TYPES, TABLEAUS, AND GODEL' S GOD

TYPES, TABLEAUS, AND GODEL' S GOD

TRENDS IN LOGIC Studia Logica Library VOLUME 13 Managing Editor Ryszard Wojcicki, Institute of Philosoph y and Sociolog y. Polish Academ y of Sciences. Warsaw, Poland Editors Daniele Mundici, Department ofcomputer Sciences, University ofmilan, Italy Ewa Orlowska, National Institute oftelecommunications, Warsaw. Poland Graham Priest, Department ofphilosophy, University ofqueensland, Brisbane, Australia Krister Segerberg, Department of Philosoph y, Uppsala University, Sweden Alasdair Urquhart, Department of Philosoph y, University of Toronto, Canada Heinrich Wansing, Institute ofphilosophy, Dresden University oftechnology, Germany SCOPE OF THE SERIES Trends in Logic is a bookseries covering essentially the same area as the journal Studia Logica - that is, contemporary formal logic and its applications and relations to other disciplines. These include artificial intelligence, informatics, cognitive science, philosophy of science, and the philosophy of language. However, this list is not exhaustive, moreover, the range of applications, comparisons and sources of inspiration is open and evolves over time. Volume Editor Heinrich Wansing The titles published in this series are listed at the end of this volume.

MELVIN FITIING Lehman College and the Graduate Center; City University of New York, U.S.A. TYPES, TABLEAUS, AND GODEL'S GOD 111... " SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-94-010-3912-3 ISBN 978-94-010-0411-4 (ebook) DOI 10.1007/978-94-010-0411-4 Printed on acid-free paper AII Rights Reserved 2002 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2002 Softcover reprint of the hardcover 1 st edition 2002 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.

Contents PREFACE Xl Part I CLASSICAL LOGIC 1. CLASSICAL LOGIC-SYNTAX 3 1 Terms and Formulas 3 2 Substitutions 8 2. CLASSICAL LOGIC-SEMANTICS 11 1 Classical Models 11 2 Truth in a Model 12 3 Problems 15 3.1 Compactness 15 3.2 Strong Completeness 16 3.3 Weak Completeness 16 3.4 And Worse 17 4 Henkin Models 19 5 Generalized Henkin Models 24 6 A Few Technical Results 29 6.1 Terms and Formulas 29 6.2 Extensional Models 29 6.3 Language Extensions 30 3. CLASSICAL LOGIC-BASIC TABLEAUS 33 1 A Different Language 33 2 Basic Tableaus 35 3 Tableau Examples 37 v

vi TYPES, TABLEAUS, AND GODEL'S GOD 4. SOUNDNESS AND COMPLETENESS 43 1 Soundness 43 2 Completeness 46 2.1 Hintikka Sets 47 2.2 Pseudo- Models 48 2.3 Substitution and Pseudo-Models 52 2.4 Hintikka Sets and Pseudo- Models 59 2.5 Pseudo- Models are Models 62 2.6 Completeness At Last 63 3 Miscellaneous Model Theory 66 5. EQUALITY 69 1 Adding Equality 69 2 Derived Rules and Tableau Examples 69 3 Soundness and Completeness 73 6. EXTENSIONALITY 77 1 Adding Extensionality 77 2 A Derived Rule and an Example 77 3 Soundness and Completeness 79 Part II MODAL LOGIC 7. MODAL LOGIC, SYNTAX AND SEMANTICS 83 1 Introduction 83 2 Types and Syntax 86 3 Constant Domains and Varying Domains 89 4 Standard Modal Models 90 5 Truth in a Model 92 6 Validity and Consequence 94 7 Examples 95 8 Related Systems 101 9 Henkin/Kripke Models 102 8. MODAL TABLEAUS 105 1 The Rules 105 1.1 Prefixes 105 1.2 Propositional Rules 107

Contents VB 1.3 Modal Rules 107 1.4 Quantifier Rules 108 1.5 Abstraction Rules 109 1.6 Atomic Rules 109 1.7 Proofs and Derivations 110 2 Tableau Examples 111 3 A Few Derived Rules 113 9. MISCELLANEOUS MATTERS 115 1 Equality 115 1.1 Equality Axioms 115 1.2 Extensionality 117 2 De Re and De Dicto 118 3 Rigidity 121 4 Stability Conditions 124 5 Definite Descriptions 125 6 Choice Functions 128 Part III ONTOLOGICAL ARGUMENTS 10. GODEL'S ARGUMENT, BACKGROUND 133 1 Introduction 133 2 Anselm 134 3 Descartes 134 4 Leibniz 137 5 GCidel 138 6 Codel's Argument, Informally 139 11. GODEL'S ARGUMENT, FORMALLY 145 1 General Plan 145 2 Positiveness 145 3 Possibly God Exists 150 4 Objections 152 5 Essence 156 6 Necessarily God Exists 160 7 Going Further 162 7.1 Monotheism 162

viii TYPES, TABLEAUS, AND GODEL 'S GOD 7.2 Positive Properties are Necessarily Instantiated 162 8 More Objections 163 9 A Solution 164 10 Anderson's Alternative 169 11 Conclusion 171 REFERENCES INDEX 173 179

Truth did not come into the world naked, but it came in types and images. One will not receive truth in any other way. The Gospel of Philip [Rob77]

Preface What's Here This is a book about intensional logic. It also provides a thorough look at higher-type classical logic, including tableaus and a completeness proof for them. It also provides a formal examination of the Godel ontological argument. These are not disparate topics. Higher-type classical logic is intensional logic with the intensional features removed, so this is a good place to start. Ontological arguments, Godel's in particular, are natural examples of intensional logic at work, so this is a good place to finish. The term formal logic covers a broad range of inventions. At one end are small, special-purpose systems; at the other are rich, expressive ones. Higher-type modal logic-intensional logic-is one of the rich ones. Originating with Carnap and Montague, it has been applied to provide a semantics for natural language, to model intensional notions, and to treat long-standing philosophical problems. Recently it has also supplied a semantic foundation for some complex database systems. But besides being rich and expressive, it is also tremendously complex, and requires patience and sympathy on the part of its students. There are two quite different reasons to be interested in a logic. There is its formal machinery for its own sake, and there is using the formal machinery to address problems from the outside world. The mechanism of higher-type modal logic is complex and requires serious mathematics to develop properly. Models are not simple to define, and tableau systems are quite elaborate. A completeness argument, to connect the two, is difficult. But, the machinery is of considerable interest, if this is the sort of thing you have a considerable interest in. Ifyou are such a reader, applications concerning the existence of God can simply be skipped. On the other hand, if philosophical applications are what you are after, the xi

xii TYPES, TABLEAUS, AND GODEL'S GOD Godel ontological argument is a prime example. If this is the kind of reader you are, much of the mathematical background can be taken on faith, so to speak. It is a rare reader who will be interested equally in both the formal and the applied aspects of intensional logic. In a sense, then, this book has no audience-there are separate audiences for different parts of it. (But I encourage these audiences to do some 'crossing over.') Ifyou are interested in ontological arguments for their own sakes, start with Part III, and pick up material from earlier chapters as it is needed. If you are interested in the mathematical details of the formal system, its semantics and its proof theory, Parts I and II will be of interest-you can skimp on reading Part III. Part I is entirely devoted to classical logic, and Part II to modal. Here is a more detailed summary. Part I presents higher-type classical logic. It begins with a discussion of syntax matters, Chapter 1. I present types in Schutte's style, rather than following Church. Types can be somewhat daunting and I've tried to make things go as smoothly as I can. Chapter 2 examines semantics in considerable detail. What are sometimes called "true" higher-order models are presented first. After this, Henkin's generalization is given, and finally a non-extensional version of Henkin models is defined. Henkin himself mentioned such models, but knowledge of them does not seem to be widespread. They are natural, and should become more familiar to the logic community-the philosophical logic community in particular. Classical higher-order tableaus are formulated in Chapter 3. These are not original here-versions can be found in several places. A number of worked out examples of tableau proofs are given, and more are in exercises. The system is best understood if used. I do not attempt a consideration of automation-the system is designed entirely for human application. There is even some discussion of why. Soundness and completeness are proved in Chapter 4. Tableaus are complete with respect to non-extensional Henkin models. The completeness argument is not original; it is, however, intricate, and detailed proofs are scarce in the literature. After the hard work has been done, equality and extensionality are easy to add using axioms, and this is done in Chapters 5 and 6. And this concludes Part I. Except for the explicit formulation of non-extensional models, the material in Part I is not original-see [Tak67, Pra68, To175, And86, Sha91, Lei94, Koh95, Man96], for example. Part II is devoted to the complications that modality brings. Chapter 7 adds the usual box and diamond to the syntax, and possible worlds

PREFACE xiii to the semantics. It is now that choices must be made, since quantified modal logic is not a thing, but a multitude. First, at ground level quantifiers could be actualist or possibilistthey can range over what actually exists at a world, or over what might exist. This corresponds to the varying domain, constant domain split familiar to many from first-order modal discussions. However, either an actualist or a possibilist approach can simulate the other. I opt for a possibilist approach, with an explicit existence predicate, because it is technically simpler. Next, we must go up the ladder of higher types. Doing so extensionally, as in classical logic, means we take subsets of the ground-level domain, subsets of these, and so on. Going up intensionally, as Montague did, means we introduce functions from possible worlds to sets of groundlevel objects, functions from possible worlds to sets of such things, and so on. What is presented here mixes the two notions-both extensional and intensional objects are present. I refer you to [FitOOb, FitOOa] for applications of these ideas to database theory-intensional and extensional objects make natural sense even in such a context. Classical tableau rules are adapted in Chapter 8, using prefixes, to produce modal systems. While the modal tableau rules are rather straightforward, they are new to the literature, and should be of interest. Since things are already quite complex, no completeness proof is given. If it were given, it would be a direct extension of the classical proof of Part 1. Using modal semantics and tableaus, in Chapter 9 I discuss the relationships between rigidity, de re and de dicto usages, and what I call Godel's stability conditions, which arise in his proof of the existence of God. I also relate all this to definite descriptions. While this is not deep material, much of it does not seem to have been noted before, and many should find it of some significance. Finally, Part III is devoted to ontological proofs. Chapter 10 gives a brief history and analysis of arguments of Anselm, Descartes, and Leibniz. This is followed by a longer, still informal, presentation of the Godel argument itself. Formal methods are applied in Chapter 11, where Godel's proof is examined in great detail. While Godel's argument is formally correct, some fundamental flaws are pointed out. One, noted by Sobel, is that it is too strong-the modal system collapses. This could be seen as showing that free will is incompatible with Godel's assumptions. Some ways out of this are explored. Another flaw is equally serious: Godel assumes as an axiom something directly equivalent to a key conclusion of his argument. The problematic axiom is related to a principle Leibniz proposed as a way of dealing with a hole he found in an ontological proof of Descartes. Descartes, Leibniz, and Godel (and

XIV TYPES, TABLEAUS, AND GODEL'S GOD also Anselm) all have proofs that stick at the same point: showing that the existence of God is possible. If the Oodel argument is what you are interested in, start with Part III, and pick up earlier material as needed. Many of the uses of the formalism are relatively intuitive. Indeed, in Codel's notes on his ontological argument, formal machinery is never discussed, yet it is possible to get a sense of what it is about anyway. How Did This Get Written? Having just completed work on a book about first-order modal logic, [FM98], a look at higher-order modal logic suggested itself. I thought I would use Godel's ontological argument as a paradigm, because it is one of the few examples I have run across that makes essential use of higherorder modal constructs. Codel's argument for the existence of God is not particularly well-known, but there is a growing body of literature on it. This literature sometimes gives formalizations of Codel's rather sketchy ideas-generally along natural deduction or axiomatic lines. My idea was, I would design a tableau system within which the argument could be formalized, and this might lead to a nice paper illustrating the use of tableau methods. First, give tableau rules, then give Godel's proof. One cannot really develop semantic tableaus without a semantics behind it. The semantics of higher-order modal logic turned out to be of considerable intricacy, far beyond what could even be sketched in a paper. Clearly, an extended discussion of the semantics for higher-order modal logic was needed before the tableau rules could be motivated. I soon realized that in presenting higher-order modal logic, I was trying to explicate ideas coming from two quite different sources. On the one hand, there are essentially modal problems, some of which already arise at the first-order level and have little to do with higher-order constructs. On the other hand, a number of higher-order modal complexities also manifest themselves in a classical setting, and can be discussed more clearly without modalities complicating things. So I decided that before modal operators were introduced, I would give a thorough presentation of a semantics and tableau system for higher-order classical logic. There are already treatments of tableau, or Gentzen, systems for higher-order classical logic in the literature, but I felt it would be useful to give things in full here. Detailed completeness proofs are hard to find, for instance. Higher-order classical logic already has its hidden pitfalls. It is common knowledge, so to speak, that "true" higher-order classical models cannot correspond to any proof procedure. Henkin models are what is needed. But a "natural" formulation of tableaus is not complete with respect to Henkin models either. This is something known to experts-

PREFACE xv it was not known to me when I started this book. A broader notion of Henkin model (also due to Henkin) is needed, a non-extensional version. Such models should be better known since they are actually quite plausible things, and address problems that, while not common in mathematics, do arise in linguistic applications of logic. In the 1960's, cut-elimination theorems were proved for higher-order classical logic, using semantic methods that relied on non-extensional models. In effect, these cut-elimination proofs concealed a completeness argument within them, but the general notion of non-extensional model was not formulated abstractly--only the specific structure constructed by the completeness argument was considered. In short, a completeness theorem was never stated, only a consequence, albeit a very important one. So I found myself required to formulate a general notion of classical non-extensional Henkin model, then prove completeness for a suitable classical tableau system. After this I could move on to discuss modality. What sort of modal features did I want? Formalizations of the G6del argument by others had generally used some version of an intensional logic, with origins in work of Carnap, [Car56], developed and applied by Montague, [Mon60, Mon68, Mon70], and formally elaborated in [GaI75]. After several preliminary attempts I decided this logic was not quite what I wanted. In it, semantically speaking, all objects are intensional. I decided I needed a logic containing both intensional and extensional objects. Of course, one could bring extensional objects into the Montague setting by identifying them with objects that are rigid, in an appropriate sense, but it seemed much more natural to have extensional objects from the start. Thus the modal logic given in the second part of this book is somewhat different from what has been previously considered. Once I had formulated the modal logic I wanted, tableau rules were easy, and I could finally formalize the G6del argument. What began as a short paper had turned into a book. My after-the-fact justification is that there are few treatments of higher-order logic at all, and fewer still of higher-order modal logic. It is a rare flower in a remote field. But it is a pretty flower. Acknowledgments An earlier draft of this work was on my web page for some time, and I was given several helpful suggestions as a result. In particular I want to thank Peter Hajek, Oliver Kutz, Paul Gilmore, and especially Howard Sobel.