), all of which can be affixed to the same predicate variables. The quantifiers (∀j F ) and (∃j F ), where F is an n-place predicate variable, will then be understood to refer to all, or some, respectively, of the n-ary “‘predicative”’ concepts that can be formed at the jth stage of the potentially infinite sequence of stages of concept-formation in question. But because open formulas representing “predicative” contexts of later stages will not be substituends of predicate variables bound by quantifiers of an earlier stage, this means that the logic of the quantifiers ∀j and ∃j must be free of existential presuppositions regarding predicate expressions, which is why the comprehension principle for this logic must be closed with respect to all the predicate variables occurring free in the comprehending formula. Thus, as applied at the jth stage, the ramified conceptualist comprehension principle that is validated in this framework is the following: (∀j G1 )...(∀j Gk )(∃j F )(∀x1 )...(∀xn )[F (x1 , ..., xn ) ↔ ϕ],
(RCCP!)
where (1) G1 , ..., Gk are all of the predicate variables occurring free in ϕ; (2) F is an n-place predicate variable not occurring free in ϕ; 17 This is because standard “predicative” logic has been associated with nominalism, and standard ramified logic is an extension of standard “predicative” logic, and not of the free “predicative” logic of constructive conceptualism.
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(3) x1 , ..., xn are distinct individual variables; and (4) ϕ is a pure ramified formula, i.e., one in which no nonlogical constants occur and in which (a) the identity sign also does not occur and (b) in which no predicate variable is bound by a quantifier of a stage > j, i.e., for all i ≥ j, neither ∀j nor ∃j occurs in ϕ. The process of concept-formation that we are describing here amounts to a type of reflective abstraction that involves a projection of previously constructed concepts onto a new plane of thought where they are reorganized under the closure conditions of new laws of concept-formation characteristic of the new stage in question. This pattern of reflective abstraction is precisely what is represented by the ramified comprehension principle (RCCP!) and the logic of constructive conceptualism. Each successive stage of concept-formation in the ramified hierarchy is generated by a disequilibrium, or conceptual tension, between the formulas that stand for the “predicative” concepts of the preceding stage, as opposed to those that do not. Thus, in order to close the “gap” between formulas that stand for “predicative” concepts and those do not, we must proceed through a potentially infinite sequence of stages of concept-formation.18 Whatever the motivation for ramification in nominalism, it is clear that what moves us on from one stage of concept-formation to the next in constructive conceptualism is a drive for closure, where all predicates stand for concepts. Such a closure cannot be realized in constructive conceptualism, of course, where the principal constraint guiding the formation of “predicative” concepts is their being specifiable by conditions that are in accord with the so-called vicious circle principle. But the particular pattern of reflective abstraction that corresponds to this constraint is not all there is to concept-formation, and, in fact, as a pattern that represents a drive for closure, it contains the seeds of its own transcendence to a new plane or level of thought where such closure is achieved. Concept-formation is not constrained by the vicious circle principle, in other words, because after reaching what Piaget calls the stage of formal operational thought, certain new patterns of concept-formation are realizable, albeit usually only in post-adolescence.19 One such pattern involves an idealized transition to a limit, where “impredicative” concept-formation becomes possible, i.e., where the restrictions imposed by the vicious-circle principle are transcended. The idealized transition to a limit, in the case of our ramified logic, is conceptually similar to, but ontologically different from, an actual transition to a limit at an infinite stage of concept-formation. This is a stage of concept-formation that, in effect, is not only the summation of all of the finite stages of the ramified hierarchy but also one that is closed with respect to itself. 18 See Cocchiarella 1986 for a more detailed discussion of this principle in constructive conceptualism. 19 Cf. Piaget 1977.
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Ontologically, of course, there cannot be an infinite stage of concept-formation, but that is not to say that an idealized transition to a limit is conceptually impossible as well. Indeed, such an idealized transition to a limit is precisely what is assumed to be possible in holistic conceptualism, and it is possible, moreover, on the basis of the pattern of reflective abstraction represented by (RCCP!*). That is, in holistic conceptualism, the drive for closure upon which the pattern of reflective abstraction of ramified constructive conceptualism is based is finally achieved, albeit only as the result of an idealized transition to a limit and not on the basis of an actual transition. In this way, conceptualism, by means of a mechanism of autoregulation that enables us to construct stronger and more complex logical systems out of weaker ones, is able to validate not only the ramified conceptualist comprehension principle but also the full, unqualified “impredicative” comprehension principle (CP) of “standard” second-order logic. There is no comparable mechanism in nominalism that can similarly lead to a validation of the impredicative comprehension principle (CP). What is inadequate about the logic of constructive conceptualism, it is important to note, is that it cannot provide an account of the kind of impredicative concept-formation that is necessary for the development and use of the theory of real numbers, and which, as a matter of cultural history, we have in fact achieved since the nineteenth century. The concept of a least upper bound, for example, or of the limit of a converging sequence of rational numbers, is an impredicative concept that was not acquired by the mathematical community until a little more than a century ago; and although, in our own time, it is not usually a part of a person’s conceptual repertoire until post-adolescence, nevertheless, with proper training and conceptual development, it is a concept that most of us can come to acquire as a cognitive capacity. Yet, notwithstanding these facts of cultural history and conceptual development, it is also a concept that cannot be accounted for from within the framework of constructive conceptualism. The constraints of the vicious circle principle, at least in the way they apply to concept-formation, simply do not conform to the facts of conceptual development in an age of advanced scientific knowledge. The validation of the full comprehension principle in holistic conceptualism, which we will hereafter refer to simply as conceptualism, does not mean that the logic of constructive conceptualism is no longer a useful part of conceptualism. What it does mean is that although all predicates stand for “predicable” concepts, not all predicates stand for “predicative” concepts, and that is a distinction we can turn to in conceptualism whenever it is relevant and useful.
4.5
The Logic of Nominalized Predicates
Is there no difference then between logical realism and holistic conceptualism as theories of predication, other than the fact that the latter presupposes a logic
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of “predicative” concepts as a proper part? Well, in fact there is a difference once we consider the import of nominalized predicates and propositional forms as abstract singular terms in the wider context of modal predicate logic. The use of nominalized predicates as abstract singular terms is not only a part of our commonsense framework, but it is also central to how both logical realism and conceptual intensional realism provide an ontological foundation for the natural numbers and other parts of mathematics. This part of logical realism is sometimes called ontological logicism. In Bertrand Russell’s form of logical realism, or ontological logicism, for example, universals are not just what predicates stand for, but also what nominalized predicates, i.e., abstract nouns, denote as objectual terms.20 Here, by nominalization we mean the transformation of a predicate phrase into an abstract noun, which is represented in logical syntax as a objectual term, i.e., the type of expression that can be substituted for first-order object variables. The following are some examples of predicate nominalizations:
is triangular is wise is just
triangularity wisdom justice
It was Plato who first recognized the ontological significance of such a transformation and who built his ontology and his account of predication around it. In nominalism, of course, abstract nouns denote nothing. In English we usually mark the transformation of a predicate into an abstract noun by adding such suffixes as ‘-ity’, ‘-ness’, or ‘hood’, as with ‘triangularity’, ‘redness’, and ‘brotherhood’. We do not need to introduce a special operator for this purpose in logical syntax, however. Rather, we need only delete the parentheses that are a part of a predicate variable or constant in its predicative role. Thus, for a monadic predicate F we would have not only formulas such as F (x), where F occurs in its predicative role, but also formulas such as G(F ), R(x, F ), where F occurs nominalized as an abstract objectual term. Note: In F (F ) and ¬F (F ), F occurs both in it predicative role and as an abstract objectual term, though in no single occurrence can it occur both as a predicate and as a objectual term. With nominalized predicates as abstract objectual terms, it is convenient to have complex predicates represented directly by using Alonzo Church’s variablebinding λ-operator. Thus, where ϕ is a formula of whatever complexity and n is a natural number, we have a complex predicate of the form [λx1 ...xn ϕ]( ), which has parentheses accompanying it in its predicative role, but which are deleted when the complex predicate is nominalized. With λ-abstracts, the comprehension principle can be stated in a stronger and more natural form as (∃F )([λx1 ...xn ϕ] = F ). 20 Cf.
Russell 1903, p. 43.
(CP∗λ )
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This form is stronger than (CP) in that it implies, but is not implied by, the latter.21 One of the rules for the new λ-operator is the rule of λ-conversion, [λx1 ...xn ϕ](a1 , ..., an ) ↔ ϕ[a1 /x1 , ..., an /xn ]
(λ-Conv∗ )
The grammar of our logical syntax is now more complicated of course. In particular, objectual terms and formulas must now be defined simultaneously. For this purpose we will speak of a meaning expression of a given type n, where n is a natural number. We will use 0 to represent the type of objectual terms (or just ‘terms’ for short), 1 to represent the type of formulas (propositional forms), and n + 1 to represent the type of n-place predicate expressions. For each natural number n, we recursively define the meaningful expressions of type n, in symbols, MEn , as follows: 1. every individual variable (or constant) is in ME0 , and every n-place predicate variable (or constant) is in both MEn+1 and ME0 ; 2. if a, b ∈ ME0 , then (a = b) ∈ ME1 ; 3. if π ∈ MEn+1 and a1 , ..., an ∈ ME0 , then π(a1 , ..., an ) ∈ ME1 ;22 4. if ϕ ∈ ME1 and x1 , ..., xn are pairwise distinct individual variables, then [λx1 ...xn ϕ] ∈ MEn+1 ; 5. if ϕ ∈ ME1 , then ¬ϕ ∈ ME1 ; 6. if ϕ, χ ∈ ME1 , then (ϕ → χ) ∈ ME1 ; 7. if ϕ ∈ ME1 and a is an individual or a predicate variable, then (∀a)ϕ ∈ ME1 ; 8. if ϕ ∈ ME1 , then [λϕ] ∈ ME0 ; and 9. if n > 1, then MEn ⊆ ME0 . By clause (9), every predicate expression without parentheses is a objectual term. This includes 0-place predicates but not formulas in general unless they are of the form [λϕ], which we take as the nominalization of ϕ, and which we read as ‘that ϕ’. For convenience, however, we shall write ‘[ϕ]’ for ‘[λϕ]’.23 It is noteworthy that this logical grammar contains what might described as the essential parts of a theory of logical form: namely, 21 The λ in (CP∗ ) indicates that a λ-abstract is part of this principle, and the ‘∗ ’ indicates λ that nominalized predicates may occur in ϕ as singular terms. 22 If n = 0, we take a , ..., a (and similarly x , ..., x ) to be the empty sequence, resulting n n 1 1 in this case in a 0-place predicate expression, which, as already noted, we take to be a formula. 23 We could require that n be greater than 1 in clause (4)—in which case [λϕ] ∈ ME would 1 not follow—and then have clause (8) state that [ϕ] ∈ ME0 when ϕ ∈ ME1 . But then general principles—such as (CP∗λ ) and (Ext∗ ) described below—that we want to apply to all n-place predicate expressions would have to be stated separately for n = 0.
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• (1) the basic forms of predication, as in F (x), R(x, y), etc.; • (2) propositional (sentential) connectives, e.g., ∧, ∨, →, and ↔; • (3) quantifiers that reach into predicate as well as subject (or argument) positions; • (4) nominalized predicates and propositional forms as abstract objectual terms. These four components correspond to fundamental features of natural language, and each needs to be accounted for in any theory of logical form underlying natural language. Now one of our goals here is to characterize a consistent second-order predicate logic with nominalized predicates and propositional forms as abstract objectual terms. This goal is important because such a logic deals with the four important features of natural language described above. Another goal is that as a framework for logical realism or (holistic) conceptualism such a logic should contain all of the theorems of “standard” second-order predicate logic as a proper part.24 This means in particular that we should retain all of the theorems of classical propositional logic, and that all instances of the comprehension principle (CP) of “standard” second-order logic—i.e., instances in which abstract objectual terms do not occur—should be provable. Initially, we will assume standard first-order predicate logic with identity as well; but, as we will see, it may be appropriate to assume “free” first-order predicate logic instead. With standard first-order predicate logic, we have by axiom (A8), (∃y)(F = y) as provable for every nominalized predicate F , and therefore also for λ-abstracts as well: (∃y)([λx1 ...xn ϕ] = y). Another consequence is that the first-order principle of universal instantiation now also applies to nominalized predicates as abstract objectual terms25 : (∀x)ϕ → ϕ[F/x]
(UI∗1 )
It would be ideal, of course, if the comprehension principle (CP∗λ ) can be assumed for all formulas ϕ, including those in which nominalized predicates and propositional forms occur as abstract objectual terms. But, if the logic is not “free of existential presuppositions” for objectual terms, such an unrestricted 24 By the valid formulas of “standard” second-order logic we mean all of the second-order formulas that are valid with respect to Henkin general models. 25 We use ‘(UI∗ )’ to label this principle, with the subscript indicated that it is a first-order 1 object quantifier thesis, and with the ‘∗ ’ to indicate that the principle applies to abstract singular terms as well object variables and constants.
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second-order logic—which is similar to the system of Gottlob Frege’s Grundgesetze 26 —is subject to Russell’s paradox of predication, and therefore cannot be assumed as a consistent principle. Thus, e.g., where ϕ represents the Russell property of being identical to a property that is not predicable of itself27 , which as a λ-abstract can be formalized as [λx(∃G)(x = G ∧ ¬G(x))], then, by the unrestricted comprehension principle (CP∗λ ), (∃F )([λx(∃G)(x = G ∧ ¬G(x))] = F )
(1)
is provable, and therefore, by Leibniz’s Law, (LL∗ ), so is the weaker form, (∃F )(∀x)[F (x) ↔ [λx(∃G)(x = G ∧ ¬G(x))](x)],
(2)
which, by λ-conversion, is equivalent to (∃F )(∀x)(F (x) ↔ (∃G)[x = G ∧ ¬G(x)]).
(3)
But by (UI∗1 ), (∀x)(F (x) ↔ (∃G)[x = G ∧ ¬G(x)]) → (F (F ) ↔ (∃G)[F = G ∧ ¬G(F )]), and, by (LL∗ ), (∃G)[F = G ∧ ¬G(F )] ↔ ¬F (F ) are also provable, and therefore, (∀x)(F (x) ↔ (∃G)[x = G ∧ ¬G(x)]) → [F (F ) ↔ ¬F (F )] is provable as well. But, by sentential logic, the consequent of this last conditional is clearly impossible, which means that ¬(∀x)(F (x) ↔ (∃G)[x = G ∧ ¬G(x)])
(4)
is provable, and therefore, by (UG2 ) and a quantifier negation law, ¬(∃F )(∀x)(F (x) ↔ (∃G)[x = G ∧ ¬G(x)]),
(5)
which contradicts (3) above. The above result is what is known as Russell’s paradox of predication. Russell himself later turned to his theory of ramified types to avoid the contradiction. Later, it was later pointed out that a theory of simple types sufficed, at least for the so-called logical paradoxes such as Russell’s. The idea of a hierarchy of types based on the fundamental asymmetry of subject and predicate is fundamentally correct, we maintain. But, as we will see, the idea can be simplified even further within a strictly second-order predicate logic with nominalized predicates as abstract objectual terms that is both consistent and equivalent to the simple theory of types. 26 Frege’s expressions for value-ranges (Wertverl¨ aufe) were his formal counterparts of predicate nominalizations, i.e., formal counterparts of expressions such as ‘the concept F ’. 27 See Russell 1903, p. 97.
4.6. SUMMARY AND CONCLUDING REMARKS
4.6
99
Summary and Concluding Remarks
• A formal ontology is based on a formal theory of predication, which in turn is based on a theory of universals. • The formal theories predication discussed in this chapter are logical realism (as a modern form of Platonism), nominalism, and conceptualism, where the latter is distinguished between a constructive conceptualism and a holistic conceptualism. • Which comprehension principle is validated in a formal theory of predication is one of its main distinguishing features. This is because a comprehension principle determines what is definable in a given applied language, and it also indicates, as in logical realism, what our existential posits are regarding universals. • Logical realism validates a full, unrestricted, impredicative comprehension principle. • Nominalism validates only a predicative comprehension principle, i.e., one in which only formulas free of bound predicate variables can occur as comprehending formulas. • Bound predicate variables can be interpreted only substitutionally in nominalism, which means that only first-order formulas (including those with free predicate variables) can be properly substituted for such variables. According to nominalism, it is the logico-grammatical roles that predicate expressions have in the logical forms of first-order predicate logic that explains their predicative nature. • Constructive conceptualism validates only a predicative comprehension principle but allows impredicative definitions. This is because, unlike nominalism, constructive conceptualism is free of “existential presuppositions” regarding free predicate constants (and free predicate variables). • Constructive conceptualism, unlike nominalism, also validates a ramified second-order predicate logic that is free of “existential presuppositions” regarding predicates. The freedom from such presuppositions indicates how conceptformation is essentially an open process. It is this open process and the conceptual tension it generates for closure that drives concept-formation through the ramified hierarchy. • The drive for closure in constructive conceptualism contains the seeds of its own transcendence to a new plane or level of thought where such closure is achieved. • The conceptualist closure of the ramified hierarchy is achieved through an idealized transition to a limit, where “impredicative” concept-formation becomes possible, and hence where the restrictions imposed by the vicious-circle principle are transcended. • The introduction of nominalized predicates as abstract singular terms marks the critical extension of second-order predicate logics. • The use of nominalized predicates as abstract singular terms is not only a part of our commonsense framework, but it is also central to how both logical
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realism and conceptual intensional realism provide an ontological foundation for the natural numbers and other parts of mathematics. • Nominalized predicates and propositional forms as abstract objectual terms are an essential part of a theory of logical form designed to represent natural language. • As a formal ontology logical realism or holistic conceptualism should contain all of the theorems of “standard” second-order predicate logic as a proper part. • With λ-abstracts to represent complex predicates, the comprehension principle can be stated in a stronger and more natural form as (CP∗λ ) with identity in place of the biconditional. • A second-order predicate logic with nominalized predicates, standard firstorder logic, and a full, unrestricted comprehension principle leads to Russell’s paradox of predication.
Chapter 5
Formal Theories of Predication Part II 5.1
Homogeneous Stratification
We saw at the end of the last chapter how Russell’s paradox of predication was generated by simply extending the consistent framework of standard secondorder predicate logic to include nominalized predicates as abstract terms. The implications of this result for logical realism as a modern form of Platonism were profound. How could mathematics be explained both ontologically and epistemologically if the formal theory of predication represented by secondorder predicate logic with nominalized predicates as abstract singular terms were inconsistent? This was the situation that confronted Russell in his 1903 Principles of Mathematics. In fact, the form of logical realism that Russell had in mind in 1903 was essentially the second-order predicate logic with nominalized predicates as abstract terms that we have described in the previous lecture. Beginning in 1903, and for some years afterwards, Russell tried to resolve his paradox in many different ways. It was not until 1908 that he settled on his theory of ramified types. Now both the theory of ramified types and the later theory of simple types avoid Russell’s paradox by setting limits on what is meaningful or significant in language. For that reason, and not without some justification, it has been severely criticized. What the theory of simple logical types does is divide the predicate expressions (and their corresponding abstract singular terms) of the second-order logic described in the previous chapter into a hierarchy of different types, and then it imposes a grammatical constraint that nominalized predicates can occur as argument- or subject-expressions only of predicates of higher types. This purely grammatical constraint excludes from the theory expressions of the form F (F ), or F ([λxF (x)]), as well as their negations, ¬F (F ), or ¬F ([λxF (x)]), which are 101
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just the types of expressions needed to generate Russell’s paradox.1 This was all that Russell needed to avoid his paradox of predication; but, as a way to avoid the so-called semantical paradoxes—such as that of the liar—Russell also divided the predicates on each level of the hierarchy of types into a ramified hierarchy of orders. The simple theory of types, which is all that is needed to avoid Russell’s paradox, is based only on the first hierarchy, whereas the ramified theory of types is based on both. These grammatical constraints are undesirable, we want to emphasize, because they exclude as meaningless many expressions that are not only grammatically correct in natural language but also intuitively meaningful, and sometimes even true (such as ‘Being a property is a property’ or ‘Being red is not red’, etc.). Fortunately, it turns out, the logical insights behind these constraints can be retained while mitigating the constraints themselves. In particular, we need impose only a constraint on λ-abstracts, namely that they be restricted to those that are homogeneously stratified in a metalinguistic sense. Here, by a metalinguistic characterization, we mean one that applies only in the metalanguage and not as a distinction between types of predicates in the object language. Retaining the same logical syntax that we described in the previous section, we say that a formula or λ-abstract ϕ is homogeneously stratif ied (or just h-stratif ied) if, and only if there is an assignment (in the metalanguage) t of natural numbers to the terms and predicate expressions occurring in ϕ (including ϕ itself if it is a λ-abstract) such that • (1) for all terms a, b, if (a = b) occurs in ϕ, then t(a) = t(b); • (2) for all n ≥ 1, all n-place predicate expressions π, and all terms a1 , ..., an , if π(a1 , ..., an ) is a formula occurring in ϕ, then (i) t(ai ) = t(aj ), for 1 ≤ i, j ≤ n, and (ii) t(π) = t(a1 ) + 1; • (3) for n ≥ 1, all objectual variables x1 , ..., xn , and formulas χ, if [λx1 ...xn χ] occurs in ϕ, then (i) t(xi ) = t(xj ), for 1 ≤ i, j ≤ n, and (ii) t([λx1 ...xn χ]) = t(x1 ) + 1; and • (4) for all formulas χ, if [χ] (i.e., [λχ]) occurs in ϕ and a1 , ..., ak are all of the terms or predicates occurring in χ, then t([χ]) ≥ max[t(a1 ), ..., t(ak) ]. The one constraint we need to impose to retain a consistent version of Russell’s earlier 1903 logical realism is that, to be grammatically well-formed, all 1 Instead of the λ-operator, Russell used a cap-notation, ϕ(ˆ x), to represent a property expression as an abstract singular term. In Russell’s notation, type theory excluded formulas of the form ϕ(ϕ(ˆ x)) and ¬ϕ(ϕ(ˆ x)), which in our notation correspond to F ([λxF (x)]) and ¬F ([λxF (x)]). In the logic described in the preceding section,
F = [λxF (x)] is assumed to be valid.
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λ-abstracts must be homogeneously stratified (in the metalinguistic sense defined). This means that formulas of the form F (F ) and ¬F (F ) are still grammatically meaningful even though they are not h-stratified. Formulas of the form F ([λxF (x)]), [λxF (x)]([λxF (x)]),
¬F ([λxF (x)]), ¬[λxF (x)]([λxF (x)])
are also grammatically well-formed so long as the λ-abstracts in these formulas are h-stratified. On the other hand, the complex predicate that is involved in Russell’s paradox, namely, [λx(∃G)(x = G ∧ ¬G(x))], is not h-stratified, because, x and G must be assigned the same number (level) for their occurrence in x = G, whereas G must also be assigned the successor of what x is assigned for their occurrence in ¬G(x). The comprehension principle (CP∗λ ) and the second-order logic of the previous section can be retained in its entirety, with the one restriction that the λ-abstracts that occur in the formulas of this logic must all be h-stratified. Because of this one restriction we will refer to the system as λHST∗ . Finally, let us note that not only is Russell’s paradox blocked in λHST∗ , but so are other logical paradoxes as well. Indeed, as we have shown elsewhere, λHST∗ is consistent relative to Zermelo set theory and equiconsistent with the simple theory of logical types.2 Also, if we were to add to λHST∗ the following axiom of extensionality, (∀x1 )...(∀xn )[F (x1 , ...x,n ) ↔ G(x1 , ...x,n )] → F = G,
(Ext∗ )
or, equivalently, because F n = [λx1 ...xn F (x1 , ...x,n )], is valid in λHST∗ (and in fact is taken as an axiom), (∀x1 )...(∀xn )[F (x1 , ...x,n ) [λx1 ...xn F (x1 , ...x,n )]
↔ =
G(x1 , ...x,n )] → [λx1 ...xn G(x1 , ...x,n )]
(Ext∗ )
then the result is equiconsistent with the set theory known as NFU (New foundations with Urelements) as well.3
Metatheorem: λHST∗ is consistent relative to Zermelo set theory; and it
is equiconsistent with the theory of simple logical types. In addition, λHST∗ + (Ext∗ ) is equiconsistent with the set theory NFU. 2 Cf.
3 See
Cocchiarella 1986. Holmes 1999 for a development of NFU.
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CHAPTER 5. FORMAL THEORIES OF PREDICATION PART II
Frege’s Logic Reconstructed
Gottlob Frege’s form of logical realism as described in his Grundgesetze was also a second-order predicate logic with nominalized predicates as abstract singular terms, and it too was subject to Russell’s paradox.4 But Frege also had a hierarchy of universals implicit in his logic, except that he assumed that all higher levels of his hierarchy beyond the second could be reflected downward into the second level, which in turn was reflected in the first level of objects, which is the situation that is implicitly represented in λHST∗ .5 In this respect, λHST∗ can also be used as a consistent reconstruction of Frege’s form of logical realism. Unlike Russell, however, Frege did not assume that what a nominalized predicate denotes as an abstract singular term is the same universal that the predicate stands for in its predicative role. In addition, Frege’s universals, which he called concepts (Begriffe) and relations, but which we will call properties and relations instead, have an unsaturated nature, and this unsaturated nature precludes the properties and relations of Frege’s ontology from being objects.6 For this reason, Frege’s universals cannot be what nominalized predicates denote as abstract singular terms. In other words, in Frege’s ontology what a predicate stands for in its predicative role is not what a nominalized predicate denotes as an abstract singular term. Why then have nominalized predicates at all? In Frege’s ontology it was not just to explain an important feature of natural language. Rather, it was a matter of “how we are to conceive of logical objects,” and numbers in particular.7 “By what means,” Frege noted, “are we justified in recognizing numbers as objects?” The answer, for Frege, was that we apprehend logical objects as the extensions of properties and relations (or concepts), and it is through the process of nominalization that we are able to achieve this. Here, it is the logical notion of a class as the extension of a property, or concept, that is involved, and not the mathematical notion of a set. Unlike a set, which has its being in its members, a class in the logical sense has its being in the property, or concept, whose extension it is.8 Now it was Frege’s commitment to an extensional logic that led him to take classes as the objects denoted by nominalized predicates. A class, after all, is the extension of a predicate as well as of the property or concept that the 4 See Cocchiarella 1987, chapter 2, section 6, and chapter 4, section 3, for a detailed defense of the claim that Frege’s extensional logic of Wertverl¨ aufe is really a logic of nominalized predicates. 5 This reflection downward in Frege’s logic is what I have referred to elsewhere as Frege’s double correlation thesis. See Cocchiarella 1987, chapter two, section 9, for a discussion of this part of Frege’s logic. Implicit in this reflection is a rejection of Cantor’s power-set theorem as applied to Frege’s logic of extensions. 6 Frege sometimes also described concepts as properties (Eigenshaften) as well. 7 Frege 1893, p. 143. 8 Cf. Frege 1979, p. 183.
5.2. FREGE’S LOGIC RECONSTRUCTED
105
predicate stands for, and it was in terms of classes and classes of classes that Frege proposed to construct the natural numbers. That, in fact, is the basis of his ontological logicism. Notationally, Frege used for this purpose the spiritus lenis, or smooth-breathing symbol, as a variable-binding operator. Thus, given a formula ϕ and a variable ε, applying the spiritus lenis resulted in an expression of the form ´ϕ(), which Frege took to be a nominalized form of the predicate represented by ϕ(ε). The smooth-breathing operator functioned in Frege’s logic in much the same way as the λ-operator does in the logics we have described, and for that reason we will continue to use the λ-operator here instead. Given Frege’s commitment to an extensional logic, then it is not just λHST∗ that we should take as a consistent reconstruction of his logic, but λHST∗ + (Ext∗ ), which, as already noted, is equiconsistent with the set theory NFU and consistent relative to Zermelo set theory. The extensionality axiom, (Ext∗ ), incidentally, is one direction of Frege’s well-known Axiom V, which was critical to the way Russell’s paradox was proved in Frege’s logic. This direction was called Basic Law Vb. The other direction, Basic Law Va, is actually an instance of Leibniz’s law in λHST∗ . That is, by (LL∗ ), Frege’s Basic Law Va, F = G → (∀x1 )...(∀xn )[F (x1 , ...x,n ) ↔ G(x1 , ...x,n )] is provable in λHST∗ , independently of (Ext∗ ), which was Frege’s Basic Law Vb. Given the consistency of λHST∗ + (Ext∗ ) (relative to Zermelo set theory), which includes Leibniz’s law, (LL∗ ), as a theorem schema, it is not Frege’s Basic Law V that was the problem for Frege so much as the way his hierarchy of universals was reflected downward into the first- and second-order levels. This was because Frege had heterogeneous, and not just homogeneous, relations in his logic, including heterogenous relations between universals and objects, such as that of predication, and these were included as part of the reflection downward of his hierarchy.9 The hierarchy consistently represented in λHST∗ , on the hand, consists only of homogeneous relations. The representation of heterogenous relations can be retained, however, by turning to an alternative reconstruction of Frege’s logic that is closely related to λHST∗ . This alternative involves replacing the standard first-order logic that is part of λHST∗ with a logic that is free of existential presuppositions regarding objectual terms, including nominalized predicates such as that corresponding to the complex predicate involved in Russell’s paradox. Now it is significant that in an appendix to his Grundgesetze Frege considered resolving Russell’s paradox by allowing that “there are cases where an unexceptional concept has no extension”.10 Here, by an “unexceptional concept” Frege had Russell’s rather exceptional concept, or property, in mind. After all, what is exceptional about the Russell concept, or property, in Frege’s logic is that it leads to a contradiction, unless, that is, we allow that it has no extension. But allowing that the Russell property has no extension in Frege’s logic requires 9 See
Cocchiarella 1987, section 9, for a more detailed discussion of this point. 1893, p. 128.
10 Frege
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allowing the nominalized form of the Russell predicate to denote nothing. That is, it requires a shift from standard first-order logic to a logic free of existential presuppositions regarding objectual terms, including especially nominalized predicates. In other words, instead of axiom (A8) of the logic of possible objects in chapter 2, namely, (∃x)(a = x), where x does not occur in a, we now have (∀x)(∃y)(x = y), where x, y are distinct object variables. In other words, the logic of possible objects is now a “free logic,” just as the logic of actual objects is a free logic. We want to use this logic as our “free” logic because although abstract object have being as values of the bound object variables, nevertheless, they do not “exist” in the concrete sense of existence, a restricted notion of being that we want to retain in an extended development in chapter 6 of the second-order logic with nominalized predicates that we are now considering. In fact, this strategy works. By adopting a free first-order logic and yet retaining the unrestricted comprehension principle (CP∗λ ), all that follows by the argument for Russell’s paradox is that there is no object corresponding to the Russell property, i.e., ¬(∃y)([λx(∃G)(x = G ∧ ¬G(x))] = y) is provable, even though, by (CP∗λ ), the Russell concept, or property, “exists” as a concept, or property; that is, even though (∃F )([λx(∃G)(x = G ∧ ¬G(x))] = F ) is also provable. Because the first-order logic is now a “free” logic, the original rule of λ-conversion must be modified as follows: [λx1 ...xn ϕ](a1 , ..., an ) ↔ (∃x1 )...(∃xn )(a1 = x1 ∧ ... ∧ an = xn ∧ ϕ), (∃/λ-Conv∗ ) where, for all i, j ≤ n, xi does not occur in aj . Revised in this way, our original second-order logic with nominalized predicates can be easily shown to be consistent. But that is because, without any further assumptions, we can no longer prove that any property or relation has an extension. That is, because the logic is free of existential presuppositions, all nominalized predicates might be denotationless, a position that a nominalist might well adopt. But in Frege’s ontological logicism some properties and relations must have extensions, and, indeed, it would be appropriate to assume that all of the properties and relations that can be represented in λHST∗ have extensions in this alternative logic. That in fact is exactly what we allow in our alternative reconstruction of Frege’s logic.
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The added assumption can be stipulated in the form of an axiom schema. But to do so we need to first define the key notion of when an expression of our logical grammar can be said to be bound to objects. Definition: If ξ is a meaningful expression of our logical grammar, i.e., ξ ∈ MEn , for some natural number n, then ξ is bound to objects if, and only if, for all predicate variables F , and all formulas ϕ ∈ ME1 , if (∀F )ϕ is a formula occurring in ξ, then for some object variable x and some formula ψ, ϕ is the formula [(∃x)(F = x) → ψ]. To be bound to objects, in other words, every predicate quantifier occurring in an expression ξ must refer only to those properties and relations (or concepts) that have objects corresponding to them, which in Frege’s logic are classes as the extensions of the properties or relations in question. The axiom schema we need for this is given as follows: (∃y)(a1 = y) ∧ ... ∧ (∃y)(ak = y) → (∃y)([λx1 ...xn ϕ] = y),
(∃/HSCP∗λ )
where, • (1) [λx1 ...xn ϕ] is h-stratified, • (2) ϕ is bound to objects, • (3) y is an object variable not occurring in ϕ, and • (4) a1 , ..., ak are all of the object or predicate variables or nonlogical constants occurring free in [λx1 ...xn ϕ].11 Because of it close similarity to our first reconstructed system, λHST∗ , we will refer to this alternative logic as HST∗λ .12 As we have shown elsewhere, HST∗λ is equiconsistent with λHST∗ , and therefore with the theory of simple types as well. It is of course also consistent relative to Zermelo set theory. Finally, let us note that although HST∗λ + (Ext∗ ) can be taken as a reconstruction of Frege’s logic and ontology, it cannot also be taken as a reconstruction of Russell’s early (1903) ontology, even without the extensionality axiom, (Ext∗ ). This is because Russell rejected Frege’s notion of unsaturatedness and assumed that nominalized predicates denoted as singular terms the same concepts and relations they stand for as predicates. In other words, unlike Frege, Russell cannot allow that some predicates stand for properties and relations (or concepts), but that as objectual terms their nominalizations denote nothing. Of course, we do have the logical system λHST∗ , which can be taken as a reconstruction of Russell’s early ontological framework. 11 We understand the conditional posited in this axiom schema to reduce to just the consequent if k = 0, i.e., if [λx1 ...xn ϕ] contains no free variables or nonlogical constants. 12 For a detailed account of all of the axioms of these systems see Appendix 1 of chapter 6, and for an account of their various properties see Cocchiarella 1986, chapter V.
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CHAPTER 5. FORMAL THEORIES OF PREDICATION PART II
Conceptual Intensional Realism
In conceptualism, predicable concepts are cognitive capacities that underlie our rule-following abilities in the use of the predicate expressions of natural language, and, as such, concepts determine the truth conditions of that use. Moreover, as capacities that can be exercised by different persons at the same time, as well as by the same person at different times, concepts cannot be objects, e.g., ideas or mental images as particular mental occurrences. In other words, as intersubjectively realizable cognitive capacities, concepts are objective and not merely subjective entities. Moreover, as essential components of predication in language and thought, concepts as cognitive capacities have an unsaturated nature, and it is this unsaturated nature that is the basis of predication in language and thought. In particular, it is the exercise of a predicable concept in a speech or mental act that informs that act with a predicable nature, a nature by means of which we characterize and relate objects in various ways. The unsaturatedness of a concept as a cognitive capacity is not the same as the unsaturatedness of a universal in Frege’s ontology. For Frege, a property (Begriff, Eigenshaft ) or relation is really a function from objects to truth values, and it is part of the nature of every function, according to Frege, even those from numbers to numbers, to be unsaturated. Predication in other words, is reduced to functionality in Frege’s ontology. In conceptualism, on the other hand, it is predication that is more fundamental than functionality. We understand what it means to say that a function assigns truth values to objects, after all, only by knowing what it means to predicate concepts, or properties and relations, of objects. The unsaturated nature of a concept is not that of a function, but of a cognitive capacity that could be exercised by different people at the same time as well as by the same person at different times, and it might in fact not even be exercised ever at all. When such a capacity is exercised , however, what results is not a truth value, whatever sort of entity that might be, but a mental event, and if expressed overtly in language then a speech-act event as well. Now we should note that conceptual thought consists not just of predicable concepts, but of referential and other types of concepts as well. Referential concepts, for example, are cognitive capacities that underlie our ability to refer (or really to purport to refer) to objects, and, as such, they too have an unsaturated cognitive structure. More importantly, referential concepts have a structure that is complementary to that of predicable concepts, so that each, when exercised or applied jointly in a judgment, mutually saturates the other, resulting thereby in an act (event) that is informed with a referential and a predicable nature. As we will explain in a later chapter, it is the complementarity of predicable and referential concepts as unsaturated cognitive structures that is the basis of the unity of our speech and acts, and which explains why a transcendental subjectivity need not be assumed as the basis of this unity. In this lecture, however, we will restrict ourselves to the predicable concepts that are the counterparts of the universals of logical realism.
5.3. CONCEPTUAL INTENSIONAL REALISM
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If predicable concepts are unsaturated cognitive structures, then what point is there in having a logic of nominalized predicates as abstract objectual terms? One point is the same as it was for Frege, namely to account for the ontology of the natural numbers as logical objects. Another is to explain the significance of nominalized predicates in natural language, including especially complex forms of predication containing infinitives, gerunds, and other abstract nouns. There is a difference with Frege’s ontology, however, in that whereas Frege was committed to an extensional logic, conceptualism, once it admits abstract objects into its ontology, is committed to an intensional logic. Instead of denoting the extensions of concepts, in other words, nominalized predicates in conceptualism denote the intensional contents of concepts, which is why we refer to this extension of conceptualism as conceptual intensional realism. Now by the intensional content of a predicable concept we understand an abstract intensional object corresponding to the truth conditions determined by the different possible applications of that concept, i.e., the conditions under which objects can be said to fall under the concept in any possible context of use, including fictional contexts. Of course, there are some predicable concepts, such as that represented by the Russell predicate, [λx(∃G)(x = G ∧ ¬G(x))], that determine truth conditions corresponding to which, logically, there can be no corresponding abstract object, on pain of contradiction. This does not mean that such a predicable concept does not determine truth conditions and therefore does not have intensional content; rather, it means only that such a content cannot be “object”-ified, i.e., there cannot be an abstract object corresponding to the content of that concept, the way there are for the contents of other concepts. The real lesson of Russell’s paradox is that some rather exceptionable, “impredicatively” constructed concepts determine truth conditions that logically cannot be “object”-ified, whereas most predicable concepts are unexceptionable in this way. In the alternative ontology of conceptual Platonism, incidentally, the abstract object corresponding to the intensional content of a predicable concept is a Platonic Form, which traditionally has also been called a property or relation— a terminology that we can allow as well in conceptual realism so long as we do not confuse these properties and relations with the natural properties and relations of conceptual natural realism. There is an important ontological difference between conceptual Platonism and conceptual intensional realism, however, despite the similarity of both to logical realism. Unlike logical realism, conceptual Platonism is an indirect and not a direct Platonism. That is, in conceptual Platonism, but not in logical realism, abstract objects are cognized only indirectly through the concepts whose correlates they are. This means that our representation of abstract objects is seen as a reflexive abstraction corresponding to the process of nominalization. In other words, even though abstract objects according to conceptual Platonism exist in a realm that transcends space, time and causality, and in that sense preexist the evolution of
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consciousness and the cognitive capacities that we exercise in thought and our use of language, nevertheless, from an epistemological point of view, no such entity can be cognized otherwise than as the correlate of a concept, i.e., as an abstract intensional object corresponding to the truth conditions determined by that concept. In conceptual intensional realism, on the other hand, all abstract objects, despite having a certain autonomy, are products of language and culture, and in that respect do not preexist the evolution of consciousness and the cognitive capacities that we exercise in language.13 They are, in other words, evolutionary products of language and culture, and therefore depend ontologically on language and culture for their “existence,” or being. Of course, abstract objects, especially numbers, are also an essential part of the means whereby further cultural development becomes possible. Nevertheless, as cultural products, the “existence” of abstract objects is primarily the result, and development of, the kind of reflexive abstraction that is represented by the process of nominalization. It was through the institutionalization of this process that abstract objects achieved a certain autonomy and, in time, became reified as objects. Abstract objects do not exist in a Platonic realm outside of space, time, and causality, on our interpretation, but are in fact the result, in effect, of an ontological projection inherent in the development and institutionalization in language of the process of nominalization. The fundamental insight into the nature of abstract objects according to conceptual intensional realism is that we are able to grasp and have knowledge of such objects as the “object”-ified truth conditions of the concepts whose contents they are, i.e., as the object correlates of those concepts. This “object”-ification of truth conditions is realized, moreover, through a kind of reflexive abstraction in which we attempt to represent what is not an object—in particular an unsaturated cognitive structure underlying our use of a predicate expression—as if it were an object. In language this reflexive abstraction is institutionalized in the rule-based linguistic process of nominalization. Finally, we note that we can take the system HST∗λ as the core part of the formal ontology of conceptual realism (or of conceptual Platonism), as well as of Frege’s form of logical realism. Of course Frege’s logical realism differs from conceptual realism in having the extensionality axiom, (Ext∗ ), as part its core as well. There are other difference as well, which we take up in the next chapter.
5.4
Hyperintensionality
There might be a problem with the systems λHST∗ and HST∗λ , it should be noted, in that they do not seem to adequately respect intentional contexts such as belief, desire, etc.—at least not if such contexts must be “hyperintensional,” 13 Cf. Popper and Eccles 1977, chap. P2, for a similar view. We should note, however, that although our view of abstract objects supports the Popper-Eccles interactionist theory of mind, it does not also depend on that theory.
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or more “fine-grained” in structure than is allowed, e.g., by logical equivalence. The claim is that logically equivalent formulas do not in general preserve truth when interchanged in intentional contexts, and that only a hyperintensional logic will suffice for that purpose. A hyperintensional logic is one whose identity criteria for intensional objects is as “fine-grained” as possible, which means, for example, that conjunctive or disjunctive complex predicates are identical only when the first and second conjunct, or disjunct, of each is identical with the first and second conjunct, or disjunct, respectively, of the other; and the same condition applies to conditionals and biconditionals, and quantifier phrases as well. The argument that neither λHST∗ nor HST∗λ can be developed as a hyperintensional logic is a result of a so-called “hyperintensional paradox”, which depends on adding the following hyperintensional assumptions as axioms to these systems:14 (Hyper1): [λx1 ...xn (Qa)ϕ] = [λx1 ...xn (Qa)ψ] → (∀a)([λx1 ...xn ϕ] = [λx1 ...xn ψ]) where Q is either ∀ or ∃ and a is a predicate or objectual variable. (Hyper2): [λx1 ...xn (ϕ ψ)] = [λx1 ...xn (ϕ ψ )] → [λx1 ...xn ϕ] = [λx1 ...xn ϕ ]∧ [λx1 ...xn ψ] = [λx1 ...xn ψ ] where is any of the sentential connectives ∧, ∨, →, or ↔. (Hyper3): (∀F )(∀G)((∀a)([λx1 ...xn F (a, x1 , ..., xn )] = [λx1 ...xn G(a, x1 , ..., xn )]) → [λx1 ...xn F (x1 , ..., xn )] = [λx1 ...xn G(x1 , ..., xn )]), where a is a predicate or objectual variable. (Hyper3) is an implausible assumption, as we explain below, and (Hyper1) is in conflict with our conceptual realist account of the copula, which we will take up in Part II.15 We do not, indeed cannot, accept these assumptions, but for now we can state the thesis in question as the following theorem. Theorem: λHST∗ + (Hyper1) + (Hyper2) + (Hyper3) and HST∗λ +(Hyper1) + (Hyper2) + (Hyper3) are inconsistent. Proof: Let H = [λy(∃F )(y = [λz(∀G)(F (G) → G(z)) ∧ ¬F (y))] 14 See Bozon 2004 for the original formulation of this paradox, which is patterned after one given for simple type theory. 15 In section 8.5 we introduce Is as a special predicate for the copula and in HST∗ take the λ following as a meaning postulate
[λxIs(x, [∃yA])] = [λx(∃yA)(x = y)]. The Is predicate is not h-stratifiable. Our use of it, however, is only for the initial level of analysis of predication in speech and mental acts; that is, we do not need its objectification or nominalization. In any case, applying (Hyper3) to this meaning postulate would yield the result that (∃yA)(x = y), which says that x is an A, is identical with x = [∃yA], where [∃yA] is the intensional content of the phrase ‘an A , which is false when x is a concrete object.
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and
H ∧ = [λz(∀G)(H(G) → G(z))].
Both H and H ∧ are h-stratified,16 and therefore well-formed in λHST∗ and “object”-fiable in HST∗λ . Suppose first that H(H ∧ ). Then, by definition of H and λ-conversion, (∃F )(H ∧ = [λz(∀G)(F (G) → G(z)] ∧ ¬F (H ∧ )). That is,
H ∧ = [λz(∀G)[F (G) → G(z)] ∧ ¬F (H ∧ )),
for some value of F , and hence by definition of H ∧ and Leibniz’s law, [λz(∀G)(H(G) → G(z))] = [λz(∀G)(F (G) → G(z))]. Therefore, by (Hyper1), (∀G)([λz(H(G) → G(z))] = [λz(F (G) → G(z))]), and hence, by (Hyper2), (∀G)([λzH(G)] = [λzF (G)] ∧ [λzG(z)] = [λzG(z)]), and in particular, (∀G)([λzH(G)] = [λzF (G)]). But then, by (Hyper3),[λzH(z)] = [λzF (z)], and therefore, by axiom (Id∗λ ) of both λHST∗ and HST∗λ ,17 it follows that (H = F ) and hence, by Leibniz’s law, ¬H(H ∧ ), contrary to our initial assumption, which is impossible. But now given ¬H(H ∧ ), it follows by definition of H and λ-conversion (or (∃/λ-Conv∗ in HST∗λ ), (∀F )(H ∧ = [λz(∀G)(F (G) → G(z))] → F (H ∧ )), and therefore by universal instantiation, H ∧ = [λz(∀G)(H(G) → G(z))] → H(H ∧ ), from which, by definition of H ∧ , it follows that H(H ∧ ), which is impossible given ¬H(H ∧ ). In other words, the above hyperintensional axioms together with the definitions of H and H ∧ lead to the following impossible result: H(H ∧ ) ↔ ¬H(H ∧ ). The above use of (Hyper3) in the inference from (∀G)([λzH(G)] = [λzF (G)]) to [λzH(z)] = [λzF (z)] indicates how implausible (Hyper3) is as a logical principle. After all, even if H(z) and F (z) are identical for all values of z that are 16 Both H and H ∧ , however, are impredicatively specified and would not be allowed in constructive conceptualism’s ramified logic, which can be developed into a hyperintensional logic. 17 See appendix 1 of chapter 6 for a description of the axioms of λHST∗ and HST∗ , λ including (Id∗λ ).
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properties, it does not follow that H(z) and F (z) are identical for all objects as values for z, i.e., that [λzH(z)] = [λzF (z)]. If we reject this use of (Hyper3), then one important step in the above proof is no longer acceptable. The supposed conclusion of this “paradox”, as we have said, is that, like the simple theory of types, neither λHST∗ nor HST∗λ can serve as the basis of a hyperintensional logic if (Hyper1)–(Hyper3) are assumed as principles of hyperintensionality. The more restrictive predicative logic of constructive conceptualism described in section five of the previous chapter can suffice for this purpose, however, because the above argument depends essentially on impredicative expressions that are not applicable in predicative logic, ramified or otherwise. Despite the implausibility of (Hyper3) for λHST∗ and HST∗λ , and of (Hyper1) for HST∗λ , the question we want to turn to here is not whether or not (Hyper1)–(Hyper3) are appropriate principles of hyperintensionality, but whether or not either logical realism or conceptual (intensional) realism must be committed to hyperintensionality. This is a relevant question because, intuitively, the most natural interpretation for a hyperintensional logic is a strictly syntactical one in which identity means identity of constituent expressions, except perhaps for rewrite of bound variables. That interpretation, of course, is inappropriate for either logical or conceptual realism. A closely related interpretation is one that assumes an ontology in which properties and relations are either absolutely simple or intrinsically complex, and in which the latter are built up from the former by means of ontologically real counterparts of the logical constants that occur in their syntactic representations in a formal ontology. We will call this a simples⊕complex ontology to distinguish it from one, such as logical atomism, in which there are simple properties and relations but no complex ones. Now a simples⊕complexes ontology does not seem appropriate for Frege’s variant of logical realism. That is because the properties—or what Frege called “concepts”—and relations of Frege’s realism are functions from objects to truth values, and as such they do not in themselves contain ontological counterparts of the logical constants as constituents. The property, or function, corresponding to the predicate [λx(F (x)∨G(x))], e.g., does not contain a disjunction operation as a constituent of the function, even though the values of the function are determined in terms of a disjunction. The function itself is “simply” a correlation of truth values with objects. A property, or function, may be represented by a complex predicate expression, in other words, without itself having an ontological complexity corresponding to that of the predicate expression. A similar observation applies, of course, to properties and relations represented by conjunctive and other complex predicate expressions. Functions in general consist only of correlations from arguments to values and, unless a function is an argument or a value of another function, it is not a constituent of that function even if its values are arguments of that function. Now Bertrand Russell in his early form of logical realism as described in his 1903 Principles of Mathematics certainly seems to have assumed an ontology of simples and complexes, where by a complex he means an object whose
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“being presupposes the being of certain other terms,” namely objects that are constituents of the complex.18 But his complexes are of only two kinds, namely, nonempty classes as ones, which are complexes in that the being of such a class presupposes the being of its members. Of course, hyperintensionality cannot be said to apply to classes as ones, which are strictly extensional entities. The other kind of complex entities recognized by Russell are propositions. “For example, ‘A differs from B ’, or ‘A’s difference from B ’, is a complex,” according to Russell, “of which the parts are A and B and difference.”19 Nothing is said in the Principles about properties or relations also being complex, and Russell’s statement that there are only two kinds of complexes, namely classes as ones and propositions, seems to exclude there being complex properties and relations. What about propositional functions? Are they not complex properties and relations? Well, certainly not in Russell’s 1903 ontology. That is because, in 1903, propositional functions are not themselves objects, or entities of any type, and therefore they could not be properties or relations. Thus, according to this early Russell, “the [propositional function] ϕ in ϕx is not a separate and distinguishable entity: it lives in the propositions of the form ϕx and cannot survive analysis.”20 On the other hand, in Russell’s later logical realism, as described in Principia Mathematica (1910–13), propositional functions do seem to be part of his ontology, whereas propositions are no longer themselves “single entities”.21 In fact, I myself have identified the propositional functions of Principia with properties and relations, though I now think an alternative interpretation is more appropriate.22 The connection of propositional functions with properties and relations was explicitly made by Russell in 1907 when he stated two “principles” that he said were “indispensable if we are to avoid contradictions and ... preserve ordinary mathematics,” principles that seem to be implicit in Principia:23 Any propositional function of x is equivalent to one assigning a property to x. Any propositional function of x and y is equivalent to one assigning a relation between x and y. Note that Russell does not say here that a propositional function of one variable is identical with a property, but only that it is equivalent to one, and similarly that a relation is only equivalent, not identical, with a propositional function with several variables. This does not mean that there are now two types of universals, namely propositional functions on the one hand and properties and relations on the other, and that the two are somehow equivalent. That would be both redundant as a formal ontology and ambiguous about the role 18 Russell
1903, §133, p. 137. §135, p. 139. 20 Ibid., §85, p.88. 21 See Cocchiarella 1987 chapter one for a discussion of this change in Russell’s views. 22 See Cocchiarella 1987, chapter 1, §§9–10 and chapter 5, §2, for the claim that the propositional functions of Principia are properties and relations. 23 Russell 1907, p. 281. 19 Ibid.,
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of universals in predication. On our alternative interpretation, however, what Russell is really doing here is taking propositional functions as expressions that stand for, or represent, properties and relations, and, although he could have been clearer about the matter, that is what he really means by equivalence in this context. Something like that is what Russell himself claims in his later book, My Philosophical Development, where he wrote that “Whitehead and I thought of a propositional function as an expression containing an undetermined variable and becoming an ordinary sentence as soon as a value is assigned to the variable: ‘x is human’, for example, becomes an ordinary sentence as soon as we substitute a proper name for ‘x’. In this view ... the propositional function is a method of making a bundle of such sentences.”24 If this view is correct, then the complexity of propositional functions is only a complexity of expression, i.e., of syntax, and not one of ontology. The fact that properties and relations are described by complex predicate expressions, in other words, does not mean that the properties and relations are themselves complex and somehow contain simple properties and relations and logical operations as constituents. Just as the union or intersection of two sets A and B does not contain the union or intersection operation as a constituent, so too a property that is represented by a complex conjunctive or disjunctive predicate expression does not itself contain the conjunction or disjunction operation as a constituent. The fact that A ∪ B contains both A and B, i.e., that A, B ⊆ A ∪ B, or that A ∩ B is contained in both A and B, i.e., that A ∩ B ⊆ A, B, does not mean that A and B are themselves constituents of A ∪ B the way, e.g., the members of A and B are constituents (members) of A and B (and of A ∪ B as well), or that A and B are constituents (members) of A ∩ B. Nor of course are the union operation, ∪, and the intersection operation, ∩, constituents of A∪B and A∩B respectively. Similarly, neither the properties F and G, nor the disjunction and conjunction operations, are real constituents of the properties [λx(F (x) ∨ G(x))] and [λx(F (x) ∧ G(x))], even though there is a logical relation based upon these operations between F and G and [λx(F (x) ∨ G(x))] and [λx(F (x) ∧ G(x))]—a relation that is the basis of an abstract fact having these properties as constituents. The only complexes in Russell’s 1910–13 logical realist ontology are facts— and events, though Russell does not clearly distinguish events from concrete, physical facts—including abstract facts having universals as constituents.25 Thus, according to Russell, “the statement ‘two and two are four’ deals exclusively with universals,” and the complex that makes it true is an abstract fact.26 Abstract facts are needed in Russell’s logical realism as the “truth makers” of the statements of logic and mathematics that contain no descriptive constants. This ontology is quite different from an ontology of sets (with or without urelements), i.e., one in which pure mathematics is reduced to set theory as opposed to logic. In set theory there are no set-theoretical facts about pure sets, i.e., sets whose transitive closures contain no urelements other than the empty set. This is be24 Russell
1959, p.124. Russell 1912, p. 137. 26 Ibid., p.105. 25 See
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cause a set has its being in its members, and a set’s being is all that is needed to account for the truth or falsehood of statements about membership in that set. In other words, the being of a set consists in its having just the members that it has, and no fact over and above the being of the set itself is needed to account for membership in that set. A property or relation (in intension), on the other hand, does not have its being in its “instances,” and therefore its being cannot of itself account for the truth of statements about objects having that property or relation. Propositions, it should be noted, are not objective truths or falsehoods in this ontology; rather, according to Russell, there are no propositions other than sentences, i.e., propositions, like propositional functions, are now only expressions. Russell’s 1910–13 logical realist ontology, in other words, is not a simples⊕complexes ontology as characterized above, and in that regard the hyperintensional assumptions (Hyper1)–(Hyper3) are inapplicable to this version of logical realism. What a complex predicate represents in this ontology is not a complex property or relation, but only another property or relation that stands in a certain logical relation between the properties and relations that are represented by the component parts of that complex predicate. The logical relation, together with these properties and relations, are constituents of an abstract fact, rather than of a complex property or relation. And abstract facts, like concrete, physical facts, are extensional entities and not hyperintensional entities. The above hyperintensional “paradox”, accordingly, does not apply to either Russell’s or Frege’s versions of logical realism. Of course, one might argue that there is still the problem of explaining hyperintensional contexts, i.e., of how a logical realist formal ontology such Frege’s or Russell’s can account for the logic of such intentional contexts as belief, desire, etc. Be that as it may, in any case one cannot reject these formal ontologies on the basis of the above hyperintensionality paradox. One might, on the other hand, reject our interpretation of propositional functions as expressions, or one might just insist on maintaining a simples⊕complexes ontology regardless of what Frege’s or Russell’s own views were. In that case, however, an explanation must be given of how a property or relation can contain a logical operation as well other properties or relations as constituents. It is not enough to simply assume that this is so without giving an ontological account of how it is possible, i.e., of how there can be such complexes.27 But then, assuming such an ontology will bring one back to the problem of how the hyperintensional paradox is to be avoided. Finally, with regard to conceptual realism, it is noteworthy that although concepts are formed, or constructed, on the basis of other concepts, concepts themselves, as cognitive capacities, do not contain other concepts or logical operations as constituents. In other words, with respect to concepts as cognitive capacities, conceptual realism is not a simples⊕complexes ontology. Of course, with respect to the intensional contents of concepts, including propositions as the contents of our speech and mental acts, the situation might well be different. 27 An algebraic or set-theoretical semantics for hyperintensionality, we should note, does not of itself amount to an ontological account.
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In other words, we might well allow intensional objects to be either simple or complex, though some account will then be needed of the sense in which they might contain objects, including the intensional counterparts of logical operations, as constituents. But even assuming that a simples⊕complexes ontology applies to the intensional objects of conceptual realism, nevertheless, the hyperintensional assumption (Hyper3) does not apply to conceptual realism as represented by λHST∗ or HST∗λ for the reason given above. Moreover, none of the above hyperintensional principles (Hyper1)–(Hyper3) apply to conceptual realism at the level of the logical forms that represent the cognitive structure of our speech and mental acts; nor can most of the steps in the above argument be taken as representing the cognitive structure of a speech or mental act. This is important because it is only on this level of analysis that hyperintensionality has a role to play. In other words, it is only on this initial level of analysis, as opposed to the second level where deductive transformations are represented, must complex predicates be given a fine-grained representation. A speech or mental act in which, e.g., being round and red, i.e., [λx(Round(x) ∧ Red(x))], is predicated of an object is not the same as a speech or mental act in which being not either not-round or not-red, i.e., [λx¬(¬Round(x) ∨ ¬Red(x))] is predicated of that object, even though the predicate expressions representing these concepts are logically equivalent. Thus, although hyperintensionality does apply on the level of analysis on which the cognitive structure of our speech and mental acts are represented, it does not apply on the level of deductive transformations, such as those involved in the above paradox. Hyperintensionality, or fine-grained structure, applies only to cognitive structure, which, in conceptual realism is represented only at the initial level of analysis. Deductive transformations, which may involve logical forms that in no sense can be taken as representing the cognitive structure of our speech and mental acts—such as the predicate forms [λz(H(G)] and [λz(F (G)] that occur in the above proof of the paradox—are represented on a second and different level of analysis. Logical forms on this level allow for a variety of transformations that show the deductive consequences of our speech and mental acts, but do not themselves represent such acts. Deductive transformations also allow for the study of the consequences of scientific hypothesis, or of mathematical theories, which in general are not intended or designed to represent features of cognition. On this level, there is no basis for allowing (Hyper1)–(Hyper3) as deductive principles, and in fact given the above argument there is good reason to reject these assumptions altogether. Finally, we should note that there are other, equally important reasons why we must distinguish an initial level of analysis regarding the cognitive structure of our speech and mental acts from a second level at which deductive transformations are allowed to occur. These other considerations have to do with conceptual realism’s theory of reference and the deactivation of the referential expressions that occur as the direct objects of transitive verbs. We will return to this issue in section nine of chapter seven.
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Summary and Concluding Remarks
• As a way of avoiding Russell’s paradox, the theory of simple logical types divides predicate expressions and their corresponding abstract singular terms into a hierarchy of different types, and then imposes a grammatical constraint that nominalized predicates can occur as argument- or subject-expressions only of predicates of higher types. • The grammatical constraints of type theory exclude as meaningless many expressions that are not only grammatically correct in natural language but also intuitively meaningful, and sometimes even true. • The logical insights of type theory, and in particular the asymmetry between predicate and subject expressions, can be retained while mitigating the grammatical constraints of the theory. Within standard second-order predicate logic with nominalized predicates as abstract singular terms we need only impose a constraint on complex predicates (λ-abstracts), namely that they be restricted to those that are homogeneously stratified in a metalinguistic sense. • By retaining the full comprehension principle (CP∗λ ) of second-order predicate logic with nominalized predicates, but excluding λ-abstracts that are not homogeneously stratified, we obtain the system λHST∗ , which is equivalent to simple type theory and consistent relative to Zermelo set theory. • λHST∗ can be taken as a consistent reconstruction of Frege’s and Russell’s early 1903 form of logical realism. • For Frege’s extensional ontology, an extensionality axiom, (Ext∗ ), can be added to λHST∗ . This extensionality axiom is Frege’s Basic Law Vb. The other direction, Basic Law Va, is an instance of Leibniz’s law in λHST∗ . Thus Frege’s basic law V is consistent in λHST∗ . • By replacing standard first-order (possibilist) logic by free logic, λ-abstracts, whether homogeneously stratified or not, can be allowed in the comprehension principle (CP∗λ ). Russell’s paradox then only shows that the λ-abstract for the Russell property, when transformed into an abstract singular term, must fail to denote. A new axiom schema, (∃/HSCP∗λ ), is added in order to include the objects denoted by nominalized λ-abstracts that are homogeneously stratified. The resulting system HST∗λ is equivalent to λHST∗ and can be taken as a better reconstruction of Frege’s ontology, but not also of Russell’s. • The difference between the systems λHST∗ and HST∗λ shows in a clear and precise way one of the important differences between Russell’s and Frege’s formal ontologies. • The concepts of conceptual realism are rule-following cognitive capacities in the use of predicate expressions and as such have an unsaturated nature similar to but also different from the unsaturated concepts and relations of Frege’s ontology. It is the unsaturated nature of a predicable concept that informs a speech or mental act with a predicable nature. • Unlike Frege’s ontology where nominalized predicates denote the extensions of concepts, in conceptual realism nominalized predicates denote the intensional content of concepts—if in fact that content can be “object”-ified. The “object”-
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ification of the intensional content of a concept is an abstract intensional object that represents the truth conditions determined by that concept. • The intensional content of most concepts can be “object”-ified, i.e., projected onto the level of objects, as abstract intensional objects. But the intensional content of the concept that the Russell predicate stands for, as well as certain others, cannot be “object”-fied, i.e., the nominalized predicates that stand for those concepts must be denotationless. • The system HST∗λ can be taken as the core part of the formal ontology for conceptual realism as well as of Frege’s form of logical realism. Frege’s logical realism differs from conceptual realism in having the extensionality axiom, (Ext∗ ), as part its core as well. • The principle of rigidity (PR), which is discussed in the next chapter, is valid in logical realism but not in conceptual realism. This is another important difference between the two formal ontologies. • Logically equivalent formulas do not in general preserve truth when interchanged in intentional contexts, and only a hyperintensional logic will suffice for that purpose. • One objection to the systems λHST∗ and HST∗λ (and the theory of simple types as well) is that they do not adequately respect the fine-grained, or “hyperintensional,” structure of such intentional contexts as belief, desire, etc. This is because formulas provably equivalent in these systems do not in general preserve truth when interchanged in such contexts. The claim is that only a hyperintensional logic will suffice for that purpose. • The argument that neither λHST∗ nor HST∗λ (nor simple type theory) cannot be considered a hyperintensional logic is a result of a so-called “hyperintensional paradox”, which depends on certain hyperintensional assumptions that result in a contradiction in λHST∗ and HST∗λ (and simple type theory). • One of these assumptions, (Hyper3), is implausible for both λHST∗ and HST∗λ , and another, (Hyper1), is implausible for HST∗λ . • A hyperintensional logic is based on either a strictly syntactical interpretation of properties and relations or an ontology in which properties and relations are either absolutely simple or intrinsically complex, and in which the latter are built up from the former by means of ontologically real counterparts of the logical constants that occur in their syntactic representation. Both interpretations are not appropriate for either Frege’s or Russell’s logical realism, and it might apply at best to the intensional objects of conceptual realism. It does not apply to predicable concepts however. • Although predicable concepts are formed, or constructed, on the basis of other concepts in conceptual realism, concepts themselves, as cognitive capacities, do not contain other concepts or logical operations as constituents. • None of the hyperintensional assumptions (Hyper1)–(Hyper3) apply to conceptual realism at the level of the logical forms that represent the cognitive structure of our speech and mental acts. • It is only on the level of the logical forms of our speech and mental acts, as opposed to the level where deductive transformations are represented, that complex predicates must be given a fine-grained representation.
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• Hyperintensionality, or fine-grained structure, applies only to cognitive structure, which, in conceptual realism is represented only at the initial level of analysis.
Chapter 6
Intensional Possible Worlds We have seen in chapter two how actualism and possibilism can be distinguished from one another in tense logic and with respect to the logic of the temporal modalities of Aristotle and Diodorus. A deeper analysis can be given, however, in terms of the second-order theories of predication of both logical realism and conceptual realism. We could, for this purpose, restrict ourselves to secondorder tense logic, which would be appropriate for conceptual realism because, as already noted, thought and communication, as forms of conceptual activity, are inextricably temporal phenomena. Tense logic, in other words, is implicitly assumed as a fundamental part of the formal ontology of conceptual realism. A causal or natural modality is also a fundamental part of conceptual (natural) realism, and with this modality comes the distinction between what is actual and what is possible in nature. That is, with a causal or natural modalitywe have a broader and perhaps an even sharper distinction between actualism and possibilism. We will take up this type of modality in chapter twelve. Logical realism does not, or at least need not, reject tense logic and the temporal modalities; nor, unlike logical atomism, must it reject a causal or natural modality. But, except for Frege’s extensional ontology, one might think that intensional versions of logical realism are implicitly, if not explicitly, committed to the logical modalities, i.e., logical necessity and possibility. But, as we have argued in chapter three, only logical atomism can provide an unproblematic account of the logical modalities, despite the fact that logical atomism is an extreme form of natural realism rather than of logical realism. Unlike logical realism, the comprehension principle (CP) is not valid in logical atomism where only simple properties and relations are the nexuses of the atomic states of affairs that make up its ontology. Every possible world in logical atomism is made up of the same simple material objects and properties and relations, as those that make up the actual world, the only difference being between those states of affairs that obtain in a given world and those that do not. Different possible worlds in logical atomism, as we have noted, amount to different permutations of being-the-case and not-being-the-case of the same atomic states of affairs, and hence the same simple material objects and properties and relations, that 121
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make up the actual world. As a result, in such an ontology we have a precise notion of what it means to refer to “all possible worlds,” and dually to “some possible world”, which are the key notions characterizing how necessity and possibility are to be understood. What results from such an account is a strictly formal notion of necessity and possibility, i.e., a notion that has no material or ontological content, as it must be in logical atomism, and which is why there can be no causal modality in the framework. This logical atomist notion of a possible world is not at all suitable for logical realism with its commitment to a realm of abstract objects and a network of complex logical relations between them. The kind of modality that an intensional form of logical realism is committed to is an ontological, rather than a strictly formal, notion of logical necessity and possibility, a notion that can be referred to as metaphysical rather than formal. A metaphysical modality, apparently, is stronger than a physical or natural modality, and yet, because it has ontological content, it is not a purely formal logical modality such as is found in logical atomism. This notion of a metaphysical modality is difficult to explicate, however, because, as we explained in §§3.6-3.7, we do not have clear and precise criteria by which to determine the semantical conditions appropriate for a logic of metaphysical necessity and possibility. What we need is some way by which to understand the key notion of a metaphysically possible world—if in fact such a notion is really different from what is possible in nature. Question: What notion of a metaphysically possible world is appropriate for logical realism?1 A similar difficulty applies to conceptual (intensional) realism insofar as one is tempted to introduce a notion of conceptual necessity or possibility. Here too we have a complex network of concepts and a realm of abstract intensional objects that cannot be accounted for within logical atomism, and for which strictly formal notions of necessity and possibility therefore cannot be given. Temporal and causal, or natural, modalities do not seem to suffice to account for the network of relations between concepts and the intensional objects that are their objectual counterparts, nor of the logico-mathematical relations between abstract objects in general. A conceptual modality based on the strictly psychological abilities of humans seems inadequate to account for these complex networks, however, and yet how otherwise to formally account for the kinds of model structures that could represent conceptually possible worlds is not all clear. We will not attempt to deal with these difficulties for logical and conceptual realism here. We will assume instead that some explication can in principle be given. We will also assume that all metaphysically possible worlds are equally 1 What is involved in answering this question is determining the appropriate conditions for sets of models as set-theoretic counterparts of possible worlds in the semantic clauses for necessity and possibility. Allowing arbitrary sets of models as possible-world counterparts, where one can add or take away a model from such a set, does nothing by way explaining what is meant by metaphysical necessity.
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accessible from other metaphysically possible worlds, so that the modal logic for metaphysical necessity contains at least the laws of S5, and similarly that all conceptually possible worlds are accessible from each other so that an S5 modal logic applies to conceptual realism as well. We will assume, in other words, that the notions of metaphysical necessity and possibility correspond, at least roughly, to the equally difficult notions of conceptual necessity and possibility. Despite their similarity in this regard, there is an important difference between the conceptual and the metaphysical modalities, and hence a difference between logical realism and conceptual realism that we will explain later. For convenience, we will use and ♦ as modal operators for both metaphysical and conceptual necessity and possibility. We will also speak of possibilism and actualism as a distinction applicable to both logical and conceptual realism. The second-order modal predicate logics with nominalized predicates as abstract singular terms that result from this addition to λHST∗ and HST∗λ are called λHST∗ and HST∗λ .2
6.1
Actualism Versus Possibilism Redux
We assume that the logic of possible objects described in chapter two has been modified in accordance with §§5.2–5.3, so that axiom (A8), namely, (∃x)(a = x), where x does not occur in a, has been changed to (∀x)(∃y)(x = y), where x, y are distinct object variables. In other words, the logic of possible objects is now a “free logic,” just as the logic of actual objects is a free logic. The difference between them is that whereas what is true of every possible object is therefore also true of every actual object, the converse does not also hold; that is, (∀x)ϕ → (∀e x)ϕ is a basic law, but the converse is not. Strictly speaking, our so-called logic of “possible objects” now deals with more than merely possible objects, i.e., objects that actually exist in some possible world or other. Here, by existence—or for emphasis, actual existence—we mean existence as a concrete object, as opposed to the being of an abstract intensional object. Abstract intensional objects do not ever exist as actual objects in this sense even though they have being and are now taken as values of the bound object variables along with all possible objects. We can formulate this difference between actual existence and being as follows: (∀F n )¬E!(F ).
(¬E!(Abst))
2 For a detailed axiomatization and discussion of λHST∗ and HST∗ see Cocchiarella λ 1986.
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Now it is noteworthy that in actualism there is no distinction between being and existence; that is, actualism is committed to there being only actually existing objects. Actualism can allow for nominalized predicates, but only as vacuous, i.e., nondenoting, objectual terms. In that case, actualism will validate something like ¬E!(Abst), but expressed in terms of an actualist predicate quantifier as described below. In possibilism, or rather in what we are now calling possibilism, there is a categorial distinction between being and possible existence, which is expressed in part by the validity of ¬E!(Abst). In fact, this particular categorial distinction is one of the many—perhaps indeterminately many—conditions needed to characterize a metaphysical, or conceptual, possible world. For convenience, however, we will continue to speak of the first-order logic of being, which includes possible existence, as simply the logic of possibilism. The actualist quantifiers ∀e and ∃e when applied to predicate variables refer to those concepts (or properties) and relations that only existing, actual objects can fall under, or have, in any given possible world.3 The following, accordingly, are valid theses of the second-order logic of possible objects: (∀e F n )ϕ ↔ (∀F n )((∀x1 )...∀xn )[F (x1 , ..., xn ) → E!(x1 ) ∧ ... ∧ E!(xn )] → ϕ), and (∃e F n )ϕ ↔ (∃F n )((∀x1 )...∀xn )[F (x1 , ..., xn ) → E!(x1 ) ∧ ... ∧ E!(xn )] ∧ ϕ). We call the concepts, or properties, and relations that only actual existing objects can fall under at any time in any possible world existence-entailing concepts (properties) and relations. Now many concepts (or properties) and relations are such that only actual existing objects can have, or fall under, them. In fact these are the more common concepts (or properties) and relations that we ordinarily apply in our commonsense framework. Thus, for example, an object cannot be red, or green, or blue, etc., at any time in a given possible world unless that object exists in that world at that time. Similarly an object cannot be a pig or a horse at a time in a world unless it exists at that time in that world; nor, we should add, can there be a winged horse or a pig that flies unless it exists. Of course, in mythology there is a winged horse, namely Pegasus, and there could as well be a pig in fiction that flies. But that is not at all the same as actually being a winged horse or an actual pig that flies. Indeed, as far as fiction goes, there can even be a story in which there is an impossible object, such as a round square.4 But a fictional or mythological horse is not a real, actually existing horse, and 3 We should keep in mind that in this section we are characterizing actualism and “possibilism” (which now includes abstract objects as well) for both logical realism and conceptual realism, even though logical realism takes predicates to stand for properties and relations, whereas conceptual realism takes predicates to stand for concepts (whether monadic or relational). In logical realism, nominalized predicates denote the same properties and relations they stand for in their role as predicates, whereas in conceptual realism nominalized predicates denote the intensional contents of the concepts they stand for in their role as predicates. 4 See, e.g., the story Romeo and Juliet in Flatland in Cocchiarella 1996, § 7.11.
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one of the tasks of a formal ontology is to account for the distinction between merely fictional and actually existing objects. Later, in a subsequent chapter we will explain how the ontology of fictional, or mythological, characters and entities such as the winged horse Pegasus, can be accounted for as intensional objects. For now it is important to distinguish actual existence from being and merely possible existence. The two main theses of actualism are: (1) quantificational reference to objects can be only to objects that actually existence, and (2) quantificational reference to properties, concepts, or relations can be only to those that “entail” existence in the above sense, i.e., the only properties, concepts or relations there are according to actualism are those that only actually existing objects can have or fall under. What this means is that in actualism the quantifiers ∀e and ∃e , must be taken as primitive symbols when applied to object or predicate variables. The following, moreover, is a basic theorem of actualism. (∀e F n )[F (x1 , ..., xn ) → E!(x1 ) ∧ ... ∧ E!(xn )]. In regard to the concept of existence, note that the statement that every object exists, i.e., (∀e x)E!(x), is a valid thesis of actualism. In possibilism, however, the same statement is false, and in fact, given the being of abstract intensional objects, none of which ever exist as actual objects, it is logically false that every object exists; that is, ¬(∀x)E!(x), is a valid thesis of possibilism as we understand it here.5 What is true in both possibilism and actualism, on the other hand, is the thesis that to exist is to possess, or fall under, an existence-entailing concept or property; that is, E!(x) ↔ (∃e F )F (x) is valid in both actualism and possibilism. In possibilism, in fact, by taking quantification over existence-entailing concepts or properties as primitive, or basic, we can define existence as follows: E! =df [λx(∃e F )F (x)]. What this definition indicates in conceptualism is that the concept of existence is an “impredicative” concept. In other words, because existence is itself an existence-entailing concept—i.e., if a thing exists, then it exists—then, the concept of existence is formed or constructed in terms of a totality to which it belongs. This, in fact, is why according to conceptual realism the concept of existence is so different from ordinary existence-entailing concepts, such as being red, or green, a horse, a tree, etc. 5 The universal and null concepts [λx(x = x)] and [λx(x = x)] are h-stratified and therefore provably have intensional objects as their object-correlates in both λHST∗ and HST∗λ .
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Identity, incidentally, coincides with indiscernibility in both possibilism and actualism. In possibilism this thesis that identity coincides with indiscernibility is formulated as follows: (∀x)(∀y)(x = y ↔ (∀F )[F (x) ↔ F (y)]). In actualism, the thesis is stated in a more restricted way, namely, as (∀e x)(∀e y)(x = y ↔ (∀e F )[F (x) ↔ F (y)]). There is another difference on this matter if we formulate the thesis without the initial quantifiers. That is, whereas in possibilism, the following x = y ↔ (∀F )[F (x) ↔ F (y)] is valid, in actualism the related formula x = y ↔ (∀e F )[F (x) ↔ F (y)] is actually false in the right-to-left direction when neither x nor y exist. In other words, if x and y do not exist, then they vacuously fall under all the same existence-entailing concepts, namely none; and yet it does not follow that that x = y. Neither Pegasus nor Bellerophon actually exist, and yet it would be false to conclude that therefore Pegasus is Bellerophon. In other words, ¬E!(x) ∧ ¬E!(y) → (∀e F )[F (x) ↔ F (y)] is valid in both possibilism and actualism even when x = y. The principle of universal instantiation for actualist quantifiers can be formulated as follows in possibilism: (∃e F n )([λx1 ...xn ϕ] = F ) → ((∀e G)ψ → ψ[ϕ/G(x1 , ..., xn )]).
(∃/UIe2 )
In actualism, where λ-abstracts cannot be nominalized6 , the formulation, as follows, is somewhat more complex regarding its antecedent condition: (∃e F n )(∀x1 )(∀x2 )...(∀xn )[F (x1 , ..., xn ) ↔ ϕ] → ((∀e G)ψ → ψ[ϕ/G(x1 , ..., xn )]).
The comprehension principle for the actualist quantifiers is not as simple as the comprehension principle (CP∗λ ) for possibilism. It amounts to a kind of Aussonderungsaxiom for existence-entailing properties, concepts and relations: (∀e Gk )(∃e F n )([λx1 ...xn (G(x1 , ..., xk ) ∧ ϕ)] = F ),
(CPeλ )
where k ≤ n, and Gk and F n are distinct predicate variables that do not occur in ϕ. In the monadic case, for example, what this principle states is that although 6 Strictly speaking, actualism can allow for nominalized predicate expressions, but only by assuming as a basic principle that they denote nothing.
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we cannot expect every open formula ϕx to represent an existence-entailing concept or property, nevertheless conjoining ϕx with an existence-entailing concept or property G(x) does represent a concept or property that entails existence. The formulation in actualism, where the nominalized λ-abstract is not allowed (as a denoting singular term), is again more complex, but we will forego those details here.7 A simpler comprehension principle that is also valid in actualism is: (∃e F n )(∀e x1 )...(∀e xn )[F (x1 , ..., xn ) ↔ ϕ]. Before concluding this section we should note that although many of our commonsense concepts are concepts that only existing objects can have, nevertheless there are some concepts, especially relational ones, that can hold between objects that do not exist in the same period of time in our world or even in the same world. All animals, for example, have ancestors whose lifespans do not overlap with their own, and yet they remain their ancestors. An acorn that we choose to crush under our feet will never grow into an oak tree in our world, and yet, as a matter of natural possibility (as based, e.g., on the many-worlds interpretation of quantum mechanics), there is a world very much like ours in which we choose not to crush the acorn but leave it to grow into an oak tree. It is only a possible, and not an actual, oak tree in our world, but there is still a relation between it and the acorn that we crushed, just as there is a ancestral relation between animals whose lifespans do not overlap.
6.2
Intensional Possible Worlds
The actual world, according to conceptual realism, consists of physical objects, states of affairs, and events of all sorts that are structured in terms of the laws of nature and the natural kinds of things there are in the world. There are intensional objects as well, to be sure, but these have being only as natural products of the evolution of consciousness, language and culture; and without language and thought they would not be at all. Abstract intensional objects, in other words, have a mode of being dependent on language and culture, and are not part of what makes up the physical world. Possible worlds are like the actual world, moreover, i.e., they consist of physical objects and events structured in terms of the laws of nature as described earlier, e.g., in §§3.6.1–3.6.2. In speaking of possible worlds here—and even of the actual world—we should be cautious to note that possible worlds are not “objects,” as when we speak of the various possible kinds of objects and events in the world. In other words, in conceptual realism possible worlds are not values of the bound object variables, and therefore they are not quantified over as objects. It is true that we seem to quantify over possible worlds in our semantical set-theoretic metalanguage. But, to be precise, what we really quantify over are set-theoretical 7 See Cocchiarella 1986, chapter III, §9, for a detailed axiomatization of actualism. For a completeness theorem for the second-order logic of actual and possible objects without abstract objects see Cocchiarella 1969.
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model structures that we call possible worlds. These set-theoretical model structures are not possible worlds in any literal metaphysical sense; rather, they are abstract mathematical objects that we use to represent the different possible situations described in the object-language of our formal ontology by means of modal operators. Unlike the model-theoretic “possible-world” parameters of the set-theoretical metalanguage, the modal operators of a formal ontology can be iterated and can occur within the scope of other occurrences of the same, or of a dual, operator. In addition, unlike the “external relations” expressed in the metalanguage between model-theoretic “possible worlds” and objects in the domains of those worlds, modal operators in the language of the formal ontology express “internal relations” between objects and the properties or concepts they fall under.8 Possible worlds in logical realism are not really different from those in conceptual realism, i.e., they are made up of the same physical objects and events, and, as in conceptual realism, they do not themselves “exist” in different possible worlds. They are not themselves a kind of concrete object in addition to the possible situations represented by the modal operators.9 It is significant, however, that there are abstract objects in logical realism that are the intensional counterparts of possible worlds. This is because, unlike the situation in conceptual realism, intensional objects have a mode of being in logical realism (Platonism) that is independent of the natural world, and even of whether or not there is a natural world, and in that sense they are independent of all possible worlds. That is, even though abstract intensional objects in general do not “exist” as concrete objects—and hence unlike the latter do not “exist” in some possible worlds and fail to “exist” in others—nevertheless, in logical realism they have a mode of being (namely, being simpliciter) that is independent of all possible worlds. These abstract intensional objects have being in a Platonic realm that is so richly structured logically that it even includes objects that are intensional counterparts of possible worlds. There are no intensional counterparts of possible worlds in conceptual realism, on the other hand, because in this framework all abstract intensional objects are ontologically grounded in terms of the human capacity for thought and concept-formation. The intensional counterparts of possible worlds in logical realism, other words, cannot be accounted for in conceptual realism because they far exceed our cognitive abilities in the construction, or projection, of such counterparts. It is in the ontological status of these kinds of entities that we begin to see an important difference between logical realism and conceptual (intensional) realism. Now there are at least two kinds of intensional objects in the ontology of logical realism that are counterparts of metaphysically possible worlds. For 8 In logical atomism, which provides the only ontology suitable for the logical modalities, all “internal” relation are strictly formal relations. 9 In Plantinga 1974, which we take to be a kind of Platonic realism, the only possible worlds considered are what we describe below as the intensional counterpart to a possible world. David Lewis, on the other hand, definitely takes possible worlds to be concrete objects. Lewis, however, is not a Platonist.
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convenience, we will call both kinds intensional possible worlds. Our first such kind is a “world” proposition; that is, a proposition P that is both possible, i.e., ♦P , and maximal in the sense that for each proposition Q, either P entails Q or P entails ¬Q, where by “entailment” we mean necessary material implication, i.e., either (P → Q) or (P → ¬Q).10 Where P is a possible-world counterpart in this sense, we read (P → Q) as ‘Q is true in P ’. This notion of an intensional possible world can be defined by the following λ-abstract: P oss-W ld1 =df [λx(∃P )(x = P ∧ ♦P ∧ (∀Q)[(P → Q) ∨ (P → ¬Q)])]. Note that this λ-abstract is a homogeneously stratified, and therefore P ossW ld1 stands for a property in logical realism. It also stands for a concept in conceptual (intensional) realism, where it is not the concept that is impossible to construct but the being of the propositions (intensional objects) that fall under the concept. This is because, as already noted, such intensional objects go beyond what we can construct by means of our cognitive abilities. In logical realism, on the other hand, there is not only such a property as a value of the bound predicate variables of both λHST∗ and HST∗λ , but, in addition, its nominalization denotes in both of these systems a value of the bound objectual variables as well. Of course, the fact that P oss-W ld1 is a well-formed predicate that stands for a property, or concept, does not mean that it must be true of anything, i.e., that there must be propositions that have this property, or fall under this concept. In fact, as already noted, any proposition that falls under this concept as an intensional object has content that so far exceeds what is cognitively possible for humans to have as an object of thought that the “existence,” or rather being, of such a proposition can in no sense be validated in conceptual realism. But then, in logical realism, propositions are Platonic entities existing independently of the world and all forms of human cognition, and therefore the possible objectual being of a proposition P such that P oss-W ld1 (P ) is not constrained in logical realism by what is cognitively possible for humans to have as an object of thought. Nevertheless, we cannot prove that there are intensional possible worlds in this sense in either λHST∗ and HST∗λ , unless some axiom is added to that effect. One such axiom would be the following, which posits the being of a proposition corresponding to each possible world (or model of the metalanguage). (∃P )[P oss-W ld1 (P ) ∧ P ].
(∃Wld1 )
Of course, one immediate consequence of (∃Wld1 ) is that some intensional possible world now obtains, i.e., (∃P )[P oss-W ld1 (P ) ∧ P ], which we can refer to as “the intensional (counterpart of the) actual world.” A criterion of adequacy for this notion of a possible world is that it should yield the type of results we find in the set-theoretic semantics for modal logic. One such result is that a proposition is true, i.e., now obtains, if, and only if, 10 See
Prior & Fine, 1977 for a discussion of this approach to intensional possible worlds.
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it is true in the actual world. This result is in fact provable on the basis of (∃Wld1 ), i.e., Q ↔ (∃P )[P oss-W ld1 (P ) ∧ P ∧ (P → Q)], is provable in both λHST∗ and HST∗λ if (∃Wld1 ) is added as an axiom. From this another appropriate result follows; namely, that a proposition Q is possible, i.e., ♦Q, if it true in some possible world; that is, ♦Q ↔ (∃P )[P oss-W ld1 (P ) ∧ (P → Q)], is provable in both λHST∗ + (∃Wld1 ) and HST∗λ + (∃Wld1 ). Finally, another appropriate consequence is that if Q and Q are true in all the same possible worlds (models), then they are necessarily equivalent; and, conversely, if they are necessarily equivalent, then they are true in all the same possible worlds; that is, (∀P )(P oss-W ld1 (P ) → [([P → Q] ↔ [P → Q ]) ↔ (Q ↔ Q )]) is provable in λHST∗ + (∃Wld1 ) and HST∗λ + (∃Wld1 ). It does not follow, however, that propositions are identical if they are true in all of the same intensional possible worlds. There is yet another notion of an intensional possible world that can have instances in logical realism but not in conceptual realism. This is the notion of an intensional possible world as a property in the sense of “the way things might have been.” David Lewis, for example, claimed that possible worlds are “ways things might have been,” but, curiously, according to Lewis the “ways that things might have been” are concrete objects, not properties.11 Robert Stalnaker, who is an actualist, pointed out, however, that “the way things are is a property or state of the world, [and] not the world itself,” as Lewis would have it.12 In other words, according to Stalnaker, “the ways things might have been” are properties. For an actualist such as Stalnaker this means that there are possible worlds qua properties, but, except for the actual world, which is concrete, they are all uninstantiated properties. This is because, according to Stalnaker, only concrete worlds could be instances of such properties, and as an actualist the only such concrete instance is the actual world. In other words, although there are possible worlds qua properties, according to Stalnaker, nevertheless there can be no possible worlds qua instances of those properties other than the actual world, because such instances would then be concrete and yet not actual, which is what actualism rejects. In logical realism, however, the situation is quite different. In the ontology of logical realism, the “ways things might have been” are properties, but they are not properties of metaphysically possible worlds as concrete objects. Rather, 11 See
Lewis 1973, p. 84. The relevant text is reprinted in Loux, 1979, p. 182. 1976, p. 228.
12 Stalnaker
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they are properties of propositions as intensional objects; in particular, they are properties of all and only the propositions that are true in the possible world in question. Now a property of all and only the propositions that are true in a possible world (model) A could not be an intensional counterpart of the world A if the extension of that property were different in different possible worlds (models). Possible worlds are different, in other words, if the propositions true in those worlds are different. What is needed is a property of propositions that does not change its extension from possible world to possible world. We will call such a property a “rigid” property, which we “define” as follows:13 Rigidn(F n )
:
(∀x1 )...(∀xn )[F (x1 , ...xn ) ∨ ¬F (x1 , ...xn )].
Note that because a rigid property will have the same extension in every possible world (model), the extension of that property can then be identified ontologically with the property itself. The type of intensional possible world that is now under consideration is that of a rigid property that describes “the ways things might have been” in a given given metaphysically possible world. It does this by rigidly holding of all and only the propositions that are true in that world. This notion can be specified by a homogeneously stratified formula, which means that the property of being an intensional possible world in this sense can be defined by means of a λ-abstract as follows: P oss-W ld2 =df [λx(∃G)(x = G∧Rigid1 (G)∧♦(∀y)[G(y) ↔ (∃P )(y = P ∧P )])]. Now, as with our first notion of an intensional possible world, the claim that there are possible worlds in this second sense is also not provable in either λHST∗ or HST∗λ , unless we add an assumption to that effect. One such assumption is the following, which says that there is such a possible-world property G, i.e., P oss-W ld2 (G), that holds in any possible world of all and only the propositions that are true in that world: (∃G)(P oss-W ld2 (G) ∧ (∀y)[G(y) ↔ T rue(y)]).
(∃Wld2 )
Here by T rue(y) we mean that y is a proposition that is the case, i.e., T rue =df [λy(∃P )(y = P ∧ P )], which, because the λ-abstract in question is homogeneously stratified, specifies a property, or concept, in both λHST∗ and HST∗λ . Now it turns out that we do not have to assume either of the new axioms, (∃Wld1 ) or (∃Wld2 ), to prove that there are intensional possible worlds in the formal ontology of logical realism in either of these two senses. Both, in 13 Rigidity can be λ-defined as a predicate in λHST∗ , but not in HST∗ , where it must λ be construed only as an abbreviation in the principle of rigidity described below. That is why we present rigidity in the present form.
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fact, can be derived in the modal systems λHST∗ and HST∗λ from what we will call the principle of rigidity, (PR), which stipulates that every property, or concept, F , is co-extensive in any possible world with a rigid property, i.e., a rigid property that can in effect be taken as the extension of F (in that world).14 We formulate the principle of rigidity as follows: (∀F n )(∃Gn )(Rigidn (G) ∧ (∀x1 )...(∀xn )[F (x1 , ...xn ) ↔ G(x1 , ...xn )]). (PR) The thesis that there is a rigid property corresponding to any given property is intuitively valid in logical realism where properties have a mode of being that is independent of our ability to conceive or form them as concepts. In conceptual realism, on the other hand, the thesis amounts to a “reducibility axiom” claiming that for any given concept or relation F we can construct a corresponding rigid concept or relation that in effect represents the extension of the concept F . Such a “reducibility axiom” is much too strong a thesis about our abilities for concept-formation. That (∃Wld2 ) is derivable from (PR) follows from the fact that T rue represents a property in these systems; that is, (∃G)(Rigid2(G) ∧ (∀y)[G(y) ↔ T rue(y)]) is provable on the basis of (CP∗λ ) in both λHST∗ +(PR) and HST∗λ +(PR), and therefore, by the rule of necessitation and obvious theses of S5 modal logic, it follows that (∃Wld2 ) is derivable from (PR). A similar argument, which we will not go into here, shows that (∃Wld1 ) is also derivable from in λHST∗ + (PR) and HST∗λ + (PR). The fact that with (PR) we can prove in both of the theories of predication λHST∗ and HST∗λ that there are intensional possible worlds in either the sense of P oss-W ld1 or P oss-W ld2 is significant in more than one respect. On the one hand it indicates the kind of ontological commitment that logical realism has as a modern form of Platonism. On the other hand, it also indicates a major kind of difference between logical realism and conceptual realism, because, unlike logical realism, the principle of rigidity is not valid in conceptual realism. What it claims about concept-formation is not cognitively realizable for humans. Nor can there be intensional possible worlds in the sense either of P oss-W ld1 or P oss-W ld2 in conceptual realism, because such intensional objects are not cognitively realizable in human thought and concept-formation. Here, with the principle of rigidity and the notion of a proposition as the intensional counterpart of a possible world, we have clear distinction between logical realism as a modern form of Platonism and conceptual realism as a modern form of conceptualism, i.e., as a form of conceptual intensional realism. Also, what these differences indicate is that the notions of metaphysical necessity 14 The idea of representing the extension of a property by means of a rigid property was first suggested by Richard Montague.A type-theoretical version of the thesis was used as a principle of “extensional comprehension” by Dan Gallin in his development of Montague’s intensional logic. See Montague 1974, p.132, and Gallin 1975, p. 77.
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133
and possibility in logical realism are not the same as the notions of conceptual necessity and possibility in conceptual realism, at least not if conceptual possibility is grounded in what is cognitively realizable in human thought and concept-formation. Logical realism: The principle of rigidity, (PR), is valid, and therefore so are (∃Wld1 ) and (∃Wld2 ). That is, there are intensional possible worlds in logical realism in the sense of P oss-W ld1 as well as of P oss-W ld2 . Conceptual realism: The principle of rigidity, (PR), is not valid, and there can be no intensional possible worlds in the sense of either P oss-W ld1 or P oss-W ld2 , because such intensional objects exceed what is cognitively realizable in human thought and conceptformation. Therefore, metaphysical necessity and possibility are the not the same as conceptual necessity and possibility.
6.3
Summary and Concluding Remarks
• How the notion of a metaphysically possible world can be characterized is problematic, and it is difficult to determine what notion of a metaphysically possible world, if any, is appropriate for logical realism. The notion of a conceptually possible world is similarly problematic, and it is similarly difficult to determine what notion of a conceptually possible world, if any, is appropriate for conceptual realism. • We assume, for convenience of comparison, that the notions of metaphysical necessity and possibility correspond, at least roughly, to the equally difficult notions of conceptual necessity and possibility, and that both can be represented by the modal logic S5. • Actualism and possibilism can be given a fuller ontological explanation in terms of a distinction between concepts, or properties and relations, that entail (concrete) existence and those that do not. • To exist is to possess, or fall under, an existence-entailing concept in conceptual realism, or an existence-entailing property in logical realism. Existence, accordingly, is an impredicative concept or property. • Abstract objects have being but do not exist, where by existence we mean concrete existence. In logical realism abstract objects “exist” independently of whether there is a world or not. In conceptual realism, abstract objects do not “exist” independently of the world, and in particular they do not “exist” independently of consciousness, language and culture. • In both logical and conceptual realism, possible worlds, including the actual world, are made up exclusively of physical objects and events. Possible worlds are not objects in either ontology and do not themselves “exist” in different worlds. • In logical realism, but not in conceptual realism, there are abstract intensional objects that are the counterparts of possible worlds. These abstract
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objects, like all abstract objects, have being only in a Platonic realm separate from all possible worlds. This is a fundamental difference between logical and conceptual realism. • There are two kinds of abstract possible-world counterparts in logical realism, both of which are called intensional possible worlds. The first kind is a world-proposition, i.e., a proposition P that is both possible and maximal in the sense that for each proposition Q, either P entails Q or P entails ¬Q. The second kind of intensional possible world is a “rigid” property that describes “the ways things might have been” in a given metaphysically possible world by rigidly holding of all and only the propositions that are true in that world. • The “existence” (or really being) of both kinds of intensional possible worlds is derivable in the modal systems λHST∗ and HST∗λ from the principle of rigidity, (PR), which stipulates that every property, or concept, F , is co-extensive in any possible world with a rigid property, i.e., a rigid property that can in effect be taken as the extension of F (in that world). • The principle of rigidity, (PR), is valid in logical realism. It is not valid in conceptual realism because the intensional objects it posits exceed what is cognitively realizable in human thought and concept-formation. This is a fundamental difference between the formal ontologies of logical and conceptual realism.
6.4
Appendix 1
The well-formulas of the logic λHST∗ are as defined in chapter five, with the proviso that only h-stratified λ-abstracts are well-formed in λHST∗ . The axioms of λHST∗ consist of those of standard first-order logic as described in chapter two together with the comprehension principle (CP∗λ )
(∃F )([λx1 ...xn ϕ] = F ),
but, again, with the understanding that the λ-abstract [λx1 ...xn ϕ] must be h-stratified. There are also an axiom regarding the distribution of a universal predicate quantifier over a conditional and an axiom regarding vacuous predicate quantifiers: (∀F )[ϕ → ψ] → [(∀F )ϕ → (∀F )ψ], (∀F )[ϕ → ψ] → [ϕ → (∀F )ψ], where F is not free in ϕ. The rule of λ-conversion is also an axiom scheme of λHST∗ , [λx1 ...xn ϕ](a1 , ..., an ) ↔ ϕ(a1 /x1 , ..., an /xn )
(λ-Conv∗ )
where each ai is free for xi , for 1 ≤ i ≤ n, as well as an axiom for rewrite of bound variables, [λx1 ...xn ϕ] = [λy1 ...yn ϕ(y1 /x1 , ..., yn /xn )], and a final axiom [λx1 ...xn F (x1 , ..., xn )] = F,
(Id∗λ )
6.4. APPENDIX 1
135
where F is an n-place predicate variable. The only inference rules of λHST∗ are modus ponens and universal generalization for both predicate and individual variables. The formulas of HST∗λ are the same as those for λHST∗ , except that λabstracts that are not h-stratified are allowed as well. The first-order axioms are those of standard first-order logic as described in chapter two, except that axiom (A8), i.e., (∃x)(a = x), where x is not a, is replaced by the free-logic axiom (∀x)(∃y)(x = y) together with the identity axiom (a = a) for all object terms a. In other words, standard first-order logic is replaced by a logic free of existential presuppositions regarding object terms. The comprehension principle (CP∗λ ) for HST∗λ is as above, except, again, non-h-stratified λ-abstracts are no longer excluded. The remaining axioms and inference rules are the same as those for λHST∗ , except for the rule of λ-conversion, which is replaced by the following free-logic version: [λx1 ...xn ϕ](a1 , ..., an ) ↔ (∃x1 )...(∃xn )(a1 = x1 ∧ ... ∧ an = xn ∧ ϕ), (∃/λ-Conv∗ ) where, for all i, j ≤ n, xi does not occur in aj . We can extend either λHST∗ or HST∗λ so as to contain a logic of existence by adding the actualist quantifier ∀e (with ∃e defined as usual) and then redefine the notion of a well-formed formula so as to include those containing the actualist quantifier as well. We also add the modal operator (and, by definition, ♦) with axioms for S5 modal logic. The axioms of first-order actualism as described in chapter two are then added as well as a distribution over conditionals axiom and a vacuous quantifier axiom for ∀e when applied to predicate variables: (∀e F )[ϕ → ψ] → [(∀e F )ϕ → (∀e F )ψ], (∀e F )[ϕ → ψ] → [ϕ → (∀e F )ψ],
where F is not free in ϕ.
There are two special axioms for the existence-entailing concepts or properties, namely, (∃e F )([λx1 ...x2 (∃e G)G(x1 , ..., xn )] = F ), where F ,G are distinct n-place predicate variables, and (∀e Gk )(∃e F n )([λx1 ...x2 (∃e G)(G(x1 , ..., xk ) ∧ ϕ)] = F ), where n ≤ k, F is a n-place predicate variable and G is a k-place predicate variable, neither of which occur free in ϕ. The first axiom stipulates, e.g., in the monadic case that falling under an existence-entailing concept, or having an existence-entailing property, is itself an existence-entailing concept or property. The second axiom is a schema that amounts in effect to an Aussonderungs axiom for existence-entailing concepts or properties and relations. The hierarchy of predicative concepts described in the previous chapter can be retained in conceptual realism in case we want to represent the impredicativity of certain concepts, such as existence defined as follows: E! = [λx(∃e F )F (x)].
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That is, for each positive integer n, we will have, ¬(∃j Gn )(E! = G), as a valid thesis for each predicative stage j. Similarly, we can note the impredicativity of the Russell concept, or property, as follows: ¬(∃j Gn )([λx(∃F )(x = F ∧ ¬F (x))] = G).
Part II
Conceptual Realism
137
Chapter 7
The Nexus of Predication A universal, we have said, is what can be predicated of things.1 But what exactly do we mean in saying that a universal can be predicated of things? In particular, how, or in what way, do universals function in the nexus of predication? In nominalism, there are no universals, and the only nexus of predication is the linguistic nexus between subject and predicate expressions (or tokens of such). What this means in nominalism is that only predicates can be true or false of things. But what are the semantic grounds for predicates to be true or false of things? Are there really no concepts as cognitive capacities involved in such grounds? What then accounts for the unity of a sentence in nominalism as opposed to a mere sequence of words? Can nominalism really explain the unity of the linguistic nexus? In logical realism, which is a modern form of Platonism, universals exist independently of language, thought, and the natural world, and even of whether or not there is a natural world. Bertrand Russell and Gottlob Frege, as we have noted, described two of the better known versions of logical realism.2 In Russell’s early form of logical realism, for example, universals are constituents of propositions, where the latter are independently real intensional objects. The nexus of predication in such a proposition, according to Russell, is a relation relating the constituents and giving the proposition “a unity” that makes it different from the sum of its constituents.3 Thus, according to Russell, “a proposition ... is essentially a unity, and when analysis has destroyed this unity, 1 Cf. Aristotle, De Interpretatione, 17a 39. Some of the material in this chapter occurred in my paper in Metalogicon, VI, 2003. Some critical points have been revised. 2 We have in mind here mainly the 1903 Russell of The Principles of Mathematics. Russell’s later turn in 1914 to logical atomism is a turn to a form of natural realism. 3 See Russell 1903, §55, p. 52. Russell is unclear in 1903 about what relation is the unity of the proposition expressed by ‘Socrates is human’ and others of this type. A solution is proposed in Cocchiarella 1987, chapter 2, §5, where this proposition is rephrased as ‘Socrates is a human being’, and where the verb ‘is’ stands for the relation of identity and ‘a human being’ stands for what Russell in 1903 called a denoting concept. The result on this analysis expresses the proposition that Socrates is identical with a human being.
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140
no enumeration of constituents will restore the proposition. The verb [i.e., the relation the verb stands for], when used as a verb, embodies the unity of the proposition ....”4 Now a relation, in Russell’s modern form of Platonism, can also occur in a proposition as a term, i.e., as one of the constituents related. That is why we can have formulas of the form R(x, R) as well as R(x, y). But then how can a relation occur as a term in some propositions and in others, and perhaps even in the same proposition, as the unifying relating relation? That is, how can a relation have a predicative nature holding the constituents of a proposition together and also be one of the objects held together by the relating relation of that proposition? This was something Russell was unable to explain.5 Frege introduced a fundamental new idea regarding the unity of a proposition and the nexus of predication.6 This was his notion of an unsaturated function, which applies to the nexus of predication in language as well as to propositions as abstract entities. On the unsaturated nature of a predicate as the nexus of predication of a sentence, Frege claimed that “this unsaturatedness ... is necessary, since otherwise the parts [of the sentence] do not hold together”.7 On the unsaturated nature of the nexus of predication of a proposition, Frege similarly claimed that “not all parts of a proposition can be complete; at least one must be ‘unsaturated’, or predicative; otherwise, they would not hold together.”8 It is the unsaturated nature of a predicate and the properties and relations it stands for that accounts for both predication in language and the unity of a proposition, according to Frege.9 Now in Frege’s ontology properties and relations of objects are functions that assign the truth values “the true” or “the false” to objects. These truth values are abstract objects, but, apparently, they are not the properties truth and falsehood that propositions have in Russell’s form of Platonism. In any case, all functions, including functions from numbers to numbers, have an unsaturated nature according to Frege. Objects, on the other hand, and only objects, have a saturated nature, and therefore functions, being unsaturated, cannot be objects. This distinction between functions and objects is fundamental in Frege’s ontology, and, as we will see, it has a counterpart in conceptualism. Predication in Frege’s ontology, as we have noted, is explained in terms of functionality, which is contrary to the usual understanding of functionality in terms of predication, i.e., in terms of many-one relations.10 But conceptually it is predication that is more fundamental than functionality. We understand what it means to say that a function assigns truth values 4 Russell 5 Ibid.
6 Frege
1903, p. 50.
used the word ‘Gedanke’ for what we are here calling a proposition. A Gedanke in Frege’s ontology is not a thought in the sense of conceptualism but an independently real intensional object expressed by a sentence. 7 Frege 1979, p. 177. 8 Frege 1952, p. 54. 9 Frege usually referred to properties (Eigenshaften) as concepts; but we will avoid that terminology here so as not confuse Frege’s realism with conceptualism. 10 Compare Russell 1903, p. 83.
7.1. PREDICATION IN NATURAL REALISM
141
to objects, for example, only by knowing what it means to predicate concepts, or properties and relations, of objects. Nevertheless, aside from this reversal of priority between predication and functionality, Frege’s real contribution to the analysis of the nexus of predication is his view of the unsaturated nature of universals as the ground of their predicative nature. Something like this view is basic to the way the nexus of predication is explained in conceptualism.
7.1
Predication in Natural Realism
Natural realism is different from logical realism, we have noted, in that for natural realism universals do not exist independently of the natural world and its causal matrix. Universals exist only in things in nature, or at least in things that could exist in nature, and whether or not a predicate stands for such a universal is strictly a factual, and not a logical, matter. Logical atomism is a form of natural realism that provides a clear and useful account of predication in reality. In particular, in the Tractatus LogicoPhilosophicus, Wittgenstein replaced Frege’s unsaturated logically real properties and relations (as functions from objects to truth values) with unsaturated “material”, i.e., natural, properties and relations as the modes of configuration of atomic states of affairs. Reality, on this account, is just the totality of atomic facts—i.e., the states of affairs that obtain in the world; and the nexus of predication of a fact is the material property or relation that is the mode of configuration of that fact (atomic state of affairs). This is similar to Russell’s theory of a relating relation as what unifies a proposition, except that instead of a proposition as an abstract intensional entity we now have facts, or states of affairs, and instead of a logically real relation we have a natural property or relation as the nexus, or mode of configuration, of such a state of affairs. Also, because natural properties and relations have an unsaturated nature as the nexuses of predication, they cannot themselves be objects in states of affairs, unlike the situation in Russell’s early Platonist ontology. One of the major flaws of logical atomism, however, is its ontology of simple material objects (bare particulars?). The idea that the complex natural world is reducible to ontologically simple objects and atomic states of affairs is a difficult, if not impossible, thesis to defend. It is even more difficult to defend the added claim, which is also made in logical atomism, that all meaning and analysis must be based on ontologically simple objects and the atomic states of affairs in which they are configured. But having natural properties and relations as modes of configuration of states affairs—i.e., as the nexuses of predication in reality—is an important and useful view. In fact, we can retain this view of natural properties and relations even though we reject the idea of simple objects. Something very much like this is exactly what we have in conceptual natural realism, where instead of the simple material objects of logical atomism we have complex physical objects as the constituents of states of affairs. Conceptual natural realism, as
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we noted in our introductory lecture, is a modern counterpart to Aristotle’s natural realism, just as conceptual intensional realism is a mitigated, modern counterpart to conceptual Platonism; and both are taken as part of what we mean by conceptual realism. Also, if we add to the logic of conceptual natural c realism the modal operator for a causal or natural necessity and also add a logic of natural kinds, then we get a modern form of Aristotelean essentialism.11 But this is a topic we will turn to and develop in more detail in a later lecture.
7.2
Conceptualism
What underlies our capacity for language and predication in language, according to conceptualism, is our capacity for thought and concept formation, a capacity that is grounded in our evolutionary history and the social and cultural environment in which we live. Predication in thought is more fundamental than predication in language, in other words, and this is so because what holds the parts of a sentence together in a speech act are the cognitive capacities that underlie predication in thought. There are two major types of cognitive capacities that characterize the nexus of predication in conceptualism. These are: (1) a referential capacity, and (2) a predicable capacity. These capacities underlie our rule-following abilities in the use of referential and predicable expressions. Predicable concepts, for example, are the cognitive capacities that underlie our abilities in the correct use of predicate expressions. When exercised in a speech or mental act, a predicable concept is what informs that act with a predicable nature—a nature by which we characterize or relate objects in a certain way. A predicate expression whose use is determined in this way is then said to stand for the concept that underlies its use. Referential concepts, on the other hand, are cognitive capacities that underlie our use of referential expressions. Referential concepts are what underlie the intentionality and directedness of our speech and mental acts. When exercised a referential concept informs a speech or mental act with a referential nature. A referential expression whose use is determined in this way is said to stand for the concept that underlies that use. Referential and predicable concepts are a kind of knowledge, more specifically a knowing how to do things with referential and predicable expressions. They are not a form of propositional knowledge, i.e., a knowledge that certain propositions about the rules of language are true, even though they underlie the rule-following behavior those rules might describe. Referential and predicable concepts are objective cognitive universals. 11 See chapter 12 for a more detailed account of conceptual natural realism as a modern form of Aristotelian essentialism.
7.2. CONCEPTUALISM
143
The objectivity of referential and predicable concepts does not consist in their being independently real universals, however; that is, they do not have the kind of objectivity universals are assumed to have in logical realism. The objectivity of referential and predicable concepts consists in their being intersubjectively realizable cognitive capacities that enable us to think and communicate with one another. As intersubjectively realizable cognitive capacities, moreover, concepts are not mental objects—e.g., they are not mental images or ideas as in the traditional conceptualism of British empiricism—though when exercised they result in objects, namely speech and mental acts, which are certain types of events. In particular, as cognitive capacities that (1) may never be exercised, or (2) that may be exercised at the same time by different people, or (3) by the same person at different times, concepts are not objects at all but have an unsaturated nature analogous to, but not the same as, the unsaturated nature concepts are said to have in Frege’s ontology. Unlike the concepts of Frege’s ontology, however, which are functions from objects to truth values, the concepts of conceptualism are cognitive capacities that when exercised result in a speech or mental act (which may be either true or false). Another important feature of predicable and referential concepts is that each has a cognitive structure that is complementary to the other—a complementarity that is similar to, but also different from, that between the functions that predicates stand for and those that quantifier phrases stand for in Frege’s ontology. In conceptualism, it is the complementarity between predicable and referential concepts that underlies the mental chemistry of language and thought. In particular, as complementary, unsaturated cognitive capacities, predicable and referential concepts mutually saturate each other when they are jointly exercised in a speech or mental act. In conceptualism, in other words, the nexus of predication is the joint exercise of a referential and a predicable concept, which interact and mutually saturate each other in a kind of mental chemistry. A judgment or basic speech act of assertion, for example, is the result of jointly exercising a referential and a predicable concept that underlie the use, respectively, of a noun phrase (NP) as grammatical subject and a verb phrase (VP) as predicate:
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CHAPTER 7. THE NEXUS OF PREDICATION S NP ... VP ⇑ (nexus of predication)
In conceptualist terms this act can be represented as follows: Assertion (Judgment) referential act
..... predicable act ⇑ (nexus of predication) (mutual saturation)
Here, of course, by a referential act we mean the result of exercising a referential concept, and by a predicable act the result of exercising a predicable concept.
7.3
Referential Concepts
Now by a referential expression, i.e., the kind of expression that stands for a referential concept, we do not mean just proper names and definite descriptions, such as ‘Socrates’ and ‘The man who assassinated Kennedy’, but any of the types of expressions that functions in natural language as grammatical subjects, which includes quantifier phrases such as ‘All citizens’, ‘Most democrats’, ‘Few voters’, ‘Every raven’, ‘Some raven’, etc.12 In fact, in conceptual realism, only a quantifier phrase has the kind of unsaturated structure that is complementary to a predicate expression the way the structure of a referential concept is complementary to that of a predicable concept. For this reason we will represent all of the different kinds of referential expressions in conceptual realism as quantifier phrases. Referential concepts are what quantifier phrases stand for in conceptual realism, just as predicable concepts are what predicate expressions stand for. Consider, for example, a judgment that every raven is black. In conceptual realism, this judgement is analyzed as the result of jointly exercising, and mutually saturating, (a) the predicable concept that the predicate phrase ‘is black’ stands for with (b) the referential concept that the referential phrase ‘Every raven’ stands for. 12 We will not deal with the logic of determiners such as ‘most’, ‘few’, ‘several’, etc., in this book. Instead we restrict ourselves to the universal and existential quantifier phrases.
7.3. REFERENTIAL CONCEPTS
145
[Every raven]N P [is black]V P (∀xRaven) ... Black(x) (∀xRaven)Black(x). A negative judgment expressed by ‘Some raven is not black’ is analyzed similarly as: [Some raven]N P [is not black]V P (∃xRaven) ... [λx¬Black(x)]( ) (∀xRaven)[λx¬Black(x)](x). The negation in this judgment is internal to the predicate, which is analyzed as the complex predicate expression [λx¬Black(x)]( ). Now what this view of referential expressions requires is that the logical grammar of conceptual realism must be expanded to include a category of common nouns, or what we instead call common names.13 Common names, as the above examples indicate, will occur as parts of referential-quantifier phrases. Actually it is not just common names that can occur as parts of quantifier phrases, but proper names as well. In other words, instead of a category of common names, what we now add to the logical grammar of conceptual realism is a category of names, which includes proper names as well as common names. The nexus of predication in conceptual realism, as we have said, is the mutual saturation of a referential act with a predicable act, which means that objectual reference, e.g., the use of a proper name as a grammatical subject, is not essentially different from general reference, such as the use of the quantifier phrases ‘Every raven’ and ‘Some raven’ in the above examples. Thus, instead of proper names and common names being different types of expressions, in conceptual realism we have just one logical category of names, with common names and proper names as two distinct subcategories. Names proper names common names What the difference is between proper names and common names is a matter we will take up in the next section in our discussion of objectual reference.14 13 We will restrict ourselves to common names that are common count nouns. The logic of mass nouns will not be covered in this book. 14 See see chapter 10, or Cocchiarella 2002, for a detailed formal description and separate development of the logic of names.
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Now in addition to complex predicates, which are accounted for by λ-abstracts, we also need to account for complex referential expressions. What we mean by a complex referential expression is a quantifier phrase containing a complex common name, i.e., a common name restricted by a defining relative clause. To syntactically generate a complex common name, we use a forward slash, ‘/’, as a binary operator on (a) expressions from the category of common names and (b) formulas as defining relative clauses. For example, by means of this operator we can symbolize the restriction of the common name ‘citizen’ to ‘citizen (who is) over eighteen’, or more briefly, ‘citizen (who is) over-18’, as follows: Citizen (who is) over 18 ↓ ↓ Citizen whox is over 18 Citizen/Over-18(x) An assertion of the sentence ‘Every citizen (who is) over eighteen is eligible to vote’ can then be symbolized as: [Every citizen (who is) over eighteen]N P [is eligible to vote]V P (∀xCitizen/Over-18(x)) Eligible-to-vote(x) (∀xCitizen/Over-18(x))Eligible-to-vote(x) Now there is a difference in conceptual realism, we should note, between an initial level at which the logical analysis of a speech or mental act of a given context is represented, and a subsequent, lower level where inferences and logical deductions can be applied to those analyses. This means that we need rules to connect the logical forms that represent speech and mental acts with the logical forms that represent the truth conditions and logical consequences of those acts in a more logically perspicuous way. For example, where the standard quantifier phrases of our previous chapters are now understood at least implicitly as containing the ultimate, superordinate common name ‘object’, i.e., where the quantifier phrases (∀x)
and
(∀xObject)
and
(∃x)
are now read as (∃xObject).
then we can connect our new way of representing speech and mental acts on the initial level of logical analysis with the more standard way on the lower, deductive level, by means of such rules as the following: (∀xA)F (x) ↔ (∀x)[(∃yA)(x = y) → F (x)]
(MP1)
7.4. SINGULAR REFERENCE AND PROPER NAMES (∃xA)F (x) ↔ (∃x)[(∃yA)(x = y) ∧ F (x)]
147 (MP2)
For example, by means of these rules we can see why the argument: (∀xA)F (x) (∃yA)(b = y) ∴ F (b) is valid in this logic. Complex referential expressions can also be decomposed by such rules so that the relative clause is exported out. The following rules suffice for this purpose: (∀xA/G(x))F (x) ↔ (∀xA)[G(x) → F (x)],
(MP3)
(∃xA/G(x))F (x) ↔ (∃xA)[G(x) ∧ F (x)].
(MP4)
Thus, with these rules we can see why the argument:
∴
(∀xA/G(x))F (x) (∃yA)(b = y) ∧ G(b) F (b)
is also valid in this logic.
7.4
Singular Reference and Proper Names
The previous examples involve forms of general reference, in particular to every raven and to some raven, respectively. This is different from most modern theories of reference, which deal exclusively with singular reference, such as in the use of a proper name to refer to someone. The sentence ‘Socrates is wise’, for example, is usually symbolized as W ise(Socrates), or more simply as F (a), where F represents the predicate ‘is wise’ and a is an objectual constant representing the proper name ‘Socrates’. Some philosophers have even argued against the whole idea of general reference, claiming that logically there can be only singular reference.15 We will turn to such arguments in the next chapter. Now, as we have noted, a proper name can be used either with or without an existential presupposition that the name denotes. As it turns out, it is conceptually more perspicuous and logically appropriate that we use the quantifiers ∃ and ∀ to indicate which type of use is being activated in a given speech or mental act. Thus, for example, we can use (∃xSocrates) to represent a referential act in which the proper name ‘Socrates’ is used with existential presupposition, i.e., with the presupposition that the name denotes. [Socrates]N P [is wise]V P (∃xSocrates) W ise(x) (∃xSocrates)W ise(x) 15 See, e.g., Geach 1980. A refutation of Geach’s arguments against general reference is given in chapter 9 and in Cocchiarella 1998.
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In this initial-level analysis, i.e., the level of analysis at which we are representing the cognitive structure of a speech or mental act, the existential quantifier phrase (∃xSocrates) indicates that the referential act that is a part of this structure is being made with the presupposition that the name ‘Socrates’ actually names an object. In a secondary level of analysis, where deductive transformations occur, both proper and common names, as we will explain in a later chapter, can be transformed into objectual terms and allowed to occur in the argument positions of predicates in place of objectual variables, i.e., aside from occurring as parts of quantifier phrases, proper and common names will later be allowed to occur as obectual terms as well. In this lower-level logical framework, the above expression is equivalent to the form it has in first-order “free” logic; i.e., the following is valid in the lower-level logical framework: (∃xSocrates)W ise(x) ↔ (∃x)[x = Socrates ∧ W ise(x)]. Note that although the right-hand side has the same truth conditions as the left, it does not represent the cognitive structure of the speech or mental act in question. What the right-hand side says is: Some object is identical with Socrates and it is wise. Now just as the existential quantifier, ∃, indicates that a proper name is being used with existential presupposition, so too the universal quantifier, ∀, indicates that the name is being used without existential presuppositions. A referential use of the proper name ‘Pegasus’, for example, might well be without an existential presupposition that the name denotes, in which case it is appropriate to represent that use as (∀xP egasus). Thus, the sentence ‘Pegasus flies’, where the name ‘Pegasus’ is not being used with existential presupposition can be symbolized as (∀xP egasus)F lies(x), which in our lower-level logical framework is equivalent to (∀x)[x = P egasus → F lies(x)]. Again, although the latter has the same truth conditions as ‘Pegasus flies’, it does not represent the cognitive structure of that speech act. Rather, what it says is, Every object is such that if it is (identical with) Pegasus, then it flies.
7.5
Definite Descriptions
Like proper names, definite descriptions can also be used to refer with, or without, existential presuppositions. For example, there can be a context in which a father who asserts, The child of mine who gets the best report card will receive a prize,
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might not in fact presuppose that just one of his children will get a report card better than the others. The father realizes, in other words, that two or more of his children might do equally well, in which case his use of the definite description is not intended to refer to exactly one child. In other words, the father’s referential act is without existential presuppositions. Logically,what the father asserts in that context has the same truth conditions as, If there is just one child of mine who gets a report card better than the others,then s/he will receive a prize. But, as we have already noted, having the same truth conditions in conceptual realism is not the same as representing the same cognitive structure of a speech or mental act. The distinction between using a definite description with and without existential presuppositions requires the introduction of two new quantifiers, ∃1 and ∀1 . For example, where A is a common name and F and G are monadic predicates, an assertion of the form, ‘The A that is F is G’ can be analyzed as follows: [The A that is F ]N P [is G]V P , (∃1 xA/F (x)) G(x) (∃1 xA/F (x))G(x) On the other hand, an assertion of the same sentence but in which the use of the definite description is without existential presupposition will be symbolized as [The A that is F ]N P [is G]V P , (∀1 xA/F (x)) G(x) (∀1 xA/F (x))G(x) In neither case, we want to emphasize, is the definite description being interpreted as a “singular” term. In this regard, our analyses are similar to Bertrand Russell’s in his famous 1905 paper, “On Denoting”. In that paper, and thereafter, Russell did not represent definite descriptions as singular terms but analyzed them in context in terms of quantifiers and formulas. Of course, Russell did not distinguish between using a definite description with, as opposed to without, existential presuppositions, but, instead, he interpreted them all as being used with existential presuppositions. Russell’s theory is easily emended, however, so as to include that distinction as well. There is a difference between our analysis and Russell’s in that our analysis represents the cognitive structure of the speech or mental act in question,
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whereas Russell’s represents only the truth conditions of that act. The two analyses are logically equivalent, but only one represents the cognitive structure of the speech or mental act. Thus, given a slight reformulation of Russell’s contextual analysis, we can formulate the equivalence as rules connecting the logical forms of the initial level of analysis, i.e., where the cognitive structure of our speech and mental acts are analyzed, with the logical forms of the lower level where truth conditions and deductive relations are represented. Thus, in the case where the definite description is used with existential presupposition, we have the following rule that connects our analysis with Russell’s: (∃1 xA/F (x))G(x) ↔ (∃xA)[(∀yA)(F (y) ↔ y = x) ∧ G(x)]. The right-hand side of this biconditional says that there is one and only one A that is an F , and that A is G. In the case where the definite description is used without existential presupposition we have the related but somewhat different rule: (∀1 xA/F (x))G(x) ↔ (∀xA)[(∀yA)(F (y) ↔ y = x) → G(x)], which says that if there is one and only one A that is F , then that A is G. It is instructive to note why it is that although Russell’s contextual analysis provides a perspicuous representation of the truth conditions of the speech or mental act in question, it does not at all represent the cognitive structure of that act. First, note that regardless of whether or not the referential act is with or without existential presuppositions, it is in either case the same predicable concept that is being exercised, a fact that is explicitly represented by the logical forms given in our analyses above for conceptual realism. On Russell’s contextual analysis, however, the predicable expressions, as represented by the bracketed formulas on the right-hand-side of each of the above biconditionals, are different. Secondly, note that the referential import of the speech or mental act in question is properly represented in either case by a complex referential expression— namely, (∃1 xA/F (x)) or (∀1 xA/F (x)) —whereas the predicable aspect is represented by a simple predicate expression—namely, G(x). The referential expressions used in Russell’s analyses, on the other hand, are the simple quantifiers phrases (∃xA) in the one case, and (∀xA) in the other, and, as just noted, the predicate aspects are represented by complex formulas. Russell’s contextual analysis is not wrong in how it represents the truth conditions of a speech or mental act in which a definite description is used as a referential expression; but, unlike the analyses that are given in conceptual realism, it does not provide an appropriate representation of the cognitive structure of that act. In conceptual realism, the distinction between logical forms that represent the cognitive structure of a speech or mental act and those that give a logically perspicuous representation of the truth conditions of that act is fundamental and involves different levels of analysis.
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The one type of logical form occurs on an initial level of analysis and is about the cognitive structure of our speech and mental acts, whereas the other occurs on a lower level where it is the truth conditions and logical consequences of those act that are perspicuously represented by logical forms.
7.6
Nominalization as Deactivation
Not all speech or mental acts are assertions in which a referential and a predicable concept are exercised. A denial, for example, that some raven is white is not an assertion in which a referential act is exercised. Similarly, in expressing a conditional, such as that if Pegasus exists, then there is a winged horse, one is not asserting either the antecedent or the consequent of the conditional. Unlike a basic assertion in which the nexus of predication is the mutual saturation of a referential and a predicable concept, no referential concept is being exercised in a conditional assertion.16 Unlike the negative judgment that some raven is not black, a denial that some raven is white, as might be expressed by the sentence ‘No raven is white’, is not an act in which reference is made to any raven, no less to every raven, even though the denial is equivalent to asserting of each raven that it is not white. Grammatically, the denial can be analyzed as follows, [That some raven is white]N P [is not the case]V P where the sentence ‘Some raven is white’ has been nominalized and transformed into a grammatical subject. In this transformation the quantifier and predicate phrases of the sentence ‘Some raven is white’ are “deactivated,” indicating that the referential and predicable concepts these phrases stand for are not being exercised. The denial is not about a raven but about the propositional content of the sentence—namely, that it is false, i.e., not the case. We could make this deactivation explicit by symbolizing the denial as, N ot-the-Case([(∃xRaven)W hite(x)]), where the brackets around the formula (∃xRaven)W hite(x) indicate that the sentence has been transformed into an abstract singular term—i.e., an expression that can occupy the position of an object variable where it (purports to) denotes the propositional content of the sentence. It is more convenient, however, to retain the usual symbolization, namely, ¬(∃xRaven)W hite(x), so long as it is clear that, unlike the equivalent sentence, (∀xRaven)¬W hite(x) 16 See Russell 1903, §38, for a similar view, and on how ‘If p, then q’ differs from ‘p; therefore q’, where in the latter case both p and q are asserted, whereas neither is asserted in the former.
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which is read (in non-idiomatic English) as ‘Every raven is such that it is not white’, no reference is being made to ravens in the speech or mental act in question. In conceptualism, as already noted, we distinguish the initial level of analysis at which a logical form represents the cognitive structure of a speech or mental act from the lower level of deductive relations at which a logically equivalent logical form gives a perspicuous representation of the truth conditions of that act. Now deactivation applies to a predicate not only when it occurs within a nominalized sentence, but also when its infinitive or gerundive form occurs in a speech act as part of a complex predicate. In other words, deactivation also applies directly to nominalized predicates occurring as parts of other predicates. Consider, for example, the predicate phrase ‘is famous’, which can be symbolized as the λ-abstract [λxF amous(x)] as well as simply by F amous( ). The λabstract is preferable as a way of representing the infinitive ‘to be famous’, which is one form of nominalization: to be famous ↓ to be an x such that x is famous ↓ [λxF amous(x)] Now the sentence ‘Sofia wants to be famous’ does not contain the active form of the predicate ‘is famous’ but only a nominalized infinitive form as a component of the complex predicate ‘wants to be famous’. When asserting this sentence we are not asserting that Sofia is famous, in other words, where the predicable concept that ‘is famous’ stands for is activated, i.e., exercised; rather, what the complex predicate ‘wants to be famous’ indicates is that the predicable concept that ‘is famous’ stands for has been deactivated. The whole sentence can be symbolized as [Sofia]N P [wants [to be famous]]V P ↓ ↓ ↓ (∃ySof ia)[λyW ants(y, [λxF amous(x)])](y) Nominalized predicates do not denote the concepts the predicates stand for in their role as predicates, as we have already noted in a previous chapter, because the latter, as cognitive capacities, have an unsaturated nature and cannot be objects. As an abstract singular term, i.e., as an objectual term that purports to denote a single abstract object, what a nominalized predicate denotes is the intensional content of the predicable concept the predicate otherwise stands for. In conceptual realism, what we mean by the intensional content of a predicable concept is the result of a projection onto the level of objects of the truth conditions determined by the concept’s application in different possible contexts of use.
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Now it is important to note here that the complex predicate [λyW ants(y, [λxF amous(x)])] does not represent a real relation between Sofia and the intensional object that the infinitive ‘to be famous’ denotes. What the complex predicate stands for is a predicable concept, which as a cognitive capacity has no more internal complexity than any other predicable concept.17 What is complex is the predicate expression and the truth conditions determined by the concept it stands for— i.e., the conditions under which the predicate can be true of someone in any given possible context of use. It is a criterion of adequacy of any theory of predication that it must account for predication even in those cases where a complex predicate contains a nominalized predicate as a proper part, as well as the more simple kinds of predication where predicates do not have an internal complexity. What this criterion indicates is one of the reasons why conceptualism alone is inadequate as a formal ontology and needs to be extended to include an intensional realism of abstract objects as the intensional contents of both denials and assertions as well as of our predicable concepts.
7.7
The Content of Referential Concepts
The fundamental insight into the nature of abstract objects, according to conceptual realism, is that we are able to grasp and have knowledge of such objects as the objectified truth conditions of the concepts whose contents they are. This “object”-ification of truth conditions is realized through a reflexive abstraction in which we attempt to represent what is not an object—e.g., an unsaturated cognitive structure underlying our use of a predicate expression—as if it were an object. In language this reflexive abstraction is institutionalized in the rulebased linguistic process of nominalization. As already noted, we do not assume an independent realm of Platonic forms in conceptual realism in order to account for abstract objects and the logic of nominalized predicates. Conceptual realism is not the same as either logical realism or conceptual Platonism. Some of the reasons why this is so are: 1. The abstract objects of conceptual realism are not entities that are predicated of things the way they are in logical realism and conceptual Platonism—i.e., they are not unsaturated entities and therefore they do not have a predicative nature in conceptual realism. 2. The abstract objects of logical realism and conceptual Platonism exist independently of the evolution of culture and consciousness, whereas in conceptual realism all abstract objects, including numbers, are products of the evolution of language and culture. Nevertheless, although they are 17 Learning how to correctly use such an expression may of course be a more difficult cognitive process than learning simpler predicate expressions.
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3. In logical realism, abstract objects are objects of direct awareness, whereas in conceptual realism all knowledge must be grounded in psychological states and processes. In other words, we cannot have knowledge of abstract objects if our grasp of them as objects must be through some form of direct awareness. According to conceptual realism we are able to grasp and have knowledge of abstract objects only as the intensional contents of the concepts that underlie reference and predication in language and thought. That is, we are able to grasp abstract objects as the “object”ified truth conditions of our concepts as cognitive capacities. The reflexive abstraction that transforms the intensional content of an unsaturated predicable concept into an abstract object is a process that is not normally achieved until post-adolescence. An even more difficult kind of reflexive abstraction also occurs at this time. It is a double reflexive abstraction that transforms the intensional content of a referential concept into a predicable concept, and then that predicable concept into an abstract object. The full process from referential concept to abstract object is doubly complex because it involves a reflexive abstraction on the result of a reflexive abstraction. Where A is a name (proper or common, and complex or simple), and Q is a quantifier (determiner), we define the predicate that is the result of the first reflexive abstraction as follows: [QxA] =df [λx(∃F )(x = F ∧ (QxA)F (x))]. In this definition the quantifier phrase (QxA) is transformed into a complex predicate (λ-abstract), which can then be nominalized in turn as an abstract singular term that purports to denote the intensional content of being a concept F such that (QxA)F (x). Consider, for example, an assertion of the sentence ‘Sofia seeks a unicorn’, which can be analyzed as follows19 : [Sofia]N P [seeks [a unicorn]]V P ↓ ↓ ↓ (∃xSof ia)[λxSeek(x, [∃yU nicorn])](x) No reference to a unicorn is being made in this assertion. Instead, the referential concept that the phrase ‘a unicorn’ stands for has been deactivated in the speech act. This deactivation is represented on the initial level of analysis by transforming the quantifier phrase into an abstract singular term denoting its intensional content. Note that the relational predicate ‘seek’ in this example is 18 See Popper 1967, p. 106, and chapter P2 of Popper & Eccles 1983 for a similar view of abstract objects. 19 This example is from Montague 1974.
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not extensional in its second argument position. What that means is that on the lower level of representing truth conditions and logical consequences, the sentence does not imply that there is a unicorn that Sofia seeks. But the different assertion that Sofia finds a unicorn, which is symbolized in an entirely similar way: SofiaN P [finds [a unicorn]]V P ↓ ↓ ↓ (∃xSof ia)[λxF ind(x, [∃yU nicorn])](x) does imply that there exists a unicorn, and moreover that it has been found by Sofia. That is, the following (∃yU nicorn)(∃xSof ia)F inds(x, y). is a logical consequence of the above sentence. Thus, even though the two different sentences, (∃xSof ia)[λxSeek(x, [∃yU nicorn])](x) (∃xSof ia)[λxF ind(x, [∃yU nicorn])](x) have the same logical form, and therefore represent essentially the same cognitive structure of a speech or mental act, even though only one of them implies that there is a unicorn. The reason why the one sentence implies that there is a unicorn and the other does not is that the relational predicate ‘find’, but not the predicate ‘seek’, is extensional in its second argument position. The extensionality of ‘find’ is represented by the following meaning postulate20 : [λxF inds(x, [∃yA])] = [λx(∃yA)F inds(x, y)]. By identity logic and λ-conversion, the following is a consequence of this meaning postulate, (∃xSof ia)[λxF inds(x, [∃yA])](x) ↔ (∃xSof ia)(∃yA)F inds(x, y) Of course, there is no meaning postulate like this for the intensional predicate ‘seek’. 20 Actually, this is but an instance of the more general meaning postulate in question, namely that for any determiner (quantifier) Q,
[λxF inds(x, [QyA])] = [λx(QyA)F inds(x, y)]. Thus if Sophia finds all, some, most, several, etc., unicorns, then, respectively, all, some, several, etc., unicorns are such that Sophia finds them.
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7.8
The Two Levels of Analysis
Our analysis of the deactivation of quantifier phrases occurring as direct objects of transitive verbs such as ‘seek’ and ‘find’ is similar to an analysis given by Richard Montague in his paper “The Proper Treatment of Quantification in Ordinary English,” except that Montague’s framework is a type-theoretical form of logical realism.21 Unfortunately, there is a problem with Montague’s analysis that seems to apply to our approach as well. The problem arises when a quantifier phrase occurring as a direct-object of a complex predicate applies to two argument positions implicit in that predicate. Consider, for example, an assertion of the sentence ‘Gino bought and ate an apple’, which has the quantifier phrase ‘an apple’ occurring as the direct object of the complex predicate ‘bought and ate’. Now the complex predicate ‘bought and ate’ implicitly has two argument positions for the direct-object expression ‘an apple’, one associated with ‘bought’, and the other associated with ‘ate’. The problem is how can we distinguish in logical syntax a nonconjunctive assertion of the form [x]N P [(bought and ate) an apple]V P from the different conjunctive assertion of [x]N P [bought an apple]V P and [x]N P [ate an apple]V P where, as the direct object, the quantifier phrase ‘an apple’ has been deactivated in each assertion. This is a problem because although the nonconjunctive sentence ‘x (bought and ate) an apple’, implies on the deductive level the conjunctive sentence ‘x bought an apple and x ate an apple’, nevertheless the two sentences are not logically equivalent, which means that the nonconjunctive sentence should not be analyzed as the conjunctive sentence. Now if we assume that the analysis of ‘x (bought and ate) an apple’, which has a deactivated occurrence of the quantifier phrase ‘an apple’ as the directobject argument of the predicate, to be a y such that x bought-and-ate y is the same as that for the complex predicate to be a y such that x bought y and x ate y. ↓ [λy(Bought(x, y) ∧ Ate(x, y))], then, the sentence ‘Gino bought and ate an apple’ would be analyzed as 21 See
Montague 1970b.
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[Gino]N P [(bought and ate) an apple]V P ↓ (∃xGino) [λx[λy(Bought(x, y) ∧ Ate(x, y))]([∃yApple])] ↓ ↓ (∃xGino)[λx[λy(Bought(x, y) ∧ Ate(x, y))]([∃yApple])](x) But then, by λ-conversion, this analysis not only implies that Gino bought an apple and Gino ate an apple, it is also implied by the latter, i.e., on this analysis the two are equivalent, which is contrary to the result we want. Intuitively, what we want is to first “reactivate” the quantifier phrase ‘an apple’, i.e., to transform [λy(Bought(x, y) ∧ Ate(x, y))]([∃yApple]) to (∃yApple)[λy(Bought(x, y) ∧ Ate(x, y))](y), before applying λ-conversion. This reactivation is justified by the fact that ‘bought’ and ‘ate’ are both extensional in their direct-object positions, i.e., because for an arbitrary common name A, [λxBought(x, [∃yA])] = [λx(∃yA)Bought(x, y)] and [λxAte(x, [∃yA])] = [λx(∃yA)Ate(x, y)] are (instances of) meaning postulates for the predicates ‘bought’ and ‘ate’.22 Then, by a rule to the effect that a complex conjunctive predicates is extensional if it is made up of two extensional predicates, It follows that the conjunctive predicate ‘bought and ate’ is also extensional; i.e., then [λx[λy(Bought(x, y)∧Ate(x, y))]([∃yA])] = [λx(∃yA)(Bought(x, y)∧Ate(x, y))] is valid as well. One way to resolve the problem, which we will adopt here, is to not only distinguish the initial level of analysis where logical forms are designed to represent the cognitive structure of a speech or mental act from the deductive level where deductive transformations such as λ-conversion can be applied, but also to mandate that before a formula can be transformed at the deductive level to determine its logical consequences we must first apply Leibniz’s law as based on the meaning postulates for the predicate expression represented at this initial 22 As
noted in the case for ‘find’, the meaning postulates apply to determiners Q in general: [λxBought(x, [QyA])] = [λx(QyA)Bought(x, y)], [λxAte(x, [QyA])] = [λx(QyA)Ate(x, y)].
In referring to meaning postulates hereafter, we will, for convenience of discussion, generally mention only the instances.
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level. Such an application is logically obligatory in order to transform the logical form in question into a form that is appropriate for deductive transformations on the lower level of analysis. As a way of understanding the rationale for this distinction note that if there were a simple verb in English, say, e.g., ‘bouate’, that is synonymous with the complex verb ‘bought and ate’ the way that ‘period of time of fourteen days’ is synonymous with ‘fortnight’, then instead of the verb phrase ‘bought and ate an apple’, we might also have the simpler phrase ‘bouate an apple’23 , so that instead of an assertion of ‘Gino bought and ate an apple’ we might have had an assertion of ‘Gino bouate an apple’, the cognitive structure of which would be represented as follows: (∃xGino)[λxBouate(x, [∃yApple])](x). We would then of course have a meaning postulate regarding the extensionality of the predicate [λxBouate(x, [∃yApple])], namely, [λxBouate(x, [∃yApple])] = [λx(∃yApple)Bouate(x, y)] in which case, by this meaning postulate, we would then have (∃xGino)[λx(∃yApple)Bouate(x, y)](x). This last, by a meaning postulate connecting ‘bouate’ with ‘bought and ate’, is equivalent to (∃xGino)[λx(∃yApple)[λy(Bought(x, y) ∧ Ate(x, y))](y)](x), which is the result we want of course for an assertion of ‘Gino bought and ate an apple’. Our position for the distinction between the levels of analysis is that what applies in a natural and plausible way for a simple predicate like ‘bouate’ should also apply for a complex predicate like ‘bought and ate’ as well. The point of this strategy is that every predicable concept expressed by a predicate or verb phrase in a speech or mental act should be considered, qua predicable concept, on a par with any other predicable concept, regardless of the complexity of the predicate or verb phrase used to express the concept in that act. The complexity, we have said, is not in the concept as a cognitive capacity, but in the predicate or verb phrase used to express that concept. There is another way of resolving the problem, incidentally; but in the end it just brings us back to this distinction we have made between the two levels of analysis. On this alternative, we would assume that the sentence ‘Gino bought and ate an apple’ is synonymous with ‘Gino bought an apple and ate 23 Of course the more likely new verb would be in the present tense corresponding to ‘buys and eats’, with ‘bouate’ as its past tense form.
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it’, which makes explicit the two direct-object positions, one occupied by the quantifier phrase ‘an apple’ and the other by the co-referential pronoun ‘it’. This alternative reading is also synonymous with ‘Gino bought an apple and ate that apple’, which makes explicit the two direct-object positions as well. Now we have given elsewhere a conceptualist analysis of co-referential pronouns in terms a variable-binding ‘that’-operator, T , as in ‘that apple’, which we symbolize as (T yApple).24 Thus, by means of the T -operator, we can symbolize ‘Gino bought an apple and ate that apple’ as follows: [Gino]N P [bought an apple and ate that apple]V P ↓ (∃xGino) [λx(Bought(x, [∃yApple]) ∧ Ate(x, [T yApple])] ↓ ↓ (∃xGino)[λx(Bought(x, [∃yApple]) ∧ Ate(x, [T yApple])](x), where both the quantifier phrase ‘an apple’ and its co-referential phrase ‘that apple’ occur deactivated in direct-object positions. Then, given that both ‘Bought’ and ‘Ate’ are extensional in their second-argument positions, the above sentence is equivalent to (∃xGino)[λx((∃yApple)Bought(x, z) ∧ (T yApple)Ate(x, z))](x). Now, by the following meaning postulate for the T -operator,25 [(∃yA)ϕy ∧ (T yA)ψy] = [(∃yA)(ϕy ∧ ψy)],
(MPT1 )
what follows by Leibniz’s law from the above sentence is (∃xGino)[λx(∃yApple)(Bought(x, z) ∧ Ate(x, z))](x), which is the result we wanted.26 This last implies, but is not equivalent to, (∃xGino)(∃yApple)Bought(x, z) ∧ (∃xGino)(∃yApple)Ate(x, z)). The problem with this alternative is that unless the same distinction and restriction is made about the different levels of analysis, the meaning postulate (MPT1 ) could be used to generate a contradiction. Suppose, e.g., we have a 24 See
Cocchiarella 1998, §7. rule says that the sentence ‘Some A is ϕ and that A is ψ’ is equivalent to ‘Some A is such that it is ϕ and ψ’. An example of the rule is the equivalence between ‘Some man broke the bank at Monte Carlo and that man died a pauper’ and ‘Some man is such that he broke the bank at Monte Carlo and he died a pauper’. To avoid problems that could otherwise arise, this rule, as we note below, must be applied before other logical operations, such as simplification to separate conjuncts. 26 Not all complex predicates like ‘bought and ate’ are extensional, incidentally. For example, in ‘Gino seeks and wants to kill a dragon’, which we might take to be synonymous with ‘Gino seeks a dragon and wants to kill that dragon’, neither ‘seeks’ nor ‘wants to kill’ are extensional. In that case neither of the deactivated quantifier phrases ‘a dragon’ and ‘that dragon’ can be reactivated in the context in question. 25 This
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context in which a sentence of the form (∃yA)F (y) ∧ (T yA)G(y) is asserted, but in which it is also true that no A that is H is G, i.e., ¬(∃yA)[H(y) ∧ G(y)], even though some A is in fact H, i.e., (∃yA)H(y). But by simplification of (T yA)G(y) from the assertion in question and conjoining it with (∃yA)H(y), we would then have, by (MPT1 ), (∃yA)[H(y) ∧ G(y)], which is impossible in the context in question. The way to resolve this problem is again to distinguish between the level of analysis at which the purpose is to represent the cognitive structure of a speech or mental act and the level of deductive transformations where rules such as those for simplifying and putting together a conjunction can be applied. Leibniz’s law and only Leibniz’s law as based on meaning postulates can be applied on the initial level of analysis, and in fact this use of Leibniz’s law is required before we can turn to the deductive consequences of our speech and mental acts. The restriction to Leibniz’s law based on meaning postulates for complex predicates, such as those for reactivating quantifier phrases that occur as arguments of extensional predicates, or those (like (MPT1 ) above) that eliminate a co-referential ‘that’-phrase, does not mean that we can replace any predicate by any logically equivalent predicate, e.g., a conjunctive predicate [λx(ϕx ∧ ψx)] by the predicate [λx¬(ϕx ∨ ψx)] obtained by replacing a conjunction by the negation of a disjunction, or even by the predicate [λx(ψx ∧ ϕx)] obtained by simply commuting the conjunctive components of the predicate. The purpose of the initial level of analysis is not only to represent the cognitive structure of a speech or mental act but also to represent the “intentional content” of that act, where by “intentional content” we mean a strong sense of intensionality that respects cognitive structure, and in that regard a sense that is stronger than logical equivalence. It is at this level, and only at this level, that issues about the identity conditions for “intentional content” arise. It is our view that “intentional content” is preserved under the transformations of predicates by meaning postulates for reactivating nominalized quantifier phrases of extensional predicates, and for transforming predicates involving co-referential ‘that’-phrases into predicates without a co-referential ‘that’-phrase. There are other transformations that preserve “intentional content” as well, moreover, such as, e.g., transforming a predicate representing a passive verb into a predicate representing the active form of that verb. But “intentional content” is not in general preserved under the relation of logical equivalence, which is why mere logical equivalence does not apply to the initial level of analysis regarding the cognitive structure of our speech and mental acts, but only to the second level of deductive consequences.
7.9
Ontology of the Natural Numbers
Stringent identity conditions for intensional objects, including the propositions that are the contents of our speech and mental acts, apply at the initial level of analysis, we have said, i.e., the level at which we are concerned with representing the cognitive structure of our speech and mental acts. Less stringent identity conditions are allowed for abstract objects on the level of strictly deductive
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transformations, however, where we are not concerned with the fine-grained features of cognitive structure. Logical equivalence, for example, and in some contexts perhaps even extensional equivalence, suffices for most of the abstract objects dealt with in mathematics, objects that initially at least are cognized as the contents of concepts. On this level abstract objects are not only products of cultural evolution, but are themselves the means by which the further evolution of culture is possible. The question of how we are to account for the ontological status of the natural numbers and our epistemological grasp of them is an important one in philosophy and cognitive science, even if not in pure mathematics itself. Significantly, it is an issue that can be explained by the kind of analyses we have given of the abstract objects that are the correlates of referential concepts. We note in particular in this regard that there are three ontological aspects in which the natural numbers have in general been described by philosophers of logic and mathematics, and each of them, it turns out, corresponds to one or another of the three types of expressions involved in the double reflexive abstraction, or double-correlation thesis, of conceptual realism. The first type of expression is that of numerical quantifier phrases. The entities associated with these expressions are usually called “quantities”, e.g., five chairs, two cats, ten people, one president, etc. Their basic use is as referential expressions, and what they stand for are referential concepts. The objectual quantities that are usually associated with these expressions, according to conceptual realism, are the abstract objects that are the correlates of the referential concepts they stand for in their basic use. That is to say that quantities, on our proposal, are the contents of certain referential concepts, namely, those expressed by numerical quantifier phrases. Now some numerical quantifier phrases do not refer to any particular kind of object, but to objects in general; and for that reason we shall call them pure numerical quantifier phrases. The ones that we are interested in here can be contextually defined on the deductive level as follows: (0x)ϕx
=
df ¬(∃x)ϕx,
(1x)ϕx
=
df (∃x)(ϕx
∧ (0y)[y = x ∧ ϕ(y/x)]),
(2x)ϕx
=
df (∃x)(ϕx
∧ (1y)[y = x ∧ ϕ(y/x)]),
etc. Under the ontological aspect of quantities, the natural numbers, according to conceptual realism, are none other than the pure quantities that are the conceptcorrelates of the referential concepts expressed by pure numerical quantifier phrases. The second type of expression is that of the cardinal number predicates, e.g., ‘has twelve instances’, or even ‘has twelve members’. The entities associated with these predicates are usually called cardinal number properties. Thus, to take one of Frege’s examples, the class of Apostles has twelve members, or equivalently, the property of being an Apostle has twelve instances. In Bertrand
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Russell’s ‘no-classes’ version of logicism, the natural numbers were in effect identified with the finite cardinal number properties. Of course, in conceptualism predicates stand for concepts and not for properties; but then in conceptual realism the properties in question are just the intensional objects denoted by the nominalized forms of those predicates as abstract singular terms. The predicates themselves can be defined in terms of the corresponding quantifier phrases as follows. (We retain the parentheses here so as to emphasize that we are defining predicates in these cases and not abstract singular terms.)27 0( ) = 1( ) =
df [λF (0x)F (x)](
), [λF (1x)F (x)]( ), df
2( ) =
df [λF (2x)F (x)](
),
etc. These definitions, as we have already noted, indicate the general way in which a predicable (first-level) concept is to be correlated with a referential (second-level) concept in accordance with the Frege’s double-correlation thesis, which corresponds in conceptual realism to a type of double abstraction. Also, in accordance with conceptual realism’s version of Frege’s thesis, note that the cardinal number properties that are the intensions or concept-correlates of the predicable concepts that the above predicates stand for are none other than the pure quantities already identified as the natural numbers. In other words, the natural numbers under either of these ontological aspects come to the same thing, as far as conceptual realism is concerned. Finally, the third and last type of expression for the natural numbers are the numerals as abstract objectual terms. These are the expressions most favored in mathematics since the entities associated with them are the natural numbers in their most simple and direct ontological aspect. But their simplicity of expression is misleading and leads to the epistemological problem of how the natural numbers are conceptually accessible to us. This problem, along with the problem of how intensional objects are in general conceptually accessible to us, is resolved in conceptual realism through its version of Frege’s double-correlation thesis. Abstract objectual terms, including numerals, according to conceptual realism, are ultimately explained on the basis of a nominalizing transformation of predicates. Thus, the numerals in particular can be defined as follows: 0
=
df
1
=
df
2
=
[λF (0x)F (x)], [λF (1x)F (x)],
[λF (2x)F (x)], etc. df
Note that on this analysis the natural numbers are epistemically accessible to us in just the same way that concept-correlates in general are—namely, through 27 Reminder: A λ-abstract of the form [λF ϕ] is an abbreviation for a λ-abstract of the form [λx(∃F )(x = F ∧ ϕ)]).
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the concepts whose correlates they are. That is, we are conceptually able to think about the natural numbers as objects because we are conceptually able to lay hold upon them as the intensions of the (predicable concepts determined by the) referential concepts that pure numerical quantifier phrases stand for. And this conceptual ability, we have said, is just what in conceptual realism is explained by Frege’s double-correlation thesis. Finally, that the natural numbers denoted by numerals as abstract singular terms turn out to be just the pure quantities that are the intensions of the referential concepts expressed by pure numerical quantifiers is a natural, and not a fortuitous, result. For it explains why children first learn about quantities, e.g., two apples, four cows, ten trees, etc., and only later, after they have formed the concept of an object simpliciter, learn about the numbers themselves. That is, children learn to think about the natural numbers as abstract objects by learning first to objectify the content of the referential concepts expressed by numerical quantifier phrases, and then, by means of the concept of an object simpliciter, they learn to think about the natural numbers as the objectified contents of the referential concepts expressed by pure numerical quantifier phrases. And that this is possible is just what is explained in conceptual realism by Frege’s double-correlation thesis.
7.10
Ontology of Fictional Objects
The natural numbers are not the only entities that can be accounted for in terms of the deactivation of referential concepts. In fictional discourse, for example, our apparent reference to the objects of a story is not a form of direct reference but of indirect reference, i.e., indirect, or deactivated, reference with respect to an implicit operator such as ‘In the story ...’. The propositions that make up the content of a story are not the same as states of affairs, which are part of the causal order of the natural world. As intensional objects, propositions enable us to construct a “bracketed world” of intensional content within which we are able to freely speculate and construct various hypotheses and theories about the natural world. In fact, whether true or false, all theories about the natural world consist of a system of propositions, which we are able to contemplate independently of whether or not there are any states of affairs in the natural world corresponding to them. In this way, as intensional objects, propositions serve to advance the development of science and technology, and thereby the further evolution of culture. Propositions make up the content of our fables and myths, and, in fact, they are the content of stories of all kinds, both true and false. In this way propositions and the abstract objects that are the contents of deactivated referential expressions also serve the literary and aesthetic purposes of culture. In reading a fictional story, for example, we are given to understand that none of the references made in the story are to be taken literally, i.e., that all of the referential expressions occurring in the sentences of the story are understood to be deactivated, by which we mean that we are dealing with the intensional
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content of those referential expressions and not with any real objects that those expressions might otherwise be used to refer to in direct discourse. The same is true of stories that are put forward as descriptions of reality—except in those cases, assuming we believe the stories in question, we indirectly re-activate the referential function of the expressions used in those stories by indicating, even if only implicitly, that the stories are to be taken as true. Here, in fact, we see the significance of the law, that φ is true ↔ φ, or, in symbols, T rue([ϕ]) ↔ ϕ, whereby an assertoric occurrence of a propositional form φ is connected with a nominalized occurrence of φ. All stories are to be interpreted in this regard as a form of indirect discourse—such as the contexts that occur within the scope of an ‘In-the-story’ operator, which often is only implicit when we read, or are being told, a story. For it is only by first understanding the content of a story that we can then raise the question of its veracity, i.e., the question of whether or not there are states of affairs in the space-time causal manifold corresponding to the propositions that make up that story. All fictional characters, on this account, are intensional objects—namely, the intensional objects that are the correlates of the referential expressions used in the fiction in question. These intensional objects are accounted for in conceptual realism, as we have said, through the deactivation of referential concepts, which logically is represented by a double correlation, first of referential concepts with predicable concepts, and then of the latter with their concept-correlates. In a specific story, say, A, both the propositions and the intensional objects involved in the referential expressions of that story may be relativized as follows, [QxS]A =df [λy(∃F )(y = F ∧ In(A, [(QxS)F (x)])], where ‘[(QxS)F (x)]’ is a nominalization of the formula ‘(QxS)F (x)’, and ‘In(A, [...])’ represents the formula-operator ‘In (the story) A, ...’. Thus, the referential expression ‘Sherlock Holmes’ will be taken to have one intensional object as its content in Conan Doyle’s novel The Hound of the Baskervilles and a different intensional object in Conan Doyle’s The Valley of Fear. Now because the singular term ‘Sherlock Holmes’ is used with existential presupposition in the fictional worlds of both novels, it is represented as having the logical form ‘(∃xSherlock-Holmes)’ in the sentences that make up the written text of those novels; and therefore the intensional objects that are the constituents of the propositions making up the content of those novels are represented by, e.g., ‘[∃xSherlock-Holmes]Baskervilles ’ and ‘[∃xSherlockHolmes]V alley ’, respectively. Though these intensional objects are not identical, they are counterparts to one another in much the sense of David Lewis’s counterpart theory. Indeed, it is here among the intensional objects of our various stories—and not among the concrete objects that exist in, and across, different
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causally possible worlds—that David Lewis’s counterpart theory has its proper application. It is the relativization of intensional objects in this way that explains the so-called “incompleteness” of fictional objects. There are many predicate expressions of English, for example, that can be meaningfully applied to humans but that are neither affirmed nor denied of the character Sherlock Holmes in any of Conan Doyle’s novels. Neither the formula ‘In(A, [(∃xSherlock-Holmes)F (x)])’ nor ‘In(A, [(∃xSherlock-Holmes)¬F (x)])’ will then be true of the concept (as a value of ‘F ’) that such a predicate might stand for, in order words, regardless which of Conan Doyle’s novels we consider as a value of ‘A’; and therefore, neither ‘[∃xSherlock-Holmes]A (F )’ nor ‘[∃xSherlock-Holmes]A ([λx¬F (x)])’ will be true as well—which is to say that, in the story A, the character Sherlock Holmes falls under neither the concept F nor its complement, and is, therefore, “incomplete” in that regard. Alexius Meinong’s impossible objects, when construed as fictional characters or objects (or as intensional objects of someone’s belief-space), are also “incomplete” in this way.28 Thus, whereas ‘The round square is round and square’ is false as a form of direct discourse, nevertheless, it could be true in a given fictional context. Suppose, for example, we construct a story called, Romeo and Juliet in Flatland , which takes place in a two-dimensional world (Flatland) at a time when two families, the Montagues and the Capulets, are having a feud. In Flatland, the Capulets, one of whom is Juliet, are all circles, and the Montagues, one of whom is Romeo, are all squares. (Juliet has curves and Romeo has angles.) Unknown to the two families, Romeo and Juliet have an affair and decide to live together in secret. In time, Juliet becomes pregnant and, given the difference in genetic makeup between Romeo and herself, she gives birth to a round square. Although Romeo and Juliet both love their baby, the round square, the two families, the Montagues and the Capulets, become enraged when they discover what has happened. They kill Romeo and Juliet, and their baby, the round square. But, not wanting it to be known that a round square—which, given the cruel social mores of Flatland society, would have been considered a monster—was born into either family, the Montagues and Capulets keep the birth, and death, of the round square a secret. They then pass it around that Romeo and Juliet were ill-starred lovers who committed suicide in despair of the open hostility between their respective families. The story ends with Romeo and Juliet being eulogized and buried together—but without their baby, the round square, whose body was cremated and reduced to ashes. As this story makes clear, we can meaningfully talk about “impossible” objects as if they were actual objects—although such talk can be true only when relativized to a context of indirect discourse, such as a story, and perhaps the belief-space of someone with inconsistent beliefs. Thus, for example, as part of the story, Romeo and Juliet in Flatland, it is true to say that the round square 28 See Meinong 1904. Also, see Parsons 1980 for a logical reconstruction of Meinong’s ontology and Cocchiarella 1982 for a discussion of Parson’s reconstruction and an alternative account to Meinong’s ontology.
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is round and square, which, formally, can be represented as follows: In(R&J-in-F latland, [(∃1 xSquare/Round(x))[λxRound(x) ∧ Square(x)](x)]). Thus, even though both [∃1 xSquare/Round(x)]([λxRound(x)]), and [∃1 xSquare/Round(x)]([λxSquare(x)]), are false regarding the intensional content of ‘The round square’ simpliciter, nevertheless, both [∃1 xSquare/Round(x)]R&J-in-F latland ([λxRound(x)]), and [∃1 xSquare/Round(x)]R&J-in-F latland ([λxSquare(x)]), are true of the intensional content of ‘The round square’ relativized to the story, Romeo and Juliet in Flatland. Nevertheless, as an object of a fictional, intensional world—as opposed to the objects of the actual world of nature—such an “impossible” object will be “incomplete” with respect to the different kinds of things that are in fact said of it in its fictional world. It is in this way that conceptual realism is able to explain the “incomplete” and “impossible” objects of Meinong’s theory of objects.29
7.11
Summary and Concluding Remarks
• In conceptual realism the nexus of predication is what accounts for the unity of a speech or mental act that is the result of jointly exercising a referential and predicable concept. • A unified account of both general and singular reference can be given in terms of this nexus. Such a unified account is possible because the category of names includes both proper and common names. • A unified account can also be given in terms of this nexus for predicate expressions that contain abstract noun phrases, such as infinitives and gerunds. • The same unified account also applies to complex predicates containing quantifier phrases as direct-object expressions of transitive verbs, such as the phrase ‘a unicorn’ in ‘Sofia seeks a unicorn’. Conceptually, the content of such a quantifier phrase and the referential concept it stands for is “object”-ified through a double reflexive abstraction that first generates a predicable concept and then the content of that concept by deactivation and nominalization. All direct objects of speech and thought are intensionalized in this way so that a parallel analysis is given for ‘Sofia finds a unicorn’ as for ‘Sofia seeks a unicorn’. 29 See Cocchiarella 1987, chapter 3, for a more detailed account of how Meinong’s theory can be reconstructed in the kind of framework we have in mind here.
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167
And yet, relations, such as F inds, that are extensional in their second argument positions can still be distinguished from those that are not, such as Seeks, by meaning postulates. • The same double reflexive abstraction explains the three different types of expressions that represent the natural number concepts, namely first, as numerical quantifier phrases, such as three men, two cows, five cups, etc., then, second, as the cardinal number predicates ‘has n instances’, or ‘has n members’, and the third as the numerals ‘1’, ‘2’, ‘3’, etc., i.e., as objectual terms that purport to name the natural numbers as abstract objects. • The deactivation of referential expressions is also involved in fictional discourse and in stories in general. The objects of fiction are none other than the intensional objects that deactivated referential expressions denote as abstract objectual terms. This account of the ontology of fictional objects explains their “incompleteness” as well as their status as intensional content. Finally, we note that there is much more involved in a conceptualist analysis of language and thought beyond our account of the nexus of predication. One such issue, which we will take up in chapter eleven, is how both proper and common names can be transformed into objectual terms occurring as denotative arguments of predicates, which is different from their referential role as parts of quantifier phrases. Such objectual terms denote “classes as many,” as opposed to sets as “classes as ones”. In addition to providing another account of “the one and the many”, classes as many also provide truth conditions for plural reference and predication. Classes as many also lead to a natural representation of the natural numbers as properties of classes as many.
Chapter 8
Medieval Logic and Conceptual Realism Reference in conceptual realism, as we have explained in the previous chapter, is not restricted to so-called singular terms, e.g., proper names and definite descriptions, but involves quantifier phrases containing common or proper names, where the former can be complex as well as simple.1 This uniform account of both singular and general reference is similar in a number of respects to the medieval suppositio theory of the 14th century. In fact, we maintain that with minor modifications the medieval theory can be reconstructed within the framework of conceptual realism. Indeed, conceptual realism provides a general philosophical framework within which we can reconstruct and explicate not just the suppositio theory as a theory of reference, but a number of other issues that were central to medieval logic as well, such as the identity theory of the copula (for categorical propositions) and the difference between real and nominal definitions. One of the benefits of such a reconstruction is that we can defend medieval logic against the kinds of arguments that Peter Geach has given against a uniform account of general and singular reference in his book Reference and Generality.2
8.1
Terminist Logic and Mental Language
The medieval logic we will be concerned with here is what came to be called “terminist logic” because it was primarily concerned with the semantics or “properties of terms” (proprietates terminorum), i.e., of adjectives and proper and 1 This chapter is based on my 2001 paper, “A Logical Reconstruction of Medieval Terminist Logic in Conceptual Realism,” in Logical Analysis and History of Philosophy, vol. 4. 2 See T.K. Scott [1966b] for criticism of Geach’s account of Ockham’s suppositio theory. A refutation of Geach’s general arguments against the kind of uniform account of reference given in conceptual realism will be presented in the next chapter.
169
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common count nouns, the latter of which we will continue to speak of as common names. These “terms” are also called categorematic expressions, as opposed to the syncagorematic logical expressions, such as the quantifier words ‘all’ and ‘some’, and the connectives ‘or’, ‘and’, ‘if then’, etc. This approach to logic began in the 13th century with such logicians as Peter of Spain, Roger Bacon, Lambert of Auxerre, and William of Sherwood. Around 1270, however, terminist logic “went into a kind of hibernation” and was replaced by an alternative movement known as “speculative grammar”.3 The hibernation ended in the early 14th century when “terminist-style semantic theory woke up again”.4 The major terminist logicians of this later period are William of Ockham, John Buridan, Walter Burley, and Gregory of Rimini. Ockham and Buridan are generally described as nominalists, but on our account really were conceptualists. Burley is said to be a realist, but his realism was really a form of conceptual natural realism (which we will describe in a later chapter). The conceptualism of the terminist logicians is clearly seen in their assumption that underlying spoken and written language there exists a mental language made up of both categorematic and syncategorematic concepts and mental propositions, or the mental acts that in conceptual realism we prefer to call judgments—or thoughts when they are only entertained and not asserted or judged.5 This “language of thought”, which is generally referred to today as Mental , was assumed to be common to all humans.6 Unlike spoken and written languages, which were said to be “conventional” languages, Mental was said to be a “natural” language, by which was meant a language somehow established by nature. Apparently, what makes Mental natural is that its categorematic concepts (mental terms) “get their signification by nature and not by convention.”7 Signification is the basic semantical relation of Mental, but it applies only to categorematic concepts, syncategorematic concepts being said not to signify at all. The signification of a categorematic concept is not an intensional object, but rather the things that fall under the concept, by which was meant, in a narrow sense, the things that now fall under the concept, but which, in a wider sense, included the things that could fall under the concept, i.e., the things that can fall under the concept, and therefore the things that did, do, or will fall under the concept as well. This distinction between narrow and wide signification was possible because our thoughts (mental acts) occur in time and, by means of tense and modal modifications, can be oriented toward the past or the future, as well as the present, and even toward what is merely possible. 3 Spade [1996], p.43. Spade gives a useful account of the history of this period. We will be relying on this text throughout this essay. 4 Ibid. 5 In conceptual realism a proposition is the intensional content of a speech or mental act, and not, as in terminist logic, the assertion or judgment made. We will allow some laxity in the use of ‘proposition’ in our account of terminist logic, however. 6 Geach in [1957], p.102, is one of the first to use ‘Mental’ this way. See Trentman [1970], Normore [1990], and Spade [1996], chapter 4, for an account of Mental. Normore [1985], p.189, explicitly refers to it as “a language of thought.” 7 Spade [1996], p.93.
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Mental is a tensed and modal language, in other words, containing among its syncategorematic concepts certain operators that correspond to the tenses and modal modifications of verbs, or what the medieval logicians called ampliation. Moreover, because mental terms can signify in a wider sense, Mental is ontologically committed to a form of possibilism, though the possible objects signified (in the wide sense) by a concept seemed to be only those that are possible in nature.8 That mental terms get their signification by nature is based on the idea that there is a “natural likeness” between concepts and the things they signify, a natural likeness that is caused, apparently, by the things signified.9 This suggests that concepts are something like images, which is much too restrictive a view in that it excludes those concepts of things that cannot be “imaged,” i.e., things other than the visible objects of the macrophysical world and for which we cannot conceive a “likeness.” Scientific concepts of things, processes and events in the microphysical world can be mathematically modelled, and in that sense “imagined,” but, because they are smaller than the wavelength of light, they cannot be “imaged” or pictured literally, and therefore there can be no “natural likeness” between our concepts and such things. Nor, of course, can there be a “natural likeness” (or a causal relation) between concepts and mathematical objects, be they numbers, sets, fields, or whatever.10 The idea of a “natural likeness” also suggests that the concepts corresponding to common names must all be sortal concepts, i.e., concepts that have identity criteria associated with them. But the concepts corresponding to the common names ‘thing’, ‘object’, ‘individual’, and even ‘physical object’, and ‘abstract object’, do not have identity criteria associated with them (the way, e.g., the common names ‘man’, ‘dog’, ‘carrot’, ‘chair’, etc., do), and yet such (nonsortal) concepts were explicitly allowed by the terminist logicians. Even the common name ‘furniture’, unlike different sorts of furniture (such as tables and chairs) and the common name ‘event’, unlike sorts of events (such as a running, kicking, or kissing) do not have identity criteria associated with them. In what sense is there a “natural likeness” between things in general and the common-name concept thing, or between physical objects and the complex common-name concept physical object ? We can conceive of a “likeness” to, or form an image of, a chair or table, i.e., of a particular sort of furniture, but we cannot form an image of furniture in general, nor can we form an image of events in general, though we can form an image of a kissing (between, say, Abelard and Heloise, or Bill and Monica). 8 See Normore [1985], p.191. It is not clear Normore would agree that all possibilia in Ockham’s and Buridan’s ontology are possible in nature, i.e. that the modality in question is a natural possibility (as opposed, e.g., to a logical or metaphysical possibility). A natural possibility and necessity seems to be what Burley had in mind, however. 9 According to Spade, “Ockham says the things a concept signifies are all such that: (a) The concept is like every one of them. (b) It is not like any one of them any more than it is like any other one; it is equally like all of them. (c) It is like any one of them more than it is like anything else—anything that is not signified by the concept.” ([1996], pp.153f) 10 It seems that mathematical objects were not accounted for at all in terminist logic. The reconstruction we give below in terms of conceptual realism provides a way of giving an account as indicated in the previous chapter.
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The notion that all categorematic concepts have a “natural similarity” with the things they signify is much too restrictive and is one of the semantic features of Mental that we do not assume applies to conceptual realism. For this reason, we will not refer hereafter to Mental as a “natural” language, but will instead follow contemporary practice and speak of historically real languages such as Latin, English, French, etc., as natural languages. We prefer the contemporary usage partly because it is contemporary, but also because we want to distinguish artificial languages (such as HST∗λ ), which are also “conventional,” from the historically developed natural languages spoken by a linguistic community. In regard to the relation between Mental and spoken (and written) languages, Ockham’s view differs somewhat from Buridan’s. According to Ockham, for example, a linguistic term used in an assertion signifies the same things that are signified by the concept corresponding to the term. Buridan, on the other hand, says that the spoken linguistic term signifies the concept corresponding to the term, and only indirectly, through the concept, signifies the things that it signifies.11 In conceptualism, we take a position similar to Ockham’s in that, e.g., a referential expression of English that occurs as the noun phrase of an assertion in English refers to the same things that are referred to by the referential concept that the expression stands for—and, in fact, the linguistic act is just the mental act expressed overtly in English.12 On the other hand, we do want to say in conceptualism that a referential expression of a spoken language stands for (stat pro) a referential concept, and similarly that a predicable expression stands for a predicable concept, that might be exercised in a given speech act, which might seem in some respects similar to what Buridan says. But then, Ockham does have a notion of subordination, which he says holds between a linguistic term and the corresponding concept, which seems to be essentially what is meant in saying that the term stands for the concept. It is Ockham’s position, in other words, that is closer to conceptualism as we understand it here. The relation between a conventional language such as Latin or English and a language of thought such as Mental is sometimes said to be analogical, especially by writers sympathetic to nominalism. Peter Geach, for example, held the “general thesis” that “language about thoughts is an analogical development of language about language,”13 and, in particular, that “the concept judging is ... an analogical extension of the concept saying.”14 Similarly, Wilfrid Sellars claimed that our view of thoughts as “inner episodes” is a theoretical construction modelled upon our view of meaningful linguistic behavior, and in particular that “concepts pertaining to the intentionality of thoughts,” such as that of reference, are “derivative from concepts pertaining to meaningful speech.”15 The 11 Cf.
Normore [1985], p.190, and Spade [1996], chapter 3. is not to say that there are no “inner episodes” of thinking, i.e. of referring and predicating, that are nonlinguistic. 13 Geach [1957], p.98. 14 Ibid., p.75 15 Sellars [1981], p. 326. Also, see “Empiricism and the Philosophy of Mind” in Sellars [1963]. 12 This
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thesis does seem to be one that Ockham followed in that some of the features of Latin are assumed to carry over into Mental. Indeed, according to Geach, “the grammar of Mental turns out to be remarkably like Latin grammar. There are nouns and verbs in Mental; nouns have cases and numbers, and verbs have voice, mood, tense, number, and person.”16 Despite his general thesis, Geach warns us not to carry the analogy too far. Ockham, in particular, according to Geach, “merely transfers features of Latin grammar to Mental, and then regards this as explaining why such features occur in Latin.”17 Except for mood and tense, Geach maintained that “the grammatical properties ascribed by Ockham to Mental words may all be easily dismissed.”18 The exception is noteworthy because tense “does enter into the content of our thoughts,” and, according to Geach, “there are modal differences between thoughts—though the moods of a natural language like Latin are a very inadequate indication of this, being cluttered with a lot of logically insignificant idiomatic uses.”19 This of course is a position very much like the one we described in chapter two. Geach’s claim that Ockham carried the analogy of thought to language too far has been attacked on several fronts.20 Nevertheless, his critics agree that the proper comparison is not of Mental with Latin but of Mental with the kind of “ideal languages” that logicians and philosophers have constructed in the twentieth century, i.e., with a logistic system from the point of view of logic as language as opposed to the view of logic as calculus.21 Thus, according to J. Trentman, “Ockham’s real criterion ... for admitting grammatical distinctions into Mental amounts to asking whether the distinctions in question would be necessary in an ideal language—ideal for a complete, true description of the world.”22 Similarly, according to Paul Spade, “mental language is to be a kind of ideal language, which has only those features it needs to enable it to discern the true from the false, to describe the world adequately and accurately.”23 Of course, an ideal language should account not only for an adequate and accurate description of the world, but also for valid reasoning about the world. In other words, Mental should be constructed as a logically ideal language, relative to which analyses of natural language sentences can be given, thereby resulting in logically perspicuous representations of the truth conditions of those sentences. The logical forms representing these truth conditions would then determine which sentences follow validly from other sentences as premises, i.e., they would then determine 16 Geach [1957], p.102. See Spade [1996], chapter 4, for a more detailed list of the “common” and “proper accidents” of nouns and verbs of Latin, but where only the “common accidents” are part of the grammar of Mental. 17 Ibid. 18 Ibid., p.103. 19 Ibid., pp.103f. 20 See Trentman [1970] and Spade [1996], chapter 4. 21 See van Heijenoort 1967 for an account of the distinction between logic as language and logic as calculus. 22 1970, p.589. 23 1996, p.110.
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the conditions for valid reasoning in terms of the recursive operations of logical syntax.24 The kind of analyses of natural language sentences intended here are what the terminist logicians called “expounding” (exponere) or “exposing” (expositio). Thus, according to Calvin Normore, “expositio is a natural result of the recognition that the surface grammar of a sentence is not always a reliable guide either to its truth conditions or its inferential connections to other sentences,” which suggests that the analysis (or exposition) of a natural language sentence (of Latin, English, etc.) should not result in another sentence of that language but in a logical form of an ideal language based on logical grammar.25 That is why the proper province of logic for Buridan, according to Normore, is “the articulation of truth conditions for grammatically complex sentences,” i.e., the process of making “the logical form of sentences explicit.”26 What is needed in terminist logic, but up until now has not been given, is a representation of Mental as a logistic system based on the view of logic as a conceptualist theory of predication. Such a system should provide a perspicuous representation of the truth conditions of our speech and mental acts, and thereby, in terms of the recursive operations of logical syntax, the validity of our arguments as well (as determined by the deductive level of the theory). Moreover, as a conceptualist theory of our speech and mental acts, the system should also provide (on the initial level) logical forms that perspicuously represent the cognitive structure of those acts, including in particular the referential and predicable concepts underlying them. It is these concepts that in one way or another correspond to what the terminist logicians called supposition. The system we have described for conceptual realism, we believe, can be used in just this way, even if some features of the system may seem to be in conflict with certain terminist theses.
8.2
Ockham’s Early Theory of Ficta
The ontology of intensional objects we have described in the previous chapter might seem to be in conflict with the nominalism commonly associated with Ockham and the terminist logicians. Ockham, for example, clearly rejected the Platonist interpretation of nominalized predicates; but that was because he associated it with a Platonist or realist theory of predication. On this theory, a person is said to be wise, for example, because he exemplifies the quality or property denoted by ‘wisdom’. That is, a predication of the form ‘x is wise’ is explained on the basis of a supposedly more basic sentence of the form ‘x 24 The recursive operations of logical syntax will generate some logical forms that do not represent propositions of Mental, however, but which are needed for the deductive machinery of the ideal language by which to prove the validity of arguments—and for the generation of those forms that do represent the propositions of Mental as well. (See §8.6 below for more on this point.) 25 Normore 1985, p.192. 26 Ibid.
8.2. OCKHAM’S EARLY THEORY OF FICTA
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exemplifies wisdom’, which means that signification in the sense in which ‘wise’ signifies wise individuals is not a basic semantic notion after all.27 This is not how intensional objects are understood in conceptual realism, however, where predication is explained in terms of the mutual saturation of a referential and predicable concept in a speech or mental act. Intensional objects, as we have said, are products of language and culture that do not preexist the evolution of consciousness, and, as such, can in no sense be the basis of a theory of predication. Something like a modal-moderate realist theory of predication is part of conceptual natural realism, but the natural kinds and properties that are part of that theory are not objects, and therefore they are not the intensional objects denoted by nominalized predicates. Natural kinds, for example, are unsaturated causal structures that are the basis of causal laws; and they become saturated only as the nexuses of states of affairs. A terminist logician, such as Ockham (reconstructed somewhat as described below), would have rejected the (empirical) posits of conceptual natural realism, although another terminist logician, such as Burley, might well have accepted them.28 Ockham, however, could have accepted the intensional objects of conceptual intensional realism, just as he once accepted ficta as intentional objects. Ockham, in other words, did accept something like our account of intensional objects in his early view of concepts as ficta.29 On this view, which is sometimes called the fictum theory, concepts are the intentional objects of acts of intellection (e.g., judgments). Ficta were not regarded as independently “real” entities, but were said to have only an “intentional being” (esse objectivum), according to which “their being is their being cognized.”30 Ficta included not only “universals,” such as humanity and triangularity, but also logical objects, such as propositions (as abstract objects), and fictitious objects, such as chimeras and goatstags, and also impossible objects, such as the round square.31 These are just the sort of objects that are accounted for in conceptual realism as intensional objects, which suggests that the latter might not really be so alien to 27 See
Loux [1974], p. 6. chapter twelve for a description of conceptual natural realism. The same logistic formulation of natural realism seems to apply to Burley, incidentally, in that Burley accepted the thesis that some concepts have a natural kind, or a natural property., corresponding to them. 29 Spade 1996, p.154, and Normore 1990, p.59, for a description of Ockham’s fictum theory. 30 Spade [1996], p.156. Marilyn McCord Adams prefers to speak of Ockham’s early view of concepts as the “objective existence theory.” She objects to calling such objects ficta, apparently because sometimes the intentional objects thought about are real and not fictitious things ([1977], pp.151f). Our concern here, however, is not with intentional objects as such, whether real or fictitious, but with Ockham’s early theory of ficta as a way of seeing why the intensional objects of conceptual realism can be accommodated in a reconstructed version of terminist logic. After all, there are real objects corresponding to some of our intensional objects, even if they are not the same as those intensional objects. 31 See Adams [1977], p.147. T.K. Scott, [1966a], p.16, notes that Gregory of Rimini argued for propositions in the modern sense of abstract objects, or what he called enunciables (enuntiabile), which were denoted by infinitive or gerundive expressions that amounted in effect to nominalized sentences. These enunciables, like the abstract objects of conceptual realism, were taken as real, but not existent, objects. 28 See
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Ockham’s ontology after all, even if he later changed his mind about ficta.32 Ockham described his earlier view of concepts as intentional objects in his Reportatio, where he made a distinction between two kinds of mental language. One was Mental as we have described it so far, and “the other was a language whose terms were the mental representations of the spoken expressions of natural language itself,” i.e., a mental language that is “posterior to spoken language,” and therefore based on spoken language.33 This view of ficta is not unlike our view of intensional objects in conceptual realism, where conceptual nominalization and the object ification, or reification, of these objects is “posterior” to linguistic nominalization both in the historical development of language and in the conceptual development of individuals in their acquisition and use of language. Ockham did give up his early theory of concepts as ficta in favor of his later theory of concepts as mental acts (intellectiones). But that does not mean that the role ficta played in explaining how we can think about “unreal” objects can now be explained by concepts as mental acts; and, in fact, we maintain they cannot, and that something like ficta—namely, the intensional objects of conceptual realism—are needed to fulfill this role. Ontologically, ficta and the intensional objects of conceptual realism are similar in the way they depend upon concepts; for just as ficta, as intentional objects, had their being in being cognized in the mental acts that Ockham later identified with concepts, so too intensional objects similarly have their being in the concepts whose intensions they are. Ockham was right in his rejection of concepts as ficta, but wrong in then rejecting ficta altogether. They have a role to play in mathematics and the semantics of fiction, and stories and theories in general as we explained in the previous chapter; and, perhaps even more importantly, as we will see, in the semantics of those concepts that intensional verbs (such as ‘seek’, ‘promise’, ‘owe’, etc.) stand for, as well as in our conceptualist theory of predication for concepts based on relations in general, including the copula. Our proposal here is to take the intensional objects of conceptual realism as a “logical reconstruction” of Ockham’s early theory of ficta as intentional objects.
8.3
Ockham’s Later Theory of Concepts
Ockham’s later theory of concepts, which is sometimes called the intellectio or mental-act theory, does not identify concepts with intentional objects, but with mental acts themselves, i.e., with actual mental occurrences.34 The commonname concept man, for example, “is the very act of thinking of men”35 ; that is, as a mental occurrence, that very act signifies all men. This theory is similar to contemporary nominalism according to which, e.g., an actual spoken linguistic token of the word ‘man’ is said be true of all men, but as a matter of convention 32 For
an analysis of the round square as a fictum, see §7.10 of the previous chapter. 1990, p.59. 34 Spade, [1996], p. 155, and Adams [1977], p. 145. 35 Spade [1996], p. 155. 33 Normore
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only, and not because there is a “natural likeness” between the word and men. One problem with the mental-act theory is how the same concept, e.g., man, can be common to all humans as a term of Mental. That is, if a person’s concept man is just a mental act (event) of that person, and one person’s mental act is never the same as another’s, or of the same person at another time, then how can the same concept man be common to different people, or to the same person at different times? The answer that was given, apparently, is that although one person’s concept man is “numerically different” from another person’s concept man—i.e. their mental acts of thinking of men are different mental events— nevertheless, the two concepts are “exact duplicates of one another.”36 But in what sense can the mental acts of two or more people, or of the same person at different times, be exact duplicates of one another? Is it because there is an assumed “natural likeness” between concepts and the things they signify, i.e., that one person’s concept man will then have a “natural likeness” with another person’s concept man (or with the same person’s concept man at a different time)? If so, then, for reasons already given against the supposed “natural likeness” between concepts and the things they signify, this is an answer not acceptable today, or at least not for conceptual realism as we understand it here. Ockham does suggest an alternative—namely, that concepts as mental acts are qualities of the mind, and in particular qualities that “exist” only when a mind is exercising the mental act in question, as in moderate realism (but restricted to qualities that inhere only in minds37 ). Different mental acts of thinking of men are then just different instances of a mind’s having the same quality.38 This version of the mental act theory is sometimes called the quality theory of concepts, according to which “the concept is a real quality inhering in the mind just like any other real property.”39 This theory might explain how the same concept can be exercised in two actual mental acts—namely, by being the same mental quality inhering in the mind or minds whose acts they are— but it doesn’t account for concepts that are in fact never exercised and that we nevertheless “tacitly know” or have in our conceptual repertoire—e.g., concepts of very large numbers or of things that we could, but in fact never, think or speak about. Also, it is not clear how a concept as a quality inhering in a mind only when it is exercised can explain how the mind can exercise that concept, nor how it might inform the act with a referential or predicable nature. Our proposal is to “reconstruct,” or replace, Ockham’s theory of concepts as mental qualities with the theory of concepts as cognitive capacities that we have described in the previous chapter for conceptual realism. Concepts in this sense do not have an “existence” independently of the more general capacity that humans have for language and thought, and yet, as capacities that 36 Ibid.,
p. 93. a moderate realism restricted to mental qualities should be acceptable, but not a moderate realism that applies to the “external” world as well, is an issue we leave to others to explain—if it can be explained at all. 38 Spade [1996], p. 155, and Adams [1977], p. 145. 39 Spade [1996], p. 155. 37 Why
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are intersubjectively realizable, they are objective in at least as strong a sense as Ockham’s notion of a “natural likeness” between concepts and what they signify—but without the problems the latter notion raises. Also, as the rulefollowing capacities underlying our use of predicate and referential expressions in natural languages, concepts have by their very nature the function of informing a speech or mental act with a predicable or referential nature; and, of course, they are the very same capacities that are exercised in the production of those speech and mental acts. Finally, the unsaturated nature of a concept explains its non-occurrent, or quasi-dispositional status—that is, its status as a capacity that could, but need not, be exercised in an appropriate context, or, that might in fact never be exercised at all.
8.4
Personal Supposition and Reference
Reference in conceptual realism is a pragmatic notion that applies only when referential concepts are exercised in speech or mental acts. Reference in terminist logic is also a pragmatic notion, but applies to the way categorematic terms are said to supposit for the things they signify when used in a speech or mental act. Both systems distinguish reference to concepts from reference to things, but only conceptual realism is explicit in distinguishing reference to concepts in terms of predicate quantifiers. Reference to concepts in terminist logic, which is called simple supposition, does not explicitly involve predicate quantifiers, but this might be a matter only of surface grammar.40 In any case, our concern here will be with reference to things, which in terminist logic is called personal supposition. Personal supposition in terminist logic is not the same as reference to things in conceptual realism, because (as we explain in the following section) categorematic terms can have personal supposition either as subjects or predicates of categorical propositions, whereas referential concepts in conceptual realism can never function as predicable concepts, nor can predicable concepts function as referential concepts. Nevertheless, except for the so-called merely confused personal supposition of predicates containing an intensional verb or modal operator (as discussed in §7 below), the personal supposition of terms in categorical propositions does coincide with a combined notion of activated and deactivated reference in conceptual realism, where the deactivated reference is involved in the truth conditions determined by a predicable concept. Both systems, moreover, give a uniform account of general and singular reference to things. As already noted in the previous chapter, referential concepts in conceptual realism, like predicable concepts, are unsaturated cognitive structures; but the structures are not the same. Rather, like the way that quantifier phrases have a structure that is complementary to predicate expressions, or the way that noun phrases are complementary to verb phrases, referential concepts and predicable concepts are cognitive structures that are complementary to one another. This 40 There is also another type of supposition, material supposition, in which a term stands for itself or other spoken or written signs. We will not deal with this type of supposition here.
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complementarity is such that when they are exercised together in a speech or mental act each saturates the other; and just as the predicable concept is what informs that act with a predicable nature, so too the referential concept is what informs the act with a referential nature. An affirmative assertion that is analyzable in terms of a noun phrase and a verb phrase (regardless of the complexity of either) is semantically analyzable, for example, in terms of an overt joint application of a referential concept with a predicable concept; and the assertion itself, as a speech act, is the result of the mutual saturation of their complementary structures in that act. It is just this sort of mutual saturation of complementary cognitive structures that constitutes the nexus of predication in conceptualism. It is also what accounts for the unity of a speech or mental act, i.e. of an assertion or judgment, a problem that Ockham, who anticipated F.H. Bradley’s infinite regress argument, was unable to resolve.41 Ockham, for example, assumed that a judgment that every man is an animal was literally made up of a universal quantifier, the concept man, the mental copula is, and the concept animal.42 But then what unifies these mental terms into a single unified mental act? A fifth mental term that “tied” these items together would need a sixth to “tie” it with the others, which in turn would need a seventh, and so on ad infinitum. That is not how a judgment or assertion is understood in conceptual realism, where concepts, as unsaturated cognitive structures, are not objects, and therefore cannot be actual constituents of a mental act (event).43 Referential concepts, as we have explained, are what the quantifier phrases of our logistic system stand for when the latter are affixed to the symbolic counterparts of names, where both proper and common names are understood to have such counterparts, just as they do in Mental, the language of thought of terminist logic. A proper name is distinguished in the system from common names by a meaning postulate to the effect that at most one thing can be referred to by that name, and that the name refers to the same thing in every possible world in which it refers to anything at all. But names, whether proper or common are different from predicate expressions, as Geach has pointed out, because they can be used in “simple acts of naming” outside the context of a sentence.44 Naming is not the same as referring, it should be emphasized, because the latter is an act that does not occur outside the (implicit if not explicit) context of a sentence used in a speech act, i.e., independently of an associated act of predicating. 41 See
Spade [1996], chapter 4, §3. p. 123. Spade points out that not all terminists agreed with Ockham, and Buridan as well, on this view of judgments or mental propositions as complexes of syncategorematic and categorematic mental terms. Gregory of Rimini and Peter of Ailly, in particular, criticized the view, and argued instead that judgments, or mental propositions, unlike the assertions of spoken language, were “structureless mental acts” that occur, as it were, all at once. This view is similar to the notion of a judgment or assertion in conceptual realism, and might well be “reconstructed” in terms of the latter. 43 On our account of predication as the mutual saturation of a referential and predicable concept, there cannot be even a first step toward Bradley’s infinite regress. 44 [1980], p.52. 42 Ibid.,
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8.5
The Identity Theory of the Copula
As noted in the previous chapter the direct object of a relational predicate is deactivated when that predicate stands for the predicable concept of a speech or mental act. This applies no less to the copula in its use to express an identity than it does to transitive verbs such as ‘seek’ and ‘find’. But, because quantifier phrases occurring within a complex predicate do not stand for the referential concepts they stand for when used as grammatical subjects, we need to distinguish a predicable concept based upon the copula from one based on strict identity. We introduce a new symbol, ‘Is’, that we will use for this purpose, and note that, like the transitive verb ‘find’, the copula Is is extensional in its range as well as in its domain, except that in this case the copula becomes a strict identity. The schematic meaning postulate for Is then is as follows: [λxIs(x, [QyS])] = [λx(Qy)(x = y)]. With ∃ as a special case of the schematic determiner Q, we have [λxIs(x, [∃yS])] = [λx(∃y)(x = y)] as a particular meaning postulate or conceptual truth regarding the copula Is. By means of this notation, we can perspicuously represent the cognitive structure of an assertion of, e.g., ‘Socrates is a man’ as follows (assuming ‘Socrates’ is being used with existential presupposition): (∃xSocrates)[λxIs(x, [∃yM an])](x), which, by λ-conversion and the above meaning postulate, is equivalent to: (∃xSocrates)(∃yM an)(x = y). This last formula, however, unlike the one above, does not represent the structure of a speech or mental act, although it does represent the same truth conditions. Something like this kind of analysis was involved in the so-called two-name, or identity, theory of the copula in terminist logic. Apparently, Ockham and other terminists thought that every affirmative categorical proposition amounted to asserting an identity between the personal suppositions of the subject and the predicate terms of the proposition, as, e.g., the suppositions of the names ‘Socrates’ and ‘man’ in an assertion of ‘Socrates is a man’. Negative judgments, on the other hand, amounted to a denial of such an identity.45 As a result, the identity theory of the copula came to be developed as a theory of the truth conditions of categorical propositions, a theory that is now referred to as the doctrine of supposition proper .46 45 See Spade [1996], p. 133, for a discussion of this “old and venerable theory” of the intellect’s “composing and dividing” of concepts in making affirmative or negative judgments, where dividing is “composing negatively.” 46 See Scott [1966], p. 30, and Spade [1996], chapter 8, §iii.
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The personal suppositions of a proper or common name are the thing(s) signified by that name, but, unlike signification, supposition is understood to be relativized to the propositional context of a speech or mental act where a quantifier might occur with the name. That is why the theory of supposition of terminist logic is really a theory of the truth conditions of categorical propositions as linguistic or mental acts. These truth conditions, as we have said, are determined by the identity theory of the copula together with the quantifiers that occur with the names.47 Thus, for example, an assertion of ‘Some man is a thief’, which on our analysis has the form (∃xM an)[λxIs(x, [∃yT hief ])](x), is equivalent, by λ-conversion and the above meaning postulate, to (∃xM an)(∃yT hief )(x = y), which indicates that the truth conditions of this assertion amount to the identity of some supposition of the term ‘man’ with a supposition of the term ‘thief’, where each supposition, as a result of the quantifiers attached to the terms, amounts to a restriction on what the terms signify.48 Similarly, an assertion of ‘Every man is an animal’, which on our analysis has the form, (∀xM an)[λxIs(x, [∃yAnimal])](x), is equivalent, by λ-conversion and the above meaning postulate, to (∀xM an)(∃yAnimal)(x = y), which indicates that the truth conditions of the assertion involve an identity between each supposition of the categorical term ‘man’ and some supposition of the categorical term ‘animal’, where, again each supposition is a restriction, as determined by the quantifiers attached to each term, of what they signify.49 The same kind of analysis also applies to categorical propositions expressed by means of a predicate adjective with the ‘is’ of predication instead of the copula. An assertion of ‘Every swan is white’, for example, which in our framework 47 Tense or modal modifications of the copula will “ampliate” the personal supposition of the terms, and in that way modify the truth conditions of the speech or mental act in question. (See, e.g., Scott 1966, p. 33, and Spade 1966, chapter 10.) For simplicity of presentation, we restrict ourselves here to present tense uses of the copula. 48 One way to construe the personal suppositions of ‘man’ and ‘thief’ here as a form of reference (as is frequently claimed in the literature) is by noting that the assertion that some man is a thief is equivalent to an assertion that some man and some thief are identical,
(∃xM an ∧ ∃yT hief )[λxy(x = y)](x, y), i.e., where a conjunctive referential concept is used involving both ‘man’ and ‘thief’. 49 To “reconstruct” these suppositions as a form of reference, we can again use a conjunctive referential concept to assert that each man and some animal are such that they are identical: (∀xM an ∧ ∃yAnimal)[λxy(x = y)](x, y), which is equivalent to our original assertion.
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is symbolized as (∀xSwan)W hite(x), is not interpreted by the terminists as an identity between swans and white, or whiteness, or whitenesses (whatever any of these might be taken to be as objects). Rather, the predicate adjective ‘white’ is interpreted as an attributive adjective, so that to say a thing is white is to say that it is a white thing.50 Predicate adjectives, in other words, were analyzed by the terminists as attributive adjectives applied to the common name ‘thing’.51 In conceptual realism, however, the common name ‘white thing’ is interpreted as the complex common name ‘thing that is white’, which is symbolized in our system as ‘T hing/W hite(x)’ (or as ‘Object/W hite(x)’). Thus, whereas the terminist logician would interpret ‘Every swan is white’ as ‘Every swan is a white thing’, we can reconstruct the terminists’ analysis as ‘Every swan is a thing that is white’, which can be symbolized as follows: (∀xSwan)[λxIs(x, [∃yT hing/W hite(y)])](x). This formula, by λ-conversion and the above meaning postulate, has the same truth conditions as (∀xSwan)(∃yT hing/W hite(y))(x = y), which, in terms of supposition theory, is to say that each supposition of the common name ‘swan’ is identical with a supposition of the (complex) common name ‘thing that is white’, or, more simply, with the common name ‘white thing’. Negative categorical sentences such as ‘No raven is white’ are interpreted as denials or negations, as we have already said. That is, to assert that no raven is white is to deny that some raven is white: ¬(∃xRaven)W hite(x), which is provably equivalent to denying that some raven is a white thing; i.e., ¬(∃xRaven)W hite(x) ↔ ¬(∃xRaven)[λxIs(x, [∃yT hing/W hite(y)])](x) is provable in our system. But denying that some raven is a white thing is equivalent, by λ-conversion and the above meaning postulate, to ¬(∃xRaven)(∃yT hing/W hite(y))(x = y), which in terms of the theory of supposition describes the truth conditions as a denial that some supposition of the common name ‘raven’ is identical with a supposition of the complex common name ‘thing that is white’, or more simply, with the common name ‘white thing’. Finally, the logical form of a negative particular categorical sentence such as ‘Some swan is not white’, which is symbolized for us as (∃xSwan)[λx¬W hite(x)](x), 50 See
Normore [1985], p. 194. is not clear if in some cases it is a common name subordinate to ‘thing’. E.g. in asserting ‘Socrates is wise’, are we asserting that Socrates is a wise man (or person), or that Socrates is a wise thing? 51 It
8.6. ASCENDING AND DESCENDING
183
is understood in terminist logic as the equivalent statement that some swan is not a white thing’, which is symbolized as (∃xSwan)[λx¬Is(x, [∃yT hing/W hite(x)])](x). This last formula, by λ-conversion and the above meaning postulate, is equivalent to (∃xSwan)(∀yT hing/W hite(x))(x = y), the truth conditions for which are that some supposition of the common name ‘swan’ is not identical with any supposition of the complex common name ‘thing that is white’.
8.6
Ascending and Descending
Supposition theory is not one theory but two. The first, supposition theory proper, is a theory of the truth conditions of categorical propositions as described in the preceding section. The second, called the doctrine of the modes of supposition, has to do with how many things a categorematic term supposits for in a given speech or mental act.52 This doctrine, despite the reference to “modes of supposition,” is not a theory about different “ways of referring.” Rather, the “modes” are just different types and subtypes of personal supposition. The two basic types are discrete and common supposition, and the purpose of the theory is to explain, or “reduce,” the latter in terms of the former.53 Common supposition is divided into determinate and confused supposition as subtypes, and the latter is further divided into the sub-subtypes of confused and distributive supposition and merely confused supposition.54 Modes of Supposition Discrete Common Determinate Confused A term is said to have discrete supposition in a categorical sentence only if it is either a proper name, a demonstrative pronoun (such as ‘this’ and ‘that’) or a common name preceded by a demonstrative pronoun (such as ‘this man’, ‘that horse’, etc.). Terms that have discrete supposition are said to be discrete terms, and categorical propositions in which they occur as the grammatical subject are said to be “singular propositions.”55 52 Paul Spade, in 1996, p. 294, was the first to propose this interpretation—though he does not explain or develop it, as we do here, in terms of the principle of descent (described in this section). T.K. Scott,in 1966, was the first to distinguish the two doctrines, but he claims that the second has to do with the elimination of quantifiers (pp. 36f). 53 Spade 1996, p. 277. 54 Spade 1996, chapter 9. 55 Spade [1996], p. 276.
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The explanation, or “reduction,” of the types (and subtypes) of common supposition in terms of discrete supposition is given in terms of a “descent” to, and sometimes an “ascent” from, singular propositions. A proper name such as ‘Socrates’ will have discrete supposition only when Socrates exists, however, which means that even an assertion of ‘Socrates is Socrates’ will be false when Socrates does not exist.56 The situation is similar but not quite the same in conceptual realism, where we distinguish between using a proper name with existential presupposition and using it without such a presupposition. Thus if, in asserting ‘Socrates is Socrates’, we are referentially using the name ‘Socrates’ with existential presupposition, the symbolic counterpart of this assertion is (∃xSocrates)[λxIs(x, [∃ySocrates])](x), in which case the assertion is false if it is asserted at a time when Socrates does not exist—or so we may assume in conceptual realism. But if we are referentially using ‘Socrates’ without existential presupposition (which is usually not the case), the symbolic counterpart of our assertion is (∀xSocrates)[λxIs(x, [∃ySocrates])](x), which is not false but vacuously true when Socrates does not exist.57 This last point, of course, has to do with universal sentences not having “existential import,” contrary to the way they were interpreted by medieval logicians. That is, in conceptual realism, and in modern logic in general, universal 56 Scott
[1966], p. 41. proper name, such as ‘Socrates’, of a concrete, as opposed to an abstract, object can be stipulated to be “existence-entailing” in the sense that if the name can be used to refer to anything, then that thing exists (as a concrete object): 57 A
(∀xSocrates)E!(x),
(PN-E!)
where E! stands for concrete existence, as opposed to being (the value of an individual variable bound by ∃). Note that the symbolic counterpart of ‘Socrates exists’, where the name ‘Socrates’ is used only with existential presupposition, is (∃xSocrates)E!(x). Denying that Socrates exists, i.e., ¬(∃xSocrates)E!(x), is then equivalent to (∀xSocrates)¬E!(x), from which, together with the above stipulation about the name ‘Socrates’, it follows that Socrates does not have being when he does not exist, e.g., ¬(∃xSocrates)(x = x). But, because (∃xSocrates)[λxIs(x, [∃ySocrates])](x) is provably equivalent to this last formula without the negation, it then follows that it too is false when Socrates does not exist. It may be preferable, however, to allow that Socrates can exist, and in that sense have being, even when he does not exist (as the terminist logicians also seemed to have assumed). In that case, instead of (PN-E!), we would assume (∀xSocrates)♦E!(x).
(PN-♦E!)
Then, assuming that whatever can exist (in the concrete sense) has being, i.e., (∀x)[♦E!(x) → (∃y)(x = y)], it would follow, given (PN-♦E!(x)), that ‘Socrates is Socrates’ is true even at a time (or world) when Socrates did not exist (in the concrete sense). But, in either case, it should be noted, because to be an abstract object is to be a thing that cannot exist (in the sense of concrete existence), we still have Abstract =df [λx¬♦E!(x)].
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conditionals—and therefore all sentences beginning with a universal quantifier phrase—are true, and not false as interpreted by the terminists, when their antecedents (subject terms) are vacuous.58 The utility of this view, both for science and natural language, has been more than justified over the last century; and we assume that its acceptance in our reconstruction of terminist logic is but a minor modification, and improvement, of this system—just as the acceptance of a logic free of existential presuppositions for proper names and definite descriptions is a minor modification, and improvement, of so-called standard first-order logic. We need only stipulate when a name S is assumed to supposit something, i.e. when something is an S, if existential import is needed to validate a descent or ascent, as it will be in some cases. Latin lacks a definite article, incidentally, which means that the terminists did not consider definite descriptions in their analyses at all.59 The existential presuppositions of demonstrative phrases is another matter, however, because such demonstrative phrases are central to the descent to (and, in some cases, ascent from) singular propositions. The term ‘man’, for example, has determinate supposition in ‘Socrates is a man’, which means that the descent (at a time when Socrates exists) to a certain disjunction of singulars, ‘Socrates is this man or Socrates is that man or ... or Socrates is that man’, is valid—and so too is the ascent from the singulars to ‘Socrates is a man’ (again, at a time when Socrates exists), and therefore from their disjunction. Because the demonstrative ‘that’ can be used a number of times in a disjunction, or conjunction, of singular propositions, we must, in our logical language, distinguish each use from the others.60 For convenience, we will use ‘T hat1 ’, ‘T hat2 ’,..., ‘T hatn ’, etc. (for each positive integer n), as variable-binding operators that operate on a name (complex or simple), resulting thereby in a quantifier phrase that can then be applied to a formula. We will also read ‘T hat1 ’ as the English ‘this’. Thus, where S is a common name (complex or simple), e.g., ‘M an/Snubnosed(x)’, then ‘(T hat2 xM an/Snubnosed(x))’, read as ‘that man 58 Note
that by the meaning postulate (MP1) of §7.3 of the previous chapter, (∀xSocrates)[λxIs(x, [∃ySocrates])](x) ↔ (∀x)[(∃ySocrates)(x = y) → Is(x, [∃ySocrates])],
is provable. The universal conditional on the right-hand side of this biconditional is vacuously true if ¬(∃ySocrates)(x = y) is true for any value of x as bound by ∀, which it is when Socrates does not exist, or so we may assume already noted. 59 This is not to say that uniqueness cannot be expressed in Latin by means other than definite descriptions. 60 Note that more than one referential concept can be exercised in a disjunction, or conjunction, of singular propositions. The detective who says, while pointing to two different men, ‘This man is the killer or that man is the killer’ is exercising two different referential concepts, expressed by ‘this man’ and ‘that man’, in his speech act (whereas, of course, as the direct object, ‘the killer’ is deactivated in both disjuncts). Similarly, when a school coach says, while pointing to certain boys in his class, ‘That boy is on team A and ... and that boy is on team A’, he using the referential expression ‘that boy’ to refer to a certain number of different boys in his conjunctive statement. Conjunctive and disjunctive statements are not basic statements, needless to say, and hence are not subject to the restriction that only one referential, and one predicable, concept can be exercised in them.
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who is snubnosed’ (or as ‘that snubnosed man’), is a quantifier phrase of our symbolic language; and, as such, the phrase stands for a referential concept. The symbolic analysis of an assertion of ‘That man who is snubnosed is wise’, which involves the mutual saturation of a referential and a predicable concept, can now be given a logically perspicuous representation as follows: (T hat2 xM an/Snubnosed(x))W ise(x). We also interpret the use of ‘this’ and ‘that’ that occur without a common name, as in ‘This is a dog’, ‘That is a man’, etc., similar to the way we interpret the objectual quantifiers (∀x) and (∃x), i.e. as implicitly containing the common name ‘thing’ (or ‘object’), as in ‘This thing is a dog’, ‘That thing is a man’, etc. Now our point about the existential presupposition of a demonstrative phrase is that when a speaker says, e.g., ‘That man is sitting’, he is presupposing that the thing he is indicating is a man, i.e., that ‘That man is a man’ is true. But if the speaker is pointing to a manikin that he has mistaken for a man, then his purported reference has failed, and both the speaker’s assertion and the sentence ‘That man is a man’ are false in such a context. A speaker’s use of a demonstrative phrase, we maintain, is equivalent to, if not synonymous with, a use with existential presupposition of a definite description; in particular, that a use of a demonstrative phrase, e.g., ‘that S’, where S is a complex or simple common name, is equivalent to using with existential presupposition the definite description ‘the S that I am indicating’.61 A sentence of the form ‘The S is an S’, where the definite description is used with existential presupposition—as, e.g., in the implicit premise of Descartes’s ontological argument, ‘The perfect being is a perfect being’—is not a valid thesis; and, because of the equivalence between demonstrative phrases and definite descriptions used with existential presuppositions, neither are sentences of the form ‘That S is an S’.62 Determinate supposition, we have said, means a descent to a certain disjunction of singulars. More specifically, a common name S has determinate supposition in a (categorical) proposition P if S occurs in P as part of a quantifier phrase and the descent from P to a disjunction of singular propositions (Q1 ∨ ... ∨ Qn ), where Qi , for 1 ≤ i ≤ n, is obtained from P by replacing the quantifier phrase containing S by ‘thati S’, is valid. To ensure validity, the disjunction must be exhaustive of all the S there are (when P is asserted). 61 We assume here that there can be no cases of correctly using a demonstrative phrase, such as ‘that man’, without existential presupposition—the way there can be cases of correctly using a definite description without existential presupposition. The truth conditions for a sentence of the form ‘Thati S is F ’ are the same as ‘There is exactly one S that I am (now) indicating and it is F ’, which can be symbolized as follows (where ‘G(z)’ abbreviates ‘I am indicating z’):
(T hati yS)F (y) ↔ (∃yS)[(∀zS)(G(z) ↔ z = y) ∧ F (y)]. 62 ‘That S is an S’ will be true in any context in which ‘Something is that S’ is true; i.e. when (∃x)(T hati yS)(x = y) is true.
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Thus, in the determinate supposition of ‘man’ in ‘Socrates is a man’ (asserted when Socrates exists), which is symbolized as (∃xSocrates)[λxIs(x, [∃yM an])](x), the descent, to be valid, cannot be to any disjunction of the form (∃xSocrates)[λxIs(x, [T hat1 yM an])](x) ∨ ... ∨ (∃xSocrates)[λxIs(x, [T hatn yM an])](x),
but only to a disjunction that is exhaustive of all of the men there are, i.e. of all the men who exist at the time of the assertion of ‘Socrates is a man’. In other words, even after having indicated a number of men by means of a demonstrative phrase, we may still not have indicated the man that in fact is Socrates if we have not exhausted all of the men there are. Implicit in such a descent, accordingly, is the assumption that we are indicating all the men there are; that is, that A thing is a man if, and only if, either it is that1 man or ... or it is thatn man. is true (at the time of assertion). A similar assumption applies to any common name, S, of concrete physical objects; that is, for some natural number n, the terminist logicians assumed that at any given time there are exactly n many things that are S.63 As a generalized version of the above thesis for the common name ‘man’, we will call this assumption, for any common name S (complex or simple) of concrete things, and any natural number n, the principle of descent for n many S, or simply P Dn (S). The principle, which in the case when there are n many S says that something is an S if, and only if, it is that1 S or that2 S, or ... or thatn S, can be symbolized as follows64 : (∀x)[Is(x, [∃yS]) ↔ Is(x, [T hat1 yS]) ∨ ... ∨ Is(x, [T hatn yS])]. This thesis, by the meaning postulate for Is, is provably equivalent to (∀x)[(∃yS)(x = y) ↔ (T hat1 yS)(x = y) ∨ ... ∨ (T hatn yS)(x = y)]. It is important to note here that the thesis of descent amounts to an explicit answer to the question of how many S there (now) are in terms of the identity theory of the copula, which as we indicated earlier is what the doctrine of the modes of supposition is really about.65 63 The restriction must be to concrete physical objects, because the thesis will be false for such abstract objects as the natural numbers, and perhaps also for concrete events (which are not physical objects). Unlike the system of conceptual realism, terminist logic gave no explanation of how such abstract objects as the natural numbers are to be accounted for. 64 The principle is taken to apply at any moment of time considered as the present. Related principles for ampliated terms are obtained by applying tense and modal operators to P Dn (S). (We ignore the assumption that T hati yS = T hatj yS, for i, j ≤ n where i = j, incidentally, because it is not needed for the inferences noted here.) 65 When n is 0, we take the right-hand side of the biconditional within the scope of (∀x) to be the formula (x = x), from which it follows that nothing is S, which is as it should be when n is 0. Note, incidentally, that when something is an S, i.e. (∃x)Is(x, [∃yS]), then, by P Dn (S) and distribution of (∃x) over ∨, each disjunct will have the form (∃x)Is(x, [T hati yS]), which, in effect, stipulates that the existential presupposition of the demontrative phrase is fulfilled, i.e. that something is thati S.
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The principle of descent for ‘man’, i.e., P Dn (M an), also validates the ascent (when Socrates exists) from any one of the singulars ‘Socrates is this man’, ..., ‘Socrates is thatn man’ to the original proposition ‘Socrates is a man’ in which ‘man’ is said to have determinate supposition. For this kind of proposition, in other words, we can validly ascend when and only when we can validly descend. Determinate supposition applies to common names occurring as subject terms as well as to common names occurring as predicate terms, as in our example of ‘Socrates is a man’. Thus, by P Dn (M an), we can validly descend from ‘Some man is running’ to ‘This man is running or that2 man is running or ... or thatn man is running’; and of course we can ascend from any one of these disjuncts—and therefore from the disjunction as well—to the sentence ‘Some man is running’. Similarly, by P Dn (M an), we can validly descend from ‘A man is not running’ to ‘This man is not running or that2 man is not running or ... or thatn man is not running’; and, again, we can similarly ascend from any, or all, of these disjuncts to ‘A man is not running’, or to ‘Some man is not running’, both of which are symbolized the same way in our system. With determinate supposition, in other words, we can validly descend on the basis of the principle of descent when, and only when we can validly ascend on the basis of that principle.66 Confused and distributive supposition is more problematic than determinate supposition, however, because, according to Ockham, a common name will have confused and distributive supposition in a categorical proposition only when one can descend to a conjunction of singulars, but cannot ascend from any one singular in the conjunction.67 The problem is that such a descent is valid only when each conjunct can be truthfully asserted if the original premise is true, and hence only when the existential presupposition of the demonstrative phrase in that conjunct is fulfilled, i.e., only when (∃x)Is(x, [T hati yS]) is true for each i such that 1 ≤ i ≤ n, where S is the common name in question and there are exactly n many S. This, as it turns out, is just the issue of existential import, but as applied to demonstrative phrases in particular. Such presuppositions were implicit in what Ockham and the terminist logicians assumed for this type of supposition. The common name ‘man’ in ‘Every man is an animal’, for example, will have confused and distributive supposition by the principle of descent P Dn (M an), but only if the existential presuppositions in question are fulfilled.68 If these 66 The argument for the general claim is based on specific examples, to be sure; but that is because the examples can be easily schematized and shown to hold in general. 67 There is a problem with this characterization when applied to the predicate of a negative particular proposition, however, because, although the descent is to a conjunction, the conjuncts will not be singular propositions. But then each conjunct can in turn be “reduced” by determinate supposition to a disjunction of singulars. Thus, on our characterization, a common name S has confused and distributive supposition in a proposition P if S occurs in P as part of a quantifier phrase and the descent from P to a conjunction (Q1 ∧ ... ∧ Qn ) is valid, where each Qi (whether singular or otherwise) is obtained from P by replacing the quantifier phrase containing S by ‘thati S’. 68 All that follows from P D (M an) and (∀xM an)Is(x, [∃yAnimal]), using (MP1) of chapn ter 6, is the conjunction that [if anything is that1 man, then it is an animal] and ... and [if
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189
presuppositions are fulfilled, then, by P Dn (M an), one can validly descend from ‘Every man is an animal’ to the conjunction ‘This man is an animal and that2 man is an animal and ... and thatn man is an animal’, i.e. to (T hat1 xM an)[λxIs(x, [∃yAnimal])](x)∧...∧ (T hatn xM an)[λxIs(x, [∃yAnimal])](x).
One cannot validly ascend, as Ockham says, from any one of these singulars to the universal ‘Every man is an animal’; but, clearly, given P Dn (M an), one can validly ascend to the universal from the whole conjunction. Confused and distributive supposition was assumed by the terminists to apply not only to the subject term of a universal affirmative, but to the subject term of a universal negative categorical proposition as well. But, again, the descent will be valid only if the existential presuppositions of the demonstrative phrases in the conjunction in question are fulfilled. Consider, for example, the same common name ‘man’, only now occurring as the subject of a universal negative categorical, such as ‘No man is running’. The confused and distributive supposition of ‘man’ in this proposition means that, by P Dn (M an), one can descend from this sentence to ‘This man is not running and that2 man is not running and ... and thatn man is not running’ (with the negation in each conjunct internal to the predicate). But, as in our previous example, the descent will be valid only when (∃x)Is(x, [T hati yS]) is true for each i such that 1 ≤ i ≤ n. In our reconstruction, we note first that ‘No man is running’ is understood as denying that some man is running, which is symbolized as ¬(∃xM an)Running (x), but which is provably equivalent to (∀xM an)¬Running(x). By this last sentence and P Dn (M an), it follows that anything that is that1 man or ... thatn man is not running, i.e. (∀x)[(T hat1 yM an)(x = y) ∨ ... ∨ (T hatn yM an)(x = y) → ¬Running(x)], and from this and (∃x)Is(x, [T hati yM an]), for 1 ≤ i ≤ n, the desired conjunction, (T hat1 yM an)[λy¬Running(x)](x) ∧ ... ∧ (T hatn yM an)[λy¬Running(x)](x) follows.69 The same argument can be made in reverse order, moreover, which means that, given P Dn (M an), we can validly ascend from the conjunction to the universal negative sentence. In other words, with the confused and distributive supposition of a common name occurring as the subject term of a universal anything is thatn man, then it is an animal], i.e. (∀x)[Is(x, [T hat1 yM an])
→
Is(x, [∃yAnimal])] ∧ ... ∧
(∀x)[Is(x, [T hatn yM an])
→
Is(x, [∃yAnimal])].
Hence, if all universals were assumed to have existential import, then the existential presuppositions of these demonstrative phrases would be fulfilled. That is why we have said that this is just the issue of existential import, but as applied to demonstrative phrases. 69 Without the assumption that (∃x)Is(x, [T hat yS]) is true, for 1 ≤ i ≤ n, all that would i follow by P Dn (M an) is the conjunction ‘[If anything is that1 man, then it is not running] and ... and [if anything is thatn man, then it is not running]’.
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affirmative or a universal negative proposition, one can validly descend to a conjunction when and only when one can also validly ascend from such a conjunction to the universal in question. There are cases of confused and distributive supposition where the issue of existential import is not relevant, i.e., where the existential presuppositions of demonstrative phrases need not be fulfilled. In particular, a common name occurring as (part of) the predicate of any negative categorical proposition will have confused and distributive supposition, regardless of the issue of existential import. The common name ‘runner’ in ‘No man is a runner’, for example, unlike the common name ‘man’, will have confused and distributive supposition independently of whether or not the existential presupposition of any of the demonstrative phrases in question is fulfilled.70 In other words, by P Dm (Runner), one can descend validly to the conjunction ‘No man is that1 runner and ... and no man is thatm runner’ without any further assumptions about existential import.71 As a denial, ‘No man is a runner’ is symbolized as ¬(∃xM an)[λxIs(x, [∃yRunner])](x), which, by quantifier negation and λconversion, is equivalent to (∀xM an)¬Is(x, [∃yRunner]). This last formula, by P Dm (Runner) and elementary transformations, implies (∀xM an)¬Is(x, [T hat1 yRunner]) ∧ ... ∧ (∀xM an)¬Is(x, [T hatm yRunner]), which, by quantifier negation, is equivalent to the conjunction ¬(∃xM an)[λxIs(x, [T hat1 yRunner])](x) ∧ ... ∧ ∧ ¬(∃xM an)[λxIs(x, [T hatm yRunner])](x). This argument can also be given in reverse order, so that one can validly ascend, by P Dm (Runner), from the conjunction to the sentence ‘No man is a runner’. In other words, with confused and distributive supposition, one can validly descend to a conjunction from a universal negative categorical proposition when and only when one can validly ascend from the conjunction to that proposition by the same principle—regardless whether the common name occurs as the subject or the predicate of the proposition, except that when it is the subject, the existential presuppositions of the demonstrative phrases in the conjunction must be fulfilled. The way up is not always the same as the way down, however. Consider, for example, the common name ‘runner’ in the negative particular proposition ‘Some man is not a runner’ (where the negation is internal to the predicate), which is symbolized as (∃xM an)[λx¬Is(x, [∃yRunner])](x). Here, the common name ‘runner’ has confused and distributive supposition, which means that one 70 Our examples come from Spade [1996], chapter 9. Ockham and other terminists assumed that ‘is running’ can be construed as ‘is a runner’. This construal is dubious; but we will accept it here as part of our reconstruction of terminist logic. A separate, alternative treatment can be given for the present participle in conceptual realism in terms of the logic of events. 71 The conjuncts in this case are not singular propositions, it might be noted; but, by P Dn (M an), each conjunct can be expanded into a conjunction of singulars of the form ‘Thatm+i man is not thati runner’, with the negation internal to the predicate.
8.7. HOW CONFUSED IS MERELY CONFUSED
191
can descend, by P Dm (Runner), to the conjunction ‘Some man is not this runner and ... and some man is not thatm runner’.72 Note that, by P Dm (Runner), (∃xM an)[λx¬Is(x, [∃yRunner])](x) implies (∃xM an)¬[Is(x, [T hat1 yRunner]) ∨ ... ∨ Is(x, [T hatm yRunner])], which, by elementary transformations, implies (but is not implied by) (∃xM an)¬Is(x, [T hat1 yRunner]) ∧ ... ∧ (∃xM an)¬Is(x, [T hatm yRunner]), which validates the descent to the conjunction ‘Some man is not this runner and ... and some man is not thatm runner. But the reverse order of this argument is not also valid, because, unlike the inference from (∃xM an)(ϕ ∧ ψ) to (∃xM an)ϕ ∧ (∃xM an)ψ, the inference from (∃xM an)ϕ ∧ (∃xM an)ψ to (∃xM an)(ϕ ∧ ψ) is not valid. Thus, despite Heraclitus, the way up is not always the same as the way down.
8.7
How Confused is Merely Confused
Merely confused supposition has been the one type of supposition that has been controversial in terminist logic. It is the one type, for example, that does not allow for a valid descent to either a conjunction or disjunction of singular propositions. Ockham’s main characterization is that a common name has merely confused supposition in a categorical proposition if one can validly descend to a “disjoint predicate.”73 The common name ‘animal’, for example, has merely confused supposition in the universal affirmative ‘Every man is an animal’, which, as already noted, is symbolized as (∀xM an)[λxIs(x, [∃yAnimal])](x). That is, by the principle of descent, P Dk (Animal), λ-conversion, and elementary transformations, we can validly descend from this proposition to ‘Every man is [this animal or that2 animal ... or thatk animal]’, which is symbolized as follows: (∀xM an)[λx(Is(x, [T hat1 yAnimal]) ∨ ... ∨ Is(x, [T hatk yAnimal]))](x). One cannot, of course, validly distribute (after λ-conversion) the universal quantifier (∀xM an) over the disjunction [Is(x, [T hat1 yAnimal]) ∨ ... ∨ Is(x, [T hatk yAnimal])] to get ‘Every man is this animal or ... or every man is thatk animal’. So, in this case no further valid “reduction” to singulars is possible. 72 Once again, the conjuncts are not singular propositions, but, by P D (M an), each conn junct can be expanded into a disjunction so that the final result is a conjunction of disjunctions of singular propositions. 73 Spade 1996, p. 284.
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Some authors have found this “reduction” to a disjunctive predicate “very odd,” as though the resulting sentence has problematic truth conditions.74 But there are many sentences in science and natural language with disjunctive predicates that are clearly unproblematic. The sentence of arithmetic, ‘Every integer is odd or even’, symbolized as (∀xInteger)[λx(Odd(x) ∨ Even(x))](x), is perfectly clear as to its truth conditions, for example, even though it is not further “reducible” to ‘Every integer is odd or every integer is even’. Similarly, ‘Every person is either male or female’ seems perfectly clear in its truth conditions, even though it is not “reducible” to ‘Every person is male or every person is female’. Ockham also thinks that merely confused supposition applies to such sentences as ‘John promises Simon a horse’.75 But then, merely confused supposition must also apply to ‘John gives Simon a horse’, because this sentence has the same logical form as ‘John promises Simon a horse’. On our analysis, the logical form of these sentences as judgments or speech acts (where ‘John’ is used with existential presupposition) is given as: (∃xJohn)[λxP romise(x, [∃ySimon], [∃zHorse])](x), and (∃xJohn)[λxGive(x, [∃ySimon], [∃zHorse])](x). Now, because ‘gives’ is an extensional verb with respect to all of its arguments, the following identity is a conceptually valid thesis of our system as a result of the meaning postulate for Give: [λxGive(x, [∃ySimon], [∃zHorse])] = [λx(∃zHorse)Give(x, [∃ySimon], z)]. But then, by the principle of descent, P Dj (Horse), we can validly descend to the disjunction, ‘John gives Simon this horse or ... or John gives Simon thatj horse’, in symbols:76 (∃xJohn)[λxGive(x, [∃ySimon], [T hat1 zHorse])](x) ∨ ... ∨ (∃xJohn)[λxGive(x, [∃ySimon], [T hatj zHorse])](x). 74 This is Paul Spade’s view in Spade1996, p. 284. Note, however, that in our analysis we have distributed the copula over ‘this animal or ... or thisk animal’. Perhaps Spade has something like Is(x, [T hat1 yAnimal ∨ ... ∨ T hatk yAnimal]) in mind as the problematic “disjunctive predicate,” where the copula has not been distributed. If so, then he has a point, because this expression is not well-formed. 75 It is not clear that Ockham thinks of an assertion of a sentence like this as a categorical proposition. If it is a categorical, then, apparently, it is to be rephrased with a copula, e.g., as ‘John is a man who promises Simon a horse’. But then the disjunction has to do with the different demonstrative phrases, ‘this man who promises Simon a horse or ... or thatn man who promises Simon a horse’, in which case the descent is by determinate supposition, and therefore unproblematic. Of course, we then still have to explain the supposition of the common name ‘horse’ in the singulars ‘John is thati man who promises Simon a horse’. 76 By the above identity, the meaning postulate for Is, and (MP2) of §6 above,
[λx(∃zHorse)Give(x, [∃ySimon], z)] = [λx(∃w)(Is(w, [∃zHorse]) ∧ Give(x, [∃ySimon], w))]
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193
This descent, however, is not really by merely confused supposition, which, according to Ockham, does not allow descent to a disjunction. Rather, the descent seems to be the same as that already described for determinate supposition.77 Nothing like this follows for ‘John promises Simon a horse’, however; and the reason is that, unlike ‘give’, ‘promise’ is not extensional in its third argument position, i.e., ‘promise’ is an intensional verb with respect to its direct object argument position. Yet Ockham maintained that ‘horse’ has merely confused supposition in this sentence, and that the descent to a disjunctive predicate, as in ‘John promises Simon this horse or ... promises Simon thatj horse’, is valid. That is, according to Ockham, the descent from ‘John promises Simon a horse’, as symbolized above, to (∃xJohn)[λx(P romise(x, [∃ySimon], [T hat1 zHorse]) ∨ ... ∨ ∨ P romise(x, [∃ySimon], [T hatj zHorse]))](x), is supposed to be valid, when, in fact, it is not valid—as many commentators have repeatedly noted over the years.78 Unlike these other commentators, however, we have a theoretical account in terms of deactivated referential expressions that explains why such a descent fails—and why it succeeds in sentences having the same logical form. Based on this account, our proposal is that a common (or proper) name that is part of a deactivated referential expression that cannot, as it were, be “activated” in a given propositional context is a name for which no “mode” of supposition should be said to apply in the context in question.
8.8
Summary and Concluding Remarks
• The framework of conceptual realism provides a logically ideal language within which to reconstruct the medieval terminist logic of the 14th century. • The terminist notion of a concept, which shifted from Ockham’s early view of a concept as an intentional object (the fictum theory) to his later view is provable; and from this and P Dj (Horse), [λx(∃zHorse)Give(x, [∃ySimon], z)]
=
[λx(∃w)([Is(w, [T hat1 zHorse]) ∨ ... ∨ ∨Is(w, [T hatj zHorse])] ∧ Give(x, [∃ySimon], w))]
follows. From this last identity and the distribution of a conjunction over a disjunction, the disjunction in question follows. 77 Essentially the same argument would show that ‘horse’ has merely confused supposition, and not determinate supposition, however, in ‘Every man gives Simon a horse’. That is, the descent, by P Dj (Horse), to ‘Every man [gives Simon this horse or ... or gives Simon thatj horse]’ is valid, whereas a “descent” to ‘Every man gives Simon this horse or ... or every man gives Simon thatj horse’ is not valid. 78 Note that, as with the distribution of the copula over ‘this animal or ... or that animal’, k we have distributed ‘promise’ over ‘this horse or ... or thatj horse’. Otherwise, the so-called “disjoint predicate”, as in [λxP romise(x, [T hat1 yHorse ∨ ... ∨ T hatj yHorse])] is not well-formed. If this is what Ockham intended, then Spade is right to “think this appeal to disjoint terms is ... a mark of desperation” (Spade 1996, p. 284).
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of a concept as a mental act (the intellectio theory), is reconstructed in this framework in terms of the notion of a concept as an unsaturated cognitive structure. • Referential and predicable concepts are unsaturated cognitive structures that mutually saturate each other in mental acts, analogous to the way that quantifier phrases and predicate expressions mutually saturate each other in language. • Intentional objects (ficta) are not rejected but are reconstructed as the objectified intensional contents of concepts, i.e., as intensional objects obtained through the process of nominalization—and in that sense as products of the evolution of language and thought. • Intensional objects are an essential part of the theory of predication of conceptual realism. In particular, the truth conditions determined by predicable concepts based on relations—including the relation the copula stands for—are characterized in part in terms of these object-ified intensional contents. It is by means of this conceptualist theory of predication that we are able to explain how the identity theory of the copula, which was basic to terminist logic, applies to categorical propositions. • Reference in conceptual realism, based on the exercise and mutual saturation of referential and predicable concepts, is not the same as supposition in terminist logic. • Nevertheless, the various “modes” or types of personal supposition are accounted for in a natural and intuitive way in terms of the theory of reference of conceptual realism. • Ockham’s application of merely confused supposition to common names occurring within the scope of an intensional verb is rejected, as it should be, but its rejection is grounded on the notion of a deactivated referential concept—a deactivation that, because of the intensionality of the context in question, cannot be “activated,” the way it can be in extensional contexts.
Chapter 9
On Geach Against General Reference Theories of reference in the 20th Century have been almost exclusively theories of singular reference, i.e., theories of the use of proper names and definite descriptions to refer to single objects. General reference by means of quantifier phrases has usually been rejected, mainly because of a confusion of pragmatics with semantics, i.e., a confusion of the referential use of quantifier phrases in speech and mental acts with the truth conditions of sentences containing those phrases.1 This confusion of pragmatics with semantics is in marked contrast with our conceptualist theory of reference (as described in chapter seven) where singular and general reference are given a unified account. It is also in contrast with medieval suppositio theories where a unified account was also given, but only in terms of categorical propositions. Bertrand Russell had a theory of general reference in his 1903 Principles of Mathematics, but he later abandoned that theory in his 1905 paper, “On Denoting”. In his later 1905 theory, Russell took ordinary proper names to be eliminable in terms of definite descriptions, which were in turn eliminable contextually in terms of quantifier phrases, and quantifier phrases were then said to be “reducible” to conjunctions and disjunctions of singular propositions. Thus, the 1905 theory, according to Russell, “gives a reduction of all propositions in which denoting phrases [i.e., quantifier phrases and definite descriptions] occur in forms in which no such phrases occur.”2 Russell did allow for a category of “logically proper names,” however, i.e., expressions such as ‘this’ and ‘that’, each of which he said “applies directly to just one object, and does not in any way describe the object to which it applies.”3 Such a category of “logically proper names” 1 This chapter is a revised and extended version of my 1998 paper, “Reference in Conceptual Realism,” in Synthese, vol. 14. 2 Russell 1956, p. 45. 3 See, e.g., Russell’s 1914 paper “On the Nature of Acquaintance”, reprinted in Russell 1956.
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figured prominently in Russell’s logical atomism, where the idea of eliminating all forms of general reference found its clearest paradigm. Indeed, this way of reducing general reference to the singular reference of logically proper names, or what came to be called “objectual constants,” was laid out explicitly by Rudolf Carnap in his state-description semantics, which he developed and applied even to quantified modal logic.4 In many ways, and however unwittingly, it is this paradigm for reducing general reference to singular reference that is now part of the so-called “new theory of direct reference” in which there is only singular reference.5 Aside from the paradigm of logical atomism as a framework for eliminating general reference, there were for many years no explicit arguments against theories of general reference, i.e., arguments that there could be only singular reference. This situation changed in 1962 when Peter Geach published his book, Reference and Generality, which was later revised and reprinted in 1980. In this book, Geach developed arguments that are supposed to apply to any theory of general reference, as well as some others that are designed to work specifically against Russell’s 1903 theory and against the medieval suppositio theories. Geach’s arguments do not work against our conceptualist theory of reference, however, as we will explain in what follows. Nor do those arguments work against the medieval suppositio theories once they are interpreted and reconstructed within conceptual realism as we have done in the previous chapter.
9.1
Geach’s Negation Argument
The only “genuine” form of reference, according to Geach, is reference by means of singular terms, and in particular in the use of proper names. One type of argument that he gives against general reference is based on negation. Consider, for example, an indicative sentence of English containing a proper name ‘a’, and let ‘f ( )’ represent the propositional context remaining when the name ‘a’ has been extracted from the sentence.6 The propositional context ‘f ( )’ is what Geach calls a predicable, which of course can be complex as well as simple. Attaching a prime to ‘f ’, as in ‘f ( )’, is then said to represent a predicable contradictory to ‘f ( )’. Geach does not explain what he means by a predicable contradictory to ‘f ( )’ other than that ‘f ( )’ and ‘f ( )’ 4 See Carnap 1946. Carnap showed that the thesis of the necessity of identity, the modal thesis of anti-essentialism, and what later came to be called the Barcan formula by some, but which really should be called the Carnap-Barcan formula, were all valid in his statedescription semantics for quantified modal logic—long before these topics became popular in the philosophical literature. Carnap’s state-description semantics also amounted to one of the first substitution interpretation of quantifiers. In fact it was Carnap who first observed that a strong completeness theorem even for modal free first-order formulas was not possible on the basis of such an interpretation for an infinite domain (see Carnap, 1938, p. 1651). 5 For a discussion and an account of the “new theory of direct reference,” see Humphreys and Fetzer, 1998. 6 We are using Geach’s terminology here where by a “propositional context” we mean the context of an indicative sentence.
9.1. GEACH’S NEGATION ARGUMENT
197
are supposed to result in contradictory sentences when a “genuine” referential expression is put in place of ‘a’. This is important because it is not clear what Geach’s argument is without begging the question about whether or not the only “genuine” form of reference is by means of proper names. Briefly, Geach’s argument is that when ‘f ( )’ and ‘f ( )’ are attached to “any proper name ‘a’ as subject, they will give us contradictory predications; but if ‘∗ A’ takes the place of ‘a’ [where ‘A’ is a sortal common name and ‘∗ A’ is a quantifier phrase of English], the propositions ‘f (∗ A)’ and ‘f (∗ A)’ will in general not be contradictories—both may be true or both false”.7 For example, ‘Some man is wise” and ‘Some man is not wise’ can both be true, whereas, according to Geach, ‘Socrates is wise’ and ‘Socrates is not wise’ cannot both be true. This shows, Geach claims, that unlike the proper name ‘Socrates’, the quantifier (noun) phrase ‘some man’ is only a “quasi subject”, not a “genuine subject”, and therefore cannot really be used as a “genuine” referential expression.8 In other words, quantifier phrases, unlike proper names, cannot be used to stand for referential concepts (or, in Geach’s terms, cannot be “genuine logical subjects”) because they do not in general yield contradictory sentences when applied to contradictory predicables. Now in formal terms the only proper interpretation for a predicate that is contradictory to a given predicate [λxϕx] is the predicate [λx¬ϕx]. Then, assuming that ‘Socrates’ is being used with existential presuppositions, the sentences ‘Socrates is wise’ and ‘Socrates is not wise’ can be symbolized as follows: (∃xSocrates)W ise(x) and (∃xSocrates)[λx¬W ise(x)](x), where in ‘Socrates is not wise’ the negation is internal to the predicate. These sentences are in fact contradictory, which is in accordance with what Geach claims—but only because the name ‘Socrates’ is being used with existential presupposition in both. That is, ‘Socrates is not wise’ is equivalent to ‘It is not the case that Socrates is wise’, which is the contradictory of ‘Socrates is wise’, only because (∃xSocrates)(∀ySocrates)(x = y) → [¬(∃xSocrates)W ise(x)
↔ (∃xSocrates)[λx¬W ise(x)](x)]
is valid, where the antecedent says in effect that ‘Socrates’ names one and only one thing. Without this assumption it does not follow that ‘Socrates is not wise’ is equivalent to ‘It is not the case that Socrates is wise’, and hence that ‘Socrates is not wise’ is the contradictory of ‘Socrates is wise’ as Geach claims. Geach’s “criterion”, or “definition” for “genuine reference,” i.e., his claim that a “genuine” referring expression will yield contradictory propositions when 7 Geach
1980, p.84. speaks of ‘referring phrases’ where we speak of referential expressions. He adopts this terminology, which he takes to be a “misnomer”, only for the purpose of describing the theories of general reference that he claims to refute (cf., e.g., op.cit, p.73). 8 Geach
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applied to contradictory predicables, is not unqualifiedly true even for proper names in other words. For example, if A is a proper name, such as ‘Pegasus’, that names nothing, and F ( ) is a monadic predicate, so that F ( ) and ¬F ( ), or [λxF (x)] and [λx¬F (x)], are contradictory “predicables,” then when A is used without existential presupposition, we can have both (∀xA)F (x) and (∀xA)¬F (x) true. In other words, the two assertions that A is F and that A is not F can both be true in a logic that is free of existential presuppositions for objectual terms. If we replace the name ‘Socrates’ in ‘Socrates is wise’ and ‘Socrates is not wise’ by ‘Some man’, so as to obtain ‘Some man is wise’ and ‘Some man is not wise’, which can be symbolized as follows, (∃xM an)W ise(x) and (∃xM an)[λx¬W ise(x)](x) then it is clear that both can be true, which is in accordance with what Geach claims. In other words, ‘It is not the case that some man is wise’ is not equivalent to ‘Some man is not wise’. But does this show that ‘Some man’ is not being used to refer to some man? Geach does not justify or explain why yielding contradictory sentences when applied to contradictory predicables is a necessary condition for “genuine” reference—except, of course, for maintaining that this is what is true of proper names, but only, as we have noted, when the proper names are being used with existential presuppositions. That referential expressions cannot be used as forms of “genuine” reference unless they function the same way as nonvacuous proper names is simply assumed, which begs the question at issue. Now where a is an objectual variable that represents the kind of symbol Geach assumes a proper name to be, what Geach implicitly assumes is that [λxϕ](a) and ¬[λxϕ](a) are contradictories when in fact they are not, or, equivalently, that ¬[λxϕ](a) and [λx¬ϕ](a) say the same thing, when in fact they do not—or at least not in a logic that is free of existential presuppositions in the use of a proper name. In other words, whereas ¬[λxϕ](a) ↔ (∀x)[x = a → ¬ϕ], and [λx¬ϕ](a) ↔ (∃x)[x = a ∧ ¬ϕ], are valid in a logic free of existential presuppositions for objectual terms, we do not also have ¬[λxϕ](a) ↔ [λx¬ϕ](a), or equivalently (∀x)[x = a → ¬ϕ] ↔ (∃x)[x = a ∧ ¬ϕ] as valid as well. It is not unqualifiedly true in such a logic that a will yield contradictory propositions when applied to contradictory predicables.
9.2. DISJUNCTION AND CONJUNCTION ARGUMENTS
199
Geach is apparently aware that his argument does not work against proper names that denote nothing; but instead of rejecting the argument he rejects the use of “empty proper names.”9 That response, however, only indicates how inadequate his theory of reference is for pragmatics, i.e., for a realistic theory of speech and mental acts. Now the condition for when a referring expression will yield contradictory propositions when applied to contradictory predicables can be given even in free logic, but it is a condition that applies to common names as well as to proper names. In particular, what is valid in a logic free of existential presuppositions is that any proper or common name A, such as ‘Socrates’ or ‘moon of the Earth’, that denotes exactly one object will yield contradictory propositions when applied to contradictory predicables. In other words, (∃xA)(∀yA)(x = y) → [¬(∃xA)ϕ ↔ (∃xA)¬ϕ] ∧ [¬(∀xA)ϕ ↔ (∀xA)¬ϕ] is valid regardless whether or not A is a proper name or a common name. But there is nothing about this result that shows that the only “genuine” referential expressions are those of the form (∃xA), where A is a name, proper or common, for which the above antecedent condition is true. Geach simply begs the question by assuming as a criterion for “genuine reference” a condition that only names that name exactly one thing satisfy. On such a criterion, of course there can be no such thing as general reference.
9.2
Disjunction and Conjunction Arguments
Geach gives a similar argument based on the observation that connectives “that join propositions may be used to join predicables” to form complex predicate expressions.10 Thus, for example, instead of making two separate assertions, such as Sofia is sick and Sofia is home in bed we could make an assertion using the complex predicate ‘sick and home in bed’, as in ‘Sofia is sick and home in bed’, which we can symbolize as: (∃xSof ia)[λx(Si ck(x) ∧ Home-In-Bed(x))](x). Similarly, instead of asserting a disjunction such as ‘Either Sofia is home or Sofia is shopping’ we could assert ‘Sofia is either home or shopping’, symbolized as (∃xSof ia)[λx(Home(x) ∨ Shopping(x))](x). 9 Ibid.,
10 Ibid.,
p. 186. p. 86.
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Now what Geach claims—or rather assumes without argument—is that “the very meaning” such connectives as ‘and’ and ‘or’ have in a complex predicate is the meaning they have as propositional connectives. That is, “by attaching a complex predicable so formed to a logical subject [i.e. to what Geach considers a “genuine” referring expression] we get the same result as we should by first attaching the several predicables to that subject, and then using the connective to join the propositions thus formed precisely as the respective predicables were joined by that connective.”11 This claim is true when restricted to nonempty proper names, at least as far as truth conditions are concerned. An assertion of ‘Sofia is home or shopping’, for example, has the same truth conditions (but not the same cognitive structure) as an assertion of ‘Either Sofia is home or Sofia is shopping’. Indeed, where a is an objectual variable representing the kind of symbol Geach takes a nonempty proper name to be, [λx(ϕ ∨ ψ)](a) ↔ [λxϕ](a) ∨ [λxψ](a) is true in conceptual realism. The same claim is not in general true when applied to a universal quantifier phrase, on the other hand—nor, of course when a is an empty proper name. The sentence ‘Every integer is odd or even’, for example, is not equivalent to ‘Every integer is odd or every integer is even’. Indeed, (∀xA)[λx(ϕ ∨ ψ)](x) ↔ (∀xA)[λxϕ](x) ∨ (∀xA)[λxψ](x) is not a valid schema in the logic of conceptual realism, whether A is a proper or a common name. But this does not show that a universal quantifier phrase cannot be used as a “genuine” referential expression; and, in particular, that there is no reference to every integer in a speech act in which someone asserts that every integer is odd or even. What it shows is that Geach’s claim is really an assumption, and hence that his argument begs the question at issue. The equivalence does hold, however, if a proper or common name A can be used to name at most one object in a “simple act of naming”12 ; i.e., (∀xA)(∀yA)(x = y) → [(∀xA)[λx(ϕ ∨ ψ)](x) ↔ (∀xA[λxϕ](x) ∨ (∀xA)[λxψ](x)] is valid in conceptual realism. And of course, we do have (∃xA)[λx(ϕ ∨ ψ)](x) ↔ (∃xA)[λxϕ](x) ∨ (∃xA)[λxψ](x) as valid, i.e., the distribution of (∃xA) over a disjunction is valid. The distribution of (∃xA) over a conjunction, on the other hand, is valid in only one direction. But why does this show that we cannot use (∃xA) to refer to an A? In other words, why should we conclude that the invalidity of (∃xA)[λxϕ](x) ∧ (∃xA)[λxψ](x) → (∃xA)[λx(ϕ ∧ ψ)](x) 11 Ibid.
12 Geach
1980, p.53.
9.3. ACTIVE VERSUS DEACTIVATED CONCEPTS
201
shows that a quantifier phrase of the form (∃xA), where A is a common name (complex or simple), cannot be used as a “genuine” referential expression? The failure of a logical equivalence does not show this except by begging the question that only proper names can be “genuine” referential expressions. It is noteworthy, moreover, that the antecedent of the above conditional, i.e. the conjunction, (∃xA)[λxϕ](x) ∧ (∃xA)[λxψ](x), does not represent a basic speech act that is analyzable in terms of a referential and a predicable expression. What it can be used to represent is a speaker’s conjunction of two assertions in each of which the same referential concept is applied. But to apply the same referential concept, especially one of the form (∃xA), in two conjoined assertions is not the same as to purport to refer to the same object or objects in those assertions, unless, of course, the referential concept in question is based on the use of a proper name. We can assert, e.g., that some republicans are honest and that some republicans are dishonest, but in doing so we do not purport to refer to the same republicans in both uses of the quantifier phrase ‘Some republicans’. Geach’s implicit assumption is that if a quantifier phrase can be used as a “genuine” referential expression, then it must refer to one and only one object, and the same object(s), moreover, whenever it is so used. In other words, a “genuine” referential expression must refer the way a nonvacuous proper name refers, which, of course, begs the question.
It is by begging the question and assuming that only proper names can be used as “genuine” referential expressions that Geach’s negation and complex-predicate arguments have any plausibility.
9.3
Active Versus Deactivated Concepts
Geach does have a more interesting type of argument that does not beg the question, but which in our conceptualist theory involves the important distinction we made in our last two chapters between active and deactivated referential concepts. In explaining this distinction, we noted that a referential concept, as a basic thesis of our theory, is never part of what informs a speech or mental act with a predicable nature, but functions only as what informs such an act with a referential nature, i.e., as what accounts for the intentionality or aboutness of that act. Every basic assertion as expressed by a noun phrase and a verb phrase is the result, in other words, of applying just one referential concept and one predicable concept. What this means is that a complex predicate expression that contains a quantifier phrase cannot be applied in such a way as to presuppose an active exercise of the referential concept that that quantifier phrase stands for. The referential concept that the quantifier phrase stands for has been “deactivated”,
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in other words, which means that the predicable concept expressed by the complex predicate that contains that quantifier phrase is formed not on the basis of the referential concept that the quantifier phrase stands for but on the basis of its intensional content instead. Now by the intensional content of a referential concept, as we explained in previous chapters, we mean the intensional content of the predicable concept based on that referential concept. Thus, where A is a proper or common name symbol, complex or simple, and Q is a quantifier symbol representing a determiner of natural language, the predicate expression that is determined by the quantifier phrase (QxA) was defined as follows13 : [QxA] =df [λF (QxA)F (x)]. This predicate expression can be nominalized, of course, in which case what it denotes is the intensional content of the predicate, and thereby, indirectly, the intensional content of the referential (quantifier) expression (QxA). As explained in our earlier lecture, we use [QxA] as an abbreviation of [λF (QxA)F (x)]. Also, it should be remembered that a referential (quantifier) expression that occurs within an abstract singular term, i.e., within a nominalized complex predicate, has been deactivated and is not used in that occurrence to represent an active exercise of the referential concept that the expression otherwise stands for as a grammatical subject. The example we gave was [Sofia]N P [seeks [a unicorn]]V P ↓ ↓ ↓ (∃xSof ia)[λxSeek(x, [∃yU nicorn])](x), where the quantifier phrase ‘a unicorn’ that occurs as part the predicate ‘seeks a unicorn’ has been deactivated. The same quantifier is also deactivated in SofiaN P [finds [a unicorn]]V P ↓ ↓ ↓ (∃xSof ia)[λxF ind(x, [∃yU nicorn])](x). But because the predicate F ind is extensional in its second argument position, then the latter sentence implies (∃yU nicorn)(∃xSof ia)F inds(x, y). The predicate Seek, on the other hand, is not extensional in its second argument position, which means that ‘Sofia seeks a unicorn’ does not imply that there is 13 The application of the λ-operator to predicate variables is understood as an abbreviated notation, which, in the monadic case, is indicated as follows:
[λF ϕ] =df [λy(∃F )(y = F ∧ ϕ)], where y does not occur free in ϕ.
9.3. ACTIVE VERSUS DEACTIVATED CONCEPTS
203
a unicorn. In other words, even though ‘Sofia seeks a unicorn’ and ‘Sofia finds a unicorn’ have the same logical form, nevertheless one implies that there is a unicorn, whereas the other does not. The difference, as we explained in our previous chapters, is that the following (instance of a) meaning postulate, [λxF inds(x, [∃yA])] = [λx(∃yA)F inds(x, y)]. is assumed for F ind, whereas no such similar meaning postulate can be assumed for Seek. This type of meaning postulate also applies to our use of the copula to express identity, as when we say that Sofia is an actress. Note that the predicable concept expressed by ‘is an actress’ in this example cannot be represented by [λx(∃yActress)(x = y)], because the quantifier phrase (∃yActress) has not been deactivated. That is, this λ-abstract is not the appropriate way to express the cognitive structure of the speech act in question. What we need here is a symbolic counterpart of the copula, e.g., Is, as a two-place predicate constant. Thus, the appropriate analysis of the speech act in question is: [Sofia]N P [is an actress]V P (∃xSof ia) [λxIs(x, [∃yActress])] (∃xSof ia)[λxIs(x, [∃yActress])](x), where the quantifier phrase (∃yActress) has been deactivated. Now of course this does not mean that we are asserting that Sofia is identical with the intensional content of being an actress, just as in asserting that Sofia seeks a unicorn we do not mean that Sofia seeks the intensional content of being a unicorn. To get at the right truth conditions for this sort of assertion, we need to assume the following as a meaning postulate for the copula Is: [λxIs(x, [∃yA])] = [λx(∃yA)(x = y)], where A is a variable having complex or simple names, proper or common, as substituends. Thus, because of this meaning postulate, the following (∃xSof ia)[λxIs(x, [∃yActress])](x) ↔ (∃xSof ia)(∃yActress)(x = y) is valid in the logic of conceptual realism.14 14 Russell, incidentally, proposed a similar analysis in his 1903 Principles, where he assumed that every proposition consists of a relation between “terms”, and that, e.g., the proposition expressed by ‘Socrates is a man’ expresses a relation between Socrates and the denoting concept a man. Presumably, the relation was not strict identity, but something like what we are representing here by Is. Of course, Russell was proposing a logical realist theory and not a conceptualist theory; and he had nothing like our distinction between active and deactivated concepts.
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9.4
Deactivation and Geach’s Arguments
In one of his arguments against general reference, Geach claims that “we cannot suppose ‘some man’ to refer to some man in one single way,” because, if it were a “genuine” referring expression, then “we should have to distinguish several types of reference—it is not easy to see how many”.15 Suppose, Geach argues, “we can say ‘some man’ refers to some man in a statement like this: (1)
Joan admires some man.
that is, a statement for which the question ‘which man?’ would be in order. Let us call this type of reference type-A. Then in a statement like the following one: (2)
Every girl admires some man.
‘some man’ must refer to some man in a different way, since the question ‘Which man?’ is plainly silly”.16 Calling the type of reference indicated in (2) type-B reference, Geach goes on to argue that we must then distinguish further types as well. The problem with this argument is that in an assertion of either (1) or (2), the referential concept that the quantifier phrase ‘some man’ stands for has been deactivated, i.e., the phrase is not being used to refer in either case. Of course, there is a difference between the two assertions in that (1) logically implies that some particular man is admired by Joan—assuming ‘Joan’ is being used with existential presupposition in this context—whereas (2) does not logically imply that some particular man is admired by every girl. This can be easily seen to be so in the logical forms representing the cognitive structures of these assertions (1 )
(∃xJoan)[λxAdmire(x, [∃yM an])](x),
and (2 )
(∀xGirl)[λxAdmire(x, [∃yM an])](x).
Now it is natural to assume that ‘admire’ is extensional in this context in its second argument position.17 That is, we take [λxAdmire(x, [QyA])] = [λx(QyA)Admire(x, y)] to be a meaning postulate representing a conceptual truth in the context in question. Then, from an instance of this postulate it can be seen that (by λconversion and commutation of existential quantifier phrases), the statement that some man is admired by Joan, which is analyzed as, (∃yM an)[λyAdmire([∃xJoan], y)](y), 15 Geach 16 Ibid.
1980, p. 32. Geach attributes this argument to Elizabeth Anscombe.
17 It is clear that Geach assumes this to be so in the context in question. In some contexts, it would seem, ‘admire’ might function as an intensional verb—as, e.g., when we say of someone that s/he admires Sherlock Holmes.
9.4. DEACTIVATION AND GEACH’S ARGUMENTS
205
or equivalently, not considering it as the form of an assertion, (∃yM an)(∃xJoan)Admire(x, y), follows validly from (1 ), which indicates why the question ‘Which man?’ is appropriate in a context in which (1) is asserted. Thesis: In general, wh-questions—i.e., ‘who’, ‘which’, ‘what’, ‘when’ and ‘where’ questions—apply only to active referential expressions, not to deactivated ones—or, as in this case, to those that could be activated as part of a statement that follows validly from a given assertion. Now what follows validly from (2 ), on the other hand, is (∀xGirl)(∃yM an)Admire(x, y), and
not (∃yM an)(∀xGirl)Admire(x, y),
which, in the form of an assertion, is equivalent to (∃yM an)[λyAdmire([∀xGirl], y)](y), that is, the statement that some man is admired by every girl. In other words, ‘some man’ is not being used in (2) to refer to some particular man; nor does (2) imply a sentence in which one might refer to some particular man. That is why the question ‘Which man?’ is inappropriate in a context in which (2) is asserted. It is simply false, on our account, to claim that there are two different types of reference in assertions of (1) and (2). The referential concept that the quantifier phrase ‘some man’ stands for has been deactivated in both assertions, which means that the phrase is not being used in those sentences to refer, no less to refer in two different ways. Another argument that Geach gives turns on his misconstruing a reflexive pronoun as “a pronoun of laziness,” i.e. as a pronoun that functions as a proxy for its grammatical antecedent and that can be replaced by that expression “without changing the force of the proposition.”18 Thus, according to Geach, “If [the quantifier phrase] ‘every man’ has reference to every man, and if a reflexive pronoun has the same reference as the subject of the verb, [then] how can ‘Every man sees every man’ be a different statement from ‘Every man sees himself’ ?”19
18 Geach 19 Ibid.,
1980, p. 151. p. 9.
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Now, it clear that Every man sees every man. and Every man sees himself. are different statements. But does this show that the quantifier phrase ‘every man’ cannot be used to refer to every man, as Geach claims? Is it really clear in this case that the reflexive pronoun ‘himself’ is functioning here as “a pronoun of laziness,” and hence can be replaced by the quantifier phrase ‘every man’ so that the result is an equivalent sentence, i.e., a sentence having the same force as the original sentence? In our theory the occurrence of the quantifier phrase ‘every man’ in the verb phrase ‘sees every man’ is not being used to refer to every man, but instead stands for a deactivated referential concept. Let us compare an assertion of ‘Every man sees every man’ with an assertion of ‘Gino sees Gino’. Assuming that ‘Gino’ is being used with existential presupposition, the logical forms representing the cognitive structures of these two assertions are as follows: (∀xM an)[λxSees(x, [∀yM an])](x), and (∃xGino)[λxSees(x, [∃yGino])](x), where the occurrences of the referential expressions ‘every man’ and ‘Gino’ after the transitive verb are deactivated and interpreted as standing for their respective intensional contents. Note that unlike the above assertions, where the λ-abstracts represent different predicable concepts, assertions of Every man sees himself. and Gino sees himself. involve an application of the same predicable concept, namely, [λxSees(x, x)]. The logical forms representing the cognitive structures of these assertions, in other words, are as follows: (∀xM an)[λxSees(x, x)](x), and
9.5. GEACH’S ARGUMENTS AGAINST COMPLEX NAMES
207
(∃xGino)[λxSees(x, x)](x). The reflexive pronoun ‘himself’ is not functioning as “a pronoun of laziness” in these assertions—even though it has “the same reference as the subject of the verb”20 . Now, if the relational concept of seeing, i.e., [λxySee(x, y)], is extensional in its second argument position, then, because ‘Gino’ is a proper name that is assumed to name exactly one object in the context in question, it follows that ‘Gino see Gino’ and ‘Gino sees himself’ are equivalent, i.e., (∃xGino)[λxSees(x, [∃yGino])](x) ↔ (∃xGino)[λxSees(x, x)](x) is provable.21 In other words, in the case of a proper name A, where A is assumed to name exactly one object in the context in question, it is true that ‘A sees A’ and ‘A sees her/himself’ are necessarily equivalent. This, of course, is not to say that as assertions, or mental acts, ‘A sees A’ and ‘A sees her/himself’ have the same cognitive structure, and in fact they have different cognitive structures as indicated by the above logical forms. On the other hand, ‘Every man sees every man’ and ‘Every man sees himself’ are not equivalent; but, contrary to Geach’s claim, this does not mean that the use of ‘every man’ as the grammatical subject of an assertion of either of these sentences does not refer to every man, even though its use as the direct object of the verb does not stand for a referential concept. Once again, Geach’s implicit assumption seems to be that a referential expression is not a “genuine” referring expression, but only a “quasi subject”, if it does not behave logically the way a nonempty proper name does.
9.5
Geach’s Arguments Against Complex Names
Some of Geach’s arguments are directed not only against referential expressions of the form ‘every A’ and ‘some A’, but also against the view that there are complex names of the form ‘A that is F ’, and hence against complex referential expressions of the form ‘every A that is F ’ and ‘some A that is F ’, which, as already noted in the previous chapters, we symbolize in our theory as (∀xA/F (x)) and (∃xA/F (x)). 20 That is, the variable x has the same value in the object (second) position of See(x, x) as it does in the subject (first) position. 21 If ‘see’ is interpreted as an extensional transitive verb in a given context, then seeing in that context does not imply knowing who or what it is that one sees. For example, Gino’s seeing Maria (in the extensional sense) does not imply that Gino knows that it is Maria he sees; and, similarly, Gino’s seeing Gino (as in a mirror or a photo) does not imply that Gino knows that he sees himself. In some contexts ‘see’ might well be interpreted as an intensional verb, where seeing implies knowing who or what one sees, and in that case, ‘Gino sees Gino’ and ‘Gino sees himself’ would then be equivalent.
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One such argument that Geach gives against complex names is based on a medieval paralogism22: Only an animal can bray; ergo, Socrates is an animal, if he can bray. But any animal, if he can bray, is a donkey. Ergo, Socrates is a donkey. Now Geach correctly observes that “we clearly cannot take ‘animal, if he can bray’ as a complex term [i.e., as a complex name] that is a legitimate reading of ‘A’ in ‘Socrates is an A; any A is a donkey; ergo, Socrates is a donkey”23 ; but he does not explain the relevance of this observation, or how this shows that a complex name like ‘animal that can bray’ is not “a genuine logical unit,”24 namely, a complex name. Apparently, Geach is confusing the complex name ‘animal that can bray’ in this argument with an expression that is not a complex name, namely, ‘animal, if he can bray’. Note that by the exportation rule (∀xA)ϕ ↔ (∀x)[(∃yA)(x = y) → ϕ],
(MP1)
mentioned in chapter seven §3, an assertion of ‘Every animal that can bray is a donkey’, which is analyzed as follows: [Every animal that can bray]N P [is a donkey]V P ↓ ↓ (∀xAnimal/Can-Bray(x)) [λxIs(x, [∃yDonkey])] ↓ ↓ (∀xAnimal/Can-Bray(x))[λxIs(x, [∃yDonkey])](x) is equivalent to an assertion of ‘Every animal, if he can bray, is a donkey’, analyzed as, [Every animal]N P [if he can bray is a donkey]V P ↓ ↓ (∀xAnimal) [λx(Can-Bray(x) → Is(x, [∃yDonkey]))](x) ↓ ↓ (∀xAnimal)[λx(Can-Bray(x) → Is(x, [∃yDonkey]))](x) In other words, the following biconditional is valid in the logic of conceptual realism: (∀xAnimal/Can-Bray(x))[λxIs(x, [∃yDonkey])](x) ↔ (∀xAnimal)[λx(Can-Bray(x) → Is(x, [∃yDonkey]))](x). 22 Geach 23 Ibid.
24 Ibid.,
1980, p. 143.
p. 142.
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Geach confuses the grammatically correct claim that in the first assertion we are referring to every animal that can bray with the grammatically incorrect claim that in the second assertion we are referring to every animal, if he can bray. That is why Geach claims that “the phrase ‘animal that can bray’ is a systematically ambiguous one,”25 when in fact it is not.
9.6
Relative Pronouns as Referential Expressions
Geach does recognize that “we cannot count this as proved” and attempts to “confirm the suggestion of ambiguity by considering another sort of medieval example.”26 This is the pair of contradictory sentences, (3)
Any man who owns a donkey feeds it.
(4)
Some man who owns a donkey does not feed it.
in which, on our account, ‘man who owns a donkey’ occurs as a complex name. Now, according to Geach, if ‘man who owns a donkey’ is a complex name, then it is “replaceable by the single word ‘donkey-owner’,” in which case (3) and (4) would become “unintelligible”.27 Of course, this sort of “replacement argument” is fallacious in that it deprives the relative pronoun ‘it’ in (3) and (4) of an antecedent, as Geach himself seems to acknowledge. He then suggests a supposedly “plausible rewording” of (3) and (4) in which ‘it’ is given an antecedent, namely, (5)
Any man who owns a donkey owns a donkey and feeds it.
(6)
Some man who owns a donkey owns a donkey and does not feed it.
But (5) and (6) are not equivalent to (3) and (4), as Geach himself notes, because, in particular, unlike (3) and (4), (5) and (6) are not contradictories in that both would be true if each man who owned a donkey had two donkeys and fed only one of them. Geach then rephrases (3) and (4) as (3 )
Any man, if he owns a donkey, [then he] feeds it.
(4 )
Some man owns a donkey and he does not feed it.
which, by the export-import meaning postulates (MP1) and (MP2) for complex referential expressions (given in chapter seven, §3), are equivalent to (3) and (4). That is, as represented by appropriate instances of those meaning postulates, (3) and (3 ), and (4) and (4 ), have the same truth conditions, even though the cognitive structures of the speech or mental acts they represent are not the same. By ignoring the distinction between logical forms that represent the cognitive structure of our speech and mental acts on the initial level of analysis and the 25 Ibid.
26 Ibid. We have changed the verb ‘beat’ in Geach’s example to ‘feed’, which in no way affects his argument, or our criticism of it. 27 Ibid., p. 144.
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logical forms that represent the logical consequences of those acts on the second level of analysis, Geach fallaciously concludes that “the complex term ‘A that is P ’ is a sort of logical mirage. The structure of a proposition in which such a complex term appears to occur can be readily seen only when we have replaced the grammatically relative pronoun by a connective followed by a pronoun; when this is done, the apparent unity of the phrase disappears.”28 What is needed here for a proper analysis of (3) and (4) is an analysis of the role the relative pronoun ‘it’ has in in these kinds of sentences, which in the literature have come to be called “donkey-sentences.” Our proposal is that relative pronouns in general, and the pronoun ‘it’ in particular, are referential expressions that are interpreted with respect to an antecedent referential expression. In particular, we maintain that the sentence (3), ‘Any man who owns a donkey feeds it’ is synonymous with, and in fact has the same cognitive structure as, the following sentence (3 ) Any man who owns a donkey feeds that donkey. or, if one prefers, the same as Any man who owns a donkey feeds it (i.e., that donkey). with the phrase ‘that donkey’ expressed, as it were, sotto voce. Now because all referential expressions are analyzed in conceptual realism as quantifier phrases, what this means is that relative pronouns are to be logically analyzed as quantifier phrases of the form ‘that A’, where A is the name, common or proper, occurring in the antecedent referential phrase relative to which the pronoun is interpreted. What we need, accordingly, is a variable-binding ‘that’-operator, T , that, when indexed by a variable, can be applied to a name A, whether complex or simple, and result in a quantifier phrase, e.g., ‘T yA’, which is read as ‘that A’. On this proposal, the cognitive structure of (3 )—and, on our proposal, therefore of (3)—can now be analyzed as: [Any man owns a donkey]NP [feeds that donkey]VP (∀xM an/Own(x, [∃yDonkey])) [λxF eeds(x, [T yDonkey])] (∀xM an/Own(x, [∃yDonkey]))[λxF eeds(x, [T yDonkey])](x) The relative pronoun ‘it’ in (3), in other words, is a proxy for the pronominal referential expression ‘that donkey’, which in this context stands for a deactivated referential concept relative to the deactivated antecedent referential concept that ‘a donkey’ stands for in the grammatical subject of (3). Now, by the export-import meaning postulate (MP1) for complex referential expressions, the above analysis, which we will call (3cog ), is equivalent to (∀xM an)[λx(Own(x, [∃yDonkey]) → F eeds(x, [T yDonkey]))](x), 28 Ibid.,
p. 145.
(3cog )
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211
which is easily seen to represent the cognitive structure of an assertion of (3 ), i.e., the sentence ‘Any man, if he owns a donkey, [then he] feeds it’. But because ‘own’ and ‘feed’ are extensional transitive verbs, the deactivated quantifier phrases ‘a donkey’ and ‘that donkey’ can be “reactivated,” in which case (3cog ) and (3cog ) are equivalent to (∀xM an)[(∃yDonkey)Own(x, y) → (T yDonkey)F eeds(x, y)], which does not represent the cognitive structure of a speech or mental act, but does represent the truth conditions of an assertion of either (3) or (3 ). We can obtain a logically more perspicuous representation of those truth conditions, moreover, by means of the following meaning postulate for the T -operator, i.e., a postulate that makes clear that it is functioning as a pronoun relative to an antecedent referential expression: [(∃yS)ϕy → (T yS)ψy] = [(∀yS)(ϕy → ψy)],
(MPT2 )
which, by Leibniz’s law implies the weaker equivalence, [(∃yS)ϕy → (T yS)ψy] ↔ [(∀yS)(ϕy → ψy)]. Thus, by means of this postulate and the preceding formula, it follows that (∀xM an)(∀yDonkey)[Owns(x, y) → F eeds(x, y)] is equivalent to (3cog ) and (3cog ), and this formula, it is clear, provides a logically perspicuous representation of their truth conditions, and hence of the truth conditions of (3) and (3 ). The meaning postulate (MPT2 ) for the ‘that’-operator explains why sentences like If someone is married, then s/he (i.e., that person) has a spouse. and If a witness lied, then s/he (i.e., that witness) committed perjury. have the truth conditions that they do, and are equivalent to Anyone who is married has a spouse. and Any witness who lied committed perjury. As noted in section 7 of the previous chapter, the T -operator is designed to be used only on the initial level of analysis regarding the cognitive structure of our speech and mental acts. It is not designed to be used on the level of deductive transformations, where such rules as simplification, adjunction, and the rewrite of bound variables might be applied. The idea is to restrict the standard transformations to just Leibniz’s law as based on meaning postulates until the occurrences of the T -operator have been eliminated.
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Turning now to a formal representation of the cognitive structure of (4), i.e., the sentence ‘Some man who owns a donkey does not feed it’, let us note first that this sentence, on our proposal, has the same cognitive structure as (4 )
Some man who owns a donkey does not feed that donkey.
Now because the negation in the verb phrase ‘does not feed it’ is internal to the predicate, we have the following as an analysis of (4 ), and therefore, on our proposal, of (4) as well: [Some man who owns a donkey]N P [does not feed that donkey]V P (∃xM an/Own(x, [∃yDonkey])) [λx[λzw¬F eeds(z, w)](x, [T yDonkey])] (∃xM an/Own(x, [∃yDonkey]))[λx[λzw¬F eeds(z, w)](x, [T yDonkey])](x) This analysis of the cognitive structure of (4 )—and hence, on our proposal, of (4) as well—can be simplified by applying the export-import meaning postulate (MP2) for complex names and the meaning postulates regarding the extensionality of ‘own’ and ‘feed’, and therefore of ‘does not feed’. In other words, by these meaning postulates, the above analysis of (4 ) and (4), which we will call (4cog ), is equivalent to29 : (∃xM an)[(∃yDonkey)Own(x, y) ∧ (T yDonkey)¬F eeds(x, y)]. Finally, the relevant meaning postulate for the T -operator in this case is the following,30 [(∃yS)ϕy ∧ (T yS)ψy] = [(∃yS)(ϕy ∧ ψy)], (MPT1 ) which, by Leibniz’s law implies [(∃yS)ϕy ∧ (T yS)ψy] ↔ [(∃yS)(ϕy ∧ ψy)], which, together with the preceding formula, shows that (4cog ) is equivalent to, and therefore has the same truth conditions as, (∃xM an)(∃yDonkey)[Own(x, y) ∧ ¬F eeds(x, y)]. This formula is easily seen to be a contradictory of the above logically perspicuous representation of the truth conditions for (3). That is, (∀xM an)(∀yDonkey)[Owns(x, y) → F eeds(x, y)] 29 Here, we should keep in mind here that we cannot proceed from the initial level of analysis regarding cognitive structure to the deductive level until all applications of Leibniz’s law as based on meaning postulates have been applied. In particular, λ-conversion is not to be applied until after the meaning postulates for extensional verbs have been applied. 30 An example of the use of this meaning postulate is one from Geach 1980, namely, ‘Some man broke the bank at Monte Carlo and that man died a pauper’, the truth conditions of which are the same as ‘Some man broke the bank at Monte Carlo and died a pauper’.
9.7. SUMMARY AND CONCLUDING REMARKS
213
and (∃xM an)(∃yDonkey)[Own(x, y) ∧ ¬F eeds(x, y)]. are contradictories in that one is equivalent to the negation of the other. We conclude, accordingly, that the sentences (3) and (4)
Any man who owns a donkey feeds it. Some man who owns a donkey does not feed it.
do in fact have the truth conditions Geach says they have, even though the expression ‘man who owns a donkey’ functions in both as a complex name, contrary to what Geach claims. In other words, the sentences (3) and (4) in no way support Geach’s claim that “the complex term ‘A that is P ’ is a sort of logical mirage” and is not “a genuine logical unit”, and that such expressions must be expanded into forms where there are no complex names at all. Nor do they show that there are inextricable difficulties with the conceptualist theory of reference we have described here.
9.7
Summary and Concluding Remarks
• The conceptualist theory of reference we described in chapter seven not only has all of the philosophically important features we listed there, but it provides as well a general framework by which to refute the claim that there can be only singular reference, and hence the claim that there can be no “genuine” form of general reference. • The idea that the only genuine form of reference is singular reference has been the dominant theory throughout the twentieth century, but that doctrine is based either on the type of arguments that Geach has given and that we have refuted here, or it is based on a confusion of pragmatics with semantics, i.e., that the analysis of the cognitive structure of our speech and mental acts is the same as the analysis of their truth conditions. • The truth conditions of sentences containing quantifier phrases are of course reducible to the atomic components of those sentences, but that does not mean that those same quantifier phrases do not stand for referential concepts. Indeed, to the contrary, general reference is a basic feature of our speech and mental acts, which is why quantifier phrases occur as grammatical subjects, or noun phrases, in natural language—and occur as such, moreover, with as great, or greater frequency, than proper names do. • The dominance of the doctrine that there can be only singular reference explains why the logical analysis of the cognitive structure of our speech and mental acts has been ignored in the analytic movement. • By giving an analysis only of the truth conditions of our speech and mental acts, the analytic movement has assumed that singular reference is the only genuine form of reference. As a result, the analytic movement ignored the problem of giving a logical analysis of the cognitive structure of our speech and
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mental acts, because in order to do so it must give an account of general as well as singular reference. • What is needed is a theory of logical forms that can represent general as well as singular reference, and in particular a theory such as we have constructed for conceptual realism. • Just as a unified account of general and singular reference was once given by medieval logicians, but only for categorical propositions, conceptual realism provides a unified account of both general and singular reference for all propositional forms combining a noun phrase with a verb phrase. • It is by such a unified account that conceptual realism can give a logical analysis of the cognitive structure of our speech and mental acts, which is the starting point for any formal ontology that is based initially on the structure of thought. It is then by means of such a unified account that an analysis of the ontological categories of reality can be given as well.
Chapter 10
Le´ sniewski’s Ontology Referential concepts in conceptual realism are based on a logic of proper and common names as parts of quantifier phrases. This conceptualist logic of names is similar to Le´sniewski’s logic of names in that the category of names in Le´sniewski’s system also contains common as well as proper names.1 Le´sniewski’s logic is different, however, in that names do not occur as parts of quantifier phrases but are of the same category as objectual variables. Le´sniewski described his logic of names as “ontology,” apparently because it was to be the initial level of a theory of types, which Le´sniewski called semantic categories.2 Le´sniewski’s general framework also included mereology, which is a theory of the relation of part to whole, and protothetic, a quantificational logic over propositions and n-ary truth-functions, for all positive integers n. Le´sniewski’s logic of names has been used for years as a framework in which to interpret and reconstruct various doctrines of medieval logic.3 We have given an alternative interpretation and reconstruction of medieval logic in terms of the framework of conceptual realism.4 It is relevant therefore to see how, or in what respect, Le´sniewski’s logic of names is similar to our conceptualist logic of names. In fact, as we will explain, Le´sniewski’s logic of names can be completely interpreted, and in that sense is reducible, to our conceptualist logic of names.5 Le´sniewski’s based his system of mereology, i.e., his logic of the relationship between parts and wholes, on his logic of names, and though the exact form of this connection is not clear it has something to do with the notion of classes in a collective sense as opposed to a distributive sense. Our conceptualist logic of names, on the other hand, is the basis of a logic of classes as many, i.e., a logic 1 This
chapter is a development of material from my 2001 paper, “A Conceptualist Interpretation of Le´sniewski’s Ontology,” History and Philosophy of Logic, vol. 22. 2 See Lejewski 1958, p. 152, Slupecki 1955 and Iwanu´ s 1973 for a description of Le´snieski’s general framework as well as his logic of names. 3 See, e.g., Henry 1972. 4 See chapter 8 and also Cocchiarella 2001 for the details of such a reconstruction. 5 See below and also Cocchiarella 2001 for a detailed proof that Le´ sniewski’s logic of names is reducible to our conceptualist logic of names.
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216
of classes that in some respects is similar to Le´sniewski’s mereology, but in other respects it is also different. Unlike the connection between Le´sniewski’s logic of names and his mereology, however, the connection between our logic of classes as many and our simple logic of names is both precise and a fundamental part of conceptual realism. We will first briefly describe Le´sniewski’s logic of names and then formulate the simple logic of names that is a fragment of our broader, more comprehensive formal ontology for conceptual realism. We will then explain how Le´sniewski’s system can be interpreted within our logic and how certain oddities of Le´sniewski’s system can be explained in terms of our logic where those oddities do not occur. We will then explain how the logic of classes as many is developed as an extension of the simple logic of names. The logic of names of Le´sniewski’s general framework and of our framework of conceptual realism provide, incidentally, another illustration, or paradigm, of how different parts or aspects of a formal ontology can be developed independently of, or even prior to, the construction of a comprehensive system all at once.
10.1
Le´ sniewski’s Logic of Names
In Le´sniewski’s logic of names, as in our conceptualist logic, there is a distinction between 1. shared, or common names, such as ‘man’, ‘horse’, ‘house’, etc., and even the ultimate superordinate common name ‘thing’, or ‘object’; 2. unshared names, i.e., names that name just one thing, such as proper names; and 3. vacuous names, i.e., names that name nothing.6 There is a categorial difference between names in Le´sniewski’s logic and names in our conceptualist logic, however. In Le´sniewski’s logic names are of the same category as the objectual variables, which means that they are legitimate substituends for those variables in first-order logic. In our conceptualist logic, names belong to a category of expressions to which quantifiers are applied and that result in quantifier phrases such as ‘every raven’, ‘some man’, ‘every citizen over eighteen’, etc. The one primitive of Le´sniewski’s logic, aside from logical constants, is the relation symbol ‘ε’ for singular inclusion, which is read as the copula ‘is (a)’, as in ‘John is a teacher’, where both ‘John’ and ‘teacher’ are names.7 Using ‘a’, ‘b’, ‘c’, etc., as objectual constants and variables for names, the basic formula of 6 See
Lewjeski’s 1958 for a detailed description of Le´sniewski’s logic of names. it was L ukasiewicz who prompted Le´sniewski to develop his logic of names when he expressed dissatisfaction with the way G. Peano used ‘∈’ for the copula in set theory. Cp. p. 414 of Rickey’s 1977. 7 Apparently,
´ 10.1. LESNIEWSKI’S LOGIC OF NAMES
217
the logic is ‘a ε b’, where either shared, unshared, or vacuous names may occur in place of ‘a’ and ‘b’. A statement of the form ‘a ε b’ is taken as true if, and only if ‘a’ names exactly one thing and that thing is also named by ‘b’, though ‘b’ might name other things as well, as in our example of ‘John is a teacher’. Identity is not a primitive logical concept of Le´sniewski’s system, as it is in our conceptualist logic, but is defined instead as follows: a = b =df a ε b ∧ b ε a. That is, ‘a = b’ is true in Le´sniewski’s logic if, and only if, ‘a’ and ‘b’ are unshared names that name the same thing. That seems like a plausible thesis, except that then, where ‘a’ is a shared or vacuous name, ‘a = a’ is false. In fact, because there are necessarily vacuous names, such as the complex common name ‘thing that is both square and not square’, the following is provable in Le´sniewski’s logic: (∃a)(a = a), which does not seem at all like a plausible thesis. Of course, this means that (∀a)(a = a) is not a valid thesis in Le´sniewski’s system. Le´sniewski does include a weak notion of identity, which is defined as follows a ◦ b =df (∀c)(c ε a ↔ c ε b), and which does not have these odd features. This notion, of course, means that a and b are co-extensive, not identical. But then, Le´sniewski insisted on his logic being extensional, and not intensional, in which case a ◦ b does amount to a kind of identity when a and b are either shared or unshared names. Of course, in that case all vacuous names, such as ‘Pegasus’ and ‘Bucephalus’ are identical in this weak sense. It also means that Le´sniewski’s ontology is not an appropriate framework for tense and modal logic, or for intensional contexts in general.8 Another valid thesis of Le´sniewski’s logic is, ϕ(c/a) → (∃a)ϕ(a), which seems counter-intuitive when ‘c’ is a vacuous name. The following, for example, would then be valid ¬(∃b)(b = P egasus) → (∃a)¬(∃b)(b = a), 8 One can intensionalize Le´ sniewski’s framework, of course, even though he himself was against such a move.
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and therefore, because the antecedent is true, then so is the consequent, which says that something is identical with nothing. Perhaps these oddities can be explained by interpreting Le´sniewski’s firstorder quantifiers substitutionally rather than as referentially. That, however, is not how Le´sniewski understood his logic of names, which, as we have said, he also called ontology.9 A logic that interprets the quantifiers of its basic level substitutionally, rather than referentially, would be an odd sort of formal ontology. Does that mean that a name would be required for every object in the universe, including, e.g., every grain of sand and every microphysical particle? We should also note that Le´sniewski’s epsilon symbol, ‘ε’, for singular inclusion should not be confused with the epsilon symbol, ‘∈’, for membership in a set. In particular, whereas the following: a ε b → a ε a, a ε b ∧ b ε c → a ε c, are both theorems of Le´sniewski’s system, both are invalid for membership in a set. Finally, the only nonlogical axiom of ontology—i.e., the only axiom in addition to the logical axioms and inference rules of first-order predicate logic without identity—assumed by Le´sniewski was the following: (∀a)(∀b)[a ε b ↔ (∃c)c ε a ∧ (∀c)(c ε a → c ε b) ∧ (∀c)(∀d)(c ε a ∧ d ε a → c ε d)]. This axiom alone does not suffice for the elementary logic of names, however, i.e., for Le´sniewski’s logic of names as formulated independently of the typetheoretic part of Le´sniewski’s framework. It has been shown, however, that adding the following two axioms to the one above does suffice10 : (∀a)(∃b)(∀c)[c ε b ↔ c ε c ∧ c ε/ a], (∀a)(∀b)(∃c)(∀d)[d ε c ↔ d ε a ∧ d ε b].
(Compl) (Conj)
Expressed in terms of our conceptualist logic, where names are taken to express name (or nominal) concepts, what these axioms stipulate is that there is a complementary name concept corresponding to any given name concept, and, similarly, that a conjunctive name concept corresponds to any two name concepts with singular inclusion taken conjunctively. 9 See,
e.g., Lejewski 1958 and K¨ ung and Canty 1979 for a discussion of this issue. Iwanu´s 1973 for a proof of this claim. Instead of the following axioms, Le´sniewski assumed a theory of definitions for constant names and name-forming functors. Le´sniewski’s theory of definitions does not always satisfy the conditions for noncreativity, however; but it was shown in Iwanu´s 1973 that with the addition of the following axioms then Le´sniewski’s theory of definitions can be proved to be noncreative. 10 See
10.2. THE SIMPLE LOGIC OF NAMES
10.2
219
The Simple Logic of Names
The simple logic of names, which, as we said, is an independent fragment of the full logic of conceptual realism, can be described as a version of an identity logic that is free of existential presuppositions regarding singular terms—i.e., free objectual variables and expressions that can be properly substituted for such. It contains both absolute and relative quantifier phrases, i.e., relative quantifier phrases such as (∀xA) and (∃xA), as well as absolute quantifier phrases such as (∀x) and (∃y), which are read as (∀xObject) and (∃yObject), respectively. We will continue to use x, y, z, etc., with or without numerical subscripts, as objectual variables, as we did in our second lecture. We will now also use A, B, C, with or without numerical subscripts, as name, or “nominal”, variables. As explained in previous chapters, complex names are formed by adjoining (so-called “defining”) relative clauses to names, and we use ‘/’, as in ‘A/ϕ’ to represent the adjunction of a formula ϕ to the name A (which may itself be complex). Thus, e.g., the quantifier phrase representing reference to a house that is brown would be symbolized as (∃xHouse/Brown(x)). We continue to take the universal quantifier, ∀, the (material) conditional sign, →, the negation sign, ¬, and the identity sign, =, as primitive logical constants, and assume the others to be defined in the usual (abbreviatory) way. The absolute quantifier phrases (∀x) and (∃x) are read as ‘Every object ’ and ‘Some object ’, or, equivalently, as ‘Everything’ and ‘Something’, respectively. That is, the absolute quantifiers are understood as implicitly containing the most general or ultimate common name ‘object’ (which we take to be synonymous with ‘thing’). The quantifier phrases (∀A) and (∃A) are taken as referring to every, or to some, name concept, respectively. Name constants are introduced in particular applications of the logic.11 Because complex names contain formulas as relative clauses, names and formulas are inductively defined simultaneously as follows12 : • (1) every name variable (or constant) is a name; • (2) for all objectual variables x, y, (x = y) is a formula; and • if ϕ, ψ are formulas, B is a name (complex or simple), and x and C are an objectual and a name variable respectively, then (3) ¬ϕ, (4) (ϕ → ψ), (5) (∀x)ϕ, (6) (∀xB)ϕ, and (7) (∀C)ϕ are formulas, and (8) B/ϕ and (9) /ϕ are names, where /ϕ is read as ‘object that is ϕ’. 11 The absolute quantifiers and the quantifiers for name concepts are understood to be relativized to a given universe of discourse in an applied form of the logic. 12 We adopt the usual informal conventions for dropping parentheses and for sometimes using brackets instead of parentheses.
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We assume the usual definitions of bondage and freedom for objectual variables and of the proper substitution of one objectual variable for another in a formula, and similarly we assume the definitions of bondage and freedom of occurrences of name variables in formulas, and the proper substitution in a formula ϕ of a name variable (or constant) B for free occurrences of a name variable C. Definition: A complex name B/ξ is free for C in ϕ with respect to an objectual variable x (as place holder) if (1) for each variable y such that (∀yC) occurs in ϕ and C is free at that occurrence, then y is free for x in B/ξ, and (2) no variable, name or objectual, other than x that is free in B/ξ becomes bound when a free occurrence of C in ϕ is replaced by an occurrence of B/ξ(y/x).13 Note: If a name B (complex or simple) is free for C in ϕ with respect to a variable x, then the proper substitution of B for C in ϕ with respect to x is represented by ϕ(B[x]/C). Among the rules, or meaning postulates, of our logic of names are four that were mentioned in our previous lecture. The first two connect relative quantifier phrases with absolute phrases, and the next two amount to export and import rules for quantifier phrases with complex names. (∀xA)ϕ ↔ (∀x)[(∃yA)(x = y) → ϕ],
(MP1)
(∃xA)ϕ ↔ (∃x)[(∃yA)(x = y) ∧ ϕ],
(MP2)
(∀xB/ϕ)ψ ↔ (∀xB)[ϕ → ψ],
(MP3)
(∃xB/ϕ)ψ ↔ (∃xB)[ϕ ∧ ψ].
(MP4)
Of course, strictly speaking, (MP2) and (MP4) are redundant because (∃xA) is taken as an abbreviation for ¬(∀xA)¬, whether A is simple or complex. For this reason, we will restate (MP1) and (MP3) as axioms 10 and 11. below. The axioms of the simple logic of names are those of the free logic of identity plus the axioms for name quantifiers: Axiom 1: All tautologous formulas; Axiom 2: (∀x)[ϕ → ψ] → [(∀x)ϕ → (∀x)ψ]; Axiom 3: (∀C)[ϕ → ψ] → [(∀C)ϕ → (∀C)ψ]; Axiom 4: (∀C)ϕ → ϕ(B[x]/C), where B is free for C in ϕ with respect to x; Axiom 5: χ → (∀C)χ,
where C is not free in χ;
Axiom 6: χ → (∀x)χ,
where x is not free in χ;
13 The use of ‘/’ in ‘ξ(y/x)’ represents the result of properly substituting y for x in ξ, and should not be confused with the use of ‘/’ to generate complex names.
10.2. THE SIMPLE LOGIC OF NAMES Axiom 7: (∀x)(∃y)(x = y),
221
where x, y are different variables;
Axiom 8: x = x; Axiom 9: x = y → (ϕ → ψ),
where ϕ, ψ are atomic formulas and ψ is obtained from ϕ by replacing an occurrence of y by x14
Axiom 10: (∀xA)ϕ ↔ (∀x)[(∃yA)(x = y) → ϕ],
where x, y are different variables;
Axiom 11: (∀xA/ψ)ϕ ↔ (∀xA)[ψ → ϕ]. We assume as primitive inference rules modus ponens (MP) and universal generalization (UG) for absolute quantifiers indexed by either an objectual or a name variable. The rule of universal generalization for relative quantifiers, if ϕ, then (∀xA)ϕ,
(UGN )
is derivable by (UG) from Axiom 10. The usual laws for interchanging provably equivalent formulas and for rewriting bound variables are easily seen to be derivable as well. The universal instantiation law in free logic for objectual variables, (∃x)(x = y) → [(∀x)ϕ → ϕ(y/x)], (∃/UI) where x, y are distinct variables and y is free for x in ϕ, is derivable by Leibniz’s law (LL), i.e., Axiom 9, (UG), Axioms 2 and 6, and tautologous transformations. The theorems that are counterparts to Axioms 10 and 11 for the existential quantifiers, namely, T1: (∃xA)ϕ ↔ (∃x)[(∃yA)(x = y) ∧ ϕ], and T2: (∃xA/ψ)ϕ ↔ (∃xA)[ψ ∧ ϕ] are also derivable by elementary transformations and the definitions for ∧ and ∃. Also, because absolute quantifiers are viewed as implicitly containing the common name ‘object’, we assume that Axiom 11 has the following schema as a special instance. T3: (∀x/ψ)ϕ ↔ (∀x)[ψ → ϕ]. The following are some obvious theorems that are easily seen to be provable. T4: (∀x)ϕ → (∀xA)ϕ. T5: (∀xA)ϕ → [(∃zA)(y = z) → ϕ(y/x)],
where y is free for x in ϕ.
T6: (∃xA)(y = x) → (∃x)(y = x). 14 As noted in our second lecture, the full version of Leibniz’s law is derivable from this axiom so long as no intensional operators are introduced into the logic.
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T7: (∀x)ϕ ↔ (∀A)(∀xA)ϕ, where A is not free in ϕ.15 Finally, the following comprehension principle (∀A)(∃B)(∀x)[(∃yB)(x = y) ↔ (∃yA)(x = y)]
(CPN )
is immediately derivable from Axiom 1, or really from the contrapositive of Axiom 1, which amounts to a form of existential generalization for name concepts. Stated as a schema derived from Axiom 4, (CPN ) can be described as follows: (∃B)(∀x)[(∃yB)(x = y) ↔ (∃yA/ϕ)(x = y)].
10.3
Consistency and Decidability
The simple logic of names that we have formulated in the previous section is both consistent and decidable.16 This follows by noting that the logic is actually equiconsistent with second-order monadic predicate logic, which is known to be consistent and decidable.17 We will not go through all of the details of showing this here, but we give a general outline of the proof.18 First, let us note that by the rule (MP3) and a simple inductive argument it can be shown that every formula of our conceptualist system in which a complex name occurs is provably equivalent to a formula in which no complex name occurs.
Metatheorem 1:
If ϕ is a formula of the simple logic of names—i.e., of the free first-order logic of identity extended to include name variables, quantification over such, and restricted quantifiers with respect to such—then there is a formula ψ in which no complex name occurs such that ϕ is provably equivalent to ψ in this logic, i.e., ϕ ↔ ψ.
Because of the above metatheorem, we can, in what follows, restrict ourselves to formulas in which no complex name occurs.19 We assume a one-to-one correlation of the name variables A, B, C, D, etc., with one-place predicate variables FA , FB , FC , FD , etc., and inductively define a translation function trs∗ from the formulas of our simple logic of names in which no complex name occurs into formulas of second-order monadic predicate logic (with identity) as follows: 1. trs∗ (x = y) = (x = y), 15 Proof: The left-to-right direction follows by T4, (UG ), quantifier laws. The rightN to-left direction follows by first universally instantiating A to thing identical to itself, i.e., /(x = x), so that by axiom 4 we have (∀A)(∀xA)ϕ → (∀x/x = x)ϕ, and, by T3, (∀x/x = x)ϕ → (∀x)[x = x → ϕ], from which, by axiom 8, (UG), and axioms 2 and 1, (∀x/x = x)ϕ → (∀x)ϕ; and from this the right-to-left direction of T7 follows. 16 It is important to keep in mind here that the only relation symbol of the system is the identity sign. 17 See Church 1956, p. 303, exercise 52.4. 18 For all of the details, see Cocchiarella 2001a. 19 We also ignore name constants and concern ourselves only with formulas in which no applied descriptive constants occur.
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2. trs∗ (¬ϕ) = ¬trs∗ (ϕ), 3. trs∗ (ϕ → ψ) = [trs∗ (ϕ) → trs∗ (ψ)], 4. trs∗ ((∀x)ϕ) = (∀x)trs∗ (ϕ), 5. trs∗ ((∀xA)ϕ) = (∀x)[FA (x) → ϕ], 6. trs∗ ((∀A)ϕ) = (∀FA )trs∗ (ϕ). It is clear that the translation under trs∗ of every theorem of our simple logic of names in which no complex name occurs becomes a theorem of secondorder monadic predicate logic. The restriction to formulas in which no complex names occur can be dropped by allowing, for each formula ϕ in which a complex name does occur, the translation function trs∗ to assign trs∗ (ψ) to ϕ, where ψ is the first formula (in terms of some alphabetic ordering) in which no complex name occurs and such that ϕ ↔ ψ. By extending trs∗ in this way, it then follows that every theorem of our conceptualist logic of names is translated into a theorem of second-order monadic predicate logic, and hence, given the known consistency of the latter, that our conceptualist system is consistent.20
Metatheorem 2:
If ϕ is a theorem of our present conceptualist logic, then trs∗ (ϕ) is a theorem of second-order monadic predicate logic. Therefore, our simple conceptualist logic is consistent. Now we can also show that our simple conceptualist logic of names is decidable by noting that every theorem of second-order monadic predicate logic can be translated into a theorem of our conceptualist logic, and hence that the one system is essentially equivalent to the other. It is well-known, of course, that second-order monadic predicate logic is decidable. Our proof that every theorem of second-order monadic predicate logic is a theorem of our conceptualist logic of names involves a translation function, trs , which translates each formula of second-order monadic predicate logic into a formula of our simple logic of names. In giving this translation function note that monadic predicate variables can be put into a one-to-one correspondence with name variables. We can, in other words, take each predicate variable to have the form FA , where A is the name variable corresponding to that predicate variable. The translation function, trs , is then defined as follows: 1. trs (x = y) = (x = y), 2. trs (FA (x)) = (∃yA)(x = y), where y is the first objectual variable other than x, 3. trs (¬ϕ) = ¬trs (ϕ), 4. trs (ϕ → ψ) = (trs (ϕ) → trs (ψ)), 20 For
the consistency of second-order monadic predicate logic, see Church 1956, p. 303.
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5. trs ((∀x)ϕ) = (∀x)trs (ϕ), 6. trs ((∀FA )ϕ) = (∀A)trs (ϕ). It is easily seen that the trs translation of every axiom of second-order monadic predicate logic is a theorem of our simple logic of names and that the rules of inference, modus ponens and the rule of universal generalization, preserve theoremhood. From these observations we conclude that we have the following metatheorem.
Metatheorem 3:
If ϕ is a theorem of second-order monadic predicate logic, then trs (ϕ) is a theorem of our present conceptualist logic.
Finally, by metatheorem 1, to show that our conceptualist logic of names is decidable, we need only show that the formulas in which no complex names occur are decidable. To show this we first prove the following metatheorem by induction on these formulas.
Metatheorem 4: If ϕ is a formula of our conceptualist logic and no complex names occur in ϕ, then ϕ ↔ trs (trs∗ (ϕ)).21 It follows, accordingly, that if ϕ is a formula of our conceptualist logic of names in which no complex names occur, then to decide whether or not ϕ is a theorem of this logic it suffices to decide whether or not trs∗ (ϕ) is a theorem of second-order monadic predicate logic. If the latter is not a theorem of second-order monadic predicate logic, then, by metatheorem 2, ϕ is not a theorem of our conceptualist logic; and if trs∗ (ϕ) is a theorem of second-order monadic predicate logic, then, by metatheorem 3, trs (trs∗ (ϕ)) is a theorem of our conceptualist logic, and therefore, by metatheorem 4, so is ϕ. Hence, by metatheorem 1, the decision problem for our conceptualist logic is reducible to that of second-order monadic predicate logic.
Metatheorem 5:
Our present conceptualist logic is both consistent and
decidable.
10.4
A Reduction of Le´ sniewski’s System
We now turn to a translation of Le´sniewski’s logic of names, as briefly described in section 2, into our conceptualist logic of names. We assume that the name 21 Proof. As noted, we prove this metatheorem by induction on the formulas of our conceptualist logic in which no complex names occur. The case for atomic formulas, which consist only of identities, is of course immediate; and for negations and conditionals, again the proof is immediate. Suppose the metatheorem holds for ϕ; then again it follows immediately that it holds for (∀A)ϕ. The only interesting case is for (∀xA)ϕ. But, by definition of trs∗ , trs∗ ((∀xA)ϕ) = (∀x)[FA (x) → trs∗ (ϕ)], and therefore, by definition of trs , trs (trs∗ ((∀xA)ϕ)) = (∀x)[(∃yA)(x = y) → trs (trs∗ (ϕ))]; and therefore, by the inductive hypothesis and (MP1), (∀xA)ϕ ↔ (∀x)[(∃yA)(x = y) → trs (trs∗ (ϕ))], which completes our proof by induction.
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variables a, b, c, d (with or with out numerical subscripts) of Le´sniewski’s logic are correlated one-to-one with the name variables A, B, C, D (with or without numerical subscripts) of our conceptualist logic, i.e., that A is correlated with a, B is correlated with b, C is correlated with c, etc. Because the only atomic formulas of the system are of the form ‘a ε b’, the following inductive definition of a translation function trs translates each formula of Le´sniewski’s logic into a formula of our conceptualist logic (with a replaced by A, b by B, etc.): 1. trs(a ε b) = (∀xA)(∀yA)(x = y) ∧ (∃xA)(∃yB)(x = y), 2. trs(¬ϕ) = ¬trs(ϕ), 3. trs(ϕ → ψ) = [trs(ϕ) → trs(ψ)], 4. trs((∀a)ϕ) = (∀A)trs(ϕ). In regard to the translation of an atomic formula of the form a ε b, note that the first conjunct, (∀xA)(∀yA)(x = y), of the translation is interpreted as saying that at most one thing is A, and therefore, because A is correlated with a, that at most one thing is a. The second conjunct, (∃xA)(∃yB)(x = y), on the other hand, says that some A is a B, and therefore, that some a is a b. The two conjuncts together are then equivalent to saying that exactly one thing is A, and hence a, and that thing is a B, i.e., a b, which is how Le´sniewski understood ‘a ε b’ as singular inclusion. Note also that where ϕ is a logical axiom of the first-order logic of Le´sniewski’s system, then trs(ϕ) is a theorem of our conceptualist logic.22 Modus ponens and (UG) also preserve validity under trs. Accordingly, to show that this interpretation amounts to a reduction of Le´sniewski’s ontology, we need only prove that trs translates the axioms of Le´sniewski’s logic into a theorem of our present system. For example, both of the axioms, (Compl) and (Conj), of Le´sniewski’s logic—one stipulating that every name has a complementary name, and the other that there is a name corresponding to the conjunction of singular inclusion in any two names—can be derived from the comprehension principle (CPN ) of our conceptualist logic as follows. 22 In fact, Axioms 1-3 are just the translations of the quantifier axioms assumed in the first-order theory for Le´sniewski’s ontology. By definition of trs, moreover, it is obvious that the translation of a tautologous formula is also a tautologous formula.
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By (CPN ) and Axiom 1, substituting for A the complex name form /¬(∃zA)(y = z) (which is read as ‘thing that is not an A’), we have (∃B)(∀x)[(∃yB)(x = y) ↔ (∃y/¬(∃zA)(y = z))(x = y)], from which, by (MP4) and Leibniz’s law, it follows that (∃B)(∀x)[(∃yB)(x = y) ↔ ¬(∃zA)(x = z)], which affirms the existence of a nominal concept that is the complement of A. Finally, by proofs similar to those given in section three, it can be shown on the basis of this last theorem that the translation of (Compl), i.e. trs(Compl), is provable in our conceptualist logic of identity. Similarly, by (rewriting B to C in) (CPN ) and Axiom 1, substituting now for A the complex name form /(∃zA)(y = z) ∧ (∃zB)(y = z) (which is read as ‘thing that is both an A and a B’), we have (∃C)(∀x)[(∃yC)(x = y) ↔ (∃y/(∃zA)(y = z) ∧ (∃zB)(y = z))(x = y)], which, by (MP4), reduces to (∃C)(∀x)[(∃yC)(x = y) ↔ (∃y)[(∃zA)(y = z) ∧ (∃zB)(y = z) ∧ y = x]], and hence, by Leibniz’s law, to (∃C)(∀x)[(∃yC)(x = y) ↔ (∃zA)(x = z) ∧ (∃zB)(x = z)], which affirms the existence of a nominal concept corresponding to the conjunction of being both an A and a B. Again, by proofs similar to those in section three, it can be shown on the basis of this last theorem that the translation of (Conj), i.e. trs(Conj), is provable in our conceptualist logic of identity. The derivation of the translation of Le´sniewski’s principal axiom, which is the only one remaining, is relatively trivial, but long on details, and we will not go into those detail here.23 In any case, we have the following metatheorem.
Metatheorem 6: If ϕ is a theorem of Le´sniewski’s (first-order) logic of names, then trs(ϕ) is a theorem of our conceptualist simple logic of names. Finally, let us turn to an explanation of the oddities of Le´sniewski’s logic of names, i.e., an explanation in terms of our translation of Le´sniewski’s logic into our simple logic of names. First, in regard to the seemingly implausible thesis, (∃a)(a = a), of Le´sniewski’s logic, note that by Le´sniewski’s definition of identity (and hence of nonidentity) (a = a) is really short for ¬(a ε a ∧ a ε a), which is equivalent to ¬(a ε a). On our conceptualist interpretation, this formula translates into ¬[(∀xA)(∀yA)(x = y) ∧ (∃xA)(∃yA)(x = y)], 23 See
Cocchiarella 2001a for those details.
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which in effect says that it is not the case that exactly one thing is an A, a thesis that is provable in our conceptualist logic when A is taken as a necessarily vacuous common name, such as ‘object that is not self-identical’, which is symbolized as ‘/(x = x)’, or more fully as ‘Object/(x = x)’. That is,
object that is not self-identical ↓ /(x = x) In other words, the translation of Le´sniewski’s thesis, (∃a)(a = a), is equivalent in our conceptualist logic of names to (∃A)¬[(∀xA)(∀yA)(x = y) ∧ (∃xA)(∃yA)(x = y)], which is provable in this logic. Note also that because (a = b) in Le´sniewski’s system means (a ε b ∧ b ε a), then the translation of (a = b) into our conceptualist logic becomes (∀xA)(∀yA)(x = y) ∧ (∃xA)(∃yB)(x = y) ∧ (∀xB)(∀yB)(x = y) ∧ (∃xB)(∃yA)(x = y),
which in effect says that exactly one thing is A and that thing is B, and that exactly one thing is B and that thing is A, a statement that is true when A and B are proper names, or unshared common names, of the same thing, and false otherwise, which is exactly how Le´sniewski understood the situation. Now the form of existential generalization that we found odd in Le´sniewski’s, namely, ϕ(c/a) → (∃a)ϕ(a), is translated into our conceptualist logic as: ϕ(C/A) → (∃A)ϕ(A), which, if C is free for A in ϕ, is provable in our simple logic of names, and yet, of course, from this it does not follow, as it does in Le´sniewski’s logic, that something is identical with nothing. Our earlier example of this oddity was the conditional ¬(∃b)(b = P egasus) → (∃a)¬(∃b)(b = a). Now what the antecedent ¬(∃b)(b = P egasus) says under our conceptualist interpretation is that there is no name concept B such that B names exactly one thing and that thing is Pegasus, which of course is true, given that Pegasus does not exist. The consequent, (∃a)¬(∃b)(b = a), on the other hand, says that for some name concept A, there is no name concept B such that A names exactly one thing and that thing is a B. That statement is in fact is true for any name A that names nothing, e.g., where A is the complex common name
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‘object that is not self-identical’. In other words, although the above formula about Pegasus seems odd as a formula in first-order logic, what it means under our conceptualist translation is not odd at all, but quite natural. In regard to the following thesis of Le´sniewski’s logic, a ε b → a ε a, note that its translation into our conceptualist logic is, (∀xA)(∀yA)(x = y)∧(∃xA)(∃yB)(x = y) → (∀xA)(∀yA)(x = y)∧(∃xA)(∃yA)(x = y),
which says that if exactly one thing is an A and that thing is a B, then exactly one thing is an A and that thing is an A, which is unproblematic. Similarly, the translation, which we will avoid writing out in full here, of Le´sniewski’s seemingly odd transitivity thesis, a ε b ∧ b ε c → a ε c, says that if exactly one thing is an A and that thing is a B and that if exactly one thing is a B—which therefore is the one thing that is an A—then exactly one thing is A and that thing is C, which again is easily seen to be valid in our conceptualist logic. Finally, putting aside Le´sniewski’s definition of identity, it is noteworthy that although ‘A = B’, unlike ‘x = y’, is not a well-formed formula of our simple logic of names, nevertheless, it will be well-formed in our next chapter where we will extend the simple logic of names to include a logic of classes as many. This extension involves a transformation of names as parts of quantifier phrases to objectual terms, i.e., terms that can be substituted for object variables. In this extended framework, as we will see, when A and B are proper names, or unshared common names, of the same thing, then A = B will be true independently of Le´sniewski’s definition of identity.
10.5
Pragmatic Uses of Proper and Common Names
The apparent oddities of Le´sniewski’s logic of names are the result of treating both proper and common names as if they were “singular terms,” i.e., expressions that can be substituends of object variables and occur as arguments (subjects) of predicates. That, in any case, is how they are understood in Le´sniewski’s elementary ontology as an applied first-order logic (without identity). Of course, that is also how proper names, but not common names, are usually analyzed in modern logic, a practice we ourselves initially followed in our lecture on tense logic. But then, before the development of free logic where proper names that denote nothing are allowed, it was sometimes also the practice to transform proper names into monadic predicates. The proper name ‘Socrates’, for example, became the monadic predicate ‘Socratizes’, which was
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true of exactly one thing, and the name ‘Pegasus’ became ‘Pegasizes’, which was true of nothing. In this way, the statement that Pegasus does not exist could be analyzed as saying that nothing Pegasizes.24 Common names, on the other hand, have usually been analyzed as, or really transformed into, monadic predicates in modern logic, both before and after the development of free logic. Of course, in our general framework of conceptual realism there are complex monadic predicates that are constructed on the basis of both proper and common names. Thus, where A is a name, proper or common, then [λx(∃yA)(x = y)] is a monadic predicate, read as ‘x is an A’ when A is a common name, and as ‘x is A’ when A is a proper name. Le´sniewski’s logic of names is viewed as odd in modern logic, as we have said, because it takes common names to be more like proper names than like monadic predicates, and in particular it represents them the way that “singular terms” are represented in modern logic. On this view, if common names were to be put in the same syntactic category as proper names, then it should be by taking both as monadic predicates. Now in our conceptualist logic, proper names and common names are in the same syntactic category, but it is not the category of monadic predicates, nor is it the category of “singular terms”. Proper and common names belong to a more general category of names, and as such they are taken as parts of quantifier phrases, i.e., phrases that stand for referential concepts. This is not at all like taking them as “singular terms,” the way they are in Le´sniewski’s logic, though, as we will explain shortly, they can be transformed into “singular terms”, i.e., terms that can be substituends of objectual variables and occupy the argument positions of predicates. The important point is that unlike the view of names in Le´sniewski’s logic, the occurrence of names as parts of quantifier phrases, i.e., of referential expressions, is an essential component of how the nexus of predication is understood in conceptual realism. In other words, having a single category of names containing both proper and common names is a basic part of our theory of reference. This does not mean that we cannot distinguish a proper name from a common name in our logic. In particular, a proper name, when introduced into an applied formal language, brings with it a meaning postulate to the effect that the name can be used to refer to at most one thing. If the language also contains tense and modal operators, then not only is it stipulated that a proper name can be used to refer to at most one thing, but that it must be the same thing at any time in each possible world in which that thing exists. Common names, on the other hand, are not introduced into an applied formal language with such a meaning postulate. Now there are other uses of proper and common names as well. Both, for example, can be used in simple acts of naming, as when a parent teaches a child 24 See,
e,g,, Quine 1960, p. 179.
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what a dog or a cat is by pointing to the animal and saying ‘dog’, or ‘cat’.25 A simple act of naming is not an assertion and does not involve the exercise of either a predicable or a referential concept. Also, names, both proper and common, can be used in greetings, or in exclamations as when someone shouts ‘Wolf!’ or ‘Fire!’, which again are not assertions and do not involve the exercise of a referential act. A common name such as ‘poison’ is also used as a label, which again is not a referential act. Nor are referential acts involved in the use of name labels that people wear at conferences. These kinds of uses of names, especially proper names or sortal common names, i.e., names that have identity criteria associated with their use, are conceptually prior to the referential use of names in sentences. But the analysis of these kinds of uses as well as the use of names in referential acts belongs to the discipline of pragmatics, and not that of semantics, which deals exclusively with denotation and truth conditions.26
10.6
Classes as Many as the Extensions of Names
Now in addition to these pragmatic uses of names there are also “denotative” uses of names as well, as when we speak of mankind, or humankind, by which we mean the totality, or entire group, of humans taken collectively—but not in the sense of a set or class as an abstract object.27 Thus, we say that Socrates is a member of mankind, as well as that Socrates is a man. Also, instead of ‘mankind’, we can use the plural of ‘man’ and say that Socrates is one among men. These in fact are transformations of the name ‘man’ into an “objectual term,” i.e., an expression that can occur as an argument of predicates. But it is not a “singular term” in the sense that it denotes a single entity, e.g., a set or a class as an abstract object. Instead of using the phrase ‘singular term’, which suggests that we are dealing with a “single” entity, a better, or less misleading, phrase is ‘objectual term’, which we have used instead. An “object” such as mankind is not a “single” entity, but a plural object, i.e., a plurality taken collectively. The transformation of ‘man’ into ‘mankind’, or ‘human’ into ‘humankind’, and ‘dog’ into ‘dogkind’, etc. is different from the nominalizing transformation of a predicate adjective, such as ‘human’, into an abstract noun—i.e., an abstract “singular term”—such as ‘humanity’, or into the gerundive phrase ‘being human’, or into the related infinitive and gerundive phrases ‘to be a man’ and ‘being a man’, all of which are represented in our logical syntax by a nominalization of the (complex) predicate phrase [λx(∃yM an)(x = y)]( ). 25 See,
e.g., Geach 1980, p. 52. and semantics are two of the three principal parts of semiotics. The third is
26 Pragmatics
syntax. 27 See, e.g., Sellars 1963, p. 253.
10.6. CLASSES AS MANY AS THE EXTENSIONS OF NAMES
231
• The transformation of a predicate, e.g., ‘is human’, into an abstract noun, ‘humanity’, results in a genuine “singular term”, i.e., a term that purports to denote a single object, albeit an abstract intensional one. • But the transformation of ‘man’ into ‘mankind’ or ‘men’, or ‘dog’ into ‘dogkind’ or ‘dogs’, does not result in a “nominal” expression that purports to denote a single object; nor does it purport to denote an abstract intensional object. • What such a noun as ‘mankind’, or ‘dogkind’, or either of the plurals ‘men’ and ‘dogs’, purports to denote is a plural object, namely, men, or dogs, taken collectively as a group—but not as a set or a class as a single object. The expressions ‘mankind’ and ‘dogkind’, are indeed “nominal” expressions, i.e., nouns, and therefore, logically, they should be represented as “objectual terms,” but not as “singular” terms in the sense of nominal expressions that denote single objects. What they denote are pluralities, i.e., plural objects. Now a plural object, such as a group of things, is what Bertrand Russell once called a class as many, as opposed to a class as one. Russell allowed that a class as many could consist of just a single object, as when a common name has just one object in its extension, in which case the class as many is the same as that one object. On the other hand, there is no class as many that is empty.28 There is more than one notion of a class, in other words, and in fact there is even more than one notion of a class in the sense of the iterative concept of a set, i.e., the concept of a set based on Cantor’s power-set theorem. The iterative concept of a set can be developed, for example, either with an axiom of foundation or an axiom of anti-foundation.29 But in neither case can there be a universal set, and yet there are set theories, such as Quine’s NF and the related set theory NFU, in which there is a universal set. The notion of a universal class is part of the traditional notion of a class as the extension of a predicable concept (Begriffsumfange), and, as we have noted in lecture four, this was how classes were understood by Frege in his Grundgetsetze. Now our point here is that classes in all of these senses are single objects, not plural objects, i.e., they are each a class as one, a single abstract object. It is not the notion of a class as one, i.e., as a single abstract object, that we are concerned with here, but the notion of a class as many, i.e., of a class as a plurality, or plural object. It is this notion that is implicitly understood as the extension of a common count noun, or what we have been calling a common name. In the development of our analysis of this notion, we will also take a class as many consisting of just one object as the extension of a nonvacuous proper name. 28 See
29 For
Russell 1903, §§69–70. a development of set theory with an anti-foundation axiom, see Aczel 1988.
´ CHAPTER 10. LESNIEWSKI’S ONTOLOGY
232
Membership in a class as many can be defined once names are allowed to be “nominalized” and occur as objectual terms. The definition is as follows: x ∈ y =df (∃A)[y = A ∧ (∃zA)(x = z)], where A is a name variable or constant. Note that in this definition, A occurs as a “nominalized”, objectual term in the conjunct ‘y = A’ as well as part of the quantifier phrase in the second conjunct ‘(∃zA)(x = z)’. An obvious theorem of the logic of classes as many, which we will develop in the next lecture, is the following, x ∈ A ↔ (∃zA)(x = z), where ‘x ∈ A’ can be read as ‘x is an A’, or ‘x is one among the A’, or ‘x is a member of the class as many of A’, or simply as ‘x is a member of A’. Now as understood by Russell, there are three important features of the notion of a class as many as the extension of a common name. These are: 1. First, that a vacuous common name, i.e., a common name that names nothing, has no extension, which is not the same as having an empty class as its extension. Thus, according to Russell, “there is no such thing as the null class, though there are null class-concepts,” i.e., common-name concepts that have no extension.30 2. Secondly, the extension of a common name that names just one thing is just that one thing. In other words, unlike the singleton sets of set theory, which are not identical with their single member, the class that is the extension of a common name that names just one thing is none other than that one thing. 3. The second feature is related to our third, namely, that unlike sets, classes as the extensions of names are literally made up of their members so that when they have more than one member they are in some sense pluralities (Vielheiten), or “plural objects,” and not things that can themselves be members of classes. 4. Thus, according to Russell, “though terms [i.e., objects] may be said to belong to ... [a] class, the class [as a plurality] must not be treated as itself a single logical subject.”31 It is a class as many that is the extension, or denotatum, of a common name, and, on our analysis, also of a proper name. On our analysis, the logic of classes as many is a direct and natural extension of the simple logic of names described earlier in this lecture. The idea is that names, both proper and common, can be transformed into, or “nominalized” as, objectual terms that can be substituends of objectual variables and occur as arguments of predicates. When so transformed, what a name denotes is its extension, which in the case 30 Russell 31 Russell
1903, §69. 1903, §70.
10.7. SUMMARY AND CONCLUDING REMARKS
233
of a common name with more than one object in its extension is a plural object, which we will also call a group. The extension of a proper name, on the other hand, is the object, if any, that the name denotes as a “singular” term. The resulting logic of classes as many is not entirely unlike the analysis given in Le´sniewski’s logic of names, where names occur only as objectual terms. In fact we can even formulate counterparts in this logic to certain of the oddities of Le´sniewski’s logic. But there is also a difference in that the counterparts of Le´sniewski’s problematic oddities are refutable, and the counterparts that are not refutable do not appear as odd but as natural consequences of an ontology with both single and plural objects.
10.7
Summary and Concluding Remarks
• The conceptualist logic of names is similar to Le´sniewski’s logic of names in that the category of names in Le´sniewski’s system contains common as well as proper names, i.e., names can be either shared (common) or unshared (proper), or they can be vacuous. • Le´sniewski’s logic is different, however, in that names do not occur as parts of quantifier phrases but are of the same category as object variables. • There are a number of oddities that are provable in Le´sniewski’s logic of names, such as (∃a)(a = a), which seems to say that some object is not identical with itself. Similarly, the law of existential generalization in Le´sniewski’s logic seems to have such odd consequences as there being something that is identical with nothing. • Both Le´sniewski’s logic of names and the conceptualist logic of names can be formulated as subsystems of their respective formal ontologies, and as subsystems both are consistent and decidable. • Le´sniewski’s logic of names is interpretable in terms of the conceptualist logic of names, and the oddities of Le´sniewski’s logic are seen as no longer odd when interpreted within the conceptualist logic. It is Le´sniewski’s apparent treatment of names as object terms that accounts for the oddities of Le´sniewski’s logic. Names in conceptualism occur primarily as parts of quantifier phrases, which explains their referential role in speech and mental acts. • But names, both proper and common, can be transformed into objectual terms in conceptual realism, and when so transformed, or “nominalized”, they are interpreted as denoting classes as many, which are not the same as sets or classes as ones. Classes as many are the extensions of names.
Chapter 11
Plurals and the Logic of Classes as Many In chapter ten we formalized the simple logic of names that is an important part of the theory of reference in conceptual realism. The category of names, it will be remembered, includes both proper and common, and complex and simple, names, all of which occur as parts of quantifiers phrases. Quantifier phrases, of course, are what stand for the referential concepts of conceptual realism. We explained in that chapter how Le´sniewski’s logic of names, which Le´sniewski called ontology, can be interpreted and reduced to our conceptualist logic of names, and how in that reduction we can explain and account for the oddities of Le´sniewski’s logic. We concluded chapter ten with observations about the “nominalization,” or transformation, of names as parts of quantifier phrases into objectual terms. What a “nominalized” name denotes as an objectual term, we said, is the extension of that name, i.e., of the concept that the name stands for in its role as part of a quantifier phrase. The extension of a name is not a set, nor a class as a “single object”. Rather, the extension of a name is a class as many, i.e., a class as a plurality that is literally made up of its members. We listed three of the central features of classes as many as originally described by Bertrand Russell in his 1903 Principles of Mathematics. These are, first, that a vacuous name—that is, a name that names nothing—has no extension, which is not the same as having an empty class as its extension. In other words, there is no empty class as many. Secondly, the extension of a name that names just one thing is none other than that one thing; that is, a class as many that has just one member is identical with that one member. In other words it is because a class as many is literally many up of its members that it is nothing if it has no members; and that is also why it is identical with its one member if it has just one member. Finally, that is also why a class as many that has more than one member is merely a plurality, or plural object, which is to say that as a 235
236 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY plurality it is not a “single object,” and therefore it cannot itself be a member of any class as many.
11.1
The Logic of Classes as Many
We begin here where left off in chapter ten, namely, with the logic of classes as many as an extension of the simple logic of names. We assume in this regard all of the axioms and theorems of the simple logic names given in that lecture. That logic consisted essentially of a free first-order logic of identity extended to include the category of names as parts of quantifiers, and where the quantifiers ∀ and ∃ can be indexed by name variables as well as objectual variables.1 Now because names can be transformed into objectual terms we need a variable-binding operator that generates complex names the way that the λoperator generates complex predicates.2 We will use the cap-notation with brackets, [ˆ xA/...x...], for this purpose. Accordingly, where A is a name, proper or common, complex or simple, we take [ˆ xA] to be a complex name, but one in which the variable x is bound. Thus, where A is a name and ϕ is a formula, [ˆ xA], [ˆ xA/ϕ], and [ˆ x/ϕ] are names in which all of the free occurrences of x in A and ϕ are bound. We read these expressions as follows: [ˆ xA] is read as ‘the class (or group) of A’, [ˆ xA/ϕ] is read as ‘the class (or group) of A that are ϕ’, and [ˆ x/ϕ] is read as ‘the class (or group) of things that are ϕ’. A formal language L is now understood as a set of predicate and name constants, instead of a set of predicates and objectual constants, as was originally described in our lecture on tense and modal logic. There will be objectual constants in a formal language as well, but they will be generated from the name constants by a “nominalizing” transformation. In our more comprehensive framework, which we are not concerned with here, objectual constants are also generated from predicate constants by the nominalizing transformation described in our fourth lecture. We extend the simultaneous inductive definition of names and formulas given in §3 of our previous lecture to include names of this complex form as well as follows: If L is a formal language, then: • (1) Every name variable or name constant in L is a name of L; • (2) if a, b are either objectual variables, name variables or name constants in L, or names of L the form [ˆ xB], where x is an objectual variable and B is a name (complex or simple) of L, then (a = b) is a formula of L; and • if ϕ, ψ are formulas of L, B is a name (complex or simple) of L, and x and C are an objectual and a name variable, respectively, then 1 Much
of the material in this chapter is based on Cocchiarella 2002. should be remembered that in free logic being a substituend of free objectual variables— i.e., being an “objectual term”—is not the same as denoting a value of the bound objectual variables. That is, in free logic some objectual terms may denote nothing. 2 It
11.1. THE LOGIC OF CLASSES AS MANY
237
(3) ¬ϕ, (4) (ϕ → ψ), (5) (∀x)ϕ, (6) (∀xB)ϕ, and (7) (∀C)ϕ are formulas of L, and (8) B/ϕ, (9) /ϕ, and (10) [ˆ xB] are names of L. Note that by definition we now have formulas of the form (∀y[ˆ xA])ϕ, as well as those of the form (∀xA)ϕ and (∀yA(y/x))ϕ as in §3 of our previous lecture. We reduce the first to the last of these forms by adding the following axiom schema to those already listed in §3 of our previous lecture, but now understood to apply to our extended notions of name and formula: Axiom 12: (∀y[ˆ xA])ϕ ↔ (∀yA(y/x))ϕ,
where y does not occur in A.
Because we are retaining the axioms and theorems of §3 of our previous lecture, our first axiom of the logic of classes as many, Axiom 12, begins where we left off, the last axiom of which was axiom 11, and the last theorem of which was T7. We might note, incidentally, that Axiom 12 is a conversion principle for complex names as parts of quantifier phrases. It is the analogue for complex names of the form [ˆ x/A] of λ-conversion for complex predicates of the form [λxϕ]. Given our understanding of the existential quantifier as defined in terms of negation and the universal quantifier, this means we also have the following as a theorem (where y is free for x in A): T8: (∃y[ˆ xA])ϕ ↔ (∃yA(y/x))ϕ. Two other axioms about the occurrence of names as objectual terms are: Axiom 13: (∃A)(A = [ˆ xB]),
where B is a name and A is a name variable that does not occur (free) in B; and
Axiom 14: A = [ˆ xA],
where A is a simple name, i.e., a name variable or constant.
Axiom 13, incidentally, is a comprehension principle for complex names, and as such is the analogue for complex names of the comprehension principle (CP∗λ ) for complex predicates. What it says is that every complex name of the form [ˆ xB] is a value of the bound name variables, and therefore stands for a name, or nominal, concept. Axiom 14 tells us that the name concept [ˆ xA] is none other than the name concept A. It is noteworthy that our earlier axiom 4 of §3 is now redundant and can be derived by Leibniz’s law, (LL∗ ), from axiom 13. That is, by (LL∗ ), C = [ˆ xB] → [ϕ → ϕ([ˆ xB]/C)], and therefore by (UG), axioms 3, 5, and tautologous transformations, (∃C)(C = [ˆ xB]) → [(∀C)ϕ → ϕ([ˆ xB]/C)], and hence, by axiom 13,
238 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY T9: (∀C)ϕ → ϕ([ˆ xB]/C). Strictly speaking, T9 is actually slightly stronger than axiom 4 in that it includes cases in which complex names occur as objectual terms, i.e., where some, or all, of the occurrences of C in ϕ may be as objectual terms, and hence where not all occurrences of [ˆ xB] in ϕ([ˆ xB]/C) can be replaced by B if B is a complex name of the form A/ψ or of the form /ψ. If C occurs in ϕ only as part of a quantifier phrase, then ϕ([ˆ xB]/C) is equivalent to ϕ(B/C) by axiom 12. We turn now to definitions of some of the concepts that are important in the logic of classes. Note that although we adopt the same symbols that are used in set theory to express membership, inclusion and proper inclusion, it should be kept in mind that the present notion of class is not that of set theory. Definition 2 x ∈ y ↔ (∃A)[y = A ∧ (∃zA)(x = z)]. Definition 3 x ⊆ y ↔ (∀z)[z ∈ x → z ∈ y]. Definition 4 x ⊂ y ↔ x ⊆ y ∧ y x. Note also that the argument for Russell’s paradox for classes does not lead to a contradiction within this system as described so far, nor will it do with the axioms yet to be listed. Rather, what it shows is that the Russell class as many does not “exist” in the sense of being the value of a bound objectual variable, which is not to say that the name concept of the Russell class does not have its own conceptual mode of being as a value of the bound name variables. Indeed, as the following definition indicates, the name, or nominal, concept of the Russell class can be defined in purely logical terms. Definition 5 Rus = [ˆ x/(∃A)(x = A ∧ x ∈ / A)]. That the Russell class as many does not “exist” as an object, i.e., as a value of the bound objectual variables, is important to note because it has been claimed that “the objective view” of plural objects, i.e., the view of them as objects (such as classes as many), is refuted by Russell’s paradox.3 The fact that the Russell class does not “exist” in the logic of classes as many is stated in the following theorem. (Proofs will be given only as footnotes.) T10: ¬(∃x)(x = Rus).4 3 See,
e.g., Schein 1993, pages 5, 15, and 32-37. By axiom 13 and identity logic, (∃A)(Rus = A), and by definition 1, Rus ∈ Rus ↔ (∃A)[Rus = A ∧ (∃xA)(x = Rus)], and therefore by Leibniz’s law, a quantifierconfinement law and tautologous transformations, Rus ∈ Rus ↔ (∃xRus)(x = Rus). But then, by definition of Rus and T8, (∃xRus)(x = Rus) ↔ (∃x/(∃A)(x = A ∧ x ∈ / A))(x = Rus), and therefore, by T1, (∃xRus)(x = Rus) ↔ (∃x)[(∃A)(x = A ∧ x ∈ / A) ∧ x = Rus], from which, by Leibniz’s law, it follows that (∃xRus)(x = Rus) ↔ (∃x)[Rus ∈ / Rus ∧ x = Rus]; and, accordingly, by quantifier-confinement laws, and tautologous transformations, (∃x)(x = Rus) → (Rus ∈ Rus ↔ Rus ∈ / Rus), from which we conclude that ¬(∃x)(x = Rus). 4 Proof.
11.1. THE LOGIC OF CLASSES AS MANY
239
What Russell’s argument shows is that not every name concept has an extension that can be “object”-ified in the sense of being the value of a bound objectual variable. Now the question arises as to whether or not we can specify a necessary and sufficient condition for when a name concept has an extension that can be “object”-ified, i.e., for when the extension of the concept can be proven to “exist” as the value of a bound objectual variable. In fact, the answer is affirmative. In other words, unlike the situation in set theory, such a condition can be specified for the notion of a class as many. An important part of this condition is Nelson Goodman’s notion of an “atom,” which, although it was intended for a strictly nominalistic framework, we can utilize for our purposes and define as follows.5 Definition 6 Atom = [ˆ x/¬(∃y)(y ⊂ x)]. This notion of an atom has nothing to do with physical atoms, of course. Rather, it corresponds in our present system approximately to the notion of an urelement, or “individual,” in set theory. We say “approximately” because in our system atoms are identical with their singletons, and hence each atom will be a member of itself. This means that not only are ordinary physical objects atoms in this sense, but so are the propositions and intensional objects denoted by nominalized sentences and predicates in the fuller system of conceptual realism. Of course, the original meaning of ‘atom’ in ancient Greek philosophy was that of being indivisible, which is exactly what was meant by ‘individual’ in medieval Latin. An atom, or individual, in other words, is a “single” object, which is apropos in that objects in our ontology are either single or plural. We will henceforth use ‘atom’ and ‘individual’ in just this sense. The following axiom (where y does not occur in A) specifies when and only when a name concept A has an extension that can be “object”-ified (as a value of the bound objectual variables). Axiom 15: (∃y)(y = [ˆ xA]) ↔ (∃xA)(x = x) ∧ (∀xA)(∃zAtom)(x = z). Stated informally, axiom 15 says that the extension of a name concept A can be “object”-ified (as a value of the bound objectual variables) if, and only if, something is an A and every A is an atom.6 An immediate consequence of this axiom, and of T8 and T1, is the following theorem schema, which stipulates exactly when an arbitrary condition ϕx has an extension that can be “object”ified. T11: (∃y)(y = [ˆ x/ϕx]) ↔ (∃x)ϕx ∧ (∀x/ϕx)(∃zAtom)(x = z). Note that where ϕx is the impossible condition (x = x), it follows from T11 that there can be no empty class, which, as already noted, is our first basic 5 See
Goodman 1956 for Goodman’s account of atoms in nominalism. something is an A is perspicuously symbolized by (∃y)(∃xA)(y = x). But because (∃xA)(x = x) ↔ (∃y)(∃xA)(y = x) is provable, we will use (∃xA)(x = x) as a shorter way of saying the same thing. 6 That
240 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY feature of the notion of a class as many. We define the empty-class concept as follows and then note that its extension, by T11, cannot “exist” (as a value of the bound objectual variables), as well as that no object can belong to it. Definition 7 Λ = [ˆ x/(x = x)]. T12a: ¬(∃x)(x = Λ). T12b: ¬(∃x)(x ∈ Λ). Finally, our last axiom concerns the second basic feature of classes as many; namely, that every atom, or individual, is identical with its singleton. In terms of a name concept A, the axiom stipulates that if at most one thing is an A and that whatever is an A is an atom, then whatever is an A is identical to the extension of A, which in that case is a singleton if in fact anything is an A. Where y does not occur in A, the axiom is as follows. Axiom 16: (∀xA)(∀yA)(x = y) ∧ (∀xA)(∃zAtom)(x = z) → (∀yA)(y = [ˆ xA]). A more explicit statement of the thesis that an atom is identical with its singleton is given in the following theorem. T13: (∃zAtom)(x = z) → x = [ˆ y /(y = x)].7 By T13, it follows that every atom is identical with the extension of some name concept, e.g., the concept of being that atom. Of course, non-atoms, i.e., plural objects, are the extensions of name concepts as well (by the definitions of Atom, ⊂, and ∈), and hence anything whatsoever is the extension of a name concept. T14: (∃zAtom)(x = z) → (∃A)(x = A). T15: ¬(∃zAtom)(x = z) → (∃A)(x = A). T16: (∃A)(x = A). Note that if A is a proper name of an ordinary, physical object (and hence an atom), then, by the meaning postulate for proper names, the antecedent of axiom 16 is true, and therefore, by axioms 16 and 14, (∀yA)(y = A). In other words, if A is a proper name of an atom, then F (A) ↔ (∀yA)F (y) is true, which in our conceptualist framework explains the role proper names have as “singular terms” (i.e., as substituends of free objectual variables) in free logic. That is, by Leibniz’s law, (UG), axioms 2 and 6, T4, a quantifier-confinement law, (∀yA)(y = A) F (A) ↔ (∀yA)F (y). 7 Proof. Where A be the nominal concept thing-that-is-identical-to x, i.e. /(y = x), then, by axiom 11 and (LL∗ ), (∀y/y = x)(∀w/w = x)(y = w), and, similarly, (∃zAtom)(x = z) → (∀y/y = x)(∃zAtom)(y = z). Therefore, by axiom 16, (∃zAtom)(x = z) → (∀y/y = x)(y = [ˆ y /(y = x]). But, by T6, (∃zAtom)(x = z) → (∃z)(z = x), and therefore by (∃/UI), T3 and axiom 8, (∃zAtom)(x = z) → x = [ˆ y /(y = x].
11.2. EXTENSIONAL IDENTITY
241
Of course, if A is a non-vacuous proper name of an ordinary object, then (∃yA)(y = A) is true as well, and hence F (A) ↔ (∃yA)F (y) as true as well. That is, (∃yA)(y = A) F (A) ↔ (∃yA)F (y). What these last results indicate is that the role proper names have as “singular terms,”— i.e., as substituends of free objectual variables that purport to denote a “single” object — in standard free logic is reducible to, and fully explainable in terms of, the role proper names have in our logic of classes as many. A consequence of T13, the definition of ∈, T8, and Leibniz’s law is the thesis that every atom is a member of itself. A similar argument, but without T13, shows that every object is a member of its singleton, which does not mean that every “real” object, i.e., every value of the bound objectual variables, is identical with its singleton.8 T17: (∀xAtom)(x ∈ x). T18: (∀x)(x ∈ [ˆ z /(z = x)]). Finally, we note that by definition of membership and Leibniz’s law an object x belongs to the extension of a name concept A if, and only if, x is an A. From this it follows that only atoms can belong to an “ object”-ified class as many, and hence that classes as many that are not atoms are not themselves members of any (real) classes as many, which is our third basic feature of classes as many. T19: x ∈ A ↔ (∃yA)(x = y). T20a: (∀x)[z ∈ x → (∃wAtom)(z = w)]. T20b: ¬(∃wAtom)(z = w) → ¬(∃x)(z ∈ x).9
11.2
Extensional Identity
The “nominalist’s dictum,” according to Nelson Goodman, is that “no two distinct things can have the same atoms.”10 Such a dictum, it would seem, should apply to classes as many as traditionally understood, regardless whether or not a more comprehensive framework containing such classes is nominalistic or not. 8 In other words, if x is a “real” class as many with more than one member, then x = [ˆ z /(z = x)], even though x ∈ [ˆ z /(z = x)]. The latter, like x s being a member of the universal class, means only that x is identical with itself. 9 Proof. By definition of ∈, z ∈ x → (∃A)[x = A ∧ (∃wA)(z = w)], and therefore, by T6, z ∈ x → (∃w)(z = w). By axiom 15, (∃y)(A = y) → (∀zA)(∃wAtom)(z = w); and therefore, by axiom 10, T19, and (LL∗ ), (∃y)(x = y) ∧ x = A → (∀z)[z ∈ A → (∃wAtom)(z = w)]. But then, by quantifier-confinement laws, T16, (LL∗ ), (∃/UI) and elementary transformations, (∃y)(x = y) → [z ∈ x → (∃wAtom)(z = w)]. Therefore, by (UG) and axiom 7, (∀x)[z ∈ x → (∃wAtom)(z = w)], which is T20a. T20b then follows by a quantifier-confinement law and tautologous transformations. 10 Goodman 1956, p. 21.
242 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY In fact, the dictum is provable here if we assume an axiom of extensionality for classes. But there is a problem with the axiom of extensionality. In particular, if the full unrestricted version of Leibniz’s law is not modified, then having an axiom of extensionality would seem to commit us to a strictly extensional framework even if it is not otherwise nominalistic. Name concepts that have the same extension at a given moment in a given possible world would then, by Leibniz’s law, be necessarily equivalent, and therefore have the same extension at all times in every possible world, which is counter-intuitive. It is hoped, for example, that the extension of a common name concept such as ‘country that is democratic’ can have more and more members in it over time. Common name concepts of animals, e.g., ‘buffalo’, certainly have different extensions over time. Some, as in the case of names of plants and animals that have become extinct have changed their extensions radically from having millions of members to now having none. The idea that common name concepts cannot have different extensions over time, no less in different possible worlds, is a consequence we do not want in our broader framework of conceptual realism. On the other hand, classes as many are extensional objects, and an axiom of extensionality that applies at least to classes as many is the only natural assumption to make in an ontology with classes as many, as in fact ours is. All objects, in other words, whether they are single or plural, are classes as many, and the idea that classes as many are not identical when they have the same members is difficult to reconcile with such an ontology. Fortunately, there is an alternative, namely, that the full version of Leibniz’s law as it applies to all contexts is to be restricted to atoms, i.e., single objects, or individuals in the ontological sense. The restricted version for extensional contexts can then still be applied to pluralities, i.e., classes as many that have more than one member. Thus, in addition to the axiom of extensionality, we will take the following as an new axiom schema of our general framework. Axiom 17: (∃zAtom)(x = z) ∧ (∃zAtom)(y = z) → [x = y → (ϕ ↔ ψ)], where ψ is obtained from ϕ by replacing one or more free occurrences of x by free occurrences of y. This axiom is redundant if we do not add any nonextensional contexts, e.g., tense or modal operators, to the logic of classes as many. The reason is because, in a strictly extensional language, the full, unrestricted version of Leibniz’s law is derivable from axiom 9.11 In other words, Axiom 9 remains in effect, but all we can prove from it is that Leibniz’s law holds for all extensional contexts. This distinction between how Leibniz’s law applies to atoms and how it applies to classes as many in general is an ontological feature of our logic in that it distinguishes the individuality of atoms from the plurality of groups. 11 As given in §3 of our previous lecture, Axiom 9 is restricted to atomic formulas. The unrestricted version is then derivable by induction over the formulas of an extensional language. See, e.g., the proof given of (LL) in our second lecture.
11.2. EXTENSIONAL IDENTITY
243
Indeed, unlike atoms, or individuals in the strict ontological sense, the identity of groups, or pluralities, i.e., classes as many with more than one member, essentially reduces to the fact that they are made up of the same members, which does not justify the full, unrestricted ontological content of Leibniz’s law. A related point about Axiom 17 is that it is an ontological thesis about the values of object variables and not about the objectual terms, e.g., name constants, that might be substituted for object variables. The validity of instantiating objectual terms for the object variables in this axiom depends on the contexts in which those variables occur and how “rigid” those objectual terms are with respect to those contexts. Proper names are assumed to be rigid with respect to tense and modal contexts, for example, but not in general with respect to belief and other cognitive-modal contexts except under special assumptions, such as knowing who, or what, the terms denote. We now include the axiom of extensionality, which we will refer to hereafter as (ext), among the axioms. Axiom 18 (ext): (∀z)[z ∈ x ↔ z ∈ y] → x = y. Goodman’s nominalistic dictum that things are identical if they have the same atoms is now provable as the following theorem. T21: (∀x)(∀y)[(∀zAtom)(z ∈ x ↔ z ∈ y) → x = y].12 Note that by T13 and the definition of ∈, whatever belongs to an atom is identical with that atom, and therefore atoms are identical if, and only if, they the have the same members. T22: (∀xAtom)[y ∈ x → y = x]. T23: (∀xAtom)(∀yAtom)[x = y ↔ (∀z)(z ∈ x ↔ z ∈ y)]. Note also that by T21 (and other theorems) it follows that everything “real”, whether it is an atom or not, has an atom in it. T24: (∀x)(∃zAtom)(z ∈ x).13 Another useful theorem is the following, which, together with T21, shows that every non-atom must have at least two atoms as members. Of course, 12 Proof. By T5, T20a, (UG), quantifier-confinement laws, and elementary transformations, (∀x)[(∀zAtom)(z ∈ x ↔ z ∈ y) → (∀z)(z ∈ x → z ∈ y)], and similarly (∀y)[(∀zAtom)(z ∈ x ↔ z ∈ y) → (∀z)(z ∈ y → z ∈ x)], from which, given (ext), T21 follows. 13 Proof. By T5 and T17, (∃zAtom)(x = z) → (∃zAtom)(z ∈ x), and hence, by contraposition and the definition of Atom, ¬(∃zAtom)(z ∈ x) → ¬(∃z[ˆ x¬(∃y)(y ⊂ x)])(x = z); and therefore, by axioms 12, 11 and elementary transformations, ¬(∃zAtom)(z ∈ x) → (∀z)[x = z → (∃y)(y ⊂ z)], from which, by (LL∗ ) and a quantifier-confinement law, it follows that ¬(∃zAtom)(z ∈ x) → [(∃z)(x = z) → (∃y)(y ⊂ x)]; and therefore, by (UG) and axioms 2 and 7, (∀x)[¬(∃zAtom)(z ∈ x) → (∃y)(y ⊂ x)]. Now, by definition of ⊂, ¬(∃zAtom)(z ∈ x)∧y ⊂ x → ¬(∃zAtom)(z ∈ y), and therefore ¬(∃zAtom)(z ∈ x)∧y ⊂ x → (∀zAtom)[z ∈ x ↔ z ∈ y], and, accordingly by (UG) and T21, (∀x)(∀y)[¬(∃zAtom)(z ∈ x) ∧ y ⊂ x → x =ex y]. But then, by definition of ⊂, (∀x)(∀y)[¬(∃zAtom)(z ∈ x) → (y ⊂ x → x ⊆ y ∧ x y)]; and therefore, by quantifier logic, (∀x)[¬(∃zAtom)(z ∈ x) → ¬(∃y)(y ⊂ x)]. Together with the above result, this shows that (∀x)[¬(∃zAtom)(z ∈ x) → (∃y)(y ⊂ x) ∧ ¬(∃y)(y ⊂ x)], from which T24 follows by quantifier logic.
244 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY conversely, any “real” class (as many) that has at least two members cannot be an atom, because then each of those members is properly contained in that class. T25: (∀x)(∀y)(y ⊂ x → (∃zAtom)[z ∈ x ∧ z ∈ / y]).14 T26: (∀x)[¬(∃yAtom)(x = y) ↔ (∃z1 /z1 ∈ x)(∃z2 /z2 ∈ x)(z1 = z2 )].15 Two consequences of the extensionality axiom, (ext), are the strict identity of a class with the class of it members and the rewrite of bound variables for class expressions. T27a: x = [ˆ z /(z ∈ x)]. T27b: [ˆ xA] = [ˆ y A(y/x)],
11.3
where y does not occur in A.16
The Universal Class
We have seen that, unlike the situation in set theory, the empty class as many does not “exist” (as a value of the bound objectual variables). But what about the universal class? In ZF, Zermelo-Fr¨ ankel set theory, there is no universal set, but in Quine’s set theory NF (New Foundations) and the related set theory, NFU (New Foundations with Urelements), there is a universal set. In our present theory, the situation is more complicated. For example, if nothing exists, then of course the universal class does not exist. But, in addition, because something exists only if an atom does, i.e., by T24 and (∃/UI), T28: (∃x)(x = x) → (∃xAtom)(x = x), it follows that the universal class does not exist if there are no atoms, i.e., individuals—which is unlike the situation in set theory where classes exist whether or not there are any urelements, i.e., individuals. As it turns out, we can also show that the universal class does not exist if there are at least two atoms. If there is just one atom, however, the situation is more problematic. 14 Proof. By quantifier logic and definition of ⊂, y ⊂ x → (∀zAtom)(z ∈ y → z ∈ x), and therefore, by (UG) and T21, (∀x)(∀y)(y ⊂ x → [(∀zAtom)(z ∈ x → z ∈ y) → x =ex y]). But then, by definition of ⊂ and =ex , (∀x)(∀y)(y ⊂ x → [(∀zAtom)(z ∈ x → z ∈ y) → x ⊆ y ∧ x y]), and hence (∀x)(∀y)(y ⊂ x → (∃zAtom)[z ∈ x ∧ z ∈ / y]). 15 Proof. By T25, (∀x)(∀y)(y ⊂ x → (∃z Atom)[z ∈ x ∧ z ∈ 1 1 1 / y]), and by T24 and (∃/UI), (∃w)(y = w) → (∃z2 Atom)(z2 ∈ y). But, by (LL∗ ) and definition of ⊂, y ⊂ x ∧ z1 ∈ / y ∧ z2 ∈ y → z2 ∈ x ∧ z1 = z2 , and therefore, by quantifier logic, (∃w)(y = w) → (∀x)[y ⊂ x → (∃z1 Atom)(∃z2 Atom)(z1 = z2 ∧ z1 ∈ x ∧ z2 ∈ x)]. Accordingly, by (UG), axiom 7, T1 and quantifier logic, (∀x)[(∃y)(y ⊂ x) → (∃z1 Atom/z1 ∈ x)(∃z2 Atom/z2 ∈ x))(z1 = z2 )]. But, by quantifier logic and definition of Atom, (∀x)[¬(∃yAtom)(x = y) → (∃y)(y ⊂ x)], from which the left-right-direction of T26 follows. The converse direction is of course trivial for the reason already noted. 16 Proof. By (∃/UI), T2, and (LL∗ ), (∃y)(z = y) → [z ∈ x → (∃y/y ∈ x)(z = y)], and therefore, by T8 and T19, (∃y)(z = y) → (z ∈ x → z ∈ [ˆ z /(z ∈ x)]), and hence, by axiom 7, (∀z)(z ∈ x → z ∈ [ˆ z /(z ∈ x)]). For the converse direction, by T19, (LL∗ ), and T8, z ∈ [ˆ z /(z ∈ x)] → (∃y/y ∈ x)(z = y); and hence z ∈ [ˆ z /(z ∈ x)] → z ∈ x. Therefore, by (UG), x = [ˆ z /(z ∈ x)]. The proof that [ˆ xA] =ex [ˆ y A(y/x)] follows from the definition of ∈ and the rewrite rule for relative quantifiers, and T27b then follows by (ext).
11.3. THE UNIVERSAL CLASS
245
First, let us define the universal class in the usual way, i.e., as the extension of the common name ‘thing that is self-identical’, and then note that whether or not the name concept thing-that-is-self-identical, i.e., [ˆ x/(x = x)], can be “object”ified (as a value of the bound objectual variables), nevertheless, everything “real” (in the sense of being the value of a bound objectual variable) is in it. Definition 8 V = [ˆ x/(x = x)]. T29: (∀x)(x ∈ V).17 Note: all that T29 really says is that everything is a thing that is self-identical. Now, by definition of ∈, nothing can belong to the empty class, i.e., x ∈ / Λ, and therefore, by Leibniz’s law, if anything at all exists, the universal class is not the empty class. T30: (∃x)(x = x) → V = Λ. But it does not follow that the universal class “exists” if anything does. Indeed, as already noted above, we can show that if there are at least two atoms, then the universal class does not exist. First, let us note that if something exists (and hence, by T28, there is an atom), then the class of atoms exists, i.e., then the name concept Atom can be “object”-ified as a value of the bound objectual variables. T31: (∃x)(x = x) → (∃y)(y = Atom).18 On the other hand, let us also note that if there are at least two atoms, then the class of atoms is not itself an atom. T32: (∃xAtom)(∃yAtom)(x = y) → ¬(∃zAtom)(z = Atom).19 By means of T32, we can now show that if there are at least two atoms, then the universal class does not “exist” (as a value of the objectual variables). 17 Proof. By axiom 8, (∀x)(∃y)(x = y) ↔ (∀x)(∃y)(y = y ∧ x = y), and therefore, by T2, (∀x)(∃y)(x = y) ↔ (∀x)(∃y/y = y)(x = y), from which T29 follows by T8, T19 and the definition of V . 18 Proof. By axiom 15, (∃xAtom)(x = x) ∧ (∀xAtom)(∃yAtom)(x = y) → (∃y)(y = Atom), from which, by T28 and quantifier logic, T31 follows. 19 Proof. By definition of ∈, T8, and elementary transformations, x = y → x ∈ / [ˆ z /(z = y)] ∧ y ∈ / [ˆ z /(z = x)], and therefore, by T13 and (LL∗ ), (∃zAtom)(x = z) ∧ (∃zAtom)(y = z) ∧ (x = y) → x ∈ / y∧y ∈ / x. By T20a, (∃zAtom)(x = z) → x ⊆ Atom, and, by T19, (∃zAtom)(y = z) → y ∈ Atom. Therefore, by definition of ⊂, (∃zAtom)(x = z) ∧ (∃zAtom)(y = z)∧y ∈ / x → x ⊂ Atom, and hence (∃zAtom)(x = z)∧(∃zAtom)(y = z)∧(x = y) → x ⊂ Atom. But, by definition of Atom, (∀x)(∀y)[x ⊂ y → ¬(∃zAtom)(z = y)], and hence, by T31, T6, and (∃/UI), (∃zAtom)(x = z) ∧ x ⊂ Atom → ¬(∃zAtom)(z = Atom). Therefore, (∃xAtom)(∃yAtom)(x = y) → ¬(∃zAtom)(z = Atom).
246 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY T33: (∃xAtom)(∃yAtom)(x = y) → ¬(∃x)(x = V).20 Finally, in regard to the question of whether or not the universal class exists if the universe consists of just one atom, i.e., just one individual, note that if that were in fact the case, then, where A is a name of that one atom, the conjunction (∃zAtom)(z = A) ∧ (∀zAtom)(z = A) would be true, and therefore the one atom A would be extensionally identical with the class of atoms, i.e., then, by T31, T21, and (ext), (A = Atom) would be true as well. Now, by T29 and T19, (∀zAtom)[z ∈ Atom ↔ z ∈ V] is provable, which, by T21 might suggest that (Atom = V) and hence (A = V) are true as well. But in order for T21 to apply in this case we need to know that V “exists,” i.e., that (∃x)(x = V) is true. So, even if there were just one atom, we still could not conclude that the universal class is extensionally identical with that one atom.
11.4
Intersection, Union, and Complementation
Let us turn now to the Boolean operations of intersection, union and complementation for classes as many. We adopt the following standard definitions of each. Definition 9 x ∪ y = [ˆ z /z ∈ x ∨ z ∈ y]. Definition 10 x ∩ y = [ˆ z /(z ∈ x ∧ z ∈ y)]. Definition 11 x ¯ = [ˆ z /z ∈ / x]. The following theorems regarding membership in the union and intersection of classes are consequences of T19 and T8. The proof of the theorem regarding membership in the complement of a class is slightly more involved. T34: (∀z)(z ∈ x ∪ y ↔ z ∈ x ∨ z ∈ y). T35: (∀z)(z ∈ x ∩ y ↔ z ∈ x ∧ z ∈ y). T36: (∀z)(z ∈ x ¯↔z∈ / x).21 Two immediate consequences of T36 and (ext) (together with T12b and T29) are that the empty class is identical with the complement of the universal class, and that the universal class is identical with the complement of the empty class. ¯ T37: Λ = V. 20 Proof. Note that by T20a and (∃/UI), (∃x)(x = V ) → (∀x)[x ∈ V → (∃yAtom)(x = y)]. But, by axiom 8, (UG), and axioms 2 and 6, (∃x)(x = V ) → (∃x)(x = x), and hence, by T31 and (∃/UI), (∃x)(x = V ) → [Atom ∈ V → (∃yAtom)(y = Atom)]. But, by T31, T29, and (∃/UI), (∃x)(x = V ) → Atom ∈ V , and hence, (∃x)(x = V ) → (∃yAtom)(y = Atom). Accordingly, by T32, (∃xAtom)(∃yAtom)(x = y) → ¬(∃x)(x = V ). 21 Proof. By definition of ∈, z ∈ x ¯ ↔ (∃A)[¯ x = A ∧ (∃yA)(z = y)], and therefore, by (LL∗ ) and T8, z ∈ x ¯ → (∃y/y ∈ / x)(z = y), and hence, by T2 and (LL∗ ), z ∈ x ¯→z∈ / x. For the converse direction, note that by T2 and (LL∗ ), (∃y)(z = y) ∧ z ∈ / x → (∃y/y ∈ / x)(z = y), and therefore, by the definitions of ∈ and x ¯, (∃y)(z = y) → [z ∈ /x→z∈x ¯], and hence by (UG) axioms 2 and 7, and elementary logic, (∀z)(z ∈ /x→z∈x ¯).
11.4. INTERSECTION, UNION, AND COMPLEMENTATION
247
¯ T38: V = Λ. In regard to the conditions for the existence of unions and intersections, we first prove a theorem that is useful in their respective proofs. T39: (∀x)[(∃z)(z ∈ x) ∧ (∀z/z ∈ x)(∃wAtom)(z = w)].22 T40: (∀x)(∀y)(∃z)(z = x ∪ y).23 The related theorem for intersection requires a qualification, because some intersections—e.g., of distinct atoms—are empty, and, the empty class as many does not exist. Clearly, the relevant qualification is that the classes being intersected have a member in common. T41: (∀x)(∀y)[(∃z)(z ∈ x ∧ z ∈ y) → (∃z)(z = x ∩ y)].24 In regard to the existence of the complement of a class as many, we first note that if some atom is not in x, and therefore, by T36, is in x ¯, then the class as many of atoms in x ¯ exists, i.e., then [ˆ z Atom/(z ∈ x ¯)] exists (as a value of the bound objectual variables). This result cannot be shown for x ¯ alone, however, because, e.g., where x = Λ, then, by T38, x¯ = V, in which case x ¯ does not exist, or at least not if there exist two or more atoms. Also, in that case [ˆ z Atom/(z ∈ x ¯)] = Atom, and therefore, by T28, [ˆ z Atom/(z ∈ x ¯)] exists even though x ¯ does not. T42: (∃zAtom)(z ∈ / x) → (∃y)(y = [ˆ z Atom/(z ∈ x ¯)]).25 Note that we can show that an atom is in [ˆ z Atom/(z ∈ x ¯)] if, and only if, it is in x ¯, but we cannot use this result (T43 below) to prove that x ¯ exists if [ˆ z Atom/(z ∈ x ¯)] exists. In particular, we cannot use T21 to prove x ¯ = [ˆ z Atom/(z ∈ x ¯)] unless we already know that both classes exist. The following theorems indicate what does in fact hold about the complement of a given class x. T43: (∀zAtom)(z ∈ [ˆ z Atom/z ∈ / x] ↔ z ∈ x ¯).26 22 Proof.
By T6, (∃/UI), (LL∗ ), and T17, (∃zAtom)(x = z) → (∃z)(z ∈ x), and, by T26, (∀x)[¬(∃zAtom)(x = z) → (∃z)(z ∈ x)]; hence, (∀x)(∃z)(z ∈ x). But then T39 follows by T20a and quantifier logic. 23 Proof. By T39 (twice), (∀x)[(∃z)(z ∈ x) ∧ (∀z/z ∈ x)(∃wAtom)(z = w)] and (∀y)[(∃z)(z ∈ y) ∧ (∀z/z ∈ y)(∃wAtom)(z = w)], and therefore, by quantifier logic, (∀x)(∀y)[(∃z)(z ∈ x ∨ z ∈ y) ∧ (∀z/z ∈ x ∨ z ∈ y)(∃wAtom)(z = w)]. Accordingly, by T11, (∀x)(∀y)(∃z1 )(z1 = [ˆ z /(z ∈ x ∨ z ∈ y)]), from which T40 follows by definition of union. 24 Proof. By T39 (twice) and elementary logic, (∀x)(∀y)(∀z/z ∈ x∧z ∈ y)(∃wAtom)(z = w)], and therefore, by T11 and the definition of ∩, (∀x)(∀y)[(∃z/z ∈ x ∧ z ∈ y) → (∃z)(z = x ∩ y)]. 25 Proof. By axiom 15, (∃zAtom)(z ∈ x ¯) ∧ (∀zAtom/z ∈ x ¯)(∃wAtom)(z = w) → (∃y)(y = [ˆ z Atom/(z ∈ x ¯)]); but, by axiom 11 and quantifier logic, (∀zAtom/z ∈ x ¯)(∃wAtom)(z = w), and therefore, by T36, (∃zAtom)(z ∈ / x) → (∃y)(y = [ˆ z Atom/(z ∈ x ¯)]). 26 Proof. By T19, (∃yAtom/y ∈ / x)(z = y) → z ∈ [ˆ y Atom/y ∈ / x]; and, by T19 and T36, (∀zAtom)(z ∈ [ˆ y Atom/y ∈ / x] → z ∈ x ¯). For the converse direction, by T36 and T2, (∀zAtom)[z ∈ x ¯ → (∃yAtom/y ∈ / x)(z = y)], and therefore, by T19, (∀zAtom)(z ∈ x ¯→ z ∈ [ˆ y Atom/y ∈ / x]).
248 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY T44: (∃y)(y = [ˆ z Atom/z ∈ / x]) ∧ (∃y)(y = x¯) → [ˆ z Atom/z ∈ / x] = x¯.27 Finally, let us consider the situation when there are a countably infinite number of individuals, i.e., single objects, in the world. Does it then follow by Cantor’s power-set theorem that there are an uncountably infinite number of groups of individuals, i.e., of classes as many of more than one member? And if so, wouldn’t that show that there is more to the notion of a group than that of a simple plurality? That is, if out of ℵ0 many individuals we can obtain 2ℵ0 many groups of individuals, then doesn’t that show that there is something abstract about groups? Perhaps. But curiously, Cantor’s proof is not provable in the logic of classes as many when applied to the class of individuals, which suggest that there is nothing abstract about plural objects after all. Indeed, as the extensions of concepts that proper and common-name concepts stand for, it would be surprising if their cardinality were to exceed that of the concepts whose extensions they are.28 We should note that a set-theoretic semantics has been constructed for the logic of classes as many, and with respect to that semantics it has been shown that the logic is consistent.29 Metatheorem: The logic of classes as many as described here is consistent.
11.5
Le´ sniewskian Theses Revisited
As we explained in our previous lecture, Le´sniewski’s logic of names is reducible to our conceptualist logic of names. On our interpretation, the oddities of Le´sniewski’s logic are seen to be a result of his representing names, both proper and common, the way singular terms are represented in modern logic. The problem was not his view that proper and common names constitute together a syntactic category of their own, because that is how names are viewed in our conceptualist logic as well. But in our conceptualist logic proper and common names function as parts of quantifier phrases, i.e., expressions that stand for referential concepts in our analysis of the nexus of predication. But if Le´sniewski’s logic of names is reducible to our conceptualist logic of names, then might not the oddities that arise in Le´sniewski’s logic also arise 27 Proof. 28 Where
By T43, T21, (∃/UI), and (ext).
the number of individuals is finite and greater than 1, i.e., where there are n many single objects in the universe, for some positive integer n > 1, then, by a simple inductive argument it can be shown that the number of classes as many of objects, single and plural, is 2n − 1, and hence that there are 2n − (n + 1) plural objects. Of course 2n − 1 > n, where n > 1. But where the number of single objects is ℵ0 , we cannot show that the number of classes as many is 2ℵ0 − 1 (which is just 2ℵ0 ). The attempt to derive a contradiction by Cantor’s argument of assuming a 1-1 mapping f of all of the classes as many of individuals into the class of all individuals fails because the Cantor diagonal class of individuals x such that x ∈ / f (x) must be known to exist (or equivalently have a member) in order to derive a contradiction. If it exists, then a contradiction follows, and what this shows is that the Cantor class as many, like the Russell class as many, doesn’t exist in the logic of classes as many (which is “free” of existential presuppositions). 29 See Appendix 1.
´ 11.5. LESNIEWSKIAN THESES REVISITED
249
when names as parts of quantifier phrases are “nominalized” and occur as objectual terms in the logic of classes as many the way they occur in Le´sniewski’s logic of names? In other words, to what extent, if any, are there any theorems in our logic of classes as many that are counterparts of the theses of Le´sniewski’s logic that struck us as odd or noteworthy? Here, by a counterpart we mean a formula that results by replacing the names in a thesis of Le´sniewski’s logic by the “nominalized”, or transformed, names of our logic of classes as many, and also, of course, replacing Le´sniewski’s epsilon ‘ε’ by our epsilon ‘∈’. First, let us consider the validity of the principle of existential generalization in Le´sniewski’s logic, i.e., ϕ(c/a) → (∃a)ϕ(a). This principle is odd, we noted, when a is a vacuous name such as ‘Pegasus’, because in that case it follows from the fact that nothing is identical with Pegasus that something is identical with nothing, which is absurd. The counterpart of this thesis in our logic of classes as many is clearly invalid. For example, the empty class as many does not “exist” in our logic, but from that it does not follow that something exists that does not exist. Indeed, it is actually disprovable, as it should be. That is, the negation of ¬(∃x)(x = Λ) → (∃y)¬(∃x)(x = y) is provable in our logic of classes as many. Another thesis of Le´sniewski’s logic that is odd is the following: (∃a)(a = a). Now, by (UG) and axiom 8, the negation of this thesis, namely (∀x)(x = x), is a theorem in our logic of classes as many. But of course stating the matter this way assumes that identity in Le´sniewski’s logic means identity simpliciter, which it doesn’t. Identity is defined in Le´sniewski’s logic, in other words, and what the above thesis really means on Le´sniewski’s definition is the following: (∃a)¬(a ε a). Now the real counterpart of this thesis in our logic of classes as many is: (∃x)¬(x ∈ x). By quantifier negation, what this formula says is that not every object belongs to itself, which because all atoms belong to themselves, means that not every object is an atom. That is not a theorem of our logic, but it would be true if in fact there were at least two atoms, in which case there would then be a group, i.e., a class as many with more than one member, which, by definition, would not be an atom, and therefore, by T20b, not a member of anything, no less of itself. Thus, although the counterpart of the above Le´sniewskian thesis is not a theorem, nevertheless it is not disprovable, and in fact it is true if there are at least two atoms, i.e., individuals in the ontological sense.
250 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY In regard to the Le´sniewskian thesis, a ε b → a ε a. we note first that the counterpart of this formula, namely, z ∈ x → z ∈ z, is refutable if there is at least one plural object, i.e., one “real” object that is not an atom. This is because every “real” object is a member of the universal class (by T29 ), even though the universal class itself is not “real” if there are at least two atoms (T33). In other words, where z is a plural object, e.g., the class as many of citizens of Italy, then even though z is a member of the universal class, i.e., z ∈ V, nevertheless z ∈ / z. That is, because z is a plural object, it is not an atom, and therefore (by T20b) z is not a member anything. Here, it should be kept in mind that even though V is not a value of the bound objectual variables, it is nevertheless a substituend of the free objectual variables. Hence, where z is a plural object, the following instance of the above formula, z∈V→z∈z is false. There is a theorem that is somewhat similar to the above counterpart of Le´sniewski’s thesis, namely, (∃x)(z ∈ x) → z ∈ z. In other words, if z belongs to something “real”, i.e., a value of the bound objectual variables, then z is an atom (by T20a) and therefore z belongs to itself (by T17). This theorem is similar to, but still not the same as, the Le´sniewskian thesis. Another theorem that is similar to, but not the same as, a thesis of Le´sniewski’s logic, is: (∀y)(∀z)[x ∈ y ∧ y ∈ z → x ∈ z]. This formula is provable because if x belongs to a “real” object y and y belongs to a “real” object z, then both x and y must be atoms (by T20a), in which case, y = [w/w ˆ = y] (by T13); and hence x = y (because x ∈ y), and therefore x ∈ z (because y ∈ z). This theorem is similar to the Le´sniewskian thesis, a ε b ∧ b ε c → a ε c, but, again, the strict counterpart of this Le´sniewskian thesis, namely, x∈y∧y ∈z →x∈z is not provable in our logic, and is refutable if there are at least two atoms. Thus, if there are two “real” atoms a and b, then y = [w/(w ˆ = a ∨ w = b)] is also “real” (by axiom 15, or T40). But then y ∈ [w/(w ˆ = y)] (by T18, even though y = [w/(w ˆ = y)]), and hence we would have a ∈ y and y ∈ [w/(w ˆ = y)], and yet a ∈ / [w/(w ˆ = y)], because a = b, and hence a = y.30 30 It
should be kept in mind that only real atoms are identical with their singletons.
11.6. GROUPS AND THE SEMANTICS OF PLURALS
11.6
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Groups and the Semantics of Plurals
One way in which the notion of a group is important is its use in determining the truth conditions of sentences that are irreducibly plural, i.e., sentences not logically equivalent to sentences that can be expressed without a plural reference to a group or plural predication about a group. An example of such a sentence is the so-called Geach-Kaplan sentence, ‘Some critics admire only each other’, or, equivalently, ‘Some critics are such that each of them admires only others of them’. Now the plural reference in this sentence is not just plural but irreducibly plural, and it cannot be logically analyzed by quantifying just over critics. The reference in this case is really to a group of critics, i.e., a class as many of critics having more than one member. The reference, moreover, is not to a set of critics, i.e., to an abstract object that is not itself a part of the physical world, but to a group of critics that is no less a part of the physical world than are the critics in the group. The difference between the group and its members is that the group, as a plural object, is ontologically founded upon its members as single objects. The reference, moreover, is not to just any class as many of critics, and in particular not to any class as many that consists of just one member. A single critic who admires no one would in effect be a class as many of critics having exactly one member, and every member of this class would vacuously satisfies the condition that he admires only other members of the class. But it is counterintuitive to claim that the sentence ‘Some critics admire only each other’ could be true only because there is a critic who admires no one. The sentence is true if, and only if, there is a group of critics every member of which admires only other members of the group. In order to formulate this sentence properly we need first to define the notion of a group, and then note that, by definition, a group will have at least two members, and hence that a group is a plural object. Definition 12 Grp = [ˆ x/(∃y)(y ⊂ x)]. T45: (∀xGrp)(∃z1 /z1 ∈ x)(∃z2 /z2 ∈ x)(z1 = z2 ). We can now represent the semantics of the sentence ‘Some critics admire only each other’ in terms of a group of critics instead of just a class as many of critics. This can be formulated as follows: [Some critics]N P [admire only each other]V P (∃xGrp/x ⊆ [ˆ y Critic]) (∀y/y ∈ x)(∀z)[Admire(y, z) → z ∈ x ∧ z = y] (∃xGrp/x ⊆ [ˆ y Critic])(∀y/y ∈ x)(∀z)[Admire(y, z) → z ∈ x ∧ z = y]. Another example of an irreducibly plural reference is ‘Some people are playing cards’, where by ‘some people’ we do not mean that at least one person
252 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY is playing cards, but that a group of people are playing cards, and that they are doing it together and not separately. The truth conditions of this sentence can be represented as follows where the argument of the predicate is irreducibly plural. [Some people]N P [are playing cards]V P ↓ ↓ (∃xGrp/x ⊆ [ˆ y P erson]) P laying-Cards(x) ↓ ↓ (∃xGrp/x ⊆ [ˆ y P erson])P laying-Cards(x) Of course, in saying that a group of people are playing cards we mean that each member of the group is playing cards, but also that the members of the group are playing cards with every other member of the group. That is, (∀xGrp/x ⊆ [ˆ yP erson])[P laying-Cards(x) → (∀y/y ∈ x)P laying-Cards(y)] and (∀xGrp/x ⊆ [ˆ y P erson])[P laying-Cards(x) → (∀y/y ∈ x)(∀z/z ∈ x/z = y)P laying-Cards-with(y, z)] are understood to be consequences of what is meant in saying that a group is playing cards. We understand the preposition ‘with’ in this last formula, to be an operator that modifies a monadic predicate and generates a binary predicate by adding one new argument position to the predicate being modified. Thus, applying this modifier to ‘x is playing cards’ we get ‘x is playing cards with y’. The added part, ‘with y’, represents a prepositional phrase of English.31 The converse, however, does not follow in either case. That is, we could have every member of a group playing cards without the group playing cards together, and we could even have every member playing cards with every other member in separate games without all of them playing cards together in a single game. Another type of referential expression that is irreducibly plural is the plural use of ‘the’, as in ‘the inhabitants of Rome’ and ‘the Greeks who fought at Thermopylae’. On our reading these expressions are to be taken as referring to the inhabitants of Rome as a group and similarly to the Greeks who fought at Thermopylae as a group. In this way the plural use of ‘the’ can be reduced to the singular ‘the’, i.e., to a definite description of a group. The singular ‘the’, as we described it in our fifth lecture, is represented by a quantifier (as are all determiners), in particular, ∃1 , where the truth conditions of an assertion of the form ‘The A is F ’ are spelled out in essentially the Russellian manner (when the definite description is used with existential presupposition). Consider now the sentence ‘The Greeks who fought at Thermopylae are heroes’, which we take to be equivalent to ‘The group of Greeks who fought 31 Note that as described here ‘with’ cannot be iterated so as to result in a three-place predicate. We assume, in this regard, that ‘playing cards with-with’ is not grammatical or logically well-formed.
11.6. GROUPS AND THE SEMANTICS OF PLURALS
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at Thermopylae are heroes’. Using F (x) for the verb phrases ‘x fought at Thermopylae’, we can semantically represent the sentence as follows: [The Greeks who fought at Thermopylae]N P [are heroes]V P [The group of Greeks who fought at Thermopylae]N P [are heroes]V P ↓ ↓ (∃1 xGrp/x = [ˆ xGreek/F (x)]) (∀y/y ∈ x)(∃zHero)(y = z) (∃1 xGrp/x = [ˆ xGreek/F (x)])[λx(∀y/y ∈ x)(∃zHero)(y = z)](x) The truth conditions of this sentence amount to there being (now, at the time of the assertion) exactly one group of Greeks who fought at Thermopylae and every member of that group is a hero, which captures the intended content of the sentence in question. We might also note that another standard formulation of the English sentence, namely, that the class as many of Greeks who fought at Thermopylae is contained within the class as many of heroes, [ˆ xGreek/F (x)] ⊆ [ˆ xHero], is a consequence of the above formulation; and, in fact, if it is assumed that [ˆ xGreek/F (x)] has at least two members and that each of its members is an atom, i.e., an individual, which in fact is the case, then the two formulations are equivalent to one another. Another type of example is plural identity, as in: Russell and Whitehead are the coauthors of PM. Here, reference is to the group consisting of Russell and Whitehead, and what is predicated of this group is that it is identical with the group consisting of those who coauthored PM (Principia Mathematica). In other words, where ‘A’ and ‘B’ are name constants for ‘Russell’ and ‘Whitehead’, a plural subject of the form ‘A and B’ is analyzed as follows: A and B ↓ The group consisting of A and B ↓ (∃1 xGrp/x = [ˆ z /(z = A ∨ z = B) Similarly, the analysis of the phrase ‘the coauthors of PM’ is to be analyzed as follows: the coauthors of PM ↓ The group of those who coauthored PM ↓ (∃1 yGrp/y = [ˆ z /Coauthored(z, P M )])
254 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY The plural identity of the two groups can then be symbolized as, z /(z = A∨z = B)(∃1 yGrp/y = [ˆ z /Coauthored(z, P M )])(x = y), (∃1 xGrp/x = [ˆ where it is the identity of two groups that is explicitly stated in the identity predicate. A similar analysis applies to the sentence The triangles that have equal sides are the triangles that have equal angles. That is, where ‘A’ is a name constant for ‘triangle’ and ‘F ’ and ‘G’ are one-place predicates for ‘has equal sides’ and ‘has equal angles’, respectively, then the two plural definite descriptions can be represented as: The triangles that have equal sides ↓ The group of triangles that have equal sides ↓ (∃1 xGrp/x = [ˆ z A/F (z)]) and with a similar analysis for ‘the group of triangles that have equal angles’, the plural identity of the two groups can be symbolized as: z A/F (z)])(∃1 yGrp/y = [ˆ z A/G(x)])(x = y), (∃1 xGrp/x = [ˆ where, again, it is the identity of two groups that is stated in the identity predicate. We should note, however, that given the axiom of extensionality, this sentence is provably equivalent to A triangle has equal sides if, and only if, it has equal angles. which can be symbolized as: (∀xA)[F (x) ↔ G(x)]. In other words, strictly speaking, the truth conditions of this sentence does not involve an irreducibly plural reference to, or predication of, groups. An example is an irreducibly plural predication is one where we predicate cardinal numbers of a group, as when we say that the Apostles are twelve. Here, the plural definite description, ‘the Apostles’ is understood to refer to the Apostles as a group, which means that we can symbolize the plural description as follows: The Apostles ↓ The group of Apostles ↓ (∃1 xGrp/x = [ˆ xApostle]) What is predicated of this group is that it has twelve members. The verb phrase ‘x has twelve members’ can be symbolized as a complex predicate as follows,
11.7. PLURAL REFERENCE AND PREDICATION
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x has twelve members ↓ [λx(∃12 y)(y ∈ x)](x) As is well-known, the numerical quantifier ∃12 is definable in first-order logic with identity, which we will not go into here.32 The important point is that this is really a plural predicate, i.e., it can be truthfully applied only to a plurality, namely a group with twelve members. The whole sentence can then be analyzed as follows: [The Apostles]N P [are twelve]V P (∃1 xGrp/x = [ˆ xApostle]) [λx(∃12 y)(y ∈ x)](x) (∃1 xGrp/x = [ˆ xApostle])[λx(∃12 y)(y ∈ x)](x) or, by λ-conversion, more simply as (∃1 xGrp/x = [ˆ xApostle])(∃12 y)(y ∈ x).
11.7
Plural Reference and Predication
The logical analyses of plural reference and predication that we have described so far are primarily analyses of the truth conditions, i.e., the semantics, of plural reference and predication. They are not analyses of the cognitive structure of plural reference and predication as part of our speech and mental acts. The question is how can we account for the cognitive structure of plural reference and predication in terms of the logical forms that we use to represent our speech and mental acts. What we propose is to formalize the pluralization of both common names and monadic predicates. We do this by means of an operator that when applied to a name results in the plural form of that name, and similarly when applied to a monadic predicate results in the plural form of that predicate. We will use the letter ‘P ’ as the symbol for this plural operator and we will represent its application to a name A or predicate F by placing the letter ‘P ’ as a superscript of the name or predicate, as in AP and F P . Thus, we now extend the simultaneous inductive definition of the meaningful (well-formed) expressions of our conceptualist framework to include the following clauses: 1. if A is a name variable or constant, then AP is a plural name variable or constant ; respectively; 32 See
chapter 7, §9, for an analysis of numerical quantifiers.
256 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY 2. if A is a name, x is an object variable, and ϕx is a formula, then [ˆ xA/ϕx]P P and [ˆ x/ϕx] are plural names; 3. if A/ϕ(x) is a (complex) name, then (A/ϕ(x))P = AP /[λxϕ(x)]P (x) and [ˆ xA/ϕ(x)]P = [ˆ xAP /[λxϕ(x)]P (x)]; 4. if F is a one-place predicate variable or constant, or of the form [λxϕ(x)], then F P is a one-place plural predicate; 5. if AP is a plural name, x is an object variable, and ϕ is a formula, then (∀xAP )ϕ and (∃xAP )ϕ are formulas. In regard to clause (5), we read, e.g., ‘(∀xM anP )’ as the plural phrase ‘all men’ and ‘(∃xM anP )’ as the plural phrase ‘some men’, and similarly ‘(∀xDog P )’ as ‘all dogs’ and ‘(∃xDog P )’ as ‘some dogs’, etc. We note that a plural name is not a name simpliciter and that unlike the latter there is no rule for the “nominalization” of plural names, i.e., their transformation into objectual terms. This is because a nominalized name (occurring as an argument of a predicate) can already be read as plural if its extension is plural, and we do not want to confuse and identify a name simpliciter with its plural form. Note also that only monadic predicates are pluralized. A two-place relation R can be pluralized in either its first- or second-argument position, or even both, by using a λ-abstract, as, e.g., [λxR(x, y)]P , [λyR(x, y)]P , [λx[λy[R(x, y)]P (y)](x)]P , respectively; and a similar observation applies to n-place predicates for n > 2. Thus, for example, we can represent an assertion of ‘Some people are playing cards with Sofia’ by pluralizing the first-argument position of the two-place predicate ‘x is playing cards with y’ as follows: [Some people]N P [are playing cards with Sofia]V P ↓ ↓ (∃xP ersonP )[λxP laying-Cards-with(x, Sof ia)]P (x) Semantically, of course, we understand the plural reference in this assertion to be to a group of people, a fact that is made explicit by assuming the following as a meaning postulate for all (nonplural) names A whether simple or complex: (∃xAP )ϕ(x) ↔ (∃xGrp/x ⊆ [ˆ y A])ϕ(x).
(MPP1)
Of course, if a group of people are playing cards with Sofia, then it follows that each person in the group is playing cards with Sofia, though, as already noted, the converse does not also hold. The one-direction implication from the plural
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to the singular can be described by assuming the following as part of the way that the monadic-predicate modifier ‘with’ operates33 : (∀x)([λxP laying-Cards-with(x, Sof ia)]P (x) → (∀y/y ∈ x)[λxP laying-Cards-with(x, Sof ia)](y)). An example where the second argument of a relation is plural is a consequence of sentence ‘Some men carry the piano downstairs’, i.e., where the consequence is that each man in the group (qua individual) carries the piano downstairs with the other men in the group (qua group or plural object). First, where F (x) is read as ‘x carries the piano downstairs’, we note that the sentence ‘Some men carry the piano downstairs’ can be analyzed as, [Some men]N P [carry the piano downstairs]V P (∃xM anP ) F P (x) (∃xM anP )F P (x) which, by the above meaning postulate for plurals, (MPP1), means that some group of men carry the piano downstairs. The consequence then is that some group of men is such that every man in the group carries the piano downstairs with the other men in the group. To analyze this, we need to represent what it means to refer to the men in the group other than a given man. For this we use the plural definite description, ‘the men in x other than z’, which can be symbolized as follows, the men in x other than z ↓ (∃1 y(M an/(y ∈ x ∧ y = z))P ) Finally, that there is a group of men such that every man in the group carries the piano downstairs with the other men in the group can now be represented as follows: (∃xGrp/x ⊆ [ˆ y M an])(∀z/z ∈ x)(∃1 y(M an/(y ∈ x ∧ y = z))P )[λyF -with(z, y)]P (y),
which, by (MPP1) as applied to (∃1 y(M an/(y ∈ x ∧ y = z))P ), reduces to (∃xGrp/x ⊆ [ˆ y M an])(∀z/z ∈ x)(∃1 yGrp/y = [ˆ y M an/y ∈ x ∧ y = z])[λyF -with(z, y)]P (y),
where the relation ‘z carries the piano downstairs with y’ is taken as plural in its second-argument position. Finally, let us turn to how the universal plural ‘All A’, the cognitive structure of which is represented by (∀xAP ), is to be semantically analyzed. Let us note first that if (∀xAP ) were taken as the logical dual of (∃xAP ), the way (∀x) 33 Note that because everything is a class as many, this condition applies even when x is an atom. In that case, of course, the condition is redundant.
258 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY is dual to (∃x), then the postulate for universal plural reference would be as follows: y A])ϕ(x). (MPP?) (∀xAP )ϕ(x) ↔ (∀xGrp/x ⊆ [ˆ Then, given that the cognitive structure of an assertion of ‘All men are mortal’ can be represented as, (∀xM anP )M ortalP (x), it would follow that, semantically, the assertion amounts to predicating mortality to every group of men, (∀xGrp/x ⊆ [ˆ yM an])M ortalP (x), which is equivalent to saying without existential presupposition that the members of the entire group of men taken collectively are mortal: (∀xGrp/x = [ˆ yM an])M ortal P (x). This formula, given that the class as many of men is in fact a group—i.e., has more than one member—is in conceptualist terms very close to what Russell claimed in his 1903 Principles, namely, that the denoting phrase ‘All men’ in the sentence ‘All men are mortal’ denotes the class as many of men, which in fact happens to be a group. But what if the class of men were to consist of exactly one man, as, e.g., at the time in the story of Genesis when Adam was first created. Presumably, the sentence ‘All men are mortal’ is true at the time in question. But is it a vacuous truth? In other words, is it true only because there is no group of men at that time but only a class as many of men having just one member? Similarly, consider the sentence ‘All moons of the earth are made of green cheese’. Presumably, this sentence is false and not vacuously true because there is no group of moons of earth but only a class as many with one member. In other words, regardless of the implicit duality of ‘All A’ with the plural ‘Some AP ’, we cannot accept the above rule (MPP?) as a meaning postulate for sentences of the form (∀xAP )ϕ(x). Yet, there is something to Russell’s claim that the phrase ‘All men’ in the sentence ‘All men are mortal’ denotes the class as many of men and differs in this regard from what ‘Every man’ denotes in ‘Every man is mortal’. In conceptualist terms, in other words, the referential concept that ‘Every man’ stands for is not the same as the referential concept that ‘All men’ stands for; nor is the predicable concept that ‘is mortal’ stands for the same as the predicable concept that ‘are mortal’ stands for.34 A judgment that all men are mortal has a different cognitive structure from a judgement that every man is mortal, even if semantically they have the same truth conditions. It is the difference in referential and predicable concepts—i.e., the difference between (∀xM anP ) and 34 This difference in predicable concepts was missed by Russell and explains why his later rejection of classes as many as what ‘All men’ denotes was based on a confusion between singular and plural predication. See Russell 1903, p. 70.
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(∀xM an), on the one hand, and M ortalP ( ) and M ortal( ) on the other—that explains why the judgments are different. Now the point of these observations is that instead of (MPP?) we can represent the difference between ‘All Ap ’ and ‘Every A’ by adopting the following meaning postulate, which takes ‘All AP ’ to refer not just to every group of A but to every class as many of A, whereas ‘Every A’ refers to each and every A taken singly: (∀xAP )ϕ(x) ↔ (∀x/x ⊆ [ˆ y A])ϕ(x).
(MPP2)
Despite this difference, however, it follows that ‘Every A is F ’ is logically equivalent to ‘All A are F ’, i.e., (∀xA)F (x) ↔ (∀xAp )F p (x) is valid in our conceptualist logic. Let us note, incidentally, that the plural verb phrase ‘are mortal’, in symbols, M ortalP , is semantically reducible to its singular form. That is, mortality can be predicated in the plural of a class as many if, and only if, every member of the class is mortal. In other words, as part of the meaning of the predicate ‘mortal’ we have the following as a meaning postulate: M ortalP (x) ↔ (∀y/y ∈ x)M ortal(y). It would be convenient, no doubt, if every plural predicate were reducible to its singular form the way M ortalP is, but that is not the case, as we noted earlier with the predicate [λxP laying-Cards-with(x, Sof ia)]P , which is plural in its first argument position. Nor is it true of the complex predicate for carrying the piano downstairs with the other members of a group, which is plural in its second argument position. The fact is that just as some references are irreducibly plural, so too some predications are irreducibly plural. Even though plural objects, i.e., groups, are ontologically founded upon the single objects that are their members, nevertheless plural objects are an irreducible part of the world as much as are single objects, i.e., individuals. What is needed for both our scientific and our commonsense frameworks is a logic that can account for plural objects and plural predication, whether in thought or in the world, no less so than it can account for single objects and singular predication. That is the logic we have presented here as a special part of the more general framework of conceptual realism.
11.8
Cardinal Numbers and Plural Quantifiers
Our earlier analysis of the sentence ‘The Apostles are twelve’ took it to have the same truth conditions as ‘The group of Apostles has twelve members’. This analysis is not quite adequate, however, in that it depends on paraphrase and does not account for the logical connection between ‘The Apostles are twelve’ and ‘The group of Apostles has twelve members’. As part of this account,
260 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY moreover, we need to explain the source of the plural ‘are’ in ‘The Apostles are twelve’. Now it is clear that the plural verb phrase ‘are twelve’ is not the plural form of the singular verb phrase ‘is twelve’, the way the phrase ‘are Apostles’ is the plural of ‘is an Apostle’, and hence that its source cannot be explained as a simple pluralization of the verb. The explanation that we propose here depends on the claim that the most basic way in which we speak and think about different numbers of things is in our use of numerical quantifier phrases, as when we say that there are twelve Apostles. This statement, we claim, can be grammatically analyzed as [twelve Apostles]N P [there are]V P , where the noun phrase ‘twelve Apostles’ is taken as the subject of the sentence, and ‘there are’, which is the plural of ‘there is’, is taken as the verb phrase. It is our contention that the source of the plural ‘are’ in ‘The Apostles are twelve’ is the ‘are’ in ‘There are twelve Apostles’, which we earlier paraphrased as ‘The group of Apostles has twelve members’. The question now is how do we logically analyze the plural predicate ‘there are’ and then logically connect the three different sentences: There are twelve Apostles, The group of Apostles has twelve members, The Apostles are twelve’. The answer, it turns out, provides another and, in our view, a better way to represent cardinal numbers than that already given in chapter six. How then we are to analyze the quantifier phrase ‘there are’ when it functions as a plural predicate? Consider, as an example, the sentence, ‘There are liberal republican senators’, which we take to be synonymous with ‘There are senators who are republican and liberal’. This sentence, let us note, is the plural form of ‘There is a senator who is republican and liberal’, which can be grammatically structured as, [a senator who is republican and liberal]N P [there is]V P , and the plural form of the sentence can be similarly grammatically structured as, [senators who are republican and liberal]N P [there are]V P . Now the logical analysis of the above singular noun phrase is, a senator who is republican and liberal ↓ (∃xSenator/Republican(x) ∧ Liberal(x)), and the logical analysis of the singular verb phrase as a predicate is
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there is ↓ [λx(∃y)(x = y)], the infinitive of which can be read as ‘to be an x such that there is something x is’. The singular form of the sentence can accordingly be formalized as: (∃xSenator/Republican(x) ∧ Liberal(x))[λx(∃y)(x = y)](x). Now what we propose is that the logical form assigned to the plural form of the sentence should be derived from the singular form by having both the referential and the predicate expressions pluralized: (∃xSenator/Republican(x) ∧ Liberal(x))P [λx(∃y)(x = y)]P (x), which, by (MP1), gives us (∃xGrp/x ⊆ [ˆ y Senator/Republican(y) ∧ Liberal(y)])[λx(∃y)(x = y)]P (x) as its semantic representation for the plural reference. In regard to the semantics of the plural predicate, we assume that a group has plural being in the sense of ‘there are’ if, and only if, each of its members has being in the sense of ‘there is’. That is, the being of a plurality reduces to—i.e., is equivalent to—the being of each of the members of that plurality: (∀xGrp)([λx(∃y)(x = y)]P (x) ↔ (∀z/z ∈ x)[λx(∃y)(x = y)](z)). The right-hand side of this biconditional is logically true and is a consequence of the free-logic axiom (∀z)(∃y)(z = y), which means that the left-hand side of the biconditional is equivalent to ‘x = x’. This means that the above formula is equivalent to (∃xGrp/x ⊆ [ˆ y Senator/Republican(y) ∧ Liberal(y)])(x = x), and hence, by the exportation thesis for complex quantifier phrases and deletion of the redundant identity conjunct, ‘x = x’, equivalent to (∃xGrp)(x ⊆ [ˆ ySenator/Republican(y) ∧ Liberal(y)]). In other words, the plural form of the sentence semantically amounts to saying that there is a group of senators who are republican and liberal, which, intuitively, is exactly what we understand the initial statement to say. Our general proposal is that other sentences with a plural ‘there are’ should be analyzed in a similar way, i.e., as being represented by the plural predicate [λx(∃y)(x = y)]P . In particular, to return to the issue in question, the sentence ‘There are twelve Apostles’, which grammatically and then logically can analyzed as follows:
262 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY There are twelve Apostles ↓ [twelve Apostles]N P [there are]V P ↓ ↓ [λx(∃y)(x = y)]P (x) (∃12 xApostles) ↓ ↓ (∃12 xApostles)[λx(∃y)(x = y)]P (x). This last formula, as noted above about the plural predicate, reduces to (∃12 xApostle)(x = x), as well as to
(∃12 x)(∃yApostle)(x = y),
which says that twelve things are Apostles, which is another way of saying that there are twelve Apostles.35 Now as we suggested in chapter seven the most basic way in which we speak and think about different numbers of things is in our use of numerical quantifier phrases, as when we say that there are two authors of PM, three people playing cards, twelve Apostles, etc. These quantifier phrases can be nominalized in conceptual realism the way that we earlier nominalized the phrase, ‘a unicorn’, which means that they are first transformed into a (complex) predicate, which in turn is then nominalized and transformed into an abstract singular term, the denotatum of which is an intensional object. This kind of transformation is based, as we explained in chapter six, on a pattern of double abstraction corresponding to Frege’s double-correlation thesis—except that Frege attached only the common name ‘object’ to his quantifiers and took other common names to be predicates. Frege, moreover, being an extensionalist, did not identify the objects correlated with his quantifiers as intensional objects, but as value-ranges (Wertverl¨ aufe). The general idea in any case is that where ∃k is a numerical quantifier, which when applied to a name A is read as ‘there are k many A’, the double-correlation thesis can be formulated as follows, (∃F )(∀A)(∀G)[(∃k xA)G(x) ↔ F ([ˆ xA/G(x)])], where instead of a value-range as the argument of the concept F corresponding to the numerical quantifier ∃k as Frege would have it, we have a class as many. In particular, taking the numerical quantifier ∃12 for ∃k , we have (∃F )(∀A)(∀G)[(∃12 xA)G(x) ↔ F ([ˆ xA/G(x)])]. Now, clearly the concept F that is posited here as provably such that 35 The application of ∃12 to the singular ‘Apostle’ is appropriate here because it says in effect that there are twelve individual Apostles, i.e., twelve individuals named by ‘Apostle’. If we used the plural, ‘ApostlesP ’, the result would amount to saying that there are twelve groups of Apostles, which is not what is intended.
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for all name concepts A and all predicable concepts G, there are twelve A that are G if, and only if, the group of A that are G falls under F is none other than the predicable concept of having twelve members, which we symbolized earlier as [λx(∃12 y)(y ∈ x)]. In other words, by the above doublecorrelation thesis we have (∀A)(∀G)[(∃12 xA)G(x) ↔ [λx(∃12 y)(y ∈ x)]([ˆ xA/G(x)])], from which, by substituting ‘Apostle’ for ‘A’ and ‘x = x’ for ‘G(x)’, and applying λ-conversion, we have (∃12 xApostle)(x = x) ↔ (∃12 y)(y ∈ [ˆ xApostle/(x = x)]), and therefore by canceling the redundant identity in [ˆ xApostle/(x = x)], xApostle]). (∃12 xApostle)(x = x) ↔ (∃12 y)(y ∈ [ˆ But, as noted above, the left-hand side of this biconditional is equivalent to our formulation of ‘There are twelve Apostles’, namely, (∃12 xApostle)[λx(∃y)(x = y)]P (x), which means that the biconditional is equivalent to 12
P
12
xApostle]). (∃ xApostle)[λx(∃y)(x = y)] (x) ↔ (∃ y)(y ∈ [ˆ Now the definite description ‘the group of Apostles’, as symbolized by (∃1 yGrp/y = [ˆ xApostle]), denotes the same group as is denoted by the abstract [ˆ xApostle], i.e., (∃1 zGrp/z = [ˆ xApostle])(z = [ˆ xApostle]) is true and provable. From this identity and the last biconditional above it follows that the cognitive structure of the statement that there are twelve Apostles is logically equivalent to that of the statement that the group of Apostles has twelve members, i.e., 12
P
12
(∃ xApostle)[λx(∃y)(x = y)] (x) ↔ (∃1 zGrp/z = [ˆ xApostle])(∃ y)(y ∈ z) is provable, which is one of the logical connections we wanted to establish. The predicable concept that we predicate in ‘The Apostles are twelve’, accordingly, is equivalent to the concept of having twelve members, a concept that every group with twelve members falls under. Thus, as a predicable concept, a cardinal number K can be defined as the predicable concept that those groups that have k many members fall under, and hence the cardinal number k itself can be identified with the “object-ified” correlate of the concept K. In other words, starting with the quantifier notion expressed by ‘There are k many A’, we obtain the predicable concept K under which a group falls if, and only if, it has k many members, and then, by “object-ifying”, or ”nominalizing”, the predicable concept K, we obtain the number k as the abstract object correlated
264 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY with the concept. It is in this way, by going through a double abstraction from a referential concept to a corresponding predicable concept and then, by the process of nominalization, to an object that we are able to grasp and understand the role of numbers as abstract objects. In particular, the predicable concept 12 can now be defined as the concept under which an object falls if, and only if, that object is a group having twelve members: 12 = [λy(∃A)(y = A ∧ (∃12 xA)(x ∈ y))]. Qua object, the number 12 is then the objectified correlate of this predicable concept. It is in this way that we are able to logically connect the sentence ‘The Apostles are twelve’ with ‘The group of Apostles has twelve members’. That is, we can now be symbolize ‘The Apostles are twelve’ as (∃1 xApostlesP )12(x), which, by (MPP1) reduces to y Apostle])12(x), (∃1 xGrp/x = [ˆ which, by definition, λ-conversion and identity logic, reduces to y Apostle])[λx(∃12 y)(y ∈ x)](x). (∃1 xGrp/x = [ˆ Thus, the three sentences, ‘The Apostles are twelve’, ‘The group of Apostles has twelve members’, and ‘There are twelve Apostles’ are each seen to be logically equivalent to one another. Each, moreover, is provably equivalent to the following simplest version of all: 12([ˆ xApostle]).
11.9
Summary and Concluding Remarks
• The logic of classes as many is an extension of the simple logic of names formalized in chapter ten. • Names, both proper and common, are transformed (or “nominalized”) in this logic into objectual terms, both simple and complex. • Russell’s paradox for classes fails in the logic of classes as many. In particular, the Russell class as many does not “exist” as an object, i.e., as a value of the bound objectual variables of this logic. • A particular axiom (namely, Axiom 15) of the logic of classes as many, specifies when and only when a name concept A has an extension that can be “object”-ified (as a value of the bound objectual variables). • Nelson Goodman’s notion of an “atom” is adopted as a way to distinguish single objects from plural objects. Atoms, i.e., single objects, are “individuals” in the traditional ontological sense, as opposed to plural objects, i.e., classes as many of two or more members.
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• The empty class as many does not “exist” as a value of the bound objectual variables in this logic (T12). • It is provable in the logic of classes as many that each atom is identical with its singleton (T13). • The logical role of “nominalized” proper names in the logic of classes as many is the same as the role of proper names in free first-order logic. Thus, the role proper names have as “singular terms” in standard free logic is reducible to, and fully explainable in terms of, the role proper names have in the logic of classes as many, and therefore in our conceptualist theory of reference. • Although the full, unrestricted version of Leibniz’s law applies to all atoms, i.e., single objects, or individuals in the ontological sense, only a version restricted to extensional contexts applies to groups, i.e., classes as many of two or more objects. This distinction between how Leibniz’s law applies to atoms and how it applies to groups is an ontological feature that distinguishes the individuality of atoms from the plurality of groups. • Nelson Goodman’s nominalistic dictum that things are identical if they have the same atoms is provable in the logic of classes as many. • If either there are no atoms or there are at least two atoms, then the universal class does not “exist” (as a value of the objectual variables). It is indeterminate whether or not the universal class exists if there is just one atom. • Cantor’s power-set theorem is provable only for finite classes in the logic of classes as many. • The oddities of Le´sniewski’s logic of names do not also hold in the logic of classes as many. • Groups are important in determining the truth conditions of sentences that are irreducibly plural. • The plural use of ‘the’ as in ‘the Greeks who fought at Thermopylae’ can be reduced to the singular ‘the’ as a definite description of a group. • Predication of cardinal numbers, as in ‘the Apostles are twelve’ can be analyzed in a natural way in terms of groups in this logic. • The logical analyses of plural reference and predication in terms of groups in the logic of classes as many amounts only to an analysis of the truth conditions of plural reference and predication, and not also to an analysis of the cognitive structure of plural reference and predication in our speech and mental acts. • The logic of classes as many is extended to included a plural operator that can be applied to names or to predicate expressions, and an analysis of the cognitive structure of plural reference and predication is given in terms of the plural operator. • The predicable concept expresssed in predicating a cardinal number n can be identified with the concept under which all and only groups with n members fall. The intensional object that results from the nominalization of this concept can be identified with the number n as an abstract object.
266 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY
11.10
Appendix 1: A Set-Theoretic Semantics
A set-theoretical semantics can be constructed for the logic of classes as many as formalized here, and the system can be shown to be consistent with respect to that construction. We will forego the proofs in what follows and just sketch out the semantics for a “standard” model of the system without indices (possible worlds, moments of time, contexts of use, etc.), and therefore one in which the axiom of extensionality is valid. Extending this semantics to one that includes indices, and hence to one in which the axiom of extensionality is not valid, is unproblematic and can be done in the usual way. By an (object) language we understand a (possibly empty) set of predicate and nominal constants. We will take ‘∈’, ‘⊆’, ‘⊂’, and ‘Atom’ defined in the logic of classes as primitive logical constants with their definitions as additional axioms of the logic. This means that the notion of a formula must be extended to include atomic formulas consisting of n-place predicate constants applied to n many singular terms, for n ∈ ω.36 Definition 13 L is a language iff L is a countable set of nominal constants (proper and common names) and predicate constants. By a set of “atoms” we understand a set that does not have the empty set as a member and no member of which has a member in common with that set. The idea is that “atoms” are to function as urelements. Thus, where D is any nonempty set, the set {d, D : d ∈ D} is a set of atoms even if D is not. Definition 14 D is a set of “atoms” iff D is a set such that 0 ∈ / D and for each d ∈ D, d ∩ D = 0. A “standard” model for a language consists of a set, possibly empty, of “atoms”, i.e., objects considered as urelements with respect to the various sets that make up the model, and an assignment of extensions to the predicate and nominal constants. The assignment to the constants is not drawn just from the set D of atoms, however, but from an extended set D+ defined as follows. Definition 15 If D is a set (of atoms), then D+ = D ∪ {X ⊆ D : X = 0 and for all d ∈ D, X = {d}}. We define the denotation function with respect to a set D as follows. Definition 16 If D is a set (empty or otherwise), then the denotation function of D, in symbols, denD , is that function with D ∪ {X : X ⊆ D+ } as domain and such that (1) for all d ∈ D, denD (d) = d, and (2) for all X ⊆ D+ , d, if X = {d}, for some (atom) d ∈ D denD (X) = . X otherwise 36 We assume our metalanguage to be ZF set theory, and we take ω to be the set of natural numbers, where for each n ∈ ω, n is the set of natural numbers less than n. Thus 0 is the empty set.
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We now define the notion of a “standard” nominal model. Definition 17 A is a nominal L-model iff L is a language and there are a set D, possibly empty, of “atoms” and a relation R such that (1) A = D, R, and (2) R is a function with L as domain and such that for each nominal constant A ∈ L, R(A) ⊆ D+ , and for each n ∈ ω, and each n-place predicate constant π ∈ L, R(π) is a set of n-tuples drawn from D+ . Note that because R(A) ⊆ D+ , where A is a nominal constant, then names of single atoms are assigned singletons of those atoms, and not the atoms themselves. This is “corrected for” in the definition of the denotation function in a model, which, in the “standard” semantics, we identify with the denotation function of the domain of atoms in the model. Definition 18 If L is a language and A = D, R is a nominal L-model, then the denotation function in A , in symbols, DenA = denD . The following metatheorem indicates some of the useful features of the denotation function. Part (4) is the semantic analogue of the axiom of extensionality in the object language, which, as already noted, is valid in this semantics.
Metatheorem 1:
If A = D, R is a nominal model, then (1) if d1 , d2 ∈ D+ , then DenA (d1 ) = DenA (d2 ) iff d1 = d2 ; (2) if X, Y ⊆ D+ and X, Y ∈ / D, then DenA (X) = DenA (Y ) iff X = Y ; (3) if d1 ∈ D, d2 ∈ D+ and for all d3 ∈ D+ [there is an X ⊆ D+ such that d3 ∈ X & DenA (d2 ) = DenA (X) only if there is a Y ⊆ D+ such that d3 ∈ Y & DenA (d1 ) = DenA (Y )], then d1 = d2 . (4) if d1 , d2 ∈ D+ : if for all d3 ∈ D+ , (there is an X ⊆ D+ such that d3 ∈ X & DenA (d1 ) = DenA (X) iff there is an Y ⊆ D+ such that d3 ∈ Y & DenA (d2 ) = DenA (Y )), then d1 = d2 . An assignment in a model of values to variables assigns objects in the union of the domain of atoms and the non-empty, nonsingleton subsets of that domain to the individual variables and subsets of the latter to the nominal variables. Definition 19 If A = D, R is a nominal L-model, then a is an assignment (of values to variables) in A iff a is a function with the set of individual and nominal variables as domain and such that (1) for each individual variable x, a(x) ∈ D , for some D ⊇ D+ , and (2) for each nominal variable A, a(A) ⊆ D+ . Thus, an assignment in a nominal model assigns values to the individual variables that are drawn from a set D that contains D+ , i.e., D+ ⊆ D , and it assigns to the nominal variables subsets of D+ . The distinction between D+ and D is required for the “free logic” aspect of the first-order part of the logic of classes as many; that is, D+ is the set of values of bound individual variables and D is the set of values of free individual variables. We next inductively define the extension of a name or formula in a model.
268 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY Definition 20 If L is a language, A = D, R is a nominal L-model, a is an assignment in A, and ξ is a name or formula of L, then the extension of ξ in A relative to a and an individual variable z (as place holder), in symbols ext(ξ, A, a, z), is defined recursively as follows: (1) if ξ is a nominal variable or constant in L, then R(ξ), if ξ is a constant in L ext(ξ, A, a, z) = ; a(ξ), if ξ is a nominal or individual variable (2) if ξ is an identity formula (a = b), where a, b are singular terms, i.e., either individual variables, nominal variables or constants in L, or names of L of the form [ˆ xB], then 1, if DenA (ext(a, A, a, z)) = DenA (ext(b, A, a, z)) ext(ξ, A, a, z) = ; 0 otherwise (3) If ξ is a ∈ b, for some singular terms a, b of L, then ext(ξ, A, a, z) = 1 if for some X ⊆ D+ , ext(a, A, a, z) ∈ X & DenA (b) = DenA (X); and otherwise ext(ξ, A, a, z) = 0; (4) If ξ is a ⊆ b, for some singular terms a, b of L, then ext(ξ, A, a, z) = 1 if for all d ∈ D+ , there is an X ⊆ D+ such that d ∈ X & DenA (a) = DenA (X) only if there is a Y ⊆ D+ such that d ∈ Y and DenA (b) = DenA (Y ); and otherwise ext(ξ, A, a, z) = 0; (5) If ξ is a ⊂ b, for some singular terms a, b of L, then ext(ξ, A, a, z) = 1 if for all d ∈ D+ , there is an X ⊆ D+ such that d ∈ X & DenA (a) = DenA (X) only if there is a Y ⊆ D+ such that d ∈ Y and DenA (b) = DenA (Y ), and yet it is not the case that for all d ∈ D+ , there is an Y ⊆ D+ such that d ∈ Y & DenA (b) = DenA (Y ), only if there is an X ⊆ D+ such that d ∈ X and DenA (a) = DenA (X); (6) if ξ is π(a1 , ..., an ), for some n-place predicate constant in L, then extA (ξ, A, a, z) = 1 if DenA (extA (a1 , A, a, z), ..., DenA(extA (an , A, a, z)) ∈ R(π); and otherwise extA (ξ, A, a, z) = 0; (7) if ξ is Atom, then ext(ξ, A, a, z) = D; (8) if ξ is ¬ϕ, for some formula ϕ of L, then 1, if ext(ϕ, A, a, z) = 0 ext(ξ, A, a, z) = ; 0, otherwise (9) if ξ is (ϕ → ψ), for some formulas ϕ, ψ of L, then 0, if ext(ϕ, A, a, z) = 1 and ext(ψ, A, a, z) = 0 ext(ξ, A, a, z) = ; 1, otherwise
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269
(10) if ξ is (∀x)ϕ, for some formula ϕ of L and individual variable x, then 1 if for all d ∈ D+ , ext(ϕ, A, a(d/x), z) = 1 ; ext(ξ, A, a, z) = 0, otherwise (11) if ξ is (∀xB)ϕ, for some formula ϕ of L, individual variable x, and name B, then (note the change in place-holder) ext(ξ, A, a, z) = 1, if for all d ∈ D + ∩ ext(B, A, a(d/x), x), ext(ϕ, A, a(d/x), x) = 1; and otherwise ext(ξ, A, a, z) = 0; (12) if ξ is (∀Cϕ), for some formula ϕ of L and nominal variable C, then 1, if for all X ⊆ D + , ext(ϕ, A, a(X/C), z) = 1 ext(ξ, A, a, z) = ; 0, otherwise (13) if ξ is B/ϕ, for some name B and formula ϕ of L, then ext(ξ, A, a, z) = {d ∈ D+ : d ∈ ext(B, A, a(d/z), z) & ext(ϕ, A, a(d/z), z) = 1}; (14) if ξ is /ϕ, for some formula ϕ of L, then ext(ξ, A, a, z) = {d ∈ D+ : d ∈ ext(ϕ, A, a(d/z), z) = 1}; and (15) if ξ is [ˆ xB], for some individual variable x and name B of L, then ext(ξ, A, a, z) = {d ∈ D+ : d ∈ ext(B, A, a(d/x), z)}. Note that in the definiens of clause 11 of this definition the place-holder variable z is replaced by the variable x. Also, although the place-holder in the definiens remains unchanged in clauses 13 and 14, the assignment is modified with respect to that place-holder. In clause 15, the place-holder is left unchanged and the assignment is modified with respect to the bound variable. Definition 21 If L is a language, ϕ is a formula of L, A is a nominal L-model and a is an assignment in A, then (1) a satisfies ϕ in A iff ext(ϕ, A, a, z) = 1, for some individual variable z (as place-holder); and (2) ϕ is true in A iff every assignment in A satisfies ϕ in A. We define logical consequence and validity in the usual way. Definition 22 If L is a language and Γ ∪ {ϕ} is a set of formulas of L, then (1) ϕ is a logical consequence of Γ, in symbols, Γ |= ϕ, iff for every L-model A and every assignment a in A, if a satisfies every formula in Γ in A, then a satisfies ϕ in A; and (2) ϕ is valid if it is a logical consequence of the empty set. The following soundness theorem leads directly to a consistency proof for the logic of classes as many plus the axiom of extensionality.
Metatheorem 2: (Soundness) If ϕ is a theorem of the logic of classes as many plus the axiom of extensionality, then ϕ is valid with respect to the above semantics.
270 CHAPTER 11. PLURALS AND THE LOGIC OF CLASSES AS MANY The consistency of the logic of classes as many (plus the axiom of extensionality) follows from the fact that {n, ω : n ∈ ω} is a set of atoms and therefore that {n, ω : n ∈ ω}+ , 0 is a nominal model, and hence that every theorem is true in {n, ω : n ∈ ω}+ , 0.
Metatheorem 3: The logic of classes as many with the axiom of extensionality added is consistent.
11.11
Appendix 2: Bell’s System M
The system M of classes as many in Bell 2000 is different from the logic described here. M is designed to show how proper (or ultimate) classes, which do not belong to other classes, can be taken to be classes as many (though Bell takes all sets to be classes as many and redefines ‘set’ in terms of certain individuals identified as “labeled” classes). Unlike the logic described here, M is not designed to provide a semantics for plural references in natural language, and it is not clear how one might use it for that purpose. Nevertheless, Bell’s system M can be translated into the logic of classes as many presented here with the result that, with the axiom of extensionality added to the latter, the translation of each axiom of M is a theorem of our present system (and hence that M is contained in the latter). M is a two-sorted first-order logic with capital letters, A, B, C, etc., for classes as many and lower-case letters x, y, z, etc., as individual variables. The logic of classes as many described here contains a two-sorted first-order (free) logic as a proper part, where the capital letters A, B, C, etc. for nominal concepts are nominalized and transformed into singular terms for classes as many. As primitive symbols, M also contains ∈ for the membership relation, the identity sign (applied to classes terms or to individual terms separately), a labeling functor λ (applied to class terms), a co-labeling functor ∗ (applied to individual terms), a monadic predicate I (applied to individual terms, where I(a) is read ‘a is an identifier’), a monadic predicate S (applied to class terms, where S(A) is read ‘A is a set’), and the abstraction operator, {x : ϕ(x)} (read as ‘the class defined by ϕ’). Our translation function τ identifies the capital and individual variables of M with the same letters in our logic of classes as many and interprets the labeling function λ as the nominalization of a nominal expression A, i.e., τ (λA) = A (nominalized). Membership in M is identified with membership in the logic of classes as many and class abstracts are similarly identified with one another We extend τ so that, in addition to the correlation of the terms of M with singular terms of our logic of classes as many, each formula of M is translated into a formula of the logic of classes as many described here, with the translation of the co-labeling functor ∗ given contextually, i.e., for formulas such as ϕ(x∗ ) in which it occurs: 1. τ (x) = x and τ (A) = A, for each individual variable x and nominal variable A; 2. τ (λa) = τ (a), where a is a class term of M (other than of the form b∗ );
11.11. APPENDIX 2: BELL’S SYSTEM M
271
3. τ ({x : ϕ(x)}) = [ˆ x/τ (ϕ(x))]; 4. τ (a ∈ b) = (τ (a) ∈ τ (b)); 5. τ (a = b) = (a = b), where a, b are either both class terms of M or both terms for individuals; 6. τ (I(a)) = (∃z)(τ (a) = z), where z is the first individual variable not occurring in a; 7. τ (S(a)) = (∃z)(τ (a) = z); 8. τ (ϕ(a∗ )) = (∃A)[τ (a) = A ∧ τ (ϕ(A/a))], where a is a term of M for an individual and A is a class variable of M that is free for a in ϕ; 9. τ (¬ϕ) = ¬τ (ϕ); and τ (ϕ → ψ) = [τ (ϕ) → τ (ψ)]; 10. τ (∀xϕ) = (∀x)τ (ϕ); and τ (∀Aϕ) = (∀A)τ (ϕ) The translation of the axiom of extensionality of M is just a version of the axiom of extensionality in the present logic of classes as many. Also, the translation of each “labeling axiom” of M is an obvious theorem of our logic: τ (S(A) ↔ I(λA)) = [(∃z)(A = z) ↔ (∃z)(A = x)], τ (I(x) ↔ S(x∗ )) = [(∃z)(x = z) ↔ (∃A)[x = A ∧ (∃z)(A = z)]], τ (S(B)) → (λB ∗ ) = B) = [(∃z)(B = z) → (∃A)[B = A ∧ A = B]], τ (I(x) → λ(x∗ ) = x) = [(∃z)(x = z) → (∃A)[x = A ∧ A = x]]. And finally the axiom of comprehension of M is also translated into a theorem of our logic: τ (y ∈ {x : ϕ(x)} ↔ ϕ(y/x)) = [(∃z[ˆ x/τ (ϕ(x))])(y = z) ↔ τ (ϕ(y/x))]. As noted, the labeling function of M that associates each class with a labeled individual corresponds to the nominalization transformation of our present logic of classes as many, and those cases where the labeled individuals are “sets” (as defined in M ) correspond to those where a nominal concept is object ified (i.e., where its nominalization is a value of the bound individual variables).
Chapter 12
The Logic of Natural Kinds 12.1
Conceptual Natural Realism
The three main theories of universals in the history of philosophy, we have noted, have been nominalism, realism, and conceptualism. In nominalism, there are no universals that predicates stand for; there is only predication in language. In conceptualism, predication in thought is what underlies predication in language, and what predicates stand for are concepts as rule-following cognitive capacities underlying our use of predicate expressions. What predicates stand for in realism are real universals that are the basis for predication in reality, i.e., for the events and states of affairs that obtain in the world. We have distinguished two types of realism in previous chapters, namely Bertrand Russell’s and Gottlob Frege’s different versions of logical realism as modern forms of Platonism, and several variants of natural realism, one of which is logical atomism. Another variant of natural realism is a modern form of Aristotle’s theory of natural kinds, or what is usually called Aristotelian essentialism. Now the relationship between conceptualism and realism is more complex than the simple kind of opposition that each has to nominalism. Conceptual intensional realism, for example, is similar to logical realism with respect to overall logical structure, and yet the two formal ontologies are different on such fundamental issues as the nature of universals and the nexus of predication. The relationship between conceptualism and natural realism, on the other hand, is even more complex. They do not, for example, have the same overall logical structure, and they also differ on the nature of universals and the nexus of predication. And yet, conceptualism and natural realism have been intimately connected with one another throughout the history of philosophy—though not always in an unproblematic way. One reason for this connection is that without some form of natural realism associated with it, conceptualism becomes an ontology restricted to the conceptual realm; and without an ontological ground in nature, conceptualism turns into a form of conceptual idealism, sometimes 273
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with, and sometimes without, a transcendental subjectivity. Conceptualism, as we understand it here, is not a form of idealism, however, but is based on a socio-biological theory of the human capacity for language, culture and thought; and therefore it must presuppose some form of natural realism as the causal ground of that capacity. On the other hand, natural realism must in turn presuppose some form of conceptualism by which to explain our capacity for language and thought, and in particular our capacity to form theories of the world and conjecture about natural properties and relations as part of the causal order. Conceptualism and natural realism, in other words, presuppose each other as part of a more general ontology, which, in one form or another, may be called conceptual natural realism.1 The connection between conceptualism and natural realism goes back at least as far as Aristotle whose doctrine of moderate realism, i.e., the doctrine that universals “exist” only in things in nature, is well-known for its opposition to Platonism, the doctrine that universals exist as abstract entities independently of concrete objects. Peter Abelard, in his Glosses on Porphyry, also dealt with the connection between the conceptual and natural orders of being. In particular, Abelard gave an account that is very much like Aristotle’s in being both conceptualist and realist. But in combining these positions, Abelard did not sharply distinguish the universals that underlie predication in thought from those that underlie predication in reality. In particular, a universal, according to Abelard, seems to exist in a double way, first as a common likeness in things, and then as a concept that exists in the human intellect through the mind’s power to abstract from our perception of things by attending to the likeness in them. What Abelard describes is a form of natural realism, where a property exists only in the causal or natural order and as a common likeness in things; and yet if those things were to cease to exist, the property would somehow still exist in the human intellect as a concept. Aristotle also seems to have described natural kinds and properties in this double way, i.e. as having a mode of being both in things and then, through an inductive abstraction (epagoge), in the mind as well. But then it is possible to interpret him otherwise, especially in his discussion in the Posterior Analytics of how concepts such as being a chimera or being a goat-stag can be formed otherwise than by abstraction. Such an alternative interpretation was in fact developed by Aquinas in his distinction between the active intellect (intellectus agens) and the receptive intellect (intellectus possibilis), which are not really two intellects but two kinds of powers or capacities of the intellect simpliciter.2 In fact, Aquinas apparently developed the central idea of conceptual natural realism, namely, that the problem of the “double existence” of universals is not an ontological problem but a problem of explaining how the same predicate can stand for, or signify, a concept in the mind on the one hand, and a natural property in nature on the other, where the natural property corresponds to, or 1 Some of the differences between these forms depends on whether a constructive or holistic conceptualism is assumed, and whether the natural realism is part of an Aristotelian essentialism or not. 2 Cf. Kenny 1969.
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is represented by, the concept. The two are not really the same universal, in other words, and do not even have the same mode of being.3 Concepts cannot literally be the same as the natural properties and relations they purport to represent, in other words, and in fact some concepts—especially those for artifacts and social conventions—do not represent any natural properties or relations at all.
12.2
The Problem with Moderate Realism
One reason why the universals of natural realism were confused with predicable concepts is that both can be designated by predicates—or, more precisely, that a predicate that stands for a concept can also be taken to stand for a natural property or relation that corresponds to that concept. A predicate can be taken to stand for a natural property or relation, in other words, as well as for a concept, but then the sense in which it stands for a property or a relation is secondary to the sense in which it stands for a concept. As Aquinas noted, the traditional problem about universals “existing in a double way” was really a matter of there being two ways in which a predicate can signify a universal, one way being primary in which the predicate stands for a concept, and the other being secondary in which the predicate stands for natural property or relation that corresponds to that concept. The sense in which a predicate stands for a concept is primary because it is the concept that determines the functional role of the predicate and the conditions under which it can be correctly used in a speech act. It is only by assuming (as a result of empirical evidence) that the truth conditions determined by the concept have a causal ground based on a natural property or relation that we can then say that the predicate also stands for a natural property or relation. In other words, even though the natural property or relation in question may in fact be the causal basis for our forming the concept, and therefore is prior in the order of being, nevertheless, the concept is prior in the order of conception. The distinction between concepts in the order of conception and natural properties and relations in the order of being does not mean that there should also be a distinction in the theory of logical forms of conceptual natural realism between predicates that stand for concepts and predicates that stand for a natural property or relation. The whole point of the double significance of a predicate is that the same predicate can stand for both a concept in the primary sense and a natural property or relation in the secondary sense. Thus, it is not that the same universal can “exist in a double way,” as Abelard assumed, first in nature and then in the mind, but rather that the same predicate can stand in a double way for both a concept and a natural property or relation—though it stands first for a concept, and then perhaps also for a natural property or relation—but only in the sense of an hypothesis about nature, which might not always be explicit, but only implicit. 3 See
Basti 2004.
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Now, just as a predicate constant can be taken to stand in double way both for a concept and a natural property or relation, so too a predicate variable can be taken in a double way to have both concepts and natural properties and relations as its values. The difference between the universals in the one order and the universals in the other is reflected not in a difference between two “types” of predicate constants and variables—where the one “type” stands for concepts and the other stands for natural properties and relations—but in the kind of reference that is made by means of predicate quantifiers, i.e., the quantifiers that can be affixed to predicate variables and that determine the conditions under which a predicate constant can be substituted for a predicate variable. In this way, the difference is reflected not in a difference of types of predicate variables to which predicate quantifiers can be affixed, but in a difference between the predicate quantifiers themselves, i.e., in the types of referential concepts the quantifiers stand for. What we need to add to the second-order conceptualist theory of logical forms described in chapter four, accordingly, are special quantifiers, ∀n and ∃n , that can be applied to predicate variables, and that, when so applied, can be used to refer to natural properties and relations. We assume, of course, that a natural property or relation is an existence-entailing property or relation, i.e., that only existing objects—where, by existence we mean concrete existence— can be characterized by a natural property or stand in a natural relation. In other words, where the monadic predicate E! stands for the concept of concrete existence, the following is assumed as an axiom4 : (∀n F j )(∀x1 )...(∀xj )[F (x1 , ..., xj ) → E!(x1 ) ∧ ... ∧ E!(xj )].
(N ⊆ E!)
The concept of concrete existence, we have said, is a logical construct, and hence there is no presumption that there is a natural property corresponding to it. In fact, as constructed within conceptual realism, it is an impredicative concept, because it is constructed, or formed, in terms of a totality to which it belongs. That is, as defined earlier, to exist is to fall under an existence-entailing concept, or, in symbols, E!(x) ↔ (∃e F )F (x). Now, because natural properties can be realized only by things that actually exist in nature, it might be thought that we could give a more specific kind of analysis in natural realism, and, in particular, that we could define concrete existence as having a natural property, and therefore as being a constituent of a fact, i.e., a state of affairs that obtains in the world: E!(x) =df (∃n F )F (x).
(E?)
4 A logical realist who wanted to distinguish natural properties and relations from properties and relations in general might assume the simpler axiom
(∀n F )(∃e G)(F = G) instead. In conceptual realism, however, concepts are values of predicate variables bound by ∀ and ∃, whereas natural properties and relations are values of predicates bound by ∀n and ∃n , and in this framework concepts are not the same type of entity as natural properties and relations.
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Such an analysis will not suffice, however, because it is possible that some existing objects might not have a natural property, but only stand to other existing objects in a natural relation. The point is that standing in a natural relation to other existing objects does not constitute having a natural property. Natural relations do not in general generate natural properties, and to claim otherwise is to confuse the conceptual order, where monadic concepts can be constructed from relational concepts, with the natural order. Properties and relations can be posited to exist in the natural order only as hypotheses about nature. This is part of what we mean when we said that conceptualism and natural realism do not have the same overall logical structure. Now Aristotle’s moderate realism as a form of natural realism can be stated as follows: (∀n F j )(∃e x1 )...(∃e xj )F (x1 , ..., xj ), (MR) where the quantifier ∃e is as described in the logic of actual (concrete) objects (in chapters two and six). What the thesis of moderate realism, (MR), says is that natural properties and relations “exist” only as components of facts, i.e., the states of affairs that obtain in the world. It is in that sense that we say that a natural property or relation “exists” only in things. Unfortunately, this is much too restrictive a view of natural realism as a scientifically acceptable ontology. At the moment of the Big Bang, for example, when the universe was first formed, there was mostly raw energy and only very few elementary particles. There were no atoms or molecules of any kind, or at least certainly not of any complex kind, all of which came later in the evolution of the universe. Consequently, many of the natural properties and relations that we now know to characterize atoms and compounds as physical complexes were not at that time realized in any objects at all. But of course that does not mean that they did not then have a real mode of being within nature’s causal matrix even at the beginning of the universe. Indeed, even today there may yet be some transuranic elements, and natural properties of such, that, as a matter of contingent fact, will never be realized in nature by any objects at all, but which, nevertheless, as a matter of a natural or causal possibility, could be realized by atoms that are generated, e.g., in a supernova, or in a very high energy accelerator. The being of such a natural property or relation does not consist of its being a property of some transuranic atom at the moment of the Big Bang, nor even, for that matter, of its being in re at some time or other in the history of the universe. Instead, its being consists of its being part of nature’s causal matrix right from the beginning of time, and therefore of its possibly being realized in nature, i.e., of its possibly being in re. The being of a natural property or relation is its possibly being in re, i.e., its being realizable in nature as a matter of a natural or causal possibility.
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Now one obvious consequence of this view of natural properties and relations is that they are not intensional objects, nor are they objects of any kind at all. This is so because if natural properties and relations were objects, then, in order to be even when they are not in things, they would have to be abstract objects. How could they be, in other words, when they are not in things, unless their being is that of an abstract object in a Platonic realm of forms, in which case they would have a mode of being that transcended the natural world and nature’s causal matrix. But natural properties and relations do not exist independently of the world and its causal matrix, even though they are not contained within the space-time causal manifold the way concrete objects are. Natural properties and relations cannot be objects, in other words, and therefore their mode of being as possibly being in re must have a different explanation. As universals that correspond to concepts as unsaturated cognitive capacities, the most plausible explanation is that they too have an unsaturated nature, albeit one that is only analogous to, and not the same as, the unsaturated predicative nature of concepts. As components of the nexus of predication in reality, which we can comprehend only by analogy with the nexus of predication in thought, natural properties and relations are unsaturated causally determinate structures that become saturated in the states of affairs that obtain in nature, and that otherwise “exist” only within nature’s causal matrix. Thus, even though natural properties and relations do not “exist in a double way,” one in nature and the other in the intellect, nevertheless, they have a mode of being as unsaturated causal structures that is analogous to that of concepts as unsaturated cognitive capacities, and hence their unsaturatedness must be understood by analogy with the unsaturated nature of concepts. In terms of the theory of logical forms of a formal ontology, where predicates signify both kinds of universals, this means that both kinds of universals are values of predicate variables, albeit variables bound by different quantifiers, namely ∀n and ∃n in the case of natural properties and relations, and ∀ and ∃ in the case of predicable concepts. Finally, we should note that just as predicable concepts do not exist independently of the general capacity humans have for language and thought, so too natural properties and relations do not exist independently of nature’s causal matrix. That is why, just as the laws of compositionality for concept-formation of predicable concepts, as characterized by the comprehension principle, (CP∗λ ), can be said to characterize the logical structure of the intellect as the basis of the human capacity for language and thought, so too the laws of nature regarding the causal connections between natural properties and relations, especially as determined by natural kinds, can be said to characterize the causal structure of the world. Thus, just as concepts have their being within the matrix of thought and concept-formation, so too natural properties and relations have their being within the matrix of the laws of nature.
12.3. MODAL MODERATE REALISM
12.3
279
Modal Moderate Realism
What is needed in the formal ontology of natural realism is a modal logic for a causal or natural necessity, or a causal or natural possibility. By a natural possibility we mean what is possible in nature, i.e., what is not precluded by the laws of nature. A natural necessity therefore is what must be so because of the laws of nature. This suggests that S5 is the appropriate modal logic for natural necessity; or to express the matter in model-theoretic terms, that the possible worlds in the multiverse that have same laws of nature constitute an equivalence class.5 Different equivalence classes of the possible worlds in the multiverse will then represent different causal matrices as determined by the laws of nature that are invariant across the worlds in those equivalence classes. As is well-known, necessity, when interpreted as invariance over each equivalence class of a set of equivalence classes of models (“possible worlds”)—i.e., where each model in any one such equivalence class is accessible from every other model in that equivalence class—results in a completeness theorem for S5 modal logic.6 One version of such a multiverse is the concordance model discussed in chapter three, §6.1, where the relation of accessibility is universal, and hence where there is but one equivalence class of possible worlds. By a causal possibility, on the other hand, we mean what can be brought about in nature through causal mechanisms of whatever natural sort, physical, biological, etc. A causal necessity then is what must be so because of its causal ground, i.e., what caused it to be so. This suggests that S4 is the appropriate modal logic to adopt, because whereas causal relations are transitive they are not also symmetric, and, as is well-known, S4 is the modal logic characterized by a transitive accessibility relation between possible worlds.7 Of course, we still assume that all of the causally accessible worlds will have the same laws of nature as our world, i.e., the causal relation does not take us to worlds that violate the laws of our universe. The many-worlds interpretation of quantum mechanics, as described in chapter three, §6.2, provides an example of a multiverse that validates S4 in this way. We will not attempt to decide here whether the appropriate modal logic for conceptual natural realism is S4 or S5. Instead, we will leave that decision to the different variants of this ontology that might be developed. These variants might differ not only in respect of which modal logic is adopted, but also in whether the first-order logic of the variant is possibilist or actualist, and also whether it allows for only a constructive conceptualism or a more comprehensive holistic conceptualism. However, because S4 is a proper part of S5, we will use S4 here without assuming that S5 is thereby precluded. In regard to notation, 5 For
an account of the different worlds in the multiverse, or megaverse, see Kaku 2005. Cocchiarella 1986, chapter III, §7, for an axiomatization and completeness theorem of natural realism based on this interpretation. 7 A model-theoretic approach to a causal modality would be based on an extension of the notion of a causally extended system of world lines as described in chapter two for the causal tenses. At each node of such a causally connected system we would have not only the actual space-time points of our universe that can be reached by a light signal, but also space-time points that are causally possible at that node as well. 6 See
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we will use c for causal necessity and ♦c for causal possibility. Now, instead of the ontological thesis of moderate realism (MR), we have the following ontological thesis of modal moderate realism as a fundamental principle of natural realism: n
c
e
e
(∀ F j )♦ (∃ x1 )...(∃ xj )F (x1 , ..., xj ).
(MMR)
Natural properties and relations “exist” not as components of actual facts, in other words, as was stipulated in the thesis of moderate realism, but as the nexuses of possible states of affairs. It is in this sense that the being of a natural property or relation is its possibly being in re. There is no general comprehension principle that is valid in natural realism, incidentally, the way that the comprehension principle (CP∗λ ) is valid for conceptual realism. Natural properties and relations are not formed, or constructed, out of other properties and relations by logical operations. But this does not mean that no natural property or relation can be specified in terms of a complex formula, i.e., a formula in which logical constants occur. What it does mean is that such a specification cannot be validated on logical grounds alone, but must be taken as a contingent hypothesis about the world. In order to consider specifying natural properties and relations in terms of complex formulas, it is convenient to have some abbreviatory notation. In particular, we can adopt some useful abbreviatory notation that simulates nominalizing predicates as objectual terms. We adopt for this purpose the following notation, which simulates a kind of identity between natural properties or relations8 : (F j ≡c Gj ) =df c (∀x1 )c ...c (∀xj )c [F (x1 , ..., xj ) ↔ G(x1 , ..., xj )] We say that ≡c represents an “identity” between natural properties and relations because, unlike concepts, natural properties and relations are “identical” when, as matter of causal necessity, they are coextensive. As part of the causal structure of the world, natural properties and relations retain their “identity” as natural properties and relations across all causally accessible worlds. As part of nature’s causal matrix, natural properties and relations are “identical” when, as a matter of causal necessity, they are co-extensive. Now the assumption that there is a natural property or relation corresponding to a given predicable concept that is represented by a complex formula ϕ, and hence by the λ-abstract [λx1 ...xj φ], can be formulated as follows9 : (∃n F j )c ([λx1 ...xj φ] ≡c F ) . 8 Only
the initial occurrence of c is needed here if the presumed modal logic is S5. The additional occurrences are needed if the logic is S4. 9 With S5, instead of S4, this axiom can be stated simply as (∃n F j ) ([λx1 ...xj φ] ≡c F ) .
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Here, we note again, that, unlike the comprehension principle of conceptual realism, such an assumption is at best only a scientific hypothesis, and as such must in principle be subject to confirmation or falsification. In this regard, there is no comprehension principle valid in natural realism other than the trivial one stipulating that every natural property or (j-ary) relation is a value of the bound (j-ary) predicate variables, i.e.10 , (∀n F j )c (∃n Gj )(F ≡c G).
12.4
Aristotelian Essentialism
Conceptual natural realism without natural kinds might be an adequate ontological framework for some philosophers of science; but to others, especially those who fall in the tradition of Aristotle and Aquinas, it is only part of a larger, more interesting ontology of Aristotelian essentialism. This is a framework that is a part of cosmology as well as of ontology. It is part of cosmology because it is based on natural kinds as causal structures, and it is part of ontology in that it determines two types of predication in reality, essential and accidental. Natural kinds—whether in the form of species or genera, and whether of natural kinds of “things,” such as plants and animals, or natural kinds of “stuff”, such as the chemical substances gold, oxygen, iron, etc., or compound substances such as water, salt, bronze, etc.—are the bases of essential predication, whereas predicable concepts and natural properties and relations are the bases of accidental, or contingent, predication. The basic assumption of this extension of natural realism is that in addition to the natural properties and relations that may correspond to some, but not all, of our predicable concepts, there are also natural kinds that may correspond to some, but not all, of our common-name concepts—especially those that are sortals, i.e., name concepts that have identity criteria associated with their use. By a natural kind we understand here a type of causal structure, or mechanism in nature, that is the basis of the powers or capacities to act, behave, function, etc., in certain determinate ways that objects belonging to that natural kind have. Natural kinds, in fact, are the causal structures, or mechanisms in nature, that underlie the causal modalities, and in particular they underlie the natural laws regarding the different natural kinds of things there are, or can be, in the world. In this ontology, natural kinds are an essential part of the internal hierarchical network of nature’s causal matrix, and in fact they constitute the more stable nodes of that hierarchical network. Now a natural kind is not a natural property or a “conjunction of natural properties,” as David Armstrong and other philosophers have claimed.11 Instead of being a “conjunction of natural properties,” a natural kind is a type of unsaturated causal structure that, when saturated, is the causal ground of 10 See Appendix 1 of this chapter for a complete axiom set for a strict actualist version of natural realism. Again the first c can be dropped if the modal logic is S5. 11 Cf. Armstrong 1978, chapter 15. In my first paper on natural kinds, Cocchiarella 1976, I did take natural kinds to be properties, but later corrected that view in Cocchiarella 1996.
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the events and states of affairs containing the natural properties that are said to “conjunctively define” that natural kind. Indeed, if a natural kind were a conjunction of natural properties, then we would need an explanation of why some conjunctions result in a natural kind whereas others do not. Why, in other words, do not all “conjunctions of natural properties” result in a natural kind if some do? If certain “conjunctions of natural properties” were to “produce,” “generate,” or result in a natural kind, whereas others do not, then that would suggest that there is more to a natural kind than just a “conjunction of natural properties.” In fact, the ontological dependence is just the opposite of what the conjunction thesis maintains, because instead of a “conjunction of natural properties” being the causal ground of a natural kind, it is the natural kind that is the causal ground of the natural properties in the “conjunction”. Moreover, there really are no “conjunctions of properties” in nature, but only causally related groups of events or states of affairs having those properties as predicable components, which, of course, we could in principle described in terms of a conjunction of sentences. In other words, as a causal structure, a natural kind has an ontological priority over the natural properties that are predicated of the objects of that kind, a priority that is part of what Aristotle means in describing natural kinds as secondary substances.12 Neither natural properties, nor so-called “conjunctions of natural properties,” on the other hand, can be described as secondary substances. Now as the causal ground of natural properties, a natural kind has a “substance-like” structure in that it is unsaturated in a way that is complementary to the unsaturated predicative structures of natural properties and relations. The nexus of predication in reality, in other words, is a kind of mutual saturation of a “substance-like” natural-kind structure, as realized by an object (or primary substance) of that kind, with a natural property or relation as a predicative structure, the result being an event or state of affairs that obtains in reality. Of course, the fact that natural kinds are unsaturated causal structures to begin with allows for there being natural kinds that in fact are not realized in nature at a given time but that could be realized, or brought about, in appropriate environmental circumstances. The transuranic elements of atomic numbers 113 and 115, for example, have only recently been realized in nature, even though for just a few fractions of a second.13 A natural kind, as an unsaturated “substance-like” causal structure, has its being in possibly being realized in things, and in that regard natural kinds can be realized in nature at different times in the evolution of the universe, or even possibly not at all. It is because the being of a natural kind is its possibly being realized in nature, that Aristotle’s problem of the fixity of species can be resolved in modal 12 Cf. 13 Cf.
Categories 2a11 . Science News, February 7, 2004, vol. 165, no. 6, p. 84.
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moderate realism. The ontological difference between natural kinds and natural properties and relations is analogous to the conceptual difference between common-name concepts and predicable concepts and the way that referential concepts based on the former may be saturated by the latter in speech and mental acts. Thus, just as a referential concept that is based upon a common-name concept can be saturated by a predicable concept in a speech or mental act, so too a natural kind, as the causal structure of an object of that kind, can be saturated by the natural properties and relations of that object, the result being a complex of events or states of affairs having that object as a constituent. Accordingly, just as a predicate expression can signify both a predicable concept and a natural property or relation, a common name can also signify or stand in a double way for both a concept and a natural kind as a causal structure. Similarly, name variables can also be given a double interpretation as well. Thus, just as the quantifiers ∀n and ∃n can be affixed to predicate variables and enable us to refer to natural properties and relations, so too we can introduce special quantifiers ∀k and ∃k , which, when affixed to name variables, enable us to refer to natural kinds. For convenience, we will assume that objectual quantifiers range over all objects, single or plural, abstract or concrete, and actual or merely possible in nature. We also assume all the distinctions we have made in previous chapters, including those that are about classes as many and membership in a class as many. Thus, e.g., where A is a common name, then x ∈ A, i.e., x belongs to A-kind, if, and only if, x is an A, i.e., x ∈ A ↔ (∃yA)(x = y). Similarly, for complex common names, e.g., A/F (y), we have x ∈ [ˆ y A/F (y)] ↔ (∃yA/F (y))(x = y). That is, x belongs to the A-kind that are F if, and only if, x is an A that is F . Now because names can be transformed into objectual terms, we can state the fact that every natural kind A is not just contingently a natural kind, but that as a node in the network of nature’s causal matrix it is necessarily so, i.e., (∀k A)c (∃k B)(A = B).
(K1)
Of course, that a common name A is co-extensive with a natural kind B, i.e., (∃k B)(A = B), does not mean that A is itself a natural kind.14 For example, assuming that the common name M an stands for a natural kind, but that the common name ‘featherless biped’, i.e., Biped/F eatheless, does not, then even though all and 14 The unrestricted form of Leibniz’s law applies, as we noted in chapter eleven, only to single objects. Where A and B are common names (or common variables), what A = B means is only that A and B are co-extensive, i.e., (∀x)[x ∈ A ↔ x ∈ B].
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only men are featherless bipeds (or so we will assume), i.e., even though it is now true that M an = [ˆ xBiped/f eatherless(x)], nevertheless it does not follow that being a featherless biped is a natural kind. That M an is a natural kind, incidentally, can be formulated as (∃k B)c (M an = B). So-called “real definitions” can be described in terms of this notation by means of a specification of the following form: xA/ϕx]), (∃k B)c (B = [ˆ where A is a natural kind genus, and B is specified as a species of A the members of which satisfy the condition ϕx. This would not be a “nominal definition,” i.e., a matter of introducing a simple common name as an abbreviation of a more complex common name; instead, it would be an hypothesis about the world, namely that there is a natural kind corresponding to the complex common name [ˆ xA/ϕx]. A “real definition” is not a definition after all; rather, it is an hypothesis that a complex common name [ˆ xA/ϕx] names a natural kind, and in particular a species of a genus A. Now there are a number of interesting laws of natural kinds that can be formulated in this formal ontology. For example, one such law is: An object belongs to a natural kind only if being of that kind is essential to it, i.e., only if it must belong to that kind whenever it exists as a real, concrete object. With E! as the predicate for concrete existence, this principle is formulated as follows: (∀k A)(∀xA)c [E!(x) → x ∈ A] . (K2) In other words, where A is a natural kind, i.e., (∃k B)c (A = B), and x is an A, i.e., x ∈ A, then x is an A whenever x exists in any causally possible world, i.e., x is essentially an A. This is the most natural way to state this principle, but it can be assumed in this form only in the modal logic S5. In S4, the principle needs to be formulated in the following somewhat stronger form, from which (K2) follows15 : (∀k A)(∀x)(♦c (x ∈ A) → c [E!(x) → x ∈ A]). 15 Because x ∈ A → ♦c (x ∈ A) is provable in S4, then (∀k A)(∀x)(x ∈ A → c [E!(x) → x ∈ A]), is a consequence the S4 version of the principle, from which (∀k A)(∀xA)c [E!(x) → x ∈ A], i.e., (K2), follows.
12.4. ARISTOTELIAN ESSENTIALISM
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Essential predication, which is represented here by the open formula, c [E!(x) → x ∈ A] is of course one of Aristotle’s two types of predication. It can be formulated, of course, as a predicate, namely, as the λ-abstract [λxc (E!(x) → x ∈ A)](x). Accidental, or contingent, predication, is represented simply as either [λx(∃yA)(x = y)](x) or F (x). Thus, given that Socrates is a teacher, but only contingently so, then this “accidental” predication is represented as follows: (∃xSocrates)[λx(∃yT eacher)(x = y)](x). Similarly, the accidental, or contingent, predication that Socrates speaks Greek can be symbolized as follows (∃xSocrates)F (x), where the predicate ‘speaks Greek’ is represented by the predicate constant F . Thus, as these examples illustrate, we have a natural and intuitive way to represent Aristotle’s two types of predication in our formal ontology. Another law of our logic is that the being of a natural kind, like that of natural properties, is its possibly being realized in nature. (∀k A)♦c (∃e x)(x ∈ A).
(K3)
The quantifier phrase ‘(∃e x)’ (‘there exists’) in (K3) can be replaced by the more general phrase ‘(∃x)’ (‘there is’), because we assume that only concrete existents belong to natural kinds. The following principle, in other words, is also an axiom of our logic of natural kinds. (∀k A)(∀x)[x ∈ A → E!(x)]
(K4)
If we adopt the following abbreviatory notation for the subordination, and proper subordination, of one kind to another, A ≤ A
