Types, Tableaus, and Gödel’s God (Trends in Logic)

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by

Melvin Fitting

TYPES, TABLEAUS, AND GO DEL'S GOD

TRENDS IN LOGIC

Studia Logica Library VOLUME 13 Managing Editor

Ryszard Wojcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland Editors

Daniele Mundici, Department of Computer Sciences, University of Milan, Italy Ewa Orlowska, National Institute of Telecommunications, Warsaw, Poland

Graham Priest, Department of Philosophy, University of Queensland, Brisbane, Australia

Krister Segerberg, Department of Philosophy, Uppsala University, Sweden

Alasdair Urquhart, Department of Philosophy, University of Toronto, Canada Heinrich Wansing, Institute of Philosophy, Dresden University of Technology, 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 FITTING Lehman College and the Graduate Center, City University of New York, U.S.A.

TYPES,••TABLEAUS, AND GO DEL'S GOD

~~~....

''

KLUWER ACADEMIC PUBLISHERS DORDRECHT/BOSTON/LONDON

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 1-4020-0604-7

Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.

Printed on acid-free paper

All Rights Reserved © 2002 Kluwer Academic Publishers 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. Printed in the Netherlands.

Contents

PREFACE Part I

Xl

CLASSICAL LOGIC

1. CLASSICAL LOGIC-SYNTAX

1 2

Terms and Formulas

3 3

Substitutions

8

2. CLASSICAL LOGIC-SEMANTICS Classical Models 1 Truth in a Model 2 Problems 3 Compactness 3.1 Strong Completeness 3.2 Weak Completeness 3.3 And Worse 3.4 Henkin Models 4 Generalized Henkin Models 5 A Few Technical Results 6 6.1 Terms and Formulas Extensional Models 6.2 Language Extensions 6.3 3. CLASSICAL LOGIC-BASIC TABLEAUS A Different Language 1 Basic Tableaus 2 Tableau Examples 3 v

11 11

12 15 15 16 16 17 19 24 29 29 29 30 33 33

35 37

vi

TYPES, TABLEAUS, AND GODEL'S GOD

4. SOUNDNESS AND COMPLETENESS Soundness 1 Completeness 2 2.1 Hintikka Sets 2.2 Pseudo-Models 2.3 Substitution and Pseudo-Models 2.4 Hintikka Sets and Pseudo-Models 2.5 Pseudo-Models are Models 2.6 Completeness At Last Miscellaneous Model Theory 3

43 43 46 47 48 52 59 62 63 66

5. EQUALITY Adding Equality 1 Derived Rules and Tableau Examples 2 Soundness and Completeness 3

69 69 69 73

6. EXTENSIONALITY 1 Adding Extensionality A Derived Rule and an Example 2 Soundness and Completeness 3

77

Part II

77 77 79

MODAL LOGIC

7. MODAL LOGIC, SYNTAX AND SEMANTICS Introduction 1 Types and Syntax 2 Constant Domains and Varying Domains 3 4 Standard Modal Models Truth in a Model 5 Validity and Consequence 6 Examples 7 Related Systems 8 Henkin/Kripke Models 9

83 83 86 89 90 92 94 95 101 102

8. MODAL TABLEAUS 1 The Rules 1.1 Prefixes Propositional Rules 1.2

105 105 105 107

vii

Contents

2

1.3 Modal Rules 1.4 Quantifier Rules 1.5 Abstraction Rules 1.6 Atomic Rules 1.7 Proofs and Derivations Tableau Examples

107 108 109 109 110 111

3

A Few Derived Rules

113

9. MISCELLANEOUS MATTERS Equality 1 1.1 Equality Axioms 1.2 Extensionality

115 115 115 117

2

De Re and De Dicta

3 4

Rigidity

118 121

Stability Conditions

124

5

Definite Descriptions

6

Choice Functions

125 128

Part III

ONTOLOGICAL ARGUMENTS

10. GODEL'S ARGUMENT, BACKGROUND Introduction 1 2 Anselm Descartes 3 Leibniz 4 Godel 5 6 Godel's Argument, Informally

133 133 134 134 137 138 139

11. GODEL'S ARGUMENT, FORMALLY

145

1

General Plan

145

2

Positiveness

145

3

Possibly God Exists

4

Objections

150 152

5 6 7

Essence Necessarily God Exists Going Further Monotheism 7.1

156 160 162 162

viii


7.2 Positive Properties are Necessarily Instantiated More Objections

162

9 10

A Solution

164

Anderson's Alternative

169

11

Conclusion

171

8

163

REFERENCES

173

INDEX

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. If you 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

Xll


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.') If you 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, Tol75, 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 I. 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

XlV


also Anselm) all have proofs that stick at the same point: showing that the existence of God is possible. If the Godel 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 Godel'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. Godel'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 Godel'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 corning 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 hr"e. 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 Godel argument by others had generally used some version of an intensional logic, with origins in work of Carnap, [Car 56], developed and applied by Montague, [Mon60, Mon68, Mon70], and formally elaborated in [Gal75]. 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 Godel 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.

I

CLASSICAL LOGIC

Chapter 1

CLASSICAL LOGIC-SYNTAX

1.

Terms and Formulas

The formulation of a higher-order logic allows some freedom-there are certain places where choices can be made. Several of these choices produce equivalent results. Before getting to the formal machinery, I informally set out my decisions on these matters. Other treatments may make different choices, but ultimately it is largely a matter of convenience that is involved. Often classical first-order logic is formulated with a rich variety of terms, built up from constant symbols and variables using function symbols. Since higher-order constructs are already complicated, I have decided to have constant symbols but not function symbols. If necessary for some purpose, it is not a major issue to add them-doing so yields a conservative extension. Higher-order logic can be formulated with or without explicit abstraction machinery. Speaking informally, one wants to make sure that every formula specifies a class, but there are two ways of making this happen. One is to assume comprehension axioms, formulas of the general form:

where cp(x1, ... , xn) is a formula with free variables as indicated. Such axioms ensure that to each formula corresponds an 'object.' The other approach is to elaborate the term-forming machinery, so that there is an explicit name for the object specified by a formula cp. This involves predicate abstraction, or >-.-abstraction:

3 M. Fitting, Types, Tableaus, and Gödel's God © Kluwer Academic Publishers 2002

4


The two approaches are equivalent in a direct way. I have chosen to use explicit abstracts for several reasons. First, axioms are not as natural when tableau systems are the proof machinery of choice. And second, predicate abstraction has already played a major role in earlier investigations of modal logic [FM98], and makes discussion of major issues considerably easier here. Finally, one can characterize higher-order formulas more-or-less the way it is done in the first-order setting, taking quantifiers and connectives as "logical constants." This is the approach of [Sch60]. Alternatively, following [Chu40], one can think of quantifiers and connectives as constants of the language, which itself is formulated in lambda-calculus style. In this book I take the first approach, though one can make arguments for the second on grounds of elegance and economy. My justification is that doing things the way that has become standard for first-order logic will be less confusing to the reader. Recently one further alternative has become available. In [Gil99, GilOl], Paul Gilmore has shown that by a relatively simple change, a system of classical higher-order logic can be developed allowing a controlled degree of impredicativity-typing rules can be relaxed to permit the formation of certain useful sentences that are not "legal" in the approach presented here. This, in turn, allows a more natural development of arithmetic in the higher-order setting. I do not follow Gilmore's approach here, but I recommend it for study. Much of what I develop carries over quite directly. So these are my choices: no function symbols, explicit predicate abstraction, quantifiers and connectives as in the first-order setting, and no impredicativity. With this out of the way I can begin presenting the formal syntactical machinery. In first-order logic, relation symbols have an arity--some are oneplace, some are two-place, and so on. In higher-order logic this simple idea gets replaced by a typing mechanism, which is considerably more complex. Terms, and certain other items, are assigned types, and rules of formation make use of these types to ensure that things fit together properly. I begin by saying what the types are. DEFINITION 1.1 (TYPE) 0 is a type. If t1, ... , tn are types, (tl, ... , tn) is a type. I generally use t, t1, t2, t', etc. to represent types.

An object of type 0 is intended to be a ground-level object-it corresponds to the designation of a constant symbol or variable in standard first-order logic. An object of type (t 1 , ... , tn) is a predicate that takes n arguments, of types t 1 , ... , tn respectively. Thus a constant symbol of type (0, 0, 0), say, would be called a three-place relation symbol in

CLASSICAL LOGIC-SYNTAX

5

standard first-order logic-it applies to three ground-level arguments. But now we can have relation symbols of types such as ( (0), (0, 0), 0), to which nothing in first-order logic corresponds. DEFINITION 1.2 (L(C)) Let C be a set of constant symbols with a type associated to each, containing at least an equality symbol =(t,t) for each type t. I denote the classical higher-order language built up from C by L( C). The rest of this section amounts to the formal characterization of L(C). For each type t I assume there are infinitely many variable symbols of that type. I generally use letters from the beginning of the Greek alphabet to represent variables, with the type written as a superscript: at, j3t, '"'/, . . . . Likewise I generally use letters from the uppercase Latin alphabet as constant symbols, again with the type written as a superscript: At, Bt, ct, Dt, .... As noted, equality is primitive, so for each type t there is a constant symbol =(t,t) of type (t, t). Often types can be inferred from context, and so superscripts will be omitted where possible, in the interests of uncluttered notation. Sometimes it is helpful to refer to the order of a term or formula-firstorder, second-order, and so on. It is simplest to define this terminology first for types themselves. DEFINITION 1.3 (ORDER) Type 0 is of order 0. Type (t1, ... , tn) has as its order the maximum of the orders of t1, ... , tn, plus one. Thus (0, 0) is of order 1, or first-order. Likewise (0, (0, 0)) is of order 2, or second-order. Types will play the fundamental role, but order provides a convenient way of referring to the maximum complexity of some construct. When I talk about the order of a constant or variable, I mean the order of its type. Likewise once formulas are defined, I may refer to the order of the formula, by which I mean the highest order of a typed part of it. Next I define the class of formulas, and their free variables. This definition is more complex than the corresponding first-order version because the notion of term cannot be defined first; both term and formula must be defined together. And to define both, I need the auxiliary notion of predicate abstract which is, itself, part of a mutual recursion involving Definitions 1.4, 1.5, and 1.6. DEFINITION 1.4 (PREDICATE ABSTRACT OF L(C)) Let .a1, ... , an.), in which case must be a formula of degree < k, so using the induction hypothesis with (2.3) we have

(v *I* A)( (>.a~, ... , an.)) = {(01, ... , On) I M 11-v,A [ai/01, ... , an/On]}= { (0~, ... , On) I M 11-v,A' [ai/01, ... , an/On]}= (v*I*A')((>.a1,··· ,an.)) Thus (2.4) holds for terms of degree ~ k. Now assume is a formula of degree k. There are several cases, depending on the form of . If is atomic, it is T( T1, ... , Tn) where T, T1, ... , Tn are all of degree ~ k. Since (2.4) holds for terms of degree< k by assumption, and for terms of degree = k by the proof above,

M 11-v,A T(T1, ... , Tn) {:}

((v *I* A)(T1), ... , (v *I* A)(Tn)) E (v *I* A)(T) {:} ((v*I*A')(T1), ... ,(v*I*A')(Tn)) E (v*I*A')(T) {:} M 11-v,A' T(T1, ... , Tn) If is a negation, conjunction, or universally quantified formula, the result follows easily using the fact that (2.3) holds for its subformulas (which are of lower degree), by the induction hypothesis. We thus have (2.3) for formulas of degree k, and this concludes the induction. •

Note on Induction Proofs The pattern of the induction proof above will recur many times, with little variation of structure. We go from terms and formulas of degrees < k to terms of degrees ~ k, and then to formulas of degrees ~ k. The Proposition above allows us to give the following extension of Definition 2.17. DEFINITION 2.19 (HENKIN MODEL) If (M,A) is a Henkin model, the proper abstraction designation function A is uniquely determined, so we

24


will say the Henkin frame M itself is a Henkin model, and write M lf-v
.a1, ... , an..ab··· ,an..-bound. (This all takes on a significant role in the next chapter.) I note the fundamental problem: even with the restrictions imposed on the additional variables, the collection of predicate abstracts of L + (C) properly extends that of L( C).

2.33 Each generalized Henkin model with respect to L(C) can be converted into a generalized Henkin model with respect to L+(c) so that truth values for formulas of L( C) are preserved.

PROPOSITION

There are two immediate consequences of this Proposition that I want to state, before I sketch its proof. First, any set S of sentences of L( C) that is satisfiable in some generalized Henkin model with respect to L( C) is also satisfiable in some generalized Henkin model with respect to L+(C). And second, any sentence of L(C) that is valid in all generalized Henkin models with respect to L +(C) is also valid in all generalized Henkin models with respect to L( C) (because an L( C) countermodel can be converted into a L+(c) countermodel).

Proof The proof basically amounts to replacing the new variables of L + (C) by some from L( C), to determine behavior of predicate abstracts. I only sketch the general outlines. Let M = (H, I,£) be a generalized Henkin frame, and let (M, A) be a generalized Henkin model with respect to L(C). Recall the notational convention: {/h / a1, . . . , f3n /an} is the substitution that replaces each f3i by the corresponding ai. Also, if v is a valuation, by v{fil/al, ... , fin/an} I mean the valuation v' such that v'(ai) = v(fii), and on other free variables, v' and v agree. Now we extend A to an abstraction designation function, A', suitable for L+(C). For each predicate abstract (A"Yl, ... ,')'k.) of L+(C), and for each valuation v with respect to L+(C), do the following. Let (31, . . . , fin be all the free variables of that are in the language L + (C) but not in L( C), and let a1, ... , an be a list of variables of L( C) of the same corresponding types, that do not occur in , free or bound. Now, set

A' (v, (>.1'1, ... , '/'k·)) = A(v{(Jl/al, ... ,fin/an}, (A')'l, ... ,')'k.{fil/al, ... ,fin/an})) It can be shown that this is a proper definition, in the sense that it does not depend on the particular choice of free variables to replace the f3i· Now it is possible to show that (M, A') is a generalized Henkin model with respect to L + (C), and truth values of sentences of L( C) evaluate the same with respect to A and A'. One must show a more general result, involving formulas with free variables. The details are messy, and I omit them. •

32


Finally, Proposition 2.33 has a kind of converse. Together they say the difference between L(C) and L+(C) doesn't matter semantically. I omit its proof altogether. PROPOSITION 2.34 A generalized Henkin model with respect to L+(c) can be converted into a generalized Henkin model with respect to L( C) so that truth values for formulas of L( C) are preserved.

Exercises EXERCISE 6.1 Give a proof of Proposition 2.30. EXERCISE 6.2 Give a proof of Proposition 2.31. EXERCISE 6.3 Supply details for a proof that each generalized Henkin frame that is extensional is isomorphic to a Henkin frame.

Chapter 3

CLASSICAL LOGIC-BASIC TABLEAUS

Several varieties of proof procedures have been developed for firstorder classical logic. Among them the semantic tableau procedure has a considerable attraction, [Smu68, Fit96]. It is intuitive, close to the intended semantics, and is automatable. For higher-order classical logic, semantic tableaus are not as often seen-most treatments in the literature are axiomatic. Among the notable exceptions are [Tol75, Smi93, Koh95, GilOl]. In fact, semantic tableaus retain much of their first-order ability to charm, and they are what I present here. Automatability becomes more problematic, however, for reasons that will become clear as we proceed. Consequently the presentation should be thought of as meant for human use, and intelligence in the construction of proofs is expected. This chapter examines what I call a basic tableau system; rules are lifted from those of first-order classical logic, and two straightforward rules for predicate abstracts are added. It is a higher-order version of the second-order system given in [Tol75]. I will show it corresponds to the generalized Henkin models from Section 5 of Chapter 2. In Chapters 5 and 6 I make additions to the system to expand its class of theorems and narrow its semantics to Henkin models.

1.

A Different Language

In creating tableau proofs I use a modified version of the language defined in Chapter 2. That is, I give tableau proofs of sentences from the original language L( C), but the proofs themselves can involve formulas from a broader language that is called L + (C). Before presenting the tableau rules, I describe the way in which the language is extended for proof purposes. 33 M. Fitting, Types, Tableaus, and Gödel's God © Kluwer Academic Publishers 2002

34


Existential quantifiers are treated at higher orders exactly as they are in the first-order case. If we know an existentially quantified formula is true, a new symbol is introduced into the language for which we say, in effect, let that be something whose value makes the formula true. As usual, newness is critical. For this purpose it is convenient to enhance the collection of free variables by adding a second kind, called parameters. 3.1 (PARAMETERS) In L(C), for each type t there is an infinite collection of free variables of that type. The language L + (C) differs from L( C) in that, for each t there is also a second infinite list of free variables of type t, called parameters, a list disjoint from that of the free variables of L( C) itself. Parameters may appear in formulas in the same way as the original list of free variables but they are never quantified or A bound. p, q, P, Q, ... are used to represent parameters. DEFINITION

Parameters appear in tableau proofs. They do not appear in the sentences being proved. Since they come from an alphabet distinct from the original free variables, an alphabet that is never quantified or A bound, we never need to worry about whether the introduction of a parameter will lead to its inadvertent capture by a quantifier or a Aintroducing them will always involve a free substitution. Thus rules that involve them can be relatively simple.

Special Terminology Technically, parameters are a special kind of free variable. But to keep terminology simple, I will continue to use the phrase free variable for the free variables of L( C) only, and when I want to include parameters in the discussion I will explicitly say so. The notion of truth in generalized Henkin models must also be adjusted to take formulas of £+(c) into account. As I have just noted, parameters are special free variables, so when dealing semantically with L + (C), valuations must be defined for parameters as well as for the free variables of L( C). Essentially, the difference between a generalized Henkin frame and a generalized Henkin model lies in the requirement that the extension of a formula appearing in a predicate abstract must correspond to the designation of that abstract, which is a member of the appropriate Henkin domain. In L + (C) there are parameters, so there are more formulas and predicate abstracts than in L( C). Then requiring that something be a generalized Henkin model with respect to L + (C) is apparently a stronger condition than requiring it be one with respect to L( C), though Section 6 establishes that this is not actually so. DEFINITION 3.2 (GROUNDED) A term or a formula of £+(C) is grounded if it contains no free variables of L( C), though it may contain parameters.


35

The notion of grounded extends the notion of closed. Specifically, a grounded formula of L + (C) that happens to be a formula of L( C) is a closed formula of L( C), and similarly for terms.

2.

Basic Tableaus

I now present the basic tableau system. It does not contain machinery for dealing with equality-that comes in Chapter 5. The rules come from [Tol75], where they were given for second-order logic. These rules, in turn, trace back to the sequent-style higher-order rules of [Pra68] and [Tak67]. All tableau proofs are proofs of sentences-closed formulas-of L( C). A tableau proof of q> is a tree that has --,q> at its root, grounded formulas of L + (C) at all nodes, is constructed following certain branch extension rules to be given below, and is closed, which means it embodies a contradiction. Such a tree intuitively says --,q> cannot happen, and so q> is valid. The branch extension rules for propositional connectives are quite straightforward and well-known. Here they are, including rules for various defined connectives. DEFINITION

3.3 (CONJUNCTIVE RULES)

XI\Y

•(X V Y)

X

-.x

X

y

--,y

--,y

•(X

=:J

Y)

For the conjunctive rules, if the formula above the line appears on a branch of a tableau, the items below the line may be added to the end of the branch. The rule for double negation is of the same nature, except that only a single added item is involved. DEFINITION

3.4

(DOUBLE NEGATION RULE)

•• x X

Next come the disjunctive rules. For these, if the formula above the line appears on a tableau branch, the end node can have two children added, labeled respectively with the two items shown below the line in the rule. In this case one says there is tableau branching.

36 DEFINITION


3.5

(DISJUNCTIVE RULES)

XVY XIY

•(X A Y)

·Xi·Y •(X = Y) ·(X ~ Y) I ·(Y ~ X)

This completes the propositional connective rules. The motivation should be intuitively obvious. For instance, if X A Y is true in a model, both X and Y are true there, and so a branch containing X A Y can be extended with X and Y. If X V Y is true in a model, one of them is true there. The corresponding tableau rule says if X V Y occurs on a branch, the branch splits using X and Y as the two cases. One or the other represents the "correct" situation. Though the universal quantifier has been taken as basic, it is convenient, and just as easy, to have tableau rules for both universal and existential quantifiers directly. To state the rules simply, I use the following convention. Suppose ( li) is a formula in which the variable at, of type t, may have free occurrences. And suppose Tt is a term of type t. Then ( Tt) is the result of carrying out the substitution {at /Tt} in ( at), replacing all free occurrences of at with occurrences of Tt. Now, here are the existential quantifier rules. 3.6 (EXISTENTIAL RULES) In the following, pt is a parameter of type t that is new to the tableau branch.

DEFINITION

(:Jat).x.-.P(x, x)}(y)] V [-.P(x, y) 1\ (>.x.-.P(x, x))(y)]} -.(:Jy){[P(p, y) 1\ -.(>.x.-.P(x, x))(y)] V [-.P(p, y) 1\ (>.x.-.P(x, x))(y)]} 4. -.{[P(p,p) 1\ -.(>.x.-.P(x, x))(p)] V [-.P(p,p) 1\ (>.x.-.P(x, x))(p)]} 5. -.[P(p,p) 1\ -.(>.x.-.P(x,x))(p)] 6.

?,p)

~

A

{Ax.~P(x,~7

~

os

-.P(p,p)

&S C,)

~

~

/

-.-.P(p,p)

11.

~

-.-.(>.x.-.P(x, x))(p) 9. (>.x.-.P(x, x))(p) 10.

8.

-.(>.x.-.P(x, x))(p) -.-.P(p,p) 15.

/

12.

-.-.P(p,p) 13. -.P(p,p) 16.

~

-.(>.x.-.P(x, x))(p)

14.

0

~ ~

0 ~

~

t3

Figure 3.1.

Tableau Proof of (\fR)(3X)(\fx)(3y){[R(x, y) 1\ •X(y)] V [•R(x, y) 1\ X(y)]}

3.

40


EXAMPLE 3.13 It is a well-known result of modal model theory that a relational frame is reflexive if and only if every instance of DP => P is valid in it. I want to give a formal version of this using the machinery of higher-order classical logic. Suppose we think of the type 0 domain of a higher-order classical model as being the set of possible worlds of a relational frame. Let us think of the atomic formula P(x) as telling us that P is true at world x, and R( x, y) as saying y is a world accessible from x. Then making use of the usual Kripke semantics, ('v'y)[R(x, y) => P(y)] corresponds to P being true at every world accessible from x, and hence to DP being true at world x, where R plays the role of the accessibility relation. Then further, saying DP => P is true at x corresponds to ('v'y)[R(x, y) => P(y)] => P(x). We want to say that if this happens at every world, and for all P, the relation R must be reflexive, and conversely. Specifically, I give a tableau proof of the following. In it, take R to be a constant symbol.

('v'x)R(x, x)

=('v'P)('v'x){('v'y)[R(x, y) => P(y)] => P(x)}

(3.3)

Actually, the implication from left to right is straightforward-! supply a tableau proof from right to left.

--,{('v'P)('v'x){('v'y)[R(x, y) => P(y)] => P(x)} => ('v'x)R(x, x)} 1. ('v'P)('v'x){(Vy)[R(x,y) => P(y)] => P(x)} 2. --,(\fx)R(x, x) 3. --,R(p, p) 4. ('v'x){('v'y)[R(x, y) => (Az.R(p, z))(y)] => (Az.R(p, z))(x)} 5. (Az.R(p,z))(y)] ::> (~)(p) 6.

(\ly)[7

--,(\fy)[R(p, y) => (Az.R(p, z))(y)] --,[R(p, q) => (Az.R(p, z))(q) 9. R(p, q) 10. --,(Az.R(p, z))(q) 11. --,R(p, q) 12.

7.

(Az.R(p, z))(p) R(p,p) 13.

8.

In this, 2 and 3 are from 1 by a conjunctive rule; 4 is from 3 by an existential rule (p is a new parameter); 5 is from 2 by a universal rule ( (Az.R(p, z)) is a grounded term); 6 is from 5 by a universal rule (p is a grounded term); 7 and 8 are from 6 by a disjunctive rule; 9 is from 7 by an existential rule (q is a new parameter); 10 and 11 are from 9 by a conjunction rule; 12 is from 11 and 13 is from 8 by abstract rules.

41


The last example is a version of the famous Knaster-Tarski theorem [Tar55]. 3.14 Let 1) be a set and let F be a function from its powerset to itself. F is called monotone provided, for each P, Q ~ 1), if P ~ Q then F(P) ~ F(Q). Theorem: any monotone function F on the powerset of 1) has a fixed point, that is, there is a set C such that F(C) =C. (Actually the Knaster-Tarski theorem says much more, but this will do for present purposes.) I now give a formalization of this theorem. Since function symbols are not available, I restate it using relation symbols, and it is not even necessary to require functionality for them. Now, (Vx)(P(x) :J Q(x)) will serve to formalize P ~ Q. If F(P, x) is used to formalize that x is in the set F(P), then (Vx)(P(x) :J Q(x)) :J (Vx)(F(P,x) :J F(Q,x)) says we have monotonicity. Then, the following embodies a version of the Knaster-Tarski theorem (F is a constant symbol). EXAMPLE

(VP)(VQ)[(Vx)(P(x) :J Q(x)) :J (Vx)(F(P, x) :J F(Q, x))] :J (3S)('v'x)(F(S, x)

= S(x))

(3.4)

I leave the construction of a tableau proof of this to you as an exercise, but I give the following hint. Let ..x.(VP).o:1, ... , O:n.) be a predicate abstract of type (t1, ... , tn). SetAH(v,(>.o:l,··· ,o:n.)) = (T+v",S) where

5 Let T =

S = { (01, ... , On) E 1iH(ti) X · · • X 1iH(tn) M lf--v,AH [o:I/01, · · · , O:n/On]}.

I

The structure (M, AH) is a pseudo-model, relative to the Hintikka set H.

The definition above has the usual complex recursive structure, with truth at the atomic level needing (v * I * AH) and hence AH, and the characterization of AH itself needing the notion of truth for formulas. Of course what makes it work is the fact that, in every case, behavior of some construct on a formula or term requires constructs involving simpler formulas and terms. The key point is, why is this called a pseudo-model, and not simply a model? The answer is, we have the characterization of the abstraction designation function backwards here. In Chapter 2 we assumed we had a function A that mapped valuations and abstracts to members of Henkin

52


domains. Then we imposed a condition that A map to the "right" members, so that our intuitions concerning abstracts would be respected. Here we made that intuitive condition the defining property, reversing the usual order of things. But now we have no guarantee that AH must assign values that are in the Henkin domains. Looking at part 5 of the definition above, it is clear that, for an abstract T of type (t1, ... , tn), E(AH(v, T)) will be a subset of 'HH(tl) x · · · x 'HH(tn), but we do not know that AH(v,T) will be a possible value, and hence a member of 'HH( (t1, ... , tn) ). In short, while quantifiers range over Henkin domains (condition 4 above), for all we know some terms-abstracts-can have values that fall outside them. As a matter of fact, it will be proved that this does not happen, but it is not obvious, and it is not easy to establish.

Exercises EXERCISE 2.1 Show that if entity E is a possible value, then E must be a possible value of T(E).

2.3

Substitution and Pseudo-Models

In this subsection valuations and substitutions are shown to be wellbehaved with respect to pseudo-models. It should be noted again that valuations always mean valuations in a pseudo-model-they map variables to members of Henkin domains, to possible values. They do not map to arbitrary entities. The proofs below are rather technical, so I begin with the statements of the two Propositions to be established, after which their proofs are given, broken into a number of Lemmas. On a first reading you might want to just read the Propositions and skip over the proofs. The first item should be compared with Proposition 2.30. PROPOSITION 4.15 Let H be a Hintikka set, let M = ('HH,I,E) be a generalized frame relative to H, and let (M, AH) be a pseudo-model relative to H. Also let v and w be valuations. 1 If v and w agree on the free variables of the term T (v*I*AH)(T) = (w*I*AH)(T). 2 If v and w agree on the free variables of the formula M II-v,AH ¢::::::} M II-w,AH ·

The second item is an analog to Proposition 2.31. Definition 2.26 is carried over to the present setting: given (M, AH), for each valuation v and substitution CJ, the valuation vu is defined by o:va = (v *I *AH) (o:o-).

SOUNDNESS AND COMPLETENESS

53

4.16 Again let H be a Hintikka set, let M = (1lH,I,£) be a generalized frame relative to H, and let (M, AH) be a pseudo-model relative to H. For any substitution CJ and valuation v:

PROPOSITION

1 If CJ is free for the term T then (v * T * AH)(TCJ) = (vu * T * AH)(T). 2 If CJ is free for the formula
····"'n(o:i) = (V*L*AH)(o:iCJa 1 , ... ,an) = (v*L*AH)(o:i) = v(o:i)· • 4.18 Let H be a Hintikka set, let M = (1tH,I,£) be a generalized frame relative to H, and let (M, AH) be a pseudo-model relative to H. For any term T of L+(C), T~ = T((v *I* AH)(T)).

LEMMA

56


Proof Suppose first that T is a predicate abstract. Then by Definition 4.14, (v *I* AH)(T) = AH(v,T) = (T*v,S) for a certain setS, and so T((v *I* AH)(T)) = T*v. If T is a variable or parameter, T*v = T(v(T)) by definition of tv", and v(T) = (v *I* AH)(T) by definition of (v * I * AH) again, for variables. If T is a constant symbol, T*v = T, and also T((v *I* AH)(T)) = T(I(T)) = T because I is an allowed interpretation. • The proof of Proposition 4.16 is by an induction on degree. Since the steps are somewhat complex, I have separated the significant parts out, in the following two Lemmas. 4.19 Let H be a Hintikka set, let M = ('HH, I,£) be a generalized frame relative to H, and let (M, AH) be a pseudo-model relative to H. Assume that for each formula of degree < k, whenever substitution C5 is free for then

LEMMA

M lf-v,A C5 {::} M lf-v",A . Then for each term T of degree k, and for each substitution C5 that is free forT, (v *I * AH) (TC5) = (vu *I * AH) (T). Proof Assume the hypothesis concerning formulas, and suppose T is a term of degree k. If k is 0, T must be a constant symbol, a variable, or a parameter. If Tis a variable or parameter, say a, then (vu *I*AH)(a) = vu (a) = (v *I* AH) (ae5). The case of a constant symbol is trivial. Now suppose k > 0, and so T must be of the form (>.a1, ... , an.), where is a formula whose degree is < k. And suppose C5 is free for (>.a1, ... , an. ). Using the definition of substitution and Definition 4.14:

(v*I*AH)((>.a1, ... ,an.)C5) = (v*I*AH)((>.a1, ... ,an.C5a 1 , ... ,an)) =AH(v,(>.a1,··· ,an.C5a 1 , ... ,aJ) =(a, S) where

a= (>.a1, ... ,an.C5a 1 , ... ,an)+v S = {(01, ... , On) E 'HH(t1) X .. · X 'HH(tn) I M lf-v,AH C5a 1 , ... ,an[al/01, · · · , an/On]}.

57


Similarly:

where

aI = ( >.a1. ... , an. )+a v S' = { (01, ... , On) E 'HH(tl) X · · · X 'HH(tn) M 11-v",AH [al/01, ... , an/On]}.

J

So, we must show a= a' and S = S'.

Part 1, a= a'. First of all,

a= (>.al, ... ,an.aal,···,an)1J = ((>.al,··· ,an.))a)1J = (>.a1, ... , an.)(a11)

(4.14) (4.15)

In this, (4.14) is by definition of substitution. Recall we are assuming that a is free for (>.a1. ... , an. ). Also, since 11 replaces variables by grounded terms, and parameters are never bound, substitution 11 is free for (>.a1, ... , an.)a. Then (4.15) follows by Theorem 1.18. So, to show a= a' it is enough to show the substitutions a11 and if are the same. Let (3 be a variable or parameter. f3(a11) = ([3a)11 by definition of composition for substitutions. And, using Definition 4.13, (3if = T(v 17 ({3)) = T((v *I* AH)(f3a)). Finally, (f3a)1J and T((v *I* AH)(f3a)) are the same, by Lemma 4.18. We thus have shown that a= a'.

Part 2, S = S'. Using Proposition 1.15 it can be assumed that a is the identity on variables and parameters that do not occur free in (>.a1, ... , an. ).

58


8= {(01, ... 0n) I M lf-v,AH <Paa 1 , ... ,an[o:I/01,··· ,o:n/On]} = { (w(o:1), ... , w(o:n)) I M lf-w,AH <Paa 1 , ... ,an where w is an 0:1, ... , O:n variant of v} = {(w(o:1), ... ,w(o:n)) I M lf-wu"l· .. ··"n,AH
of L+(C), if <J>"V E H then M lf-v,AH <J>.

Proof Both parts of the theorem are shown together by a simultaneous induction on degree. Assume they hold for formulas and terms of degree < k. It will first be shown that item 1 holds for terms of degree k; then it will be shown that item 2 holds for formulas of degree k. Part 1. Let T be a term of degree k. If k happens to be 0, T is a constant symbol, variable, or parameter. If T is a constant symbol A, A"V =A, and (v *I* AH)(A) = I(A), which is a possible value of A because I is an allowed interpretation. If Tis a variable or parameter, a, (v *I* AH) (a) = v( a) is some possible value E because v is a valuation in the pseudo-model. But then a"V = T(·v(a)) = T(E), and E is a possible value of T(E) by Exercise 2.1. Now suppose T = (.Aa1, ... , an.). Then (v*I*AH)(r) = AH(v, r) = (r"V, S) where S = { (Ot, ... , On) E HH(ti) =

X .. ·X

HH(tn)

I

M lf-v,AH <J>[ai/01, .. · , an/On]} { (w(ai), ... , w(an)) I M lf-w,AH where w is an a1, ... , an variant of v}.

I show (r"V, S) is a possible value of r"V relative to H. To do this it must be shown that if E1 is a possible value for T1, ... , En is a possible value for Tn, then


2 •(71f)(71, ... , 7n) E H implies (E1, ... , En)

61

fl. S.

I show the first of these; the second is similar. So assume E1 is a possible value for 71, . . . , En is a possible value for 7n, and (71f)(71,··. ,7n) E H. That is, [(>.a1, ... ,an.)1f](71, ... ,7n) E H. By definition of substitution we have (.Xa1, · · · , an.+v a 1 , ... ,an)(71, · · · , 7n) E H. Since H is a Hintikka set, it follows that (Definition 4.6, part 7) [+v a1, ... ,an]{al/71, ... , an/7n} E H. Since 71, ... , 7n are grounded terms, they do not contain any of a1, ... , an free. Now, let w be the a1. ... , an-variant of v such that w(a 1) = E 1, ... , w(an) =En. Since Ei is a possible value for the grounded term 7i it follows that ai w = 7i. And if j3 f= a1, ... , an then j3w = j31J. Then [ tv a 1 , ... ,an]{ al/71, ... , an/Tn} = w so

w E H. must be of lower degree than (>.a1, ... ,an.), that is, k, so the induction hypothesis applies and

M 11-w,AH . Then (w(a1), ... , w(an)) E S, so (E1, ... , En) E S, which is what we wanted. This concludes the induction step for terms. Part 2. Let be a formula of degree k. By the induction hypothesis the result holds for formulas and terms of degree < k, and by part 1 of the proof it also holds for terms of degree k. Now we have several cases, depending on the form of . I only present a few of them. Suppose is 7o(71, ... , 7n) and [To(71, ... , 7n)]tv E H. That is,

Each 7i is of degree ::; k so by the induction hypothesis, each (v * I * AH)(7i) is a possible value for 7i1f. It follows immediately from the definition of possible value (Definition 4.9) and the definition of£ (Definition 4.8) that

62


and so

Suppose
1'(72))] 3. [(71 = 72) ::::> (\f/)('y(7I) ::::> 1'(72))] 4. [('v'r)('y(7I) ::::> 1'(72)) ::::> (71 = 72)] 5.

•(71

/~

=

72) 6. (\f/)('y(7I) ::::> /(72)) 7. (Aa. (Aa.(a))(72) 8.

/ •(Aa. r(Q)) ~ (P = Q)] 3. (Vr)('y(P) ~ 1 (Q)) 4. --,(P = Q) 5. (AX.--,(X = Q))(P) ~ (AX.--,(X = Q))(Q) 6.

/

--,(AX.--,(X = Q))(P) --,--,(P = Q) 9.

7.

~

(AX.--,(X = Q))(Q) --,( Q = Q) 10. (Q = Q) 11.

8.

Here 2 is from 1, and 3 is from 2 by an existential rule (P and Q are new parameters of the appropriate type); 4 and 5 are from 3 by a conjunctive rule; 6 is from 4 by a universal rule, using the grounded term (AX.--,(X = Q)); 7 and 8 are from 6 by a disjunction rule; 9 and 10 are from 7 and 8 by abstract rules; 11 is the derived reflexivity rule. Though the derived tableau rules for equality allow us to prove the axioms, it does not follow they are their equivalent. To establish that, we would need to have a cut elimination theorem for the tableau system with the equality rules. And the way to prove cut elimination is to first have a completeness proof. I conjecture that such a completeness result is provable, but I don't know how to do it.

Exercises 2.1 Prove the following characterization of equality-it says it is the smallest reflexive relation. EXERCISE

(Vx)(Vy){(x = y) EXERCISE

= (VR)[(Vz)R(z,z)

~

R(x,y)]}

2.2 Give a tableau derivation of the following from EQ.

(Va)(V,B)[(a =,B) ~ (Vr)(a('y)

= ,8(1))]

More generally, one can do the same with the following.

EQUALITY

3.

73

Soundness and Completeness The results of this section combine to prove the following.

THEOREM 5.6 Let
.X.X(x))(P) (>.X.X(x))(Q)] :::> (>.X, X, Y.X(X) :::> X(Y))(P,P,Q)

= =

•{(Vx) [(>.X.X(x))(P) (>.X.X(x))(Q)] :::> (>.X, X, Y.X(X) :::> X(Y))(P, P, Q)} 1. (\fx) [(>.X.X(x))(P) (>.X.X(x))(Q)] 2. •(>.X,X,Y.X(X) :::> X(Y))(P,P,Q) 3. --, [P(P) :::> P(Q)] P(P) 5. •P(Q) 6.

4.

~~ P=Q

--, [P(p) = Q(p)] 7. (.XX.X(p))(P) (.XY.Y(p))(Q) 10.

=

P(Q)

8. 9.

In this, 2 and 3 are from 1 by a conjunctive rule; 4 is from 3 by an abstract rule; 5 and 6 are from 4 by a conjunctive rule. Now I apply the extensionality rule. Take 71 to be P and 72 to be Q, both of which are grounded, and take p to be a new parameter. We get a split to 7 and 8. Item 9 is from 5 and 8 by substitutivity, and the right branch is closed. Item 10 is from 2 by a universal rule. The left branch can be continued to closure. I leave this to you.

Exercises EXERCISE 2.1 Give a proof of formula (3.1) from Example 3.12. EXERCISE 2.2 Show that the rule contained in Definition 6.2 is, in fact, a derived rule, using EXT.

79

EXTENSIONALITY

3.

Soundness and Completeness

I sketch a proof that the sentences having tableau proofs using EQ U EXT as axioms are exactly the sentences valid in normal Henkin models (and similarly for derivability as well). Soundness takes very little work. It just amounts to the observation that all members of EQ U EXT are valid in normal Henkin models. Completeness also takes very little work. Using results of Chapter 4, if a sentence If? does not have a tableau proof using EQ U EXT as axioms, there is a generalized Henkin model in which If? is false, but in which all of EQ U EXT are true. I'll show it follows that If? is false in a normal Henkin model. Say (M, A), where M = (1t, I,£), is a generalized Henkin model in which the members of EQ U EXT are true but If? is false. Since the members of EQ are true, by results of Chapter 5 we can take (M, A) to be normal. I claim it is also extensional in the sense of Definition 2.32, that is, if £(0) = £(0') then 0 = 0', where 0 and O' are objects in the model domain. I now show this. Suppose £(0) = £(0'), where 0 and O' are of type (t) for simplicity (the general case is similar). The following is a member of EXT (in it, a and {3 are of type (t), and 1 is of type t) (Va)(V,B){('v'!)[a(/)

= ,8(/)] :) [a= ,8]}

and so this sentence is true in (M, A). Let v be a valuation such that = 0 and v(f3) = 0'. Then

v(a)

But since £(0) = £(0') it is easy to see we also have

and so M lf-v,A a= ,6. Since (M,A) is normal, it follows that v(a) = v({3), that is, 0 = 0'. Since (M, A) is extensional, it is isomorphic to a Henkin model, as was shown in Section 6. And trivially, isomorphism preserves sentence truth.

II

MODAL LOGIC

Chapter 7

MODAL LOGIC SYNTAX AND SEMANTICS

1.

Introduction

The second part of this book investigates a logic of intensions and extensions, using a possible world semantics. For purposes of background discussion in this section, I will assume you have some general familiarity with possible worlds at least informally. Technical details are postponed till after that. First, a point about terminology. The intensional/extensional distinction is an old one. Unfortunately, the word "extensionality" has already been given a technical meaning in Part I, where Henkin models that did or did not satisfy the extensionality axioms were considered. The use of "extension" in this part, while related, is not the same. I briefly tried using the word "denotation" here, but finally it seemed unnatural, and I resigned myself to using the word "extension" after all. As a matter of fact, the Axioms of Extensionality will be assumed throughout Part II for those terms that will be called extensional, so any confusion of meanings between the classical and the modal settings should be minimal. The machinery in Part I had no place for intensions-meanings. In a normal Henkin model, if terms intended to denote the morning star and the evening star have the same extension, as they do in the real world, they are equal, and so share all properties. They cannot be distinguished. Montague and his students, notably Gallin, developed a purely intensional logic. In this, extensions could only be handled indirectlyin some sense an extension could be an intension that did not vary with circumstances. While this can be made to work, it treats extensions as second class objects, and leads to a rather complicated development.

83 M. Fitting, Types, Tableaus, and Gödel's God © Kluwer Academic Publishers 2002

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What is presented here is a modification of the Montague/Gallin approach, in which both extensions and intensions are first class objects. What are the underlying intuitions? An extensional object will be much as it was in Part I: a set or relation in the usual sense. The added construct is that of intensional object, or concept, and this is treated in the Carnap tradition. A phrase like, say, "the royal family of England," has a meaning, an intension. At any particular moment, that meaning can be used to determine a particular set of people, constituting its extension. But that extension will vary with time. For other phrases, there may be different mechanisms for determining extensions as circumstances vary. The one thing common to all such intensional phrases is that they, somehow, induce mappings from circumstances to extensions. Abstracting to the minimum useful structure, in a possible world model an intensional object will be a function from possible worlds to extensional objects. Here is an example using the terminology just introduced. Suppose we take possible worlds as people, with an 85 accessibility relation~every person is accessible to every other person. And suppose the ground-level domain is a bunch of real-world objects. Any one person will classify some of those objects as being red. Because of differences in vision, and perhaps culture, this classification may vary from person to person. Nonetheless, there is a common concept of red, or else communication would not be possible. We can identify it with the function that maps each person to the set of objects that person classifies as red. And similarly for other colors. In addition, each person has a notion of color, though this too may vary from person to person. One person may think of ultra-violet as a color, another not. We can think of the color concept as a mapping from persons to the set of colors for that person. If we assert that red is a color for a particular person, we mean the red concept is in the extension of the color concept for that person. The extension of the red concept for that person plays no role for this purpose. Sometimes extensions are needed too. Certainly if we ask someone whether or not some object is red, the extension of the red concept, for that person, is needed to answer the question. Here is another example in this direction. Assume the word "tall" has a definite, non-fuzzy, meaning. Say everybody gets together and votes on which people are tall, or say there is a tallness czar who decides to whom the adjective applies. The key point is that the meaning of "tall," even though precise, drifts with time. Average height of the general population has increased over the last several generations, so someone who once was considered tall might not be considered so today.

MODAL LOGIC, SYNTAX AND SEMANTICS

85

Now suppose I say, "Someday everybody will be tall." There is more than one ambiguity here. On the one hand I might mean that at some point in the future, everybody then alive will be a tall person. On the other hand I might mean that everybody now alive will grow, and so at some point everybody now alive will be a tall person. Let us now read modal operators temporally, so that OX informally means that X is true and will remain true, and OX means that X either is true or will be true at some point in the future. Also, let us use T(x) as a tallness predicate. (The examples that follow assume an actualist reading of the quantifiers, and eventually I will adopt a version of a possibilist reading. For present purposes, this is a point of no fundamental importance. For now, think in terms of varying domain models, with quantifiers ranging over different domains at different worlds.) The two readings of the sentence are easily expressed as follows.

('v'x)OT(x) 0(\lx)T(x)

(7.1) (7.2)

Formula (7.1) refers to those alive now, and says at some point they will all be tall. Formula (7.2) refers to those alive at some point in the future and asserts, of them, that they will be tall. All this is standard; the problem is with the adjective "tall." Do we mean that at some point in the future everybody (read either way) will be tall as they use the word in the future, or as we use the word now? If we interpret things intensionally, T(x) at a possible world would be understood according to that world's meaning of tall. There is no way, using the present machinery, to formalize the assertion that, at some point in the future, everybody will be tall as we understand the term. But this is what is most likely meant if someone says, "Someday everybody will be tall." Here is another example, one that goes the other way. Suppose a member of the Republican Party, call him R, says, "necessarily the proposed tax cut is a good thing." Suppose we take as the possible worlds of a model the collection of all Republicans, and assume a sentence is true at a world if that Republican thinks the sentence true. (We assume Republicans are entirely rational, so we don't have to worry about contradictory beliefs.) Let us now take OX to mean that every Republican thinks X is the case, which means X is necessary for Republicans. (Technically, this gives us an 85 modality.) How do we formalize the sentence above? Let c be a constant symbol whose intended meaning is, "the proposed tax cut," and let G be a "goodness" predicate. Then OG(c) seems reasonable as a formalization. What should it mean to say it is true for R?

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One possibility is that R means every Republican thinks the tax cut is good, as R understands the word good. This may not be what was meant. After all, R might consider something good only if it personally benefitted him. Another Republican might think something good if it eventually benefitted the poor. Such a Republican probably would not think a tax cut good simply because it benefitted R but he might believe it would eventually benefit the poor, and so would be good in his own sense. Probably R is saying that every Republican thinks a tax cut is good, for his own personal reasons. The notion of what is good can vary from Republican to Republican, provided they all agree that the proposed tax cut is a good thing. But the mere fact that we can consider more than one reading tells us that a simple formalization like DG(c) is not sufficient. Here will be presented a logic of both intension and extension, of both sense and reference. In one of the examples above, color is an intensional object. It is a function from persons to sets of concepts like red, blue, and so on. As such, it is the same function for each person. The extension of color for a particular person is the color function evaluated at that person, and thus it is a particular set of concepts, such as red but not infra-red, and so on, quite possibly different from person to person. We need a logic in which both intensions and extensions are first-class objects. The machinery for doing this makes for complicated looking formulas. But I point out, in everyday discourse all the machinery is present but hidden-we infer it from our knowledge of what we think must have been meant. Formalization naturally requires complex machinery-it is making explicit what our minds do automatically.

2.

Types and Syntax

Now begins the formal treatment, starting with the notion of type. I want it to include the types of classical logic, as defined in Section 1. I also want it to include the purely intensional types of the Montague tradition, as given in [Gal75]. DEFINITION 7.1 (TYPE) The notion of a type, extensional and inten-

sional, is given by the following.

1 0 is an extensional type. 2 If t1, ... , tn are types, extensional or intensional, (t1, ... , tn) is an extensional type. 3 If t is an extensional type, jt is an intensional type. A type is an intensional or an extensional type.


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The ideas behind the definition above are these. As usual, 0 is to be the type of ground-level objects, unanalyzed "things." The type (t1, ... , tn) is intended to be analogous to types in part I. The type jt is the new piece of machinery-an object of such a type will be a function on possible worlds. Recall the example involving colors from Section 1; it can be used to give a sense of how these types are intended to be applied. In that example, real-world objects are those of type 0. A set of real-world objects is of type (0) so, for instance, the set of objects some particular person considers red is of this type; this is the extension of red for that person. The intensional object red, mapping each person to that person's set ofred objects, is of type j(O). A set of such intensional objects is of type (j(O)), so for a particular person, that person's set of colors is of this type-the extension of color for that person. Finally, the intensional object color, mapping each person to that person's set of colors, is an object of type i(j(O)). For another example, assume possible worlds are possible situations, and the ground-level objects include people. In each particular situation, there is a tallest person in the world. The tallest person, in each situation, is an object of type 0. The tallest person concept is an object of type jO-it associates with each possible world the tallest person in that possible world. As a final example example, suppose t is an extensional type, so that jt is intensional. The two-place relation: the intensional object X of type jt has the extensional object y of type t as its extension, is a relation of type (jt, t). The language of Part I must be expanded to allow for modality. Just as classically, C is a set of constant symbols of various types, containing at least an equality symbol =(t,t) for each type t, though the set of types is now larger. Note that the equality symbols themselves are of extensional type. Using them we can form the intensional terms (>.x, y.x = y) and (>.x, y.D(x = y)), as needed. We also have variables of each type. There is one new piece of machinery, an operator l, which plays a role in term formation. As usual, terms and formulas must be defined together in a mutual recursion. DEFINITION

7.2

(TERM OF

L(C)) Terms are characterized as follows.

1 A constant symbol or variable of L( C) of type t is a term of L( C) of type t. If it is a constant symbol, it has no free variable occurrences. If it is a variable, it has one free variable occurrence, itself. 2 If is a formula of L( C) and 0:1, ... , O:n is a sequence of distinct variables of types t1, ... , tn respectively, then (>..o:1, ... , O:n.) is a

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term of L( C) of the intensional type j (t1, ... , tn). Its free variable occurrences are the free variable occurrences of , except for occurrences of the variables a1, ... , an. 3 If 7 is a term of L(C) of type jt then 17 is a term of type t. It has the same free variable occurrences that 7 has. The predicate abstract (Aa1, ... , an. ) is of type j(t1, ... , tn) above, and not of type (t1, ... , tn), essentially because can vary its meaning from world to world, and so (Aa1, ... , an.) itself is world dependent. Case 3 above makes use of what may be called an extension-of operator, converting a term of an intensional type to a term of the corresponding extensional one. Continuing with the color example, suppose r is the intensional notion of red, of type j(O), mapping each person to that person's set of red objects. Then for a particular person, 1r is that person's set of red objects-the extension of r for that person, and an extensional object of type (0). Of course the symbols j and 1 were chosen to suggest their roles-in a sense 1 'cancels' j. Nonetheless, 1 is a symbol of the language, while j occurs in the metalanguage, as part of the typing mechanism. DEFINITION 7.3 (MODAL FORMULA OF L(C)) mula of L( C) is as follows:

The definition of for-

1 If 7 is a term of either type (t1, ... , tn) or type i (t1, ... , tn), and 71, ... , 7n is a sequence of terms of types t1, ... , tn respectively, then 7(71, ... , 7n) is a formula {atomic) of L(C). The free variable occurrences in it are the free variable occurrences of 7, 71, ... , 7n· 2 If is a formula of L( C) so is -,q>. The free variable occurrences of -,q> are those of . 3 If and \lT are formulas of L( C) so is ( 1\ \lf). The free variable occurrences of ( 1\ \lf) are those of together with those of \lf.

4 If is a formula of L(C) and a is a variable then (\fa) is a formula of L( C). The free variable occurrences of (\fa ) are those of , except for occurrences of a. 5 If is a formula of L( C) so is D. The free variable occurrences of D are those of . Item 1 above needs some comment, and again the example concerning colors should help make things clear. Suppose r is the intensional notion of red, of type j(O). And suppose cis an extensional notion of color,


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the set of colors for a particular person-call the person George. Also let C be the intensional version of color, mapping each person to that person's extension of color. cis of type (i(O) ), and Cis of type j(j(O)) I take both C(r) and c(r) to be atomic formulas. If we ask whether they are true for George, no matter which formula we use, we are asking if r is a color for George. But if we ask whether they are true for Natasha, we are asking different questions. C(r) is true for Natasha if r is a color for Natasha, while c(r) is true for Natasha if r is a color for George. No matter which, both c(r) and C(r) make sense, and are considered well-formed. I use 0 to abbreviate ...,o..., in the usual way, or I tacitly treat it as primitive, as is convenient at the time. And of course other propositional connectives and the existential quantifier will be introduced as needed. Likewise outer parentheses will often be dropped.

3.

Constant Domains and Varying Domains

Should quantifiers range over what does exist, or over what might exist? That is, should they be actualist or possibilist? This is really a first-order question. A flying horse may or may not exist. In the world of mythology, such a being does exist. In the present world, it does not. But the property of being a flying horse does not exist in some worlds and lack existence in others. In the present world nothing has the flying-horse property, but that does not mean the property itself is non-existent. Thus actual/possible existence issues really concern type 0 objects, so the discussion that follows assumes a first-order setting. As presented in [HC96] and [FM98], the distinction between actualist and possibilist quantification can be seen to be that between varying domain modal models and constant domain ones. In a varying domain modal model, one can think of the domain associated with a world as what actually exists at that world, and it is this domain that a quantifier ranges over when interpreted at that world. In a constant domain model one can think of the common domain as representing what does or could exist, and this is the same from world to world. Of course a choice between constant and varying domain models makes a substantial difference: both the Barcan formula and its converse are valid in a constant domain setting, but neither is in a varying domain one. As it happens, while a choice between constant and varying domain models makes a difference technically, at a deeper level such a choice is essentially an arbitrary one. If we choose varying domains as basic, we can restrict attention to constant domain models by requiring the Barcan formula and its converse to hold. (Technically this requirement involves an infinite set of formulas, but if equality is available a single formula will

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do.) Thus when using actualist quantification, we can still determine constant domain validity. The other direction is even easier. If we have possibilist-constant domain-:-quantification we can also determine varying domain validity. And on this topic I present a somewhat more detailed discussion. Suppose quantification is taken in a possibilist sense--domains are constant. Nonetheless, at each world we can intuitively divide the common domain into what 'actually' exists at that world and what does not. Introduce a predicate symbol E of type j(O) for this purpose. At a particular world, E(x) is true if x has as its value an object one thinks of as existing at that world, and is false otherwise. Then the effect of varying domain quantification can be had by relativising all quantifiers to E. That is, replace (Vx)r.p by (Vx)(E(x) :) r.p) and replace (3x)r.p by (3x)(E(x) 1\ r.p). What we get, at least intuitively, simulates an actualist version of quantification. All this can be turned into a formal result. Suppose we denote the relativization of a first-order formula r.p, as described above, by r.pE. It can be shown that r.p is valid in all varying domain models if and only if r.pE is valid in all constant domain models. Possibilist quantification can simulate actualist quantification. I note in passing that [Coc69] actually has two kinds of quantifiers, corresponding to actualist and possibilist, though it is observed that a quantifier relativization of the sort described above could be used instead. The discussion above was in a first-order setting. As observed earlier, when higher types are present the actualist/possibilist distinction is only an issue for type 0 objects. I have made the choice to use possibilist type 0 quantifiers. The justification is that, first, such quantifiers are easier to work with, and second, they can simulate actualist quantifiers, so nothing is lost. When I say they are easier to work with, I mean that both the semantics and the tableau rules are simpler. So there is considerable gain, and no loss. Officially, from now on the formal language will be assumed to contain a special constant symbol, E, of type j (0), which will be understood informally as an existence predicate.

4.

Standard Modal Models

I begin the formal presentation of semantics for higher-order modal logic with the modal analog of standard models. The new piece of semantical machinery added to that for classical logic is the possible world structure.


91

DEFINITION 7.4 (KRIPKE FRAME) A Kripke frame is a structure (Q, R). In it, g is a non-empty set (of possible worlds), and R is a binary relation on g (called accessibility). An augmented frame is a structure (Q, R, V) where (Q, R) is a frame, and V is a non-empty set, the (ground-level) domain. The notion of a Kripke frame should be familiar from propositional modal logic treatments, and I do not elaborate on it. As usual, different restrictions on R give rise to different modal logics. The only two I will be interested in are K, for which there are no restrictions on R, and 85, for which R is an equivalence relation. Note that the ground-level domain, V, is not world dependent, since the choice was to take type-0 quantification as possibilist and not actualist. Next I say what the objects of each type are, relative to a choice of ground-level domain. This is analogous to what was done in Part I, in Definition 2.1. To make things easier to state, I use some standard notation from set theory. The first item is something that was used before, but I include it here for completeness sake. 1 For sets A1, ... , An, A1 x · · · x An is the collection of all n-tuples of the form (a1, ... , an), where a1 E A1, ... , an E An. The 1-tuple (a) is generally identified with a. 2 For a set A, P(A) is the power set of A, the collection of all subsets of A.

3 For sets A and B, A B is the function space, the set of all functions from B to A. DEFINITION 7.5 (OBJECTS, EXTENSIONAL AND INTENSIONAL) Let g be a non-empty set (of possible worlds) and let V be a non-empty set (the ground-level domain). For each type t, I define the collection [t, V, Q], of objects of type t with respect to V and Q, as follows.

1 [O,V,Q] =V. 2 [(t1, ... , tn), V, Q]

= 'P([t1, V, Q] X"· X [tn, V, Q]).

3 [jt, V, Q] = [t, V, Q]g. 0 is an object of type t if 0 E [t, V, Q]. 0 is an intensional or extensional object according to whether its type is intensional or extensional. As before, 0 is used, with or without subscripts, to stand for objects.

Now the final notion of the section.

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DEFINITION 7.6 (MODAL MODEL) A (higher-order) modal model for L(C) is a structure M = (Q, R, V,I), where (Q, R, V) is an augmented frame and I is an interpretation .. The interpretation I must meet the following conditions.

1 If At is a constant symbol of type t, I(At) is an object of type t, that is, I(At) E [t, V, Q]. 2 If =(t,t) is an equality constant symbol, I( =(t,t)) is the equality relation on [t, V,Q].

5.

Truth in a Model

In this section I say how truth is to be assigned to formulas, at worlds, in models, and how values should be assigned to terms. I lead up to a proper definition after a few preliminary notions. DEFINITION 7.7 ((MODAL) VALUATION) The mapping v is a modal valuation in the modal model M = (9, R, V,I) if v assigns to each variable at of type t some object of type t, that is, v( at) E [t, V, Q]. The notion of a variant valuation is defined exactly as classically.

A term like lr is intended to designate the extension of the intensional object designated by T. To determine this a context is needed-the designation of T where, under what circumstances? The notation I'll use for a designation function is (v *I* f)(T), where vis a valuation, I is an interpretation, and r is a context, a possible world. (In fact the context only matters for terms of the form lT.) In specifying the designation of a term, the predicate abstract case requires information about formula truth. This is more complex than classically, again because a context must be specified-truth under what circumstances, in which possible world. The notation for this is a modification of what was used earlier. I'll write M, r lf-v
being true at world r of g in model M with respect to v, denoted M, r 11-v 4>, is characterized as follows. 1 For an atomic formula T( Tl, . .. , Tn), {a) lfT is of an extensional type, M,r 11-v T(TI, ... ,Tn) provided ((v*I*r)(TI), ... ,(v*I*r)(Tn)) E (v*I*r)(T).

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TYPES, TABLEAUS, AND GODEL'S GOD (b) If T is of an intensional type, M, r lf-v T(T1, ... 'Tn) provided M' r If-v (1 T) (71' . . . ' Tn). This reduces things to the previous case.

2 M, r If- v --,Y aY:=>X

8.5 (DOUBLE NEGATION RULE) For any prefix a,

a••X aX DEFINITION

8.6 (DISJUNCTIVE RULES) For any prefix a,

aXVY aX laY

a

aX::)Y

a

I

a .x a Y

a •(X 1\ Y) ·X a ,y

I

•(X

a •(X ::) Y)

= Y)

I a ·(Y ::) X)

This completes the classical connective rules. The motivation should be intuitively obvious. For instance, if X 1\ Y is true at a world named by a, both X and Y are true there, and so a branch containing a X 1\ Y can be extended with a X and a Y.

1.3

Modal Rules

Naturally the rules for modalities differ between the two logics we are considering. It is here that the structure of prefixes plays a role. The idea is, if OX is true at a world, X is true at some accessible world, and we can introduce a name-prefix-for this world. The name should be a new one, and the prefix structure should reflect the fact that it is accessible from the world at which OX is true.

8. 7 (POSSIBILITY RULES FOR K) If the prefix a.n is new to the branch,

DEFINITION

a OX a.nX

a•DX a.n •X


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DEFINITION 8.8 (POSSIBILITY RULES FOR

85) If the positive integer

n is new to the branch, aOX nX

a•DX n•X

Notice that for both logics there is a newness condition. This implicitly treats 0 as a· kind of existential quantifier. Correspondingly, the following rules treat 0 as a version of the universal quantifier. 8.9 (NECESSITY RULES FORK) If the prefix a.n already occurs on the branch,

DEFINITION

a

OX

a.nX

a•OX a.n•X

8.10 (NECESSITY RULES FOR 85) For any positive integer n that already occurs on the branch,

DEFINITION

a OX nX

a•OX n•X

Many examples of the application of these propositional and modal rules can be found in [FM98J. I do not give any here. Rather, tableau examples will be given after the full higher-type system has been introduced.

1.4

Quantifier Rules

For the existential quantifier rules parameters must be introduced, just as in the classical case. Thus proofs of sentences of L( C) are forced to be in the larger language L + (C). 8.11 (EXISTENTIAL RULES) In the following, pt is a parameter of type t that is new to the tableau branch.

DEFINITION

a (:l( oJ) a cf>(pt)

a •(Vo:t)cf>( o:t) a -,cf>(pt)

Terms of the form lT may vary their denotation from world to world of a model, because the extension of the intensional term T can change from world to world. Such terms should not be used when instantiating a universally quantified formula.

8.12 (RELATIVIZED is a relativized term.

DEFINITION

term,

lT

TERM)

If T is a grounded intensional

MODAL TABLEAUS

109

DEFINITION 8.13 (UNIVERSAL RULES)

In the following, 7t is any grounded term of type t that is not relativized.

a ('v'at).X.DX(lp))(t(>.x.O(lp = x)))

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MISCELLANEOUS MATTERS

1 •(.AX.DX(lp))(l(.Ax.O(lp = x))) 1 ·D(.Ax.O(lp = x))I(lp) 2. 1.1·(.Ax.O(lP = :t))I(lP) 3. 1.1•(Ax.O(lp = x))I(pu) 4. 1 •O(lP=pu) 5. 1.1•(1P = Pu) 6. 1.1•(Pu = Pu) 7. 1.1 (pu = Pu) 8.

1.

In this 2 is from 1 by the derived unsubscripted abstraction rule; 3 is from 2 by a possibility rule; 4 is from 3 by extensional predication; 5 is from 4 by a predicate abstract rule; 6 is from 5 by a necessity rule; 7 is from 6 by extensional predication; 8 is by the derived reflexivity rule.

1.2

Extensionality

Extensionality can, of course, be imposed by assuming the Extensionality Axioms of Chapter 6, Definition 6.1, as global assumptions. The trouble is, doing so for intensional terms yields undesirable results, as the following shows. 9.2 Assume the Extensionality Axioms apply to intensional terms. If a and f3 are of intensional type j(t), then the following is valid.

PROPOSITION

(Va)(V,B)[(la =l/3) ~ (a= ,B)]

The proof of this is left to you. It is almost immediate, using the Intensional Predication Rules. The problem with this result is, it tells us that if two intensional objects happen to coincide in extension at some world, then they are identical and hence coincide at every world. Clearly this is undesirable, so extensionality for intensional terms is not assumed. If two intensional objects agree in extension at every possible world of a model they are, in fact, the same. Saying this requires a quantification over possible worlds, which we cannot do. The following is as close as we can come.

9.3 (EXTENSIONALITY FOR INTENSIONAL TERMS) For a and f3 of the same intensional type,

DEFINITION

(Va)(V,B)[D(la

=l/3)

~(a= ,B)]

I will assume this at some points, but I will be explicit when. For extensional terms, the extensionality axioms pose no difficulty and will always be assumed. Let me make this official.

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Extensionality Assumptions From now on, the extensionality axioms will be assumed for extensional terms as global assumptions. For intensional terms extensionality, Definition 9.3, will only be assumed if explicitly stated. I restate the extensionality axioms here for convenience. 9.4 (EXTENSIONALITY FOR EXTENSIONAL TERMS) Each sentence of the following form is an extensionality axiom, where a and {3 are of type (t1, ... , tn), 11 is of type t1, ... , In is of type tn. DEFINITION

('v'a)('v'{3){('v'11) · · · ('v'ln)[a(/'1, ·. · , In)= {3(/'1, ... , In)] :J [a= {3]} In Chapter 6 a derived tableau rule for extensionality was given, assuming the extensionality axioms. Once again, it is still a derived rule for modal tableaus. Here is a statement of it.

Extensionality Rule For grounded, non-relativized extensional terms T1 and T2, and for parameters P1, ... , Pn that are new to the branch,

(}--, [T1(P1, · · · ,pn)

2.

= T2(P1, · · · ,Pn)JI (} (T1 = T2)

De Re and De Dicto

Loosely speaking, asserting the necessary truth of a sentence is a de dicta usage of necessity; for example, "it is necessary that the President of the United States is a citizen of the United States." This asserts the necessary truth of the sentence, "the President of the United States is a citizen of the United States." For this to be the case, it must be so under all circumstances, no matter who is President, and since being a citizen of the United States is a requirement for the Presidency, this is the case. Ascribing to an object a necessary property is a de re usage; for example, "it is a necessary truth, of the President of the United States, that he is at least 50 years old." This asserts, of the President, that he is and always will be at least 50 years old. Since the President, at the time of writing, is Bill Clinton, and he is at the moment 53 years old and will never be younger than this, this assertion is correct. But since the Constitution of the United States only requires that a President be at least 35, the assertion may not be true in the future, for a different President. If an object is identified using an intensional term, it makes a serious difference whether that term is used in a de dicta or a de re context, as the examples involving the Presidency illustrate. In this section the formal relationship between the two notions is explored. As will be seen over the next several sections, this also relates to other interesting concepts that have been part of historic philosophical discourse.

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In the next few paragraphs, f3 is of some extensional type t, and Tis of the corresponding intensional type jt. Consider the expression (.Xf3.D(f3))(1 T), where (r) :) cf>(lT))(,6))(lT) 11. 1.2 •(A,6.(A/.cf>(r) :) cf>(lT))(,6))(1T) 12. 1.2 •(A/.cf>(r) :) cf>(lT))(Tl.2) 13. 1.2 •[cf>(TL2) :) cf>(lT)) 14. 1.2 cf>( TL2) 15. 1.2 --,cf>(lT) 16. 1.2 cf>1.2(T1.2) 17. 1.2 --,cf>L2(71.2) 18. Item 11 is from 10 by a disjunctive rule (recall, this is the left branch); 12 is from 11 by a possibility rule; 13 is from 12 and 14 is from 13 by an unsubscripted abstract rule; 15 and 16 are from 14 by a conjunctive rule; 17 is from 15 and 18 is from 16 by a derived intensional predication rule. The branch is closed because of 17 and 18. Now I show the right branch, below item 10. 1 (.A,6.0(A/.cf>(r) :) cf>(lT))(,6))(1T) 1 D(.A,.cf>('y) :) cf>(lT))(Tl) 20. 1.1 (A/.cf>(r) :) cf>(lT))(Tl) 21. 1.1 cf>(Tl) :) cf>(lT) 22.

/ 1.1-.cf>(Tl)

19.

~

23. 1.1 cf>(lT) 24. 1.1 cf>u (Tu) 25. 1.1 •cf>u (Tu) 26.

In this part, 19 is from 10 by a disjunctive rule; 20 is from 19 by an unsubscripted abstract rule; 21 is from 20 by a necessity rule; 22 is from 21 by an unsubscripted abstract rule; 23 and 24 are from 22 by a disjunctive rule; 25 is from 24 and 26 is from 7 by a derived intensional predication rule. Closure is by 8 and 23, and by 25 and 26. •

Exercises EXERCISE 2.1 Give the tableau proof needed to complete the argument for Proposition 9.6.

3.

Rigidity

In [Kri80] the philosophical ramifications of the notion of rigidity are discussed at some length, with a key claim being that names are rigid. The setting is first-order modal logic, treated informally. A term is taken to be rigid if it designates the same thing in all possible worlds. In [FM98]

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we modified this notion somewhat so that a formal investigation could more readily be carried out-we called a term rigid if it designated the same thing in any two possible worlds that were related by accessibility. The idea is that the behavior of a term in an unrelated world should have no "visible" effect. It is this modified notion of rigidity that is used here, and it will be seen that it can be expressed directly if equality is available. (Whether models are standard, Henkin, or generalized Henkin does not matter for what we are about to do, only that they are normal.) For the rest of this section, normality is assumed.

9. 7 The intensional term T is rigid in a normal model if the following is valid in it.

DEFINITION

It is easy to see that the formula asserting rigidity of T is true at a world r of a normal model if and only if, at each world accessible from r, T designates the same object that it designates at r itself. Thus asserting validity for the rigidity formula indeed captures the notion of rigidity for terms that we have in mind. If an intensional term is rigid, it does not matter in which possible world we determine its designation. But then, if both necessitation and designation by a rigid intensional term are involved in the same formula, it should not matter whether we determine what the term designates before or after we move to alternative worlds when taking necessitation into account. In other words, for rigid intensional terms the de re/ de dicto distinction should vanish. In fact it does, and as it happens, the converse is also the case. The following is a higher order version of a first order argument from [FM98].

9.8 In K, the intensional term T is rigid if and only if the de re/de dicto distinction vanishes, that is, if and only if any (and hence all) parts of Proposition 9. 6 hold.

PROPOSITION

Proof This is shown by proving two implications, using tableau rules for K including rules for equality. Let A be the formula (A,6.0(,8 =lT))(lT) and let B be the formula ('v'a)[D(A,6.a(,B))(T) :J (A,6.0a(,6))(T)]. A says T is rigid, while B says de dicto implies de re for T. I first give a tableau proof of A :J B.

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1 •(A :J B) 1. 1 (,\,6.0(,6 =17))(17) 2. 1 •(Va)[0(,\,6.a(,6))(17) :J (>.,6.0a(,6)}(17)] 1 •[0(>.,6.(,6))(17) :J (>.,6.0({3))(17)] 4. 1 0(,\,6.(,6))(17) 5. 1 •(A,6.0(J3)}(17) 6. 1 ·D( 71) 7. 1.1•(71) 8. 1.1 (,\,6.({3))(17) 9. 1.1 ( 71.1) 10. 1 0(71 =17) 11. 1.1 (71 =17) 12. 1.1 71 = 71.1 13. 1.1 • (71.1) 14.

3.

In this tableau, 2 and 3 are from 1 by a conjunctive rule; 4 is from 3 by an existential rule, with as a new (intensional) parameter; 5 and 6 are from 4 by a conjunctive rule; 7 is from 6 by a derived unsubscripted abstract rule; 8 is from 7 by a possibility rule; 9 is from 5 by a necessity rule; 10 is from 9 and 11 is from 2 by a derived unsubscripted abstract rule; 12 is from 11 by a necessity rule; 13 is from 12 by a derived unsubscripted abstract rule; and 14 is from 8 and 13 by a derived substitutivity rule for equality. Finally I give a tableau proof of B :=:> A. 1 •(B :J A) 1. 1 (Va)[D(>.,6.a(,6)}(17) :J (>.,6.0a(,6))(l7)] 2. 1 •(A,6.0(,6 =17)}(17) 3. 1 0(,\,6.(Af'. 17 = 1)(,6)}(17) :J (,\,6.0(Af'. 17 = /)(,6)}(17) 1 •0(71 =17) 5. 1.1•(71 =17) 6. 1.1•(71 = 71.1) 7.

/

~

4.

1 •0(>.,6.(Af'. 17 = /)(,6))(17) 1 (,\,6.0(Af'. 17 = 1)(,6)}(17) 8. 14. D(>.,. 17 = ,)( 71) 15. 1.2 •(A,6.(Af'. 17 = 1)(,6))(17) 9. 1 1.1 (Af'.l7=f'}(71) 16. 1.2•(Af'.l7 = 1)(71.2) 10. 1.1 (17=71) 17. 1.2 •(17 = 71.2) 11. 1.1 71.1 = 71 18. 1.2 ·(71.2 = 71.2) 12. 1.1•(71 = 71) 19. 1.2 71.2 = 71.2 13. 1.1 71 = 71 20.

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• In this, 2 and 3 are from 1 by a conjunctive rule; 4 is from 2 by a universal rule, instantiating with the term (>."f. l T = 'Y); 5 is from 3 by an unsubscripted abstract rule; 6 is from 5 by a possibility rule; 7 is from 6 by an unsubscripted abstract rule; 8 and 14 are from 4 by a disjunctive rule; 9 is from 8 by a possibility rule; 10 is from 9, and 11 is from 10 by an unsubscripted abstract rule; 12 is from 11 by an extensional predication rule; 13 is by reflexivity; 15 is from 14 by an unsubscripted abstract rule; 16 is from 15 by a necessity rule; 17 is from 16 by an unsubscripted abstract rule; 18 is from 17 by an extensional predication rule; 19 is from 7 and 18 by substitutivity; and 20 is by reflexivity.

4.

Stability Conditions

In his ontological argument Godel makes essential use of what he called "positiveness," which is a property of properties of things. Hedoes not define the notion, instead he makes various axiomatic assumptions concerning it. Among these are: if a property is positive, it is necessarily so; and if a property is not positive, it is necessarily not positive. (His justification for these was the cryptic remark, "because it follows from the nature of the property.") Suppose we use the secondorder constant symbol P to represent positiveness, and take it to be of type j (i (0)). Godel stated his conditions more or less as follows, with quantifiers implied: P(X) ::J DP(X) and •P(X) ::J D•P(X). The second of these is equivalent to ()P(X) ::J P(X), and this form will be used in what follows. Positiveness is a second-order notion, but Godel's conditions can be extended to other orders as well. I call the resulting notion stability, which is not terminology that Godel used.

DEFINITION 9.9 (STABILITY) LetT be a term of type j(t). T satisfies the stability conditions in a model provided the following are valid in that model.

(Va)[T(a) :::> DT(a)] (Va)[()T(a) ::J T(a)] The stability conditions come in pairs. In S5, however, these pairs collapse.

PROPOSITION 9.10 In S5, (\la)[T(a) ::J DT(a)] and (\la)[()T(a) :::> T(a)] are equivalent.

Proof Suppose (Va)[T(a) ::J DT(a)]. Contraposition gives (Va)[•DT(a) ::J •T(a)]. From necessitation and converse Barcan, (Va)D[•DT(a) :::>


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•T(a)], and so ('v'a)[D•DT(a) :J D•T(a)], equivalently, ('v'a)[DO•T(a) :J D•T(a)]. But in S5, X :J DOX is valid, hence we have ('v'a)[•T(a) :J D•T( a)]. By contraposition again, ('v'a)[·D•T(a) :J ••T( a)], and hence ('v'a)[OT(a) :J T(a)]. The converse direction is similar. • In the stability conditions, T is being predicated of other things. On the other hand, to say T is rigid, or that the de re /de dicta distinction vanishes for T, involves other things being predicated of T. Here is the fundamental connection between stability and earlier items. THEOREM 9.11 An intensional term T is rigid if and only if it satisfies the stability conditions.

Proof This is most easily established using tableaus. And it is a good workout. I leave it to you to supply the details. •

Exercises EXERCISE 4.1 Complete the proof of Theorem 9.11 by giving appropriate closed tableaus. Recall that extensionality is assumed for extensional terms, and we have the derived extensionality rule given in Definition 6.2.

5.

Definite Descriptions

As is well-known, Russell treated definite descriptions by translating them away, [Rus05]. His familiar example, "The King of France is bald," is handled by eliminating the definite description, "the King of France," in context, to produce the sentence "exactly one thing Kings France, and that thing is bald." It is also possible to treat definite descriptions as first-class terms, making them a primitive part of the language. In [FM98] we showed how both of these approaches extend to first-order modal logic. Further extending this dual treatment to higherorder modal logic adds greatly to the complexity, so I confine things to a Russell-style version here. Suppose we have a formula .x.-.B(x))( 1y.K(y)) and •(>.x.B(x)) (1y.K(y) ). There is a similar distinction to be made between (>.x.DB(x))(1y.K(y)) and D(>.x.B(x))(1y.K(y)) since definite descriptions generally act nonrigidly, and so the de rej de dicta issue arises. Note that in all the examples above, scope of a definite description was indicated by the use of a predicate abstract. Now (>.x.DB(x))(1y.K(y)) is atomic, as are (>.x.B(x))(1y.K(y)) and (>.x.-.B(x))(1y.K(y)). It is enough for us to specify how definite descriptions behave in atomic contexts, and everything else follows automatically. But even at the atomic level, a definite description can occur in a variety of ways. For instance, in To(TI) either, or both, of To and TI could be descriptions. There are several ways of dealing with this, all of which lead to equivalent results. I'll use a Russell-style translation directly in the simplest case, and reduce other situations to that.

9.13 (DESCRIPTIONS IN ATOMIC CONTEXT) Let m.

(:3'Y)Do:(h)]

Informally, the axiom says that if, at each world the set of things such that o: is non-empty-0(:3f3)o:(f3)-then there is a choice function 'Y that picks out something such that o: at each world-(::l"f)Do:(h). I give one example of a Choice Axiom application. Suppose o: is an extensional variable, and m. designates in every possible world. That is, in each possible world, the is meaningful. Then, plausibly, there should be an intensional object that, in each world, designates the thing that is the of that world-that is, the term ?(.D(Ao:.)(t() should also designate. More loosely, the concept should also designate. Recall, Definition 9.12 says what it means for a definite description to designate, and since (A(.0(Ao:.)(K))("7) = D(Ao:.)(t'f]), things can be simplified a little.

9.16 Assume the Choice Axiom (Definition 9.15} and Extensionality for Intensional Terms (Definition 9.3). Assume a, {3, and 8 are of extensional type t, and 'Y and "7 are of type jt. The following is valid in all K models.

PROPOSITION

D(:3{3)(V8)[(Ao:.)(8)

= ({3 =

8)]

::::>

(:3'Y)(V"7)[D(Ao:.)(t"7)

Proof Assume D(:3f3)(V8)[(Ao:.)(8)

= ('Y =

17)]

= ({3

= 8)] is true at a possible world. I show that (:3'Y)(V7])[D(Ao:.)(1"7) = ('Y = 17)] must also be true

there. Start with

D(:3f3)(V8)[(Ao:.)(8)

= ({3 =

(9.3)

8)]

which is equivalent to

0(:3{3){ (Ao:.)(f3)

1\

(V8)[(Ao:.)(8)

::::>

({3

= 8)]}.

Instantiating the universal quantifier in the choice axiom with (A"7.(Ao:.)("7) (9.4) implies

1\

(V8)[(Ao:.)(8)

::::>

("7

= 8)])

(9.4)

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(:l!)D{(>,a.,a.,a.,a.,a.,a.,a.,a.,a.,a.,a..a.,a.

DE(g).

(10.1)

Given (10.1), using the rule of necessitation, we have the following.

D[E(g)

::::>

DE(g)]

(10.2)

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From (10.2), using the K principle D(P :::) Q) :::) (()P :::) OQ) we have the next implication. OE(g) :::) ODE(g)

(10.3)

Finally we use something peculiar to 85 (and some slightly weaker logics, a point of no importance here). The principle needed is ODP :::) DP, and so from (10.3) we have the following. OE(g) :::) DE(g)

(10.4)

We thus have a proof that God's existence is necessary, if possible. And, again following Descartes loosely, God's existence is possible because possibility is identified with conceivability, and we may take it for granted that God is conceivable. Russell's treatment of definite descriptions applies quite well in a modal setting-Chapter 9, Section 5. The use of g above was an informal way of avoiding a formal definite description-note that I gave no real prooffor (10.1). Let us recast the argument using definite descriptionsthe necessarily existent being is m.DE(a) and I assume g is an abbreviation for this type-0 term. Now (10.1) unabbreviates to the following. E(10~.DE(a))

:::) DE(1a.DE(a)).

(10.5)

This is not a valid formula of K, but that logic is too weak anyway, given the step from (10.3) to (10.4) above. But (10.5) is valid in 85, a fact I leave to you as an exercise. In fact, using 85, the argument above is entirely correct! The real problem with the Descartes argument lies in the assumption that God's existence is possible. In 85 both OE(g) :::) E(g) and E(g) :::) OE(g) are trivially valid. Since OE(g) :::) DE(g) has been shown to be valid, we have the equivalence of E(g), OE(g), and DE(g)! Thus, assuming God's existence is possible is simply equivalent to assuming God exists. This is an interesting conclusion for its own sake, but as an argument for the existence of God, it is unconvincing.

Exercises EXERCISE 3.1 Give an 85 tableau proof of the following, where P and

Q are type-(0) constant symbols. P(m.DQ(a)) :::) DQ(m.DQ(a)) From this it follows that (10.5) is valid in 85.

GODEL'S ARGUMENT, BACKGROUND

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EXERCISE 3.2 Construct a model to show

E(m.DE(a))

~

DE(1a.OE(a)).

is not valid inK. EXERCISE 3.3 Formula 10.5 can also be written as

(.X,8.E(,8))(1a.OE(a))

~

D(.X,B.E(,B))(m.OE(a))

which, by the previous exercise, is not K valid. Show the following variant is valid (a K tableau proof is probably easiest).

(.X,8.E(,8))(1a.DE(a))

~

(.X,B.OE(,B))(m.OE(a))

K formula of Exercise 3.3 can not be used in a Descartes-style argument. EXERCISE 3.4 Show why the valid

4.

Leibniz

Leibniz (1646 - 1716) partly accepted the Descartes argument from The Meditations, mentioned in the previous section. But he also clearly identified the critical issue: one must establish the possibility of God's existence. The following is from Two Notations for Discussion with Spinoza, [Lei56]. Descartes' reasoning about the existence of a most perfect being assumed that such a being can be conceived or is possible. If it is granted that there is such a concept, it follows at once that this being exists, because we set up this very concept in such a way that it at once contains existence. But it is asked whether it is in our power to set up such a being, or whether such a concept has reality and can be conceived clearly and distinctly, without contradiction. For opponents will say that such a concept of a most perfect being, or a being which exists through its essence, is a chimera. Nor does it suffice for Descartes to appeal to experience and allege that he experiences this very concept in himself, clearly and distinctly. This is not to complete the demonstration but to break it off, unless he shows a way in which others can also arrive at an experience of this kind. For whenever we inject experience into our demonstrations, we ought to show how others can produce the same experience, unless we are trying to convince them solely through our own authority.

Leibniz's remedy amounted to an attempt to prove that God's existence is possible, where God is defined to be the being having all perfections-again a maximality notion. Intuitively, a perfection is an atomic property that is, in some sense, good to have, positive. Leibniz based his proof on the compatibility of all perfections, from which he took it to follow that all perfections could reside in a being-God's

138


existence is possible. Here is another quote from Two Notations for Discussion with Spinoza, [Lei56]. By a perfection I mean every simple quality which is positive and absolute or which expresses whatever it expresses without any limits. But because a quality of this kind is simple, it is unanalyzable or indefinable .... From this it is not difficult to show that all perfections are compatible with each other or can be in the same subject.

Leibniz goes on to provide a detailed proof of the compatibility of all perfections, though it is not a proof in any modern sense. Indeed, it is not clear how a proper proof could be given at all, using the vague notion of perfection presented above. I omit his proof here. The point for us is that, as we will see, precisely this point is central to Godel's argument as well.

5.

Godel

Godel (1906- 1978) was heir to the profound developments in mathematics of the late nineteenth and early twentieth centuries, which often involved moves to greater degrees of abstraction. In particular, he was influenced by David Hilbert and his school. In the tradition of Hilbert's book, Foundations of Geometry, Godel avoided Leibniz's problems completely, by going around them. It is as if he said, "I don't know what a perfection is, but based on my understanding of it intuitively, it must have certain properties," and he proceeded to write out a list of axioms. This neatly divides his ontological argument into two parts. First, based on your understanding, do you accept the axioms. This is an issue of personal intuitions and is not, itself, subject to proof. Second, does the desired conclusion follow from the axioms. This is an issue of rigor and the use of formal methods, and is what will primarily concern us here. Godel's particular version of the argument is a direct descendent of that of Leibniz, which in turn derives from one of Descartes. These arguments all have a two-part structure: prove God's existence is necessary, if possible; and prove God's existence is possible. Godel worked on his ontological argument over many years. According to [Ada95], there is a partial version in his papers dated about 1941. In 1970, believing he would die soon, Godel showed his proof to Dana Scott. In fact Godel did not die until1978, but he never published on the matter. Information about the proof spread via a seminar conducted by Dana Scott, and his slightly different version became public knowledge. Godel's proof appeared in print in [Sob87], based on a few pages of Godel's handwritten notes. Scott also wrote some brief notes, based on his conversation with Godel, and [Sob87] provides these as well. In fact, [Sob87] has served as something of a Bible (pun intended) for the


139

Godel ontological argument. Finally the publication of Godel's collected works has brought a definitive version before the public, [G70]. Still, the notion of a definitive version is rather elusive in this case. Godel's manuscript provides almost no explanation or motivation. It amounts to an invitation to others to elaborate. Godel's argument is modal and at least second-order, since in his definition of God there is an explicit quantification over properties. Work on the Kripke semantics of modal logic was relatively new at the time Godel wrote his notes, and the complexity of quantification in modal contexts was perhaps not well appreciated. Consequently, the exact logic Godel had in mind is unclear. Subsequently several people took up the challenge of putting the Godel argument on a firm foundation and exposing any hidden assumptions. People have generally used the second-order modal logic of [Coc69], sometimes rather informally. [Sob87], playing Gaunilo to Godel's Anselm, showed the argument could be applied to prove more than one would want. Sobel's discussion has been greatly extended in [SobOl], Chapter 4; Chapter 3 is also relevant here. [AG96] showed that one could view a part of the argument not as second-order, but as third-order. Many others contributed, among which I mention [And90, Haj96b]. Postings on the internet are, by nature, somewhat ephemeral, but interesting discussions of the Godel argument, intended for a general audience, can be found at [SmaOl] as well as at [OppOl]. In addition, there are [Opp96b] and [SobOl]. The present chapter and the next can be thought of as part of the continuing tradition of explicating Godel.

6.

Godel's Argument, Informally

Before we get to precise details in the next Chapter, it would be good to run through Godel's argument informally to establish the general outline, since it is considerably more complex than the versions we have seen to this point. To begin with, Godel takes over the notion of perfection, but with some changes. For Leibniz, perfections were atomic properties, and any combination of them was compatible and thus could apply to some object. They could be freely combined, a little like the atomic facts about the world that one finds in Wittgenstein's Tractatus. Since this is the case, why not form a new collection, consisting of all the various combinations of perfections, each combination of which Leibniz considers possible. Godel found it convenient to do this, and used the term positiveness for the resulting notion. Thus we should think of a positive property, in Godel's sense, as some conjunction of perfections in Leib-

140


niz's sense. At least, I am assuming this to be the case-Godel says nothing explicit about the matter. The most notable difference between Godel and Leibniz is that, where Leibniz tried to use what are essentially informal notions in a rigorous way, Godel introduces formal axioms concerning them. Here are Godel's axioms (or their equivalents), and his argument, set forth in everyday English. A formalized version will be found in the next Chapter. The Godel argument has the familiar two-part structure: God's existence is possible; and God's existence is necessary, if possible. I'll take these in order. I'll begin with the axioms for positiveness. The first is rather strong. (I have made no attempt to follow Godel's numbering of axioms and propositions, and in some cases I have adopted equivalents or elaborations of what Godel used.) INFORMAL AXIOM 1

Exactly one of a property or its complement is pos-

itive. It follows that there must be positive properties. If we call a property that is not positive negative, it also follows that there are negative properties. By Informal Axiom 1, a negative property can also be described as one whose complement is positive. Suppose we say property P entails property Q if, necessarily, everything having P also has Q. INFORMAL AXIOM 2

Any property entailed by a positive property is pos-

itive.

This brings us to our first interesting result. 1 Any positive property is possibly instantiated. That is, if P is positive, it is possible that something has property

INFORMAL PROPOSITION

P. Proof Suppose P is positive. Let N be some negative property (the complement of P will do). It cannot be that P entails N, or else N would be positive. So it is not necessary that everything having P has N, that is, it is possible that something has P without having N. So it is possible that something hasP. • Leibniz attempted a proof that "all perfections are compatible with each other or can be in the same subject," that is, having all perfections is a possibly instantiated property. Godel instead simply takes


141

the following as an axiom-it is an immediate consequence, using Informal Proposition 1, that having all positive properties is a possibly instantiated property. INFORMAL AXIOM 3 The conjunction of any collection of positive properties is positive. This is a problematic axiom, in part because there are infinitely many positive properties, and we cannot form an infinite conjunction (unless we are willing to allow an infinitary language). There are ways around this, but there is a deeper problem as well-we will see that this axiom is equivalent to Godel's desired conclusion (given Godel's other assumptions). But further discussion of this point must wait till later on. For now we adopt the axiom and work with it in an informal sense. Now Godel defines God, or rather, defines the property of being Godlike, essentially the same way Leibniz did. INFORMAL DEFINITION 2 A God is any being that has every positive property. This gives us part one of the argument rather easily. INFORMAL PROPOSITION 3 It is possible that a God exists.

Proof By Informal Axiom 3, the conjunction of all positive properties is a positive property. But by Definition 2, this property-maximal positiveness-is what makes one a God. Since the property is positive, it is possibly instantiated, by Informal Proposition 1. • There are also a few technical assumptions concerning positiveness, whose role is not apparent in the informal presentation given here. Their significance will be seen when we come to the formalization in the next Chapter. Here is one. INFORMAL AXIOM 4 Any positive property is necessarily so, and any negative property is necessarily so. Now we move on to the second part of the argument, showing God's existence is necessary, if possible. Here Godel's proof is quite different from that of Descartes, and rather ingenious. To carry out the argument, Godel introduces a pair of notions that are of interest in their own right. INFORMAL DEFINITION 4 A property G is the essence of an object g if: 1 g has property G;

142


2 G entails every property of g.

Strictly speaking, in the definition above I should have said an essence rather than the essence, but it is an easy argument that essences are unique, if they exist at all. Very simply, if an object g had two essences, P and Q, each would be a property of g by part 1, and then each would entail the other by part 2. Godel does not, in general, assume that objects have essences, but for an object that happens to be a God, there is a clear candidate for the essence. INFORMAL PROPOSITION 5 If g is a God, the essence of g is being a God.

Proof Let's state what we must show a little more precisely. Suppose G is the conjunction of all positive properties, so having property G is what it means to be a God. It must be shown that if an object g has property G, then G is the essence of g. Suppose g has property G. Then automatically we have part 1 of Informal Definition 4. Suppose also that P is some property of g. By Informal Axiom 1, if P were not positive its complement would be. Since g has all positive properties, g then would have the property complementary toP. Since we are assuming g has P itself, we would have a contradiction. It follows that P must be positive. Since G is the conjunction of all positive properties, clearly G entails P. Since P was arbitrary, G entails every property of g, and we have part 2 of Informal Definition 4. • Here is the second of Godel's two new notions. INFORMAL DEFINITION 6 An object g has the property of necessarily existing if the essence of g is necessarily instantiated. And here is the last of G6del's axioms. INFORMAL AXIOM 5 Necessary existence, itself, is a positive property. INFORMAL PROPOSITION 7 If a God exists, a God exists necessarily.

Proof Suppose a God exists, say object g is a God. Then g has all positive properties, and these include necessary existence by Informal Axiom 5. Then the essence of g is necessarily instantiated, by Informal Definition 6. But the essence of g is being a God, by Informal Proposition 5. Thus the property of being a God is necessarily instantiated .

•

143


Now we present the second part of the ontological proof. INFORMAL PROPOSITION 8 If it is possible that a God exists, it is necessary that a God exists (assuming the logic is 85). :::::> Q is valid, so is OQ. Then by Informal Proposition 7, if it is possible that a God

Proof In any modal logic at least as strong as K, if P

OP

:::::>

exists, it is possibly necessary that a God exists. In 85, ODP valid, and the conclusion follows. •

:::::>

DP is

Finally, by Informal Propositions 3 and 8, we have our conclusion. INFORMAL THEOREM 9 Assuming all the axioms, and assuming the underlying logic is 85, a God necessarily exists. One final remark before moving on. I've been referring to a God, rather than to the God. As a matter of fact uniqueness is easy to establish, provided we make use of Leibniz's condition that having the same properties ensures identity. Let G be the property of being Godlikethe maximal positive property-and suppose both g1 and g2 possess this property. By Informal Proposition 5, G must be the essence of both g1 and g2. Now, if P is any property of g1, G must entail P, by part 2 of Informal Definition 4. Since G is a property of g2, by part 1 of the same Informal Definition, P must also be a property of g2. Similarly, any property of g2 must be a property of g1. Since g1 and g2 have the same properties, they are identical. This concludes the informal presentation of Godel's ontological argument. It is clear it is of a more complex nature than those that historically preceded it. But an informal presentation is simply not enough. God is in the details, so to speak, and details demand a formal approach. In the next Chapter I'll go through the argument again, more slowly, working things through in the intensional logic developed earlier in Part II.

Exercises EXERCISE 6.1 Show that only God can have a positive essence. (This exercise is due to Ioachim Teodora Adelaida of Bucharest.)

Chapter 11

..

GODEL'S ARGUMENT, FORMALLY

1.

General Plan

The last Chapter ended with an informal presentation of Godel's argument. This one is devoted to a formalized version. I'll also consider some objections and modifications. There are two kinds of objections. One amounts to saying that Godel committed the same fallacy Descartes did: assuming something equivalent to God's existence. Nonetheless, again as in the Descartes case, much of the argument is of interest even if it falls short of establishing the desired conclusion. The second kind of objection is that Godel's axioms are too strong, and lead to a collapse of the modal system involved. Various extensions and modifications of Godel's axioms have been proposed, to avoid this modal collapse. I'll discuss these, and propose a modification of my own. Now down to details, with the proof of God's possible existence coming first. I will not try to match the numbering of the informal axioms in the last chapter, but I will refer to them when appropriate.

2.

Positiveness

God, if one exists, will be taken to be an object of type 0. We are interested in the intensional properties of this object, properties of type j(O). Among these properties are the ones Godel calls positive, and which we can think of as conjunctive combinations of Leibniz's perfections. At least that is how I understand positiveness. Godel's ideas on the subject are given almost no explanation in his manuscript-here is what is said, using the translation of [G70]. 145 M. Fitting, Types, Tableaus, and Gödel's God © Kluwer Academic Publishers 2002

146


Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world). Only then [are] the axioms true. It may also mean pure 'attribution' as opposed to 'privation' (or containing privation).

This is not something I profess to understand. But what is significant is that, rather than attempting to define positiveness, Godel characterized it axiomatically. In this section I present his basic axioms concerning the notion, and I explore some of their consequences. DEFINITION 11.1 (POSITIVE) A constant symbol P of type j (j (0)) zs designated to represent positiveness. It is an intensional property of intensional properties. Informally, P is positive if we have P(P). It is convenient to introduce the following abbreviation. DEFINITION 11.2 (NEGATIVE) If T is a term of type j(O), take short for (Ax.•T(x)). Call T negative if •T is positive.

•T

as

Loosely, at a world in a model, •T denotes the complement of whatever T denotes. It is easy to check formally that T = •( •T), given extensionality for intensional terms, Definition 9.3. Godel assumes that, for each P, exactly one of it or its negation must be positive. Godel's axiom (which he actually stated using exclusive-or) can be broken into two implications. Here they have been formulated as two separate axioms, since they play different roles. AXIOM 11.3 (FORMALIZING INFORMAL AXIOM 1) A (VX)[P(•X) :J •P(X)] B (VX)[•P(X) :J P(•X)] Of these, Axiom 11.3A is certainly plausible: contradictory items should not both be positive. But Axiom 11.3B is more problematic: it says one of a property or its complement must be positive. We might think of the notion of a maximal consistent set of formulas-familiar from the Lindenbaum/Henkin approach to proving classical completeness-as suggestive of what Godel had in mind. There are some cryptic remarks of Godel relating disjunctive normal forms and positiveness, but these have not served as aids to my understanding. At any rate, these are the basic assumptions. The next assumption concerning positiveness is a monotonicity condition: a property that is entailed by a positive property is, itself, positive. Here it is, more or less as Godel gave it.

[P(X)

1\

D(Vx)(X(x) :J Y(x))] :J P(Y)

147


In this formula, x is a free variable of type 0. For us, type-0 quantification is possibilist, while for Godel it must have been actualist. I am assuming this because his conclusion, that God exists, is stated using an existential quantifier, and a possibilist quantifier would have been too weak for the purpose. For us, existence must be made explicit using the existence predicate E, relativizing the ('v'x) quantifier to E. Since this relativization comes up frequently, it is best to make an official definition. DEFINITION

11.4

(EXISTENTIAL RELATIVIZATION)

('v'Ex) abbrevi-

ates (Vx)[E(x) :J ], and (3Ex) abbreviates (3x)[E(x) 1\ ].

11.5 (FORMALIZING INFORMAL AXIOM 2) In the following, x is of type 0, X and Y are of type i(O).

AXIOM

('v'X)(VY){[P(X) 1\ D('v'Ex)(X(x) :J Y(x))] :J P(Y)} At one point in his proof, Godel asserts that (>.x.x = x) must be positive if anything is, and (>.x.•x = x) must be negative. This is easy to see: P( (>.x.x = x)) is valid if anything is positive because anything strictly implies a validity, and we have Axiom 11.5. The assertion that (>.x.•x = x) is negative is equivalent to the assertion that (>.x.x = x) is positive. We thus have the following consequences of Axiom 11.5. PROPOSITION

11.6 Assuming Axiom 11.5:

1 (3X)P(X) :J P((>.x.x = x));

2 (3X)P(X) :J P(•(Ax.•x = x)). PROPOSITION

11.7 Assuming Axioms 11.3A and 11.5: (3X)P(X) :J •P( (>.x.•x = x) ).

Now we have a result from which the possible existence of God will follow immediately, given one more key assumption about positiveness. PROPOSITION

11.8

(FORMALIZING INFORMAL PROPOSITION

1)

Assuming Axioms 11.3A and 11.5, ('v'X){P(X) :J 0(3Ex)X(x)}.

Proof The idea has already been explained, in the proof of Informal Proposition 1 in Section 6. This time I give a formal tableau, which is displayed in Figure 11.1. In it use is made of one of the Propositions above. Item 1 negates the proposition in unabbreviated form. Item 2 is from 1 by an existential rule (with P as a new parameter); 3 and 4 are from 2 by a conjunctive rule; 5 is Axiom 1; 6 is from 5 and 7 is from 6 by universal rules; 8 and 9 are from 7 by a disjunctive rule; 10 and 11

148


are from 8 by a disjunctive rule; 12 is from 11 by a possibility rule; 13 is from 12 by an existential rule (with pas a new parameter, and some tinkering with E); 14 and 15 are from 13 by a conjunctive rule; 16 is from 4 by a necessity rule; 17 is from 16 by a universal rule (and some tinkering with E again); 18 is Proposition 11.7; 19 and 20 are from 18 by a disjunctive rule; 21 is from 19 by a universal rule. • Leibniz attempted to prove that perfections are mutually compatible, basing his proof on the idea that perfections can only be purely positive qualities and so none can negate the others. For Godel, rather than proving any two perfections could apply to the same object, Godel assumes the positive properties are closed under conjunction. This turns out to be a critical assumption. In stating the assumption, read X 1\ Y as abbreviating (.Xx.X(x) 1\ Y(x)). AXIOM 11.9 (FORMALIZING INFORMAL AXIOM 3) (VX)(VY){[P(X) 1\ P(Y)] ::::> P(X 1\ Y)} Godel immediately adds that this axiom should hold for any number of summands. Of course one can deal with a finite number of them by repeated use of Axiom 11.9 as stated-the serious issue is that of an infinite number, which Godel needs. [AG96] gives a version of the axiom which directly postulates that the conjunction of any collection of positive properties is positive. Note that it is a third-order axiom. For reading ease I use the following two abbreviations. 1 Z applies only to positive properties (Z, like P, is of type j(j(O))):

pos(Z) {::} (\iX)[Z(X)

::::>

P(X)]

2 X applies to those objects which possess exactly the properties falling under Z-roughly, X is the (necessary) intersection of Z. (In this, Z is of type i(j(O)), X is of type j(O), and x is of type 0.)

(X intersection of Z) {::} D(\ix){X(x)

= (W)[Z(Y)

::::>

Y(x)]}

AXIOM 11.10 (ALSO FORMALIZING INFORMAL AXIOM 3) (\iZ){pos(Z) ::::> (\iX)[(Xintersection of Z) ::::> P(X)]}. Axiom 11.10 implies Axiom 11.9. I leave the verification to you. I'll finish this section with two technical assumptions that Godel makes "because it follows from the nature of the property." I don't understand this terse explanation, but here are the assumptions. (\iX)[P(X)

::::>

(\iX)[•P(X)

::::>

DP(X)] 0-.P(X)]


u

149

150


If the underlying logic is just K, equivalence of these two assumptions follows from Axioms 11.3A and 11.3B. And if the underlying logic is 85, as it must be for part of Godel's argument, equivalence also follows by Proposition 9.10. Consequently the version used here can be simplified. AXIOM 11.11 (FORMALIZING INFORMAL AXIOM 4) ('v'X)[P(X) :J DP(X)].

P has been taken to be an intensional object, of type j(j(O)). Axiom 11.11 and Theorem 9.11 tells us that Pis rigid. In effect the intensionality of P is illusory-since it is rigid it could just as well have been an extensional object of type (j(O)).

Exercises EXERCISE 2.1 Give a tableau proof that •(.Ax.•(x = x)) = (.Ax.x More generally, show that for a type (0) term T, •(•T) = T.

= x).

EXERCISE 2.2 Show that ('v'X)[•P(X) :J D•P(X)] follows from Axiom 11.11 together with Axioms 11.3A and 11.3B. EXERCISE 2.3 Show Axiom 11.10 implies Axiom 11.9. Hint: use equality.

3.

Possibly God Exists

Godel defines something to be Godlike if it possesses all positive properties. DEFINITION 11.12 (FORMALIZING INFORMAL DEFINITION 2) G is the following type j(O) term, where Y is type j(O). (.Ax.('v'Y)[P(Y) :J Y(x)]). Given certain earlier assumptions, anything having all positive properties can only have positive properties. Perhaps the easiest way to state this formally is to introduce a second notion of Godlikeness, and prove equivalence. DEFINITION 11.13 (ALSO FORMALIZING INFORMAL DEFINITION 2) G* is the type j(O) term

(>.x.(W)[P(Y)

= Y(x)]).

The following result is easily proved; I leave it to you as an exercise.

151


PROPOSITION 11.14 Assume Axiom 11.3B, ('v'X)[•P(X) :J P(•X)].

InK, with this assumption, ('v'x)[G(x)

=G*(x)].

Axiom 11.3B is a little problematic, but it is essential to the Proposition above. If, eventually, we show something having property G exists, and G and G* are equivalent, we will know that something having property G* exists. But the converse is also the case: if something having property G* exists, Axiom 11.3B is the case, even if the existence in question is possibilist. Here is a formal statement of this. Once again I leave the proof to you. PROPOSITION 11.15 InK, (3x)G*(x) :J (VX)[•P(X) :J 'P(•X)]. Now we can show that God's existence is possible. Godel assumes the conjunction of any family of positive properties is positive. Since G* is, in effect, the conjunction of all positive properties, it must be positive, and hence so must G be. PROPOSITION 11.16 InK Axiom 11.10 implies P(G). Once again I leave the formal verification to you. What must be shown is the following.

('v'Z)('v'X){[pos(Z)

1\

(X intersection of Z)] :J P(X)} :J P(G)

Essentially, this is the case because, as is easy to verify, we have each of

pos(P) and (G intersection of P). Now the possibility of God's existence is easy. In fact, it can be proved with an actualist quantifier, though only the weaker possibilist version is really needed for the rest of the argument. THEOREM 11.17 Assume Axioms 11.3A, 11.5, and 11.10. InK both

of the following are consequences. 0(3Ex)G(x) and 0(3x)G(x). Proof By Proposition 11.8,

('v'X){P(X) :J 0(3Ex)X(x)}, hence trivially,

('v'X){P(X) :J 0(3x)X(x)}. By the Proposition above, P(G). The result is immediate. • Note that the full strength of Proposition 11.8 was not really needed for the possibilist conclusion. In fact, if we modify Axiom 11.5 so that quantification is possibilist,

(VX)(W){[P(X) 1\ D('v'x)(X(x) :J Y(x))J :J P(Y)}

152


we would still be able to prove Proposition 11.8 in the weaker form

(\fX){P(X) :J 0(:3x)X(x)} and the Godel proof would still go through.

Exercises EXERCISE 3.1 Give a tableau proof that G entails any positive property: (\fX){P(X) :J D(\fy)[G(y) :J X(y)]}. You will need Axiom 11.11. EXERCISE 3.2 Give a tableau proof for Proposition 11.14. EXERCISE 3.3 Give a tableau proof for Proposition 11.15. EXERCISE 3.4 Give a tableau proof for Proposition 11.16. EXERCISE 3.5 Give a tableau proof of

(\fZ)(\fX){[pos(Z)

4.

1\

(X intersection of Z)] :J P(X)} :J P(G).

Objections

Godel replaced Leibniz's attempted proof of the compatibility of perfections by an outright assumption, given here as Axiom 11.10. Dana Scott, apparently noting that the only use Godel makes of this Axiom is to show being Godlike is positive, proposed taking P( G) itself as an axiom. Indeed, Scott maintains that the Godel argument really amounts to an elaborate begging of the question-God's existence is simply being assumed in an indirect way. In fact, it is precisely at the present point in the argument that Scott's claim can be localized. Godel's assumption that the family of positive properties is closed under conjunction turns out to be equivalent to the possibility of God's existence, a point also made in [SobOl]. We will see, later on, Godel's proof that God's existence is necessary, if possible, is correct. It is substantially different from that of Descartes, and has many points of intrinsic interest. What is curious is that the proof as a whole breaks down at precisely the same point as that of Descartes: God's possible existence is simply assumed, though in a disguised form. The rest of this section provides a formal proof of the claims just made. Enough tableau proofs have been given in full, by now, so that abbreviations can be introduced as an aid to presentation. Before giving the main result of this section, I introduce some simple conventions for shortening displayed tableau derivations.

153


If a X and a X :J Y occur on a branch, a Y can be added. Schematically, aX aX :J Y aY

The justification for this is as follows. aX 1. a X :J Y

a •X 3.

2.

4.

aY

The left branch is closed, and the branch below 4 continues as if we had used the derived rule. Here are a few more derived rules, whose justification I leave to you. aX a (X 1\ Y) :J Z aY :J Z aX aX=:Y aY

a

('v'o:1) · · · ('v'o:n) P(X 1\ Y)} 2. 1 •{[P(A) 1\ P(B)] :::> P(A 1\ B)} 3. 1 P(A) 1\ P(B) 4. 1 •P(A 1\ B) 5. 1 P(A) 6. 1 P(B) 7. 1 ('v'X)('v'Y){[P(X) 1\ D('v'Ex)(X(x) :::> Y(x))] :::> P(Y)} 8. 1 [P(G) 1\ D('v'Ex)(G(x) :::>(A 1\ B)(x))] :::> P(A 1\ B) 9. 1 D('v'Ex)(G(x) :::>(A 1\ B)(x)) :::> P(A 1\ B) 10. 1 •D('v'Ex)(G(x) :::>(A 1\ B)(x)) 11. 1.1•(\:IEx)(G(x) :::>(A 1\ B))(x) 12. 1.1•(\:lx)[E(x) :::> (G(x) :::> (A 1\ B)(x))] 13. 1.1-,[E(c) :::> (G(c) :::>(A 1\ B)(c))] 14. 1.1 E(c) 15. 1.1•(G(c) :::> (A 1\ B)( c)) 16. 1.1 G(c) 17. 1.1-,(AAB)(c) 18. 1 ('v'X)[P(X) :::> DP(X)] 19. 1 P(A) :::> DP(A) 20. 1 P(B) :::> DP(B) 21. 1 DP(A) 22. 1 DP(B) 23. 1.1 P(A) 24. 1.1 P(B) 25. 1.1 (>.x.('v'Y)[P(Y) :::> Y(x)])(c) 26. 1.1 ('v'Y)[P(Y) :::> Y(c)] 27. 1.1 P(A) :::> A(c) 28. 1.1 P(B) :::> B(c) 29. 1.1 A(c) 30. 1.1 B(c) 31. 1.1•(.\x.A(x) 1\ B(x))(c) 32. 1.1•[A(c) 1\ B(c)] 33.

/~

1.1·A(c)

34. 1.1•B(c)

Figure 11.2.

35.

Proof that item 2 implies Axiom 11.9

155

156


rule; 11 is from 5 and 10 by a derived rule; 12 is from 11 by a possibility rule; 13 is 12 unabbreviated; 14 is from 13 by an existential rule; 15 and 16 are from 14, and 17 and 18 are from 16 by conjunctive rules; 19 is Axiom 11.11; 20 and 21 are from 19 by universal rules; 22 is from 6 and 20, and 23 is from 7 and 21, by derived rules; 24 is from 22 and 25 is from 23 by necessity rules; 26 is 17 unabbreviated; 27 is from 26 by an abstraction rule; 28 and 29 are from 27 by universal rules; 30 is from 24 and 28, and 31 is from 25 and 29 by derived rules; 32 is 18 unabbreviated; 33 is from 32 by an abstraction rule; 34 and 35 are from 33 by a disjunctive rule. •

Exercises EXERCISE 4.1 Give a tableau proof showing that 0(3x)G(x) implies Axiom 11.10.

5.

Essence

Even though we ran into the old Descartes problem with half of the Godel argument, we should not abandon the enterprise. The other half contains interesting concepts and arguments. This is the half in which it is shown that God's existence is necessary, if possible. For starters, Godel defines a notion of essence that plays a central role, and is of interest in its own right. [Haz98] makes a case for calling Godel's notion character, reserving the term essence for something else. I follow Godel's terminology. The essence of something, x, is a property that entails every property that x possesses. Godel says it as follows.

cp Ess x = (\17/!){7/J(x)

~

0(\fy)[cp(y)

~

7/J(y)]}

As just given, it does not follow that the essence of x must be a property that x possesses. Dana Scott assumed this was simply a slip on the part of Godel, and inserted a conjunct cp(x) into the definition. I will follow him in this.

cp Ess x

= cp(x) 1\ (V7j!){7j!(x)

~

D('v'y)[cp(y)

~

7/J(y)]}

Godel states cp Ess x as a formula rather than a term-in the version in this book an explicit predicate abstract is used. Also, I assume the type-0 quantifier that appears is actualist, and so in my version the existence predicate, E, must appear. £(P, q) is intended to assert that P is the essence of q. DEFINITION 11.19 (ESSENCE, FORMALIZING INFORMAL DEF. 4) £ abbreviates the following type i (i (0), 0) term, in which Z is of type

157


i(O) and w is of type 0: (.XY,x.Y(x) A ('v'Z){Z(x):) O('v'Ew)[Y(w):) Z(w)]}) The property of being Godlike was defined earlier, Definition 11.12. A central fact about Godlikeness, from Godel's notes, is that it is the essence of any being that is Godlike. 11.20 (FORMALIZING INFORMAL PROPOSITION 5) Assume Axioms 11.3B and 11.11. InK the following is provable. (Note that x is of type 0.)

THEOREM

('v'x)[G(x) :) £( G, x)]. Rather than giving a direct proof, if we use Proposition 11.14 it follows from a similar result concerning G*, provided Axiom 11.3B is assumed. Since such a result has a somewhat simpler proof, this is what is actually shown.

THEOREM

11.21 InK the following is provable, assuming Axiom 11.11.

(\fx)[G*(x):) £(G*,x)]. Proof Here is a closed K tableau to establish the theorem. 1 •('v'x )[G* (x) :) £( G*, x )] 1. 1 -.., [G* (g) :) £ (G*, g)] 2. 1 G*(g) 3. 1-.E(G*,g) 4. 1-.{G*(g) A ('v'Z){Z(g):) D('v'Ew)[G*(w):) Z(w)]}}

5.

/~ 1-.G*(g)

6. 1•(\IZ){Z(g):) O('v'Ew)[G*(w):) Z(w)]}

7.

Item 2 is from 1 by an existential rule, with g a new parameter; 3 and 4 are from 2 by a conjunction rule; 5 is from 4 by a derived unsubscripted abstract rule; 6 and 7 are from 5 by a disjunction rule. The left branch is closed. I continue with the right branch, below item 7.

158


1 •{Q(g) :J D(VEw)[G*(w) :J Q(w)]} 1 Q(g) 9. 1 ·D(VEw)[G*(w) j Q(w)] 10. 1.1·(V'Ew)[G*(w) :J Q(w)] 11. 1.1•{E(a) :J [G*(a) :J Q(a)]} 12. 1.1 E(a) 13. 1.1--,[G*(a) :J Q(a)] 14. 1.1 G*(a) 15. 1.1 •Q( a) 16. 1 (VY)[P(Y) = Y(g)] 17. 1 P(Q) = Q(g) 18. 1 P(Q) 19. 1.1 (VY)[P(Y) = Y(a)] 20. 1.1 P(Q) Q(a) 21. 1 (VY)[P(Y) :J DP(Y)] 22. 1 P( Q) :J DP( Q) 23. 1 DP(Q) 24. 1.1 P(Q) 25. 1.1 Q(a) 26.

8.

=

Item 8 is from 7 by an existential rule, with Q a new parameter; 9 and 10 are from 8 by a conjunction rule; 11 is from 10 by a possibility rule; 12 is from 11 by an existential rule; 13 and 14 are from 12 by a conjunctive rule, as are 15 and 16 from 14; 17 is from 3 by a derived unsubscripted abstract rule; 18 is from 17 by a universal rule; 19 is from 9 and 18 by an earlier derived rule; 20 is from 15 by a derived unsubscripted abstract rule; 21 is from 20 by a universal rule; 22 is Axiom 11.11; 23 is from 22 by a universal rule; 24 is from 19 and 23 by a derived rule; 25 is from 24 by a necessity rule; 26 is from 21 and 25 by a derived rule. The branch is closed by 16 and 26. • In the notes Dana Scott made when Godel showed him his proof, there are two observations concerning essences. One is that something can have only one essence. The other is that an essence must be a complete characterization. Here are versions of these results. I begin by showing that any two essences of the same thing are necessarily equivalent. THEOREM

11.22 Assume the modal logic is K. The following is prov-

able.

(VX)(VY)(Vz){[t'(X, z)

1\

t'(Y, z)] :J D(VEw)[X(w)

= Y(w)]}

Proof The idea behind the proof is straightforward. If P and Q are essences of the same object, each must entail the other. I give a tableau

159


proof mainly to provide another example of such. It starts by negating the formula, applying existential rules three times, introducing new parameters P, Q, and a, then applying various propositional rules. Omitting all this, we get to items 1 - 3 below.

1 E(P, a) 1. 1 E(Q, a) 2. 1•D('v'Ew)[P(w) Q(w)] 3. 1 P(a) 4. 1 ('v'Z)[Z(a) :J D('v'Ew)[P(w) :J Z(w)]] 1 Q(a) 6. 1 ('v'Z)[Z(a) :J D('v'Ew)[Q(w) :J Z(w)Jl 1 Q(a) :J D('v'Ew)[P(w) :J Q(w)] 8. 1 P(a) :J D('v'Ew)[Q(w) :J P(w)] 9.

=

1•Q(a)

5. 7.

/~

10. 1 D('v'Ew)[P(w) :J Q(w)]

1•P(a)

11.

/~

12. 1 D('v'Ew)[Q(w) :J P(w)]

13.

Items 4 and 5 are from 1 by an abstraction rule (and a propositional rule), 6 and 7 are from 2 the same way; 8 is from 5 and 9 is from 7 by universal rules; 10 and 11 are from 8, and 12 and 13 are from 9 by disjunction rules. The left branch is closed, by 6 and 10. The middle branch is closed by 4 and 12. I continue with the rightmost branch, below item 13.

1.1•('v'Ew)[P(w) = Q(w)] 14. 1.1·{E(b) :J [P(b) = Q(b)]} 15. 1.1 E(b) 16. 1.1·[P(b) = Q(b)] 17.

/~

1.1 P(b) 1.1·Q(b)

18. 1.1·P(b) 19. 1.1 Q(b)

20. 21.

Item 14 is from 3 by a possibility rule; 15 is from 14 by an existential rule; 16 and 17 are from 15 by a conjunction rule; 18, 19, 20, 21 are from 17 by successive propositional rules. I show how the left branch can be continued to closure; the right branch has a symmetric construction which I omit.

160


1.1 (VEw)[P(w) :J Q(w)] 1.1 E(b) :J [P(b) :J Q(b)]

22. 23.

/~

1.1--,E(b)

24.

1.1 P(b) :J Q(b)

25.

/~

1.1--,P(b)

26.

1.1 Q(b)

27.

Item 22 is from 11 by a necessitation rule; 23 is from 22 by a universal rule; 24 and 25 are from 23 by a disjunction rule, as are 26 and 27 from 25. The left branch is closed by 16 and 24, the middle branch is closed by 18 and 26, and the right branch is closed by 19 and 27. • Now, here is the second of Scott's observations: if X is the essence of y, only y can have X as a property. THEOREM 11.23 Assume the modal logic is K, including equality. The following is valid. (VX)(Vy){ £(X, y) :J D(VEz)[X(z) :J (y = z)]}

This can be proved using tableaus-! leave it to you as an exercise.

Exercises EXERCISE 5.1 Give a tableau proof for Theorem 11.23. Hint: for a parameter c, one can consider the property of being, or not being, c, that is, (.Xx.x =c) and (.Xx.x #c). Either property can be used in the proof. EXERCISE 5.2 Give a tableau proof to establish Theorem 11.20 directly, without using G*.

6.

Necessarily God Exists

In this section I present a version of Godel's argument that God's possible existence implies His necessary existence. It begins with the introduction of an auxiliary notion that Godel calls necessary existence. DEFINITION 11.24 (NECESSARY EXISTENCE) (Formalizing Informal Definition 6) N abbreviates the following type i(O) term:

(.Xx.(VY)[£(Y, x)

:J

0(3Ez)Y(z)]).


161

The idea is, something has the property N of necessary existence provided any essence of it is necessarily instantiated. Godel now makes a crucial assumption: necessary existence is positive. AXIOM

11.25 (FORMALIZING INFORMAL AXIOM 5)

'P(N). Given this final axiom, Godel shows that if (some) God exists, that existence cannot be contingent. An informal sketch of the proof was given in Section 6 of Chapter 10, and it can be turned into a formal proof-see Informal Propositions 7 and 8. I will leave the details as exercises, since you have seen lots of worked out tableaus now. Here is a proper statement of Godel's result, with all the assumptions explicitly stated. Nate that the necessary actualist existence of God follows from His possibilist existence. 11.26 (FORMALIZING INFORMAL PROPOSITION 7) Assume Axioms 11.3B, 11.11, and 11.25. In the logic K,

THEOREM

(3x)G(x) ::) D(3Ex)G(x). I leave it to you to prove this, using the informal sketch as a guide. Now Godel's argument can be completed. 11.27 (FORMALIZING INFORMAL PROPOSITION 8) Assume Axioms 11.3B, 11.11, and 11.25. In the logic 85,

THEOREM

0(3x)G(x) ::) D(3Ex)G(x). Proof From Theorem 11.26,

(3x)G(x) ::) D(3Ex)G(x). By necessitation,

D[(3x)G(x) ::) D(3Ex)G(x)]. By the K validity D(A::) B)::) (OA::) OB),

0(3x)G(x) ::) OD(3Ex)G(x). Finally, in 85, ODA::) DA, so we conclude

0(3x)G(x) ::) D(3Ex)G(x) .

• Now we are at the end of the argument. COROLLARY

11.28 Assume all the Axioms. In the logic 85,

D(3Ex)G(x). Proof By Theorems 11.27 and 11.17. •

162


Exercises EXERCISE 6.1 Give a tableau proof to show Theorem 11.26. Use various earlier results as assumptions in the tableau.

7.

Going Further

Godel's axioms admit more consequences than just those of the ontological argument. In this section a few of them are presented.

7.1

Monotheism

Does there exist exactly one God? The following says "yes." You are asked to prove it, as Exercise 7.1. PROPOSITION 11.29 (3x)('v'y)[G(y) =: (y = x)j. This Proposition has a curious Corollary. Since type-0 quantification is possibilist, it makes sense to ask if there are gods that happen to be non-existent. But Corollary 11.28 tells us there is an existent God, and the Proposition above tells us it is the only one God, existent or not. Consequently we have the following. COROLLARY 11.30 ('v'x)[G(x) :J E(x)j. Proposition 11.29 tells us we can apply the machinery of definite descriptions. By Definition 9.12, 1x.(W)[P(Y) :J Y(x)] always designates, and consequently so does 1x.G(x). Proposition 9.14 tells us this will be a rigid designator provided G(x) is stable. It follows from Sobel's argument in Section 8 that it, and everything else, is. But alternative versions of Godel's axioms have been proposed-! will discuss some below-and using them the stability of G(x) does not seem to be the case. That is, it seems to be compatible with the axioms of Godel (as modified by others) that, while the existence of God is necessary, a particular being that is God need not be God necessarily. If this is not the case, and a proof has been missed, I invite the reader to correct the situation.

7.2

Positive Properties are Necessarily Instantiated

Proposition 11.8 says that positive properties are possibly instantiated. In [Sob87], it is observed that a consequence of Corollary 11.28 is that every positive property is necessarily instantiated. PROPOSITION 11.31 ('v'X){'P(X) :J 0(3Ex)X(x)}. I leave the easy proof of this to you.


163

Exercises EXERCISE 7.1 Give a tableau proof for Proposition 11.29. Hint: you will need Corollary 11.28, Theorem 11.20, and Theorem 11.23. EXERCISE 7.2 Provide a tableau proof for Proposition 11.31. Hint: by

Corollary 11.28, a Godlike being necessarily exists. Such a being has all positive properties, so every positive property is instantiated. Now, build this into a tableau.

8.

More Objections

In Section 4 we saw that one of Godel's Axioms was equivalent to the possible existence of God. Other objections have been raised that are equally as serious. Chapter 4 of [SobOl] discusses problems with Axiom 11.25, that necessary existence is positive. I do not take this point up here. But also in [SobOl], and earlier in [Sob87], it was argued that Godel's axiom system is so strong it implies that whatever is the case is so of necessity, Q ::J DQ. In other words, the modal system collapses. In still other, more controversial, words, there is no free will. Roughly speaking, the idea of Sobel's proof is this. God, having all positive properties, must possess the property of having any given truth be the case. Since God's existence is necessary, anything that is a truth must necessarily be a truth. Here is a version of the argument given by Sobel. For simplicity, assume Q is a formula that contains no free variables. By Theorem 11.20,

(\fx)[G(x) ::J £( G, x)].

(11.1)

Using the definition of£, we have as a consequence

(\fx){G(x) ::J ('v'Z){Z(x) ::J D('v'Ew)[G(w) ::J Z(w)]}}.

(11.2)

There is a minor nuisance to deal with. In the formula (11.2) I would like to instantiate the quantifier ('v'Z) with Q, but this is not a 'legal' term, so instead I use the term (>..y.Q) to instantiate. In it, y is of type 0, and so (>..y.Q) is of type i(O). We get the following consequence.

(\fx){G(x) ::J {(>..y.Q)(x) ::J D('v'Ew)[G(w) ::J (>..y.Q)(w)]}}.

(11.3)

Now to undo the technicality just introduced, note that since y does not occur free in Q, (>..y.Q)(x) = (>..y.Q)(w) = Q, and so we have

(\fx){G(x) ::J {Q ::J D('v'Ew)[G(w) ::J Q]}}.

(11.4)

164


Since x does not occur free in the consequent, (11.4) is equivalent to the following:

(3x)G(x) :::> {Q :::> D('v'Ew)(G(w) :::> Q)}.

(11.5)

We have Corollary 11.28, from which

(3x)G(x)

(11.6)

follows. Then from (11.5) and (11.6) we have

Q :::> D('v'Ew) (G( w) :::> Q).

(11.7)

Since Q has no free variables, (11.7) is equivalent to the following:

Q :::> D[(3Ew)G(w) :::> Q].

(11.8)

Using the distributivity of D over implication, (11.8) gives us

Q :::> [D(3Ew)G(w) :::> DQ].

(11.9)

Finally (11.9), and Corollary 11.28 again, give the intended result,

Q :::> DQ.

(11.10)

Most people have taken this as a counter to Godel's argument-if the axioms are strong enough to admit such a consequence, something is wrong. In the next two sections I explore some ways out of the difficulty.

9.

A Solution

Sobel's demonstration that the Godel axioms imply no free will rather takes the fun out of things. In this section I propose one solution to the problem. I don't profess to understand its implications fully. I am presenting it to the reader, hoping for comments and insights in return. Throughout, it has been assumed that Godel had in mind intensional properties when talking about positiveness and essence. But, suppose not-suppose extensional properties were intended. I reformulate Godel's argument under this alternative interpretation. It is one way of solving the problem Sobel raised.

165


In this section only I will take P to be a constant symbol of type

i((O)). Axiom 11.5 gets replaced with the following. Revised Axiom 11.5 In the formula below, xis of type 0, and X and Y are of type (0).

(VX)(VY){[P(X) A D(V'Ex)(X(x)

:J

Y(x))]

:J

P(Y)}

Note that this has the same form as Axiom 11.5, but the types of variables X and Y are now extensional rather than intensional. This will be the general pattern for changes. The definition of negative, for instance, is modified as follows. For a term T of type (0), take •T as short for l(.Xx.•T(x)). Then Axioms 11.3A and 11.3B, 11.10, and 11.11, all have their original form, but with variables changed from intensional to extensional type. The analog of Proposition 11.8 still holds, but with extensional variables involved.

(VX){P(X)

:J

0(3Ex)X(x)}

Analogs of G and G* are defined in the expected way. G is the following type j(O) term, where Y is type (0) and, as noted before, Pis of type j((O)).

(.Xx.(VY)[P(Y)

:J

Y(x)])

Likewise G* is the type i(O) term

(.Xx.(W)[P(Y)

= Y(x)]).

One can still prove (Vx)[G(x) = G*(x)]. Essence must be redefined, but again it is only variable types that are changed. £ now abbreviates the following type j( (0), 0) term, in which Z is of type (0) and w is of type 0:

(.XY, x.Y(x) A (VZ){Z(x)

:J

D(V'Ew)[Y(w)

:J

Z(w)]})

Theorem 11.21 plays an essential role in the Godel proof, and it too continues to hold, in a slightly modified form:

(Vx)[G*(x) :J £(1G*, x)] .

.

I leave the proof of this to you-it is similar to the earlier one. Of course we must modify the definition of Necessary Existence, to use the revised version of essence, and Axiom 11.25 as well, to use the modified definition of Necessary Existence. For this section, N abbreviates the following type j(O) term, in which Y is of type (0):

(>.x.('VY)[E(Y, x) :J D(3Ez)Y(z)).

166


Revised Axiom 11.25 is P(lN), where N is as just modified. With this established, the rest of Godel's argument carries over directly, giving us the following.

0(:3E z) {tG*) (z) The final step is the easy proof that this implies the desired 0(:3Ez)G*(z), and hence O(::JEz)G(z), and I leave this to you. So, we have the conclusion of Godel's argument. Finally, here is a model, adapted from [And90], that shows Sobel's continuation no longer applies. EXAMPLE 11.32 Construct a standard S5 model as follows. There are two possible worlds, call them r and D.. The accessibility relation always holds. The type-0 domain is the set {a, b}. Since this is a standard model, the remaining types are fully determined. The existence predicate, E, is interpreted to have extension {a, b} at r and {a} at D.. Informally, all type-0 objects exist at r, but only a exists at D.. Call a type-(0) object positive if it applies to a. Interpret P so that at each world its extension is the collection of positive type-(0) objects; that is, at each world P designates {{a}, {a, b}}. This finishes the definition of the model. I leave the following facts about it for you to verify.

1 The designation of G in this model is rigid, with {a} as its extension at both worlds. 2 The designation of £ is also rigid, with extension { ({a}, a), ({b}, b)} at each world. Loosely, the essence of a is {a} and the essence of b is

{b}. 3 The designation of N is also rigid, with extension {a} at each world. 4 All the Axioms are valid, as modified in this section. Now take Q to be the closed formula (::JEx)(::JEy)-,(x = y). Since it asserts two objects actually exist, it is true at r, but not at D., and hence Q :J OQ is not true at r. We now know that Sobel's argument must break down in the present system, but it is instructive to try to reproduce the earlier proof, and see just where things go wrong. The attempted argument takes on a rather formidable appearance--you might want to skip to the last paragraph and read the conclusion, before going through the details.

167


We try to prove Q ::) DQ, starting more or less as we did before.

(Vx)[G(i) ::) £(lG, x)]

(11.11)

which, unabbreviated, is

(Vx)[G(x) ::) (.XY,x.Y(x) 1\ (VZ){Z(x)::) D(VEw)[Y(w)::) Z(w)]})(1G,x)]

(11. 12 )

where Y and Z are of type (0), unlike in (11.2) where they were of type

j(O). The variable xis of type 0, and it is easy to show the following simpler formula is a consequence of (11.12).

(\ix)[G(x) ::)

(.XY.Y(x) 1\ (VZ){Z(x) ::) D(VEw)[Y(w) ::) Z(w)]})(1G)]

(11.13)

From this we trivially get the following.

(Vx)[G(x) ::) (.XY.(VZ){Z(x)::) D(VEw)[Y(w)::) Z(w)]})(1G)]

(11.14)

Next, in the argument of Section 8, we instantiated the quantifier (V Z) with the term (.Xy.Q). Of course we cannot do that now, since (.Xy.Q) is an intensional term, while the present quantifier (V Z) is extensional. Apply the extension-of operator, getting 1(-Xy.Q), and use this instead. But universal instantiation involving relativized terms is a little tricky. If 1T is a relativized term of the same type as Z, (VZ)cp(Z) ::) cp(l T) is not generally valid. What is valid is ('v'Z)cp(Z) ::) (.XZ.cp(Z))(1T). So what we get from formula (11.14) when we instantiate the quantifier is the following consequence.

(Vx)[G(x) ::) (.XY, Z.Z(x) ::) D(VEw)[Y(w) ::) Z(w)])(1G, 1(-Xy.Q) )]

(11.15)

Distributing the abstraction, this is equivalent to the following.

(Vx ){ G(x) ::) [(.XY, Z.Z(x))(lG, 1(-Xy.Q))::) (.XY, Z.D('v'Ew)(Y(w)::) Z(w)))(lG, 1(-Xy.Q))]}

(11.16)

168


The variable x does not occur free in (>.y.Q) and Y does not occur in Z(x), so (>.Y,Z.Z(x))(lG,l(>.y.Q)) is simply equivalent to Q, and (11.16) reduces to the following.

('v'x){G(x) :J

[Q :J (>.Y, Z.D('v'Ew)(Y(w) :J Z(w)))(lG, l(>.y.Q) )]}

(11.17)

From this we get

(3x)G(x) :J

[Q :J (>.Y, Z.D('v'Ew)(Y(w) :J Z(w)))(lG, l(>.y.Q) )]

(11.18)

and since we have (3x)G(x), we also have

Q :J (>.Y, Z.D('v'Ew)(Y(w) :J Z(w)))(lG, l(>.y.Q) ).

(11.19)

Since Q has no free variables, (11.19) can be shown to be equivalent to the following (where a constant symbol a has been introduced to keep formula formation correct).

Q :J (>.Y, Z.D((3Ew)Y(w) :J Z(a)))(lG, l(>.y.Q)).

(11.20)

Using the distributivity of 0 over implication, (11.20) gives us

Q :J (>.Y, Z.D(3Ew)Y(w) :J DZ(a))(lG, l(>.y.Q)).

(11.21)

From (11.21) we get

Q :J[(>.Y, Z.D(3Ew)Y(w))(lG, l(>.y.Q)) :J (>.Y, Z.DZ(a))(lG, l(>.y.Q) )].

(11.22)

Because Z has no free occurrences in D(3Ew)Y(w) and Y has no free occurrences in Z(a), (11.22) can be simplified to Q :J[(>.YD(3Ew)Y(w))(lG) :J

(>.Z.DZ(a))(l(>.y.Q) )].

(11.23)

I don't know the status of (>.Y.D(3Ew)Y(w))(lG), that is, whether or not it follows from the axioms used in this section. It does hold provided


169

G is rigid, so in particular, it holds in the model of Example 11.32. Consequently, in settings like that model (11.23) reduces to the following.

Q:) (>.Z.DZ(a))(l(>.y.Q)).

(11.24)

But (>.Z.DZ(a))(l (.>.y.Q)) is not equivalent to DQ, and that's an end of it. Expressing the essential idea of (.>.Z.DZ(a))(l(.>.y.Q)) with somewhat informal notation, we might write it as (>.Z.DZ)(lQ), and so what has been established, assuming rigidity of G, is

Q :) (.>.Z.DZ)(lQ)

(11.25)

and this is quite different from Q :) DQ. In the abstract, the variable Z is assigned the current version of Q-its truth value in the present world. Perhaps an example will make clear what is happening. Suppose it is the case, in the real world, that it is raining-take this as Q. If we had validity of Q :) DQ, it would necessarily be raining-DQ-and so in every alternative world, it would be raining. But what we have is Q :) (.>.Z.DZ)(lQ), and since Q is assumed to hold in the real world, we conclude (>.Z.DZ)(lQ). This conclusion asserts something more like: if it is raining in the real world, then in every alternative world it is true that it is raining in the real world. As it happens, this is trivially correct, and says nothing about whether or not it is raining in any alternative world.

10.

Anderson's Alternative

One solution to the objection Sobel raised has been presented. In [And90] some different, quite reasonable, modifications to the Godel axioms are proposed that also manage to avoid Sobel's conclusion. For this section I return to the use of intensional variables. Axiom 11.3B is something of a problem. Essentially it says, everything must be either positive or negative. As Anderson observes, why can't something be indifferent? Anderson drops Axiom 11.3B. The most fundamental change, however, is elsewhere. Definition 11.12 and its alternative, Definition 11.13, are discarded. Instead there is a requirement that a Godlike being have positive properties necessarily. DEFINITION 11.33 (GODLIKE, ANDERSON VERSION) GA is the type j (0) term

(.Xx.(W)[P(Y) := DY(x)]). There is a corresponding change in the definitions of essence and necessary existence. Definition 11.19 gets replaced by the following

170


DEFINITION 11.34 (ESSENCE, ANDERSON VERSION) the following type i(i(O), 0) term:

[A abbreviates

(.XY, x.(VZ){DZ(x) =: D(VEw)[Y(w) :J Z(w)]}) Notice several key things about this definition. The Scott addition, that the essence of an object actually apply to the object, is dropped. A necessity operator has been introduced that was not present in the definition of £. And finally, an implication in the definition of £ has been replaced by an equivalence. The definition of necessary existence, Definition 11.24, is replaced by a version of the same form, except that Anderson's definition of essence is used in place of that of Godel. DEFINITION 11.35 (NECESSARY EXISTENCE, ANDERSON VERSION)

NA abbreviates the following type j(O) term: (.Xx.(VY)[£A(Y, x) :J D(3Ez)Y(z)). Now, what happens to earlier reasoning? Of course Proposition 11.8 still holds, since Axioms 11.3A and 11.5 remain unaffected. Theorem 11.20 turns into the following. THEOREM 11.36 In S5 the following is provable.

(Vx)[GA(x) :J eA(GA,x)]. I leave it to you to verify the theorem, using tableaus say. Next, Anderson replaces Axiom 11.25 with a corresponding version asserting that his modification of necessary existence is positive.

AXIOM 11.37 (ANDERSON'S VERSION OF 11.25) P(NA). Now Theorem 11.26 turns into the following. THEOREM 11.38 Assume Axioms 11.11 and 11.37. In the logic S5,

(3x)GA(x) :J 0(3Ex)GA(x). Once again, I leave the proof to you. These are the main items. The rest of the ontological argument goes through as before. At the end, we have the following. THEOREM 11.39 Assume all the Axioms 11.3A, 11.5, 11.10, 11.11, and 11.37. In the logic S5,

171


Thus the desired necessary existence follows, and with one fewer axiom (though with more complex definitions). And a model, closely related to the one given in the previous section, can be constructed to show that these axioms do not yield Sobel's undesirable conclusion-see [And90] for details.

Exercises 10.1 Supply a tableau argument for Theorem 11.36. Do the same for Theorem 11.38. EXERCISE

11.

Conclusion

Godel's proof, and criticisms of it, have inspired interesting work. Some was mentioned above. More remains to be done. Here I briefly summarize some directions that might profitably be explored. [Haj95] studies the role of the comprehension axioms-work that is summarized in [Haj96b]. Completely general comprehension axioms are implicit in my presentation, they are present as the assumption that every abstract has a meaning. Hajek confines things to a second-order intensional logic, augmented with one third-order constant to handle positiveness. In this setting Hajek introduces what he calls a cautious comprehension schema:

('v'x)[G(x) :J (D(x) V 0-.(x))] :J (:3Y)D('v'x)[Y(x)

=(x)].

Hajek shows that Godel's axioms do not lead to a proof of Q :J DQ, provided cautious comprehension replaces full comprehension, but the necessary existence of God still can be concluded. Hajek refutes a claim by Magari, [Mag88], that a subset of Godel's axiom system is sufficient for the ontological argument. But he also shows Magari's claim does apply to Anderson's system. And he shows that Godel's axioms, with cautious comprehension, can be interpreted in Anderson's system, with full comprehension. The results of Hajek assume an underlying model with constant domains but no existence predicate, and only intensional properties. It is not clear what happens if these assumptions are modified. In Section 7, some further consequences of Godel's axioms were discussed. I don't know what happens to these when the axioms are modified in the various ways suggested here and in the previous two sections. Nor do I know the relationships, if any, between the extensional-property approach I suggested, and Anderson's version. Finally, and most entertainingly, I refer you to an examination of ontological arguments and counter-arguments in the form of a series of

172


puzzles, in [Smu83], Chapter 10. You should find this fun, and a bit of a relief after the rather heavy going of the book you just finished.

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[Fit96]

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[Fit98]

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North-

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Index

abstraction designation function, 21, 26, 103 proper, 22, 27 accessibility, 91 Anderson, C. A., 169-171 Anselm, 134

E-complete, 63 entity, 48, 49 equality, 115 equality axioms, 69, 115 essence, 141, 142, 156, 170 evaluation at a prefix, 109 existential relativization, 147 extensional object, 84, 91 extensionality, 117 assumptions, 118 axioms, 77 for extensional terms, 118 for intensional terms, 117

cautious comprehension, 171 character, 156 choice axiom, 129 choice function, 128-130 closed, 37, 110 compact, 15, 66 complete, 46, 73 strong, 16 weak, 16 composition, 10 comprehension axiom, 3 concept, 84 consequence, 14, 28, 95 consistent, 63 maximal, 63 constant domain, 89 constant symbol, 5, 87 continuum hypothesis, 17 cut rule, 67 cut-elimination, 66

finite support, 8 formula, 6 modal, 88 prefixed, 106 truth, 13, 22, 26, 93, 104 frame augmented, 91 extensional, 30 Henkin, 20 generalized, 25 relative generalized, 50 Henkin/Kripke, 103 Kripke, 91 free, 9 free variable, 6

de dicto, 118-121 de re, 118-121 Dedekind, R., 15 defined at, 125 definite description, 125-128 degree, 8 Descartes, R., 134-136, 152, 156 description designation, 125 designates, 125 domain, 91 domain function Henkin, 20, 103

Gaunilo, 134 global assumption, 95, 111 Gi:idel, K., 138-143, 145, 147, 148, 150, 152, 156, 158, 162-164, 166, 171 grounded, 34, 106 Hajek, P., 171 Henkin domain

179

180 relative, 50 Hintikka set, 47 impredicativity, 4 inconsistent, 63 intensional object, 84, 91 interpretation, 11, 30, 51, 73, 92, 103 allowed, 50 K, 105

L(C), 5 .X abstraction, 3 Leibniz, G., 137-140, 145, 148, 152 Lindstrom, P., 68 local assumption, 95, 111 Liiwenheim-Skolem, 66, 68 Magari, R., 171 model classical, 12 extensional, 30 general, 19 generalized Henkin, 28, 104 Henkin, 19, 22, 23 Henkin/Kripke, 104 modal, 91 standard, 24 monotheism, 162 necessary existence, 142, 160, 170 negative, 141, 146 non-rigid, 102 normal, 25 order, 5 parameter, 34, 108 perfection, 137-139 positive, 138-142, 145, 146, 162 possible value, 49 possible world, 91 predicate abstract, 5 predicate abstraction, 3 prefix, 105 pseudo-model, 47, 48, 51 quantification actualist, 89, 91 possibilist, 89, 91 rigid, 121-124 rule abstract, 37, 109 branch extension, 35 conjunctive, 35, 107 derived

TYPES, TABLEAUS, AND GO DEL'S GOD closure, 113 extensionality, 77 intensional predication, 113 reflexivity, 70 subscripted abstract, 114 substitutivity, 70 unsubscripted abstract, 114 disjunctive, 36, 107 double negation, 35, 107 existential, 36, 108 extensional, 118 extensional predication, 110 intensional predication, 109 necessity, 108 possibility, 107, 108 reflexivity, 115 substitutivity, 115 universal, 36, 109 world shift, 110 Russell, B., 125, 126, 136

85, 105 satisfiability, 14, 28 Scott, D., 138, 152, 156, 158 sentence, 6 Sobel, J. H., 163, 164, 166, 171 sound, 43, 46, 73 stability, 124-125 substitution, 8 free, 9 tableau, 33 basic, 35 derivation, 37, 111 prefixed, 105 proof, 37, 110 satisfiable, 43 term, 6, 87 denotation, 12, 21, 26 designation, 93, 103 relativized, 108 type, 4, 86 extensional, 86 Gallin/Montague, 102 intensional, 86 relation, 11 validity, 14, 28, 94 valuation, 12, 20, 26, 92 variable, 5 variant, 12 varying domain, 89 Wittgenstein, L., 139 world independent, 110 Zermelo-Fraenkel set theory, 17

TRENDS IN LOGIC 1.

G. Schurz: The Is-Ought Problem. An Investigation in Philosophical Logic. 1997 ISBN 0-7923-4410-3

2.

E. Ejerhed and S. Lindstrom (eds.): Logic, Action and Cognition. Essays in Philosophical Logic. 1997 ISBN 0-7923-4560-6

3.

H. Wansing: Displaying Modal Logic. 1998

ISBN 0-7923-5205-X

4.

P. Hajek: Metamathematics of Fuzzy Logic. 1998

ISBN 0-7923-5238-6

5.

H.J. Ohlbach and U. Reyle (eds.): Logic, Language and Reasoning. Essays in Honour ofDov Gabbay. 1999 ISBN 0-7923-5687-X

6.

K. Dosen: Cut Elimination in Categories. 2000

7.

ISBN 0-7923-5720-5

R.L.O. Cignoli, I.M.L. D' Ottaviano and D. Mundici: Algebraic Foundations ofmanyISBN 0-7923-6009-5

valued Reasoning. 2000 8.

E.P. Klement, R. Mesiar and E. Pap: Triangular Norms. 2000 ISBN 0-7923-6416-3

9.

V.F. Hendricks: The Convergence of Scientific Knowledge. A View From the Limit. 2001 ISBN 0-7923-6929-7

10.

J. Czelakowski: Protoalgebraic Logics. 2001

11.

G. Gerla: Fuzzy Logic. Mathematical Tools for Approximate Reasoning. 2001 ISBN 0-7923-6941-6

12.

M. Fitting: Types, Tableaus, and Godel's God. 2002

ISBN 0-7923-6940-8

ISBN 1-4020-0604-7

KLUWER ACADEMIC PUBLISHERS - DORDRECHT I BOSTON I LONDON

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