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

TYPES, TABLEAUS, AND GODEL'S GOD TRENDS IN LOGIC Studia Logica Library VOLUME 13 Managing Editor Ryszard Wojcicki, Ins...

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TYPES, TABLEAUS, AND GODEL'S GOD

TRENDS IN LOGIC Studia Logica Library VOLUME 13 Managing Editor Ryszard Wojcicki, Institute of Philosoph y and Sociolog y. Polish Academ y of Sciences. Warsaw, Poland Editors Daniele Mundici, Department 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 Philosoph y, Uppsala University, Sweden Alasdair Urquhart, Department of Philosoph y, 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 GODEL'S GOD ••

a.r...

"

KLUWER ACADEMIC PUBLISHERS DORDRECHTI BOSTON I 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 1 Classical Models Truth in a Model 2 Problems 3 Compactness 3.1 Strong Completeness 3.2 Weak Completeness 3.3 3.4 And Worse Henkin Models 4 Generalized Henkin Models 5 A Few Technical Results 6 6.1 Terms and Formulas 6.2 Extensional Models Language Extensions 6.3

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

3. CLASSICAL LOGIC-BASIC TABLEAUS A Different Language 1 Basic Tableaus 2 Tableau Examples 3

33 33 35 37

v

11 11

vi


4. SOUNDNESS AND COMPLETENESS Soundness 1 2 Completeness Hintikka Sets 2.1 Pseudo- Models 2.2 Substitution and Pseudo-Models 2.3 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 Adding Extensionality 1 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 Standard Modal Models 4 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 The Rules 1 Prefixes 1.1 Propositional Rules 1.2

105 105 105 107

Contents

VB

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 1 Equality 1.1 Equality Axioms 1.2 Extensionality 2

3 4 5 6

De Re and De Dicto Rigidity Stability Conditions Definite Descriptions Choice Functions

115 115 115 117 118 121 124 125 128

Part III ONTOLOGICAL ARGUMENTS 10. GODEL'S ARGUMENT, BACKGROUND 1 Introduction 2 Anselm 3 Descartes 4 Leibniz GCidel 5 Codel's Argument, Informally 6

133 133 134 134 137 138 139

11. GODEL'S ARGUMENT, FORMALLY General Plan 1

145 145 145 150 152 156 160 162 162

2

3 4 5 6 7

Positiveness Possibly God Exists Objections Essence Necessarily God Exists Going Further 7.1 Monotheism

TYPES, TABLEAUS, AND GODEL 'S GOD

viii 7.2

Positive Properties are Necessarily Instantiated

162

8

More Objections

163

9

A Solution

164

10 11

Anderson's Alternative Conclusion

169 171

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

xii


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, To175, And86, Sha91, Lei94, Koh95, Man96], for example . Part II is devoted to the complications that modality brings. Chapter 7 adds the usual box and diamond to the syntax, and possible worlds

PREFACE

xiii

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

XIV


also Anselm) all have proofs that stick at the same point: showing that the existence of God is possible. If the Oodel argument is what you are interested in, start with Part III, and pick up earlier material as needed. Many of the uses of the formalism are relatively intuitive. Indeed, in Codel's notes on his ontological argument, formal machinery is never discussed, yet it is possible to get a sense of what it is about anyway.

How Did This Get Written? Having just completed work on a book about first-order modal logic, [FM98] , a look at higher-order modal logic suggested itself. I thought I would use Godel's ontological argument as a paradigm, because it is one of the few examples I have run across that makes essential use of higherorder modal constructs. Codel 's argument for the existence of God is not particularly well-known, but there is a growing body of literature on it. This literature sometimes gives formalizations of Codel's rather sketchy ideas-generally along natural deduction or axiomatic lines. My idea was, I would design a tableau system within which the argument could be formalized, and this might lead to a nice paper illustrating the use of tableau methods. First, give tableau rules, then give Godel's proof. One cannot really develop semantic tableaus without a semantics behind it . The semantics of higher-order modal logic turned out to be of considerable intricacy, far beyond what could even be sketched in a paper. Clearly, an extended discussion of the semantics for higher-order modal logic was needed before the tableau rules could be motivated. I soon realized that in presenting higher-order modal logic, I was trying to explicate ideas coming from two quite different sources. On the one hand, there are essentially modal problems, some of which already arise at the first-order level and have little to do with higher-order constructs. On the other hand, a number of higher-order modal complexities also manifest themselves in a classical setting, and can be discussed more clearly without modalities complicating things. So I decided that before modal operators were introduced, I would give a thorough presentation of a semantics and tableau system for higher-order classical logic. There are already treatments of tableau, or Gentzen, systems for higher-order classical logic in the literature, but I felt it would be useful to give things in full here. Detailed completeness proofs are hard to find, for instance. Higher-order classical logic already has its hidden pitfalls. It is common knowledge, so to speak, that "true" higher-order classical models cannot correspond to any proof procedure. Henkin models are what is needed. But a "natural" formulation of tableaus is not complete with respect to Henkin models either. This is something known to experts-

PREFACE

xv

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

Acknowledgments An earlier draft of this work was on my web page for some time, and I was given several helpful suggestions as a result. In particular I want to thank Peter Hajek, Oliver Kutz, Paul Gilmore, and especially Howard Sobel.

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(Xl , '" ,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 ip, This involves predicate abstraction, or A-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 [Gi199, GilD1]' 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. Iftl ,'" , t n are types, (tl, ' " , t n ) is a type. 1 generally use t, tl , 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 (tl ,' " ,t n ) is a predicate that takes n arguments, of types is , ... , t« 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: ci, f3t, "l, .... Likewise I generally use letters from the uppercase Latin alphabet as constant symbols, again with the type written as a superscript: At, Bt , o, tr, .... 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 O. Type (tl, '" ,tn ) has as its order the maximum of the orders oi t«, . . . , 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 constru ct . 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
.aI, ... , an ., except for occurrences of the variables a1, ... , an' DEFINITION 1.5 (TERM OF L(C)) Terms of each type are characterized as follows.

1 A constant symbol of L( C) or variable is a term of L( C). 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 A predicate abstract of L( C) is a term of L( C). occurrences were defined above.

Its free variable

T is used, with and without subscripts, to stand for terms. DEFINITION 1.6 (FORMULA OF L( C)) The notion of formula is given as follows.

1 If T is a term of type (t1,'" ,tn) , and T1 , ... , Tn is a sequence of terms oftypestl, ... , t n respectively, thenT(T1, '" ,Tn) is a formula (atomic) of L(C) . The free variable occurrences in it are the free variable occurrences of T, T1, .. . , Tn . 2 If q> is a formula of L( C) so is ,q>. The free variable occurrences of ,
A w). The free variable occurrences of (q> A w) are those of q> together with those of w.

4 Ifq> is a formula of L(C) and a is a variable then (Va)q> is a formula of L(C) . The free variable occurrences of (Va)
.X((O» .X((O»(X(O»))(p((O))) is a formula. Only X(O) is free. 4 [(>.X((O» .X((O»(X(O»))(p((O))) :J X(O)(xO)] is a formula. The only free variable occurrences are those of X(O) and xO . 5 ('v'X(O»)[(>.X((O».X((O»(X(O»))(p((O))) :J X(O) (xO)] is a formula . The only free variable occurrence is that of xO. 6 (>'XO.(V'X(O») [(>.X((O».X((O» (X(O»))(p((O))) :J X(O)(xO)]) is a predicate abstract. It has no free variable occurrences, and is of type (0). The type machinery is needed to guarantee that what is written is wellformed . Now that the exercise above has been gone t hrough, I will display the predicate abstract without sup erscripts, as

(>.x.(V'X)[(>'X.X(X ))(P) :J X(x)]) ,

8


leaving types to be inferred, or explained in words, as necessary. In first-order logic, facts about formulas and terms are often proved by induction based on complexity. And complexity for a formula is often measured by the number of logical connectives and quantifiers, or what amounts to the same thing, by how "far away" the formula is from being atomic . In a higher-order setting these notions diverge. 1.10 (DEGREE) By the degree of a formula or term is meant the number of propositional connectives, quantifiers, and lambdasymbols it contains.

DEFINITION

Note that since an atomic formula can involve terms containing predicate abstracts which, in turn, involve other formulas, the degree of an atomic formula need not be 0, as in the first-order case.

2.

Substitutions

Formulas can contain free variables, and terms that are very complex can be substituted for them. The notion of substitution is a fundamental one, and this section is devoted to it. In a general way, I follow the treatment in [Fit96]. 1.11 (SUBSTITUTION) A substitution is a mapping from the set of variables to the set of terms of L( C) such that variables of type t map to terms of type t .

DEFINITION

I generally denote substitutions by 0", with and without subscripts. Also I generally write xa rather than O"(x) . Most concern is with substitutions having fin ite support , that is, they are the identity on all but a finite number of variables. A special notation is used for the finite support substitution that maps each a i to Ti and is the identity otherwise: {aI/TI , . .. , an/ Tn}. The action of substitutions on variables is readily extended to terms and formulas generally. 1.12 For a substitution 0", by 0"a1 ,... ,an is meant the substitution that is like 0" except that it is the identity on aI, . . . ,an .

DEFINITION

1.13 Let recursively as follows .

DEFINITION

1 AO"

=A

0"

be a substitution. The action of 0" is extended

for a constant symbol A .

2 (),al, .. . , a n. if 0- is free for q> .

4 0- is free for (q> A \lJ) if 0- is free for q> and 0- is free for \lJ . 50-is free for (V'a)q> if O-a is free for q>, and if f3 is a free variable of (V'a)q> then f30- does not contain a free. With the action of substitutions extended to all terms, composition of substitutions is easily defined.

Let 0-1 and 0-2 be substitutions. Their composition is the mapping defined by: a(o-IO"2) = (ao-I)0-2 ' for variables a .

DEFINITION 1.17 (SUBSTITUTION COMPOSITION)

It is not generally the case that r(ala2) = (ral)a2 , for terms r, and similarly for formulas . But it is when appropriate freeness conditions are imposed. THEOREM 1.18

Substitution is "compositional" under the following cir-

cumstances. 1 If 0-1 is free for the formula q>, and 0-2 is free for the formula q>o-I, then (q>0-1)0-2 = q>(o-IO"2)'

2 If 0-1 is free for the term r, and 0-2 is free for the term rO-l, then (ro-I)0-2 = r(0-10-2)' The proof of this is essentially the same as in the first-order setting. Rather than giving it here, I refer you to the proof of Theorem 5.2.13 in [Fit96J.

Exercises EXERCISE 2.1 Prove Proposition 1.15 by induction on degree. Conclude that if q> is a sentence then for every substitution 0-.

Chapter 2

CLASSICAL LOGIC-SEMANTICS

1.

Classical Models

Defining the semantics of any higher-order logic is relatively complicated. Since modalities add special complexities, it is fortunate I can discuss underlying classical issues before bringing them into the picture. In this Chapter the "real" notion of higher-order model is defined first, and truth in them is characterized. Then Henkin's modification of these models is considered-sometimes these are called general models-as well as a non-extensional version of them. I don't want just syntactic objects, terms, to have types. I want sets and relations to have them too. After all, we think of terms as designating sets and relations , and we want type information to move back and forth between syntactic object and its designation. DEFINITION 2.1 (RELATION TYPES) Let 8 be a non-empty set .

For

each type t the collection [t,8] is defined as follows.

1 [0,8] = 8 .

2 [(tl ,' " ,tn), 8] is the collection of all subsets of [tl , 8] x . .. x [tn , 8] .

o

is an object of type t over 8 if 0 E [t, 8] . 0 is systematically used, with or without subscripts, to stand for objects in this sense .

For example, a member of [(0,0),8] is a subset of 8 x 8, and in standard first-order logic it would simply be called a two-place relation on 8. But now relations ofrelations are allowed, and even more complex things as well, so terminology gets more complicated . A classical model consists of an underlying domain , thought of as the "ground level objects," and an interpretation, assigning some denota11 M. Fitting, Types, Tableaus, and Gödel's God © Kluwer Academic Publishers 2002

12


tion in the model to each constant symbol of the language. But that denotation must be consistent with type information. DEFINITION 2.2 (CLASSICAL MODEL) A higher-order classical model for L( C) is a structure M = (V, I), where V is a non-empty set called the domain of the model, and I is a mapping, the interpretation, meeting the following conditions.

1 If A is a constant symbol of L(C) of type t , I(A) E [t, V].

2 If = is the equality constant symbol of type (t, t) then I( =) is the equality relation on [t, V].

2.

Truth in a Model

Assume M = (V,I) is a classical model for a language L(C). It is time to say which sentences of the language, or more generally, which formulas with free variables, are true in M . This is symbolized by M If-v } Again let M = ('D ,I) be a classical model, and let v be a valuation in it. The notion of formula cI> of £(C) being true in model M with respect to v, denoted M If-v cI>, is characterized as follows.

DEFINITION 2.6 (TRUTH OF A FORMULA)

1 For terms T, T1, ... , Tn, M If-v T(T1,'" ,Tn) provided ((v * I) (T1), ... ,(v * I)(Tn)) E (v * I)(T).

2 M If-v

....,CI> if it is not the case that

M If-v cI>.

3 M If-v cI> 1\ 'lJ if M If-v cI> and M If-v 'lJ.

4M

If-v (\f0:)cI> if M If-v' cI> for every a-variant v' ofv.

There is an alternative notation that makes evaluating the truth of formulas in models somewhat easier. DEFINITION

2.7

(SPECIAL NOTATION)

Suppose v is a valuation, and W

is the 0:1, . .. , O:n variant ofv such that w(o:t} = 0 1, . .. , w(O:n) = On. Then, if M If- w cI> this may be symbolized by

Now part 3 of Definition 2.5 can be restated as follows.

3 (v*I)((>'O:l, .. . ,00n.cI>}) = {(01,' .. , On) I M If-v cI>[0:I/ 01,' .. ,00njOn]} Likewise part 4 of Definition 2.6 becomes 4 M If-v (\f0:)cI> if M If-v cI>[o:jO] for every object 0 of the same type as

0:. Defined symbols like D and 3 have their expected behavior, which are explicitly stated below. Alternately, this can be considered an extension of the definition above . 5 M If-v cI> V 'lJ if M If-v cI> or M If-v 'lJ. 6 M If-v cI> :) 'lJ if M If-v cI> implies M If-v 'lJ.

TYPES, TABLEA US, AND GODEL 'S GOD

14

7 M If-v ~ == \lI if M If-v e iff M If-v \lI. 8 M If-v (3a)~ if M If-v' ~ for some a-variant v' of v ; equivalently if M If-v ~[a/Ol for some object 0 of the same type as a.

As in first-order logic, if ~ has no free variables, M If-v ~ holds for some v if and only if it holds for every v . Thus for sentences (closed formulas) , truth in a model does not depend on a choice of valuation. DEFINITION 2.8 (VALIDITY, SATISFIABILITY , CONSEQUENCE) be a formula and S be a set of formulas.

Let ~

1 ~ is valid if M If-v ~ for every classical model M and valuation v.

2 S is satisfiable if there is some model M and some valuation v such that M If-v 'P for every 'P E S.

3

~

is a consequence of S provided, for every model M and every valuation v , if M If-v 'P for all 'P E S , then M If-v ~ .

The definitions above are of some complexity. Here is an example to help clarify their workings. EXAMPLE 2.9 This example shows a formula that is valid and involves equality. In it , C is a constant symbol of type O. The expression (>'X.(3x)X( x)) is a predicate abstract of type ((0)) , where X is of type (0) and x is of type O. Intuitively it is the "being instantiated" predicate. Likewise the expression (>.x.x = c) is a predicate abstract of type (0), where x and e are of type O. Intuitively this is the "being e" predicate. Since this predicate is, in fact , instantiated (by what ever e designates) , the first predicate abstract correctly applies to it . That is, one should have the validity of the following. (>'X.(3x)X(x))((>.x.x = c))

(2.1)

I now verify this validity. Suppose there is a model M = (V, I). I show the formula is true in M with respect to an arbitrary valuation v . To do this, I investigate the behavior, in M, of parts of the formula, building up to the whole thing. First, recalling that the interpretation of an equality symbol is by the equality relation of the appropriate type, we have the following. (v * I) ((>.x.x

= c)) = {o I M = {o =

I0

II-v (x

= I(c)}

{I(e)}

= c)[x/ o]}

15


We also have the following.

(v *I)((>'X.(3x)X(x))) = {O I M II-v (3x)X(x)[XjO]) = {O I M II-v X(x)[XjO,xjo] for some o} = {O I 0 E 0 for some o}

°

= {O I # 0} Now we have (2.1) because

M II-v (>'X.(3x)X(x))((>.x.x = c)) ¢? (v * I)( (>.x .x = c)) E (v * I)( (>'X.(3x)X(x))) ¢? {I(c)} E {O I 0# 0} . You might try verifying, in a similar way, the validity of the following.

-,(>'X.(3x)X(x)) ((>.x .-,(x = x)))

3.

Problems

First-order classical logic has many nice features that do not carry over to higher-order versions. This is well-known, and partly accounts for the general emphasis on first-order. I sketch a few of the higher-order problems here .

3.1

Compactness

The compactness theorem for first-order logic says a set of formulas is satisfiable if every finite subset is. This is a fundamental tool for the construction of models of various kinds-non-standard models of analysis, for instance. The higher-order analog does not hold , and counter-examples are easy to come by. Here is one. The Dedekind characterization of infinity is: a set is infinite if it can be put into a 1-1 correspondence with a proper subset. Consequently, a set is finite if any 1-1 mapping from it to itself can not be to a proper subset , i.e. must be onto . This can be said easily, as a second-order formula. Since function symbols are not available, I make do with relation symbols in the usual way-the following formula is true in a model if and only if the domain of the model is finite. (\fX)[(function(X)

1\ one-one(X)) :J

onto(X)]

In (2.2) the following abbreviations are used. function(X) one-one(X) onto(X)

(\fx)(3y)(\fz)[X(x, z) == (z = y)] for for (\fx)(\fy)(\fz){[X(x, z) 1\ X(y, z)] :J (x = y)} for (\fy)(3x)X(x, y)

(2.2)

16

TYPES, TABLEAUS, AND CODEL'S COD

Also, define the following infinite list of formulas, where x ates .(x = V). A2 = (3XI)(3x2) [Xl i= X2] A3 = (3XI)(3x2)(3x3)[(XI

i= y abbrevi-

i= X2) 1\ (Xl i= X3) 1\ (X2 i= X3)]

So An is true in a model if and only if the domain of the model contains at least n members . Now, the set consisting of (2.2) and all of A 2, A3, ... , is certainly not satisfiable, but every finite subset is, so compactness fails. (In firstorder classical logic this example turns around, and shows finiteness has no first-order characterization.)

3.2

Strong Completeness

A proof procedure is said to be (sound and) strongly complete if has a derivation from a set 8 exactly when is a logical consequence of 8. Classical first-order logic has many proof procedures that are strongly complete for it, but there is no such proof procedure for higherorder logic. To see this, one doesn't need an exact definition of proof procedure-it is enough that proofs be finite objects. Let 8 be the set of formulas defined in Section 3.1, a set that is not satisfiable though every finite subset is. And let .1 be 1\ • , for some formula . The formula .1 is a logical consequence of 8, since it is true in every model in which the members of 8 are true, namely none. If there were a strongly complete proof procedure, .1 would have a derivation from 8. That derivation, being a finite object, could only use a finite subset of 8, say 80. Then L would be a logical consequence of 80, and so 80 could not be satisfiable (otherwise there would be a model in which .1 was true). But every finite subset of 8 is satisfiable . Conclusion: no strongly complete proof procedure can exist for higher-order classical logic.

3.3

Weak Completeness

A proof procedure is (sound and) weakly complete if it proves exactly the valid formulas. A strongly complete proof procedure is automatically weakly complete (just use the empty set of premises) . Higher-order classical logic does not even possess a weakly complete proof procedure. To show this the Incompleteness Theorem can be used. The idea is to write a single formula that characterizes the natural numbers-a second-order formula will do. One needs a constant symbol


17

of type 0 to represent the number 0 and, to thoroughly overload notation, I use 0 for this. Also a successor function is needed, but since we do not have function symbols in this language, it is simulated with a relation symbol S, technically a constant" symbol of type (0,0). In addition to the abbreviations of Section 3.1, the following is needed.

O-exclude(S) for (V'x)--,S(x, O) inductive-set(P, S) for P(O) A (V'x)[P(x) :J (3y)(S(x, y) A P(y))] induction (S) for (V'P) [inductive-set(P, S) :J (V'x)P(x)] Now, let integer(S) be the formula function(S) A one-one(S) A O-exclude(S) A induction(S) It is not hard to show that integer( S) is true in a model (V, I) if and only if the domain V is (isomorphic to) the natural numbers, with I(S) as successor. Consequently for any sentence of arithmetic, is true of the natural numbers if and only if integer(S) :J is valid. It is a standard requirement that the set of (Godel numbers of) theorems of a proof procedure must be recursively enumerable, so if there were a weakly complete proof procedure for higher-order classical logic, the set of valid formulas would be recursively enumerable. The recursive enumerability of the following set would then be an easy consequence: the set of sentences such that integer(S) :J is valid. But, as noted above, this is just the set of true sentences of arithmetic, and this is not a recursively enumerable set. Conclusion: no weakly complete proof procedure can exist for higher-order classical logic.

3.4

And Worse

I have been discussing higher-order classical logic, particularly its models, using conventional informal mathematics of the sort that every mathematician applies in papers and books. But certain areas of mathematics-certainly formal logic is among them-are close to foundational issues, and one needs to be careful. It is generally understood that informal mathematics can be formalized in set theory, and this is commonly taken to be Zermelo-Fraenkel set theory, or a variant of it. Let us suppose, for the time being, that the development so far has been within such a framework. One of the famous problems associated with set theory is Cantor's continuum hypothesis . It is the statement that there are no sets intermediate in size between a countable set and its powerset . A little more formally, it says:

18

TYPES, TABLEAUS, AND GODEL'S GOD Let X be a set , and let P(X) be its powerset. If X is countable, then any infinite subset Y of P(X) either is in a 1-1 correspondence with X, or is in a 1-1 correspondence with P(X).

(The generalized continuum hypothesis is the natural extension of this to uncountable infinite sets as well, but the simple continuum hypothesis will do for present purposes.) Now, a difficulty for set theory is this: the continuum hypothesis has been proved to be undecidable on the basis of the generally accepted axioms for Zermelo-F'raenkel set theory. That is (assuming the axioms for set theory are consistent) there is a model of the Zermelo-F'raenkel axioms in which the continuum hypothesis is true, and there is another in which it is false. The problem for us is that the continuum hypothesis can be stated as a sentence of higher-order c1assicallogic. I briefly sketch how. First, one can say the domain of a model is countable by saying there is a relation that orders it isomorphically to the natural numbers. Using a formula from Section 3.3, the following will do: (3a(0,0))integer(a(0,0)). Next, one can identify a subset of the domain with an object of type (0). Then the collection of all subsets of the domain is an object of type ((0)), so the following says there is a powerset for the domain of a model: (3,8( (0))) (V, (0)),8((0))(,(0)).

Having shown how to start, I leave the rest of the details to you. Write a sentence saying: if the domain is countable then there is a powerset for the domain and, for every infinite subset of that powerset, either there is a 1-1 correspondence between it and the domain, or there is a 1-1 correspondence between it and the powerset. You can say a set is infinite using the negation of a formula from Section 3.1. And the existence of a 1-1 correspondence amounts to the existence of a binary relation meeting certain appropriate conditions . Let us call the sentence that is the higher-order formalization of the continuum hypothesis CH. Now, the real problem is: is the sentence CH valid or not? There are the following not very palatable options. 1 Assume the foundation for informal mathematics is Zermelo-F'raenkel set theory, formulated axiomatically. In this case neither CH nor its negation can be shown to be valid, since the continuum hypothesis is consistent with, but independent of, the Zermelo-F'raenkel axioms. 2 Assume that informal mathematics is being done in some particular model for the Zermelo-F'raenkel axioms. In this case, CH is definitely valid, or its negation is, but it depends on which Zermelo-F'raenkel model is being considered .


19

3 Assume that higher-order classical logic itself supplies the theoretical foundations for mathematics. In this case CH either is valid or its negation is, but which is it? I have reached perhaps the most basic difficulty of all with classical higher-order logic. Not only is there no proof procedure that will allow us to prove every valid formula, the very status of validity for some important formulas is unclear.

4.

Henkin Models

As we saw in the previous section, higher-order classical logic is difficult to work with . Indeed, difficulties already appear at the second-order level. Not only does it lack a complete proof procedure, but the very notion of validity touches on profound foundational issues. Nonetheless, there are several sound proof procedures for the logic-any formula that has a proof must be valid, though not every valid formula will have a proof. So, there are certainly fragments of higher-order logic that we can hope to make use of. In a sense, too many formulas of higher-order classical logic are valid, so no proof procedure can be adequate to prove them all. Henkin broadened the notion of higher-order model [Hen50] in a natural way, which will be described shortly. With this broader notion there are more models, hence fewer valid formulas, since there are more candidates for counter-models. Henkin called his extension of the semantics general models-I will call them Henkin models . Henkin's idea seems straightforward, after years of getting used to it. Given a domain 7), a universal quantifier whose variable is of type 0, (Vz}, ranges over the members of 7) . If we have a universal quantifier, (\IX) , whose variable is of type (O), it ranges over the collection of properties of 7), or equivalently, over the subsets of 7). The problem of just what subsets an infinite set has is actually a deep one. The independence of Cantor's continuum hypothesis is one manifestation of this problem. Methods for establishing consistency and independence results in set theory can be used to produce models with considerable variation in the powerset of an infinite set. Henkin essentially said that, instead of trying to work with all subsets of 7), we should work with enough of them, that is, we should take (\IX) as ranging over some collection of subsets of D, not necessarily all of them, but containing enough to satisfy natural closure properties. Think of the collection as being intermediate between all subsets and all definable subsets. In a higher-order model as defined earlier, there is a domain , 7), and this determines the range of quantification for each type. Specifically,

20

TYPES, TABLEA US, AND GODEL'S GOD

we thought of a quantifier (Vat) as ranging over the members of [t, V]. This time around a function is introduced, which I call a Henkin domain function and denote by H, explicitly giving us the range for each quantifier type. Then Henkin frames are defined. This basic machinery is needed before it can be specified what it means to have enough sets available at each type. DEFINITION 2.10 (HENKIN DOMAIN FUNCTION) H is a Henkin domain function if H is a junction whose domain is the collection of types and, for each type (tl, " . , t n ) , H( (tl' . . . ,tn ) ) is some non-empty collection of subsets ofH(tl)

x . .. x H(t n ) .

Sets of the form H(t) are called Henkin domains. The key point is allowing some of the subsets of H(tl) x . .. x H(tn)-the definition of [t, V] had the word all at the corresponding point. Obviously the function H(t) = [t, V] is a Henkin domain function. In fact, if H is any Henkin domain function, and H(O) = V , then for every type t, H(t) ~ [t, V], with equality holding at t = O. DEFINITION 2.11 (HENKIN FRAME) The structure M = (H ,T) is a Henkin frame for a language L(G) if it meets the following conditions. 1 H is a Henkin domain junction.

2 If A is a constant symbol of L(G) of type t, T(A) E H(t). 3 I(=(t,t» is the equality relation on H(t) for each type t .

The notion of valuation must be suitably restricted, of course. DEFINITION 2.12 (VALUATION) v is a valuation in a Henkin frame M = (H, T) if v maps each variable of type t to some member of H( t). Now, what will make a Henkin frame into a Henkin model? Let's try a first attempt at a characterization. (This is not the "official" one, however. That will come later.) Definition 2.5, for the meaning of a term, carries over almost word for word to a Henkin frame M. Also Definition 2.6, for truth in a model, carries over to M, with one restrictive change. Item 4, the universal quantifier condition, gets replaced with the following. 4'. Let M = (H ,I) be a Henkin frame and let at be a variable of type t. M I~v (Vat)~ if M I~v ~[at jot] for every o' E H(t), or equivalently, if M If-v'
'0:1 , ... ,00n. of L(C) is true in model M with respect to v and A, denoted M If-v,A cI>, if the following holds. DEFINITION

1 For terms T, T1, . . . .r«, M If-v,A T(T1,'" ,Tn) provided ((v * I * A)(T1),'" , (v * I * A)(Tn)) E (v * I * A)(T). 2 M If-vIA -,CI> if it is not the case that M lf-v,A cI>. 3 M If-v,A cI> 1\ 1IJ if M If-v,A cI> and M If-v,A 1IJ.

4M

lf-v,A (Vat)cI> if M lf-v,A cI>[a t /0] for every 0 E H(t).

Now we can impose a requirement that designations of predicate abstracts be "correct ." 2.16 (PROPER ABSTRACTION DESIGNATION FUNCTION) Let M = (H, I) be a Henkin frame and let A be an abstraction designation function in it, with respect to L(C). A is proper provided the following is the case. For each predicate abstract (Aa1, . .. , an .cI» (with ai of type ti) and for each valuation v we have DEFINITION

(v*I*A)((Aa1 , ... ,an .cI») =

{(01 , ... , On) E H(t1)

X . •. X

H(tn) I M lf-v,A cI>[aIf01 " " , an/On]}.

DEFINITION 2.17 (HENKIN MODEL) Let M be a Henkin frame, and let A be an abstraction designation function in M. If A is proper, (M, A) is a Henkin model.

For a given Henkin frame M it may be the case that no proper abstraction designation function exists . But, if one does exist it must be unique. 2.18 Let M = (H,I) be a Henkin frame and let both A and A' be proper abstraction designation functions , with respect to L(C). Then A= A' .

PROPOSITION

Proof The following two items are shown simultaneously, by induction on degree (Definition 1.10). From this the Proposition follows immediately. M lf-v,A cI> ¢:> M lf-v,A' cI> (v*I*A)(T) = (v*I*A')(T)

(2.3) (2.4)

23


Suppose (2.3) and (2.4) are known for formulas and terms whose degree is < k. It will be shown they hold for degree k too, beginning with (2.4). Suppose 7 is a term of degree k. Since k could be 0, 7 could be a constant symbol or a variable. If it is a constant symbol, (v*I *A)(7) = I( 7) = (v * I * A') (7). Similarly if 7 is a variable. Finally, 7 could be a predicate abstract, ('xa1, '" ,an ..al, ' " , an..al , ' " , a n.1It). In a Henkin model, if and 1It are equivalent formulas, they will be true of the same objects and so the two predicate abstracts will designate the same thing, since they have the same extensions . But now we are explicitly allowing predicate abstracts having the same extension to denote different objects. Still, we don't want the designation of objects by predicate abstracts to be entirely arbitrary-I will require equi-designation under circumstances of "structural similarity." DEFINITION 2.26 Let M be a generalized Henkin frame (or a Henkin frame), and let A be an abstraction designation junction in it. For each valuation v and substitution a, define a new valuation o" by:

Thus va assigns to a variable a the "meaning" of the term aa. 2.27 (PROPER ABSTRACTION DESIGNATION FUNCTION) Let M = (11. ,I, £) be a generalized Henkin frame and let A be an abstraction designation junction in it, with respect to L( C) . A is proper provided, for each predicate abstract (>.al,' " ,an .'al, .. . ,an."'Yl, ' " ,'Yk.{f3I! o.l, ... ,(3n/o.n}))

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 (3i . 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 [ToI75, Smi93, Koh95 , GilOl] . In fact, semantic tableaus retain much of their first-order ability to char m, 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 [ToI75]. 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 syst em 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. 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 >. bound. p, q, P, Q, ... are used to represent parameters. DEFINITION 3.1 (PARAMETERS)

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 >. bound, we never need to worry about whether the introduction of a parameter will lead to its inadvertent capture by a quantifier or a >.introducing 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 variabl e 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 L+(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. D EFI NITION 3.2 (GROUNDED) A term or a formula of L+(C) is grounded if it contains no free variables of L(C) , though it may contain parameters.

CLASSICAL LOGIC-BASIC TABLEA US

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 is a tree that has --, 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 --, cannot happen, and so 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)

XAY

--,(X ~ Y)

X:==Y

X Y

--,y

X

X~Y y~X

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 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

TYPES, TABLEAUS, AND CODEL 'S COD

3.5

(DISJUNCTIVE RULES)

-,(X 1\ Y)

XVY

TTY

-,X

I -,y

-,(X == Y) --,--'-'(X=-=--~-=-=Y:-;-) 1--,""7:(y'=-~---=-=X"--) This completes the propositional connective rules. The motivation should be intuitively obvious. For instance, if X 1\ Y is true in a model , both X and Yare true there, and so a branch containing X 1\ Yean 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 ~(at) is a formula in which the variable 0/, of type t, may have free occurrences. And suppose T t is a term of type t. Then ~(Tt) is the result of carrying out the substitution {at ITt} in ll'(at), replacing all free occurrences of at with occurrences of Tt . Now, here are the existential quantifier rules.

In the following, pt is a parameter of type t that is new to the tableau branch.

DEFINITION 3.6 (EXISTENTIAL RULES)

(3at)~(at)

~(pt)

-,(Vat)ll'(at) -,ll'(pt)

The rules above embody the familiar notion of existential instantiation. Since the convention is that parameters are never quantified or Abound, we don't have to worry about accidental variable capture. More precisely, in the rules above, the substitution {at Ipt} is free for the formula -.y.R(x,y)) represents the set coded by x. Then the following second-order sentence does the job. EXAMPLE

(VR)(3X)(Vx)""[(>-.y .R(x, y)) = X]

(3.1)

This formulation contains equality. I have not given rules for equality yet, so I give an alternative formulation that does not involve it .

(VR) (3X) (Vx)(3y){[R(x, y) A ...,X(y)] V [...,R(x ,y) A X(y)]}

(3 .2)

I give a proof of (3.2). It is contained in Figure 3.1. In it, 2 is from 1 by an existential rule (P is a new parameter); 3 is from 2 by a universal rule ((>-.x....,P(x, x)) is a grounded term); 4 is from 3 by an existential rule (p is another new parameter) ; 5 is from 4 by a universal rule (p is a grounded term) ; 6 and 7 are from 5 by a conjunction rule; 8 and 9 are from 6 by a disjunction rule; 10 is from 9 by double negation; 11 and 12 are from 7 by a disjunction rule, as are 13 and 14; 15 is from 12 by an abstract rule, as is 16 from 10. Closure is by 8 and 11, 8 and 15, 13 and 16, and 10 and 14. A key feature in the tableau proof of (3.2) is the use of (>-.x ....,P(x, x)) in an application of a universal rule. This , in fact, is the heart of diagonal arguments and amounts to looking at the collection of things that do not belong to the set they code. The choice of such abstracts at key points of proofs is the distilled essence of mathematical thinking-everything else is mechanical. It is the need for such choices that stands in the way of fully automating higher-order proof search. Next is an example that comes out of propositional modal logic. Some knowledge of Kripke semantics will be needed in order to understand the background explanation, though not the tableau proof. See [HC96, pp 188-190] for a fuller treatment.

CLASSICAL LOGIC-BASIC TABLEA US

J J

'1l

5?

39

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 OP => 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, (Vy) [R(x, y) => P(y)] corresponds to P being true at every world accessible from x , and hence to OP being true at world x, where R plays the role of th e accessibility relation. Then further , saying OP => P is true at x corresponds to (Vy)[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.

(Vx)R(x, x) == (VP)(Vx){(Vy)[R(x,y) => P(y)] => P(x)}

(3.3)

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

-,((VP)(Vx){(Vy)[R(x,y) => P(y)] => P(x)} => (Vx)R(x,x)} I. (VP)(Vx){(Vy)[R(x ,y) => P(y)] => P(x)} 2. -,(Vx)R(x, x) 3. -,R(p,p) 4. (Vx){(Vy)[R(x,y) => (Az.R(p , z)}(y)] => (Az.R(p , z)}(x)} 5. (Az.R(p, z))(y)] :J (),~}(P) 6.

('v'Y)[7

-,(Vy)[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) II. -,R(p, q) 12.

7. (Az.R(p, z))(p) 8. R(p,p) 13.

In thi s, 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.

CLASSICAL LOGIC-BASIC TABLEAUS

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)(Vx)(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 (P,x) abbreviate the formula (Vy)(F(P,y) :J P(y)) :J P(x) . An appropriate term to consider during a universal rule application is: (AX.(VP)(P, x)). A comment on the hint above. Rewriting (Vy)(F(P,y) :J P(y)) using conventional function notation: it says F(P) ~ P. Then (P, x) says that x belongs to a set P if P meets the condition F(P) ~ P. Then further, (VP) (P,x) says that x is in niP I F(P) ~ P}. So finally, (AX.(VP) (P,x)) represents the set niP I F(P) ~ P} itself. In the most common proof of the Knaster-Tarski theorem, one proceeds by showing this set, in fact , is a fixed point of F . Example 3.14 once again illustrates a fundamental point about higherorder tableaus. They mechanize routine steps, but do not substitute for mathematical insight. The choice of which predicate abstract to use during an application of a universal rule really contains , in distilled form, the essence of a standard mathematical argument. The problem of what choice to make when instantiating a universal quantifier also arises in first-order logic, but there is a way around it-one uses free variables when instantiating, then one determines later which values to choose for them [Fit96]. This last step, picking values, involves unification, the solving of equations involving first-order terms. There are several unification algorithms to do this, all of which accomplish

42


the following: given two terms, if there is a choice of values for their free variables that makes the terms identical, the algorithm finds the most general such choice; and if the terms cannot be made identical, the algorithm reports this fact. Unification is at the heart of every first-order theorem prover. If we attempt a similar strategy in automating higher-order logic, we immediately run into an obstacle at this point. The problem of unification for higher-order terms is undecidable! This was shown for thirdorder terms in [Hue73], and improved to show unification for secondorder terms is already undecidable , in [Gol81]. This does not mean the situation is completely hopeless. While first-order unification is decidable, and second-order is not , still there is a kind of semi-decision procedure, [Hue75]. Two free-variable tableau systems for higher-order classical logic, using unification, are presented in [Koh95] . The use of higher-order unification in this way traces back to resolution work of [And71] and [Hue72] . But finally, technical issues aside, we always come back to the observation made above: the choice of predicate abstract to use in instantiating a universally quantified formula often embodies the mathematical "essence" of a proof. Too much should not be expected from the purely mechanical.

Exercises EXERCISE 3.1 Extending the ideas of Example 3.13, give tableau proofs of the following. 1 (symmetry)

('v'x) ('v'y)[R(x, y) => R(y, x)] == ('v'P)('v'x){(3y)[R(x, y) A ('v'z) (R(y, z) => P(z))] => P(x)} 2 (transitivity)

('v'x) ('v'y) ('v'z)[(R(x ,y) A R(y ,z)) => R(x , z)] == ('v'P)('v'x){('v'y)[R(x, y) => P(y)] => ('v'y) ('v'z)[(R(x, y) A R(y, z)) => P(z)]} EXERCISE 3.2 Give the tableau proof to complete Example 3.14. EXERCISE 3.3 Continuing with Example 3.14, the set n{p I F(P) ~ P} is not only a fixed point of monotonic F , it is the smallest one. Dually, u{P IPS;;; F(P)} is also a fixed point , the largest one. Give a tableau proof of (3.4) based on this idea.

Chapter 4

SOUNDNESS AND COMPLETENESS

This chapter contains a proof that the basic tableau rules are sound and complet e with respect to generalized Henkin models. Soundness is by the "usual" argument, is straightforward, and is what I begin with. Completeness is something else altogether. For that I use the ideas developed simultaneously in [Tak67, Pra68], where they were applied to give a non-constructive proof of a cut eliminat ion theorem.

1.

Soundness

Soundness means that any sentence having a tableau proof must be valid. Tableau soundness arguments follow the same pattern for all logics: some not ion of satisfiability is defined for tableaus; then satisfiability is shown to be preserved by each tableau rule application. Note that in the following, L + (C) is used rather than L(C), because formulas of the larger language L +(C) can occur in tableaus. DEFINITION 4.1 (TABLEAU SATISFIABILITV) A tableau branch is satisfiable if the set of formulas on it is satisfiable in a generalized Henkin model for L+(C) (see Definition 2.29). A tableau is satisfiable if some branch is satisfiable. Now, two key facts about these notions easily give us soundness. For the first, a closed tableau branch contains some formula and its negation, hence cannot be satisfiable. Since a closed tableau has every branch closed, we immediately have the following. LEMMA

4.2 A closed tableau cannot be satisfiable.

The second key fact takes more work to prove, but the work is spread over several cases, each of which is rather simple. 43 M. Fitting, Types, Tableaus, and Gödel's God © Kluwer Academic Publishers 2002

44


4.3 If a branch extension rule is applied to a satisfiable tableau, the result is another satisfiable tableau.

LEMMA

Proof Suppose T is a satisfiable tableau. Then it has some satisfiable branch, say B. Also suppose some branch extension rule is applied to T to produce a new tableau, T'. It must be shown that T' is satisfiable. The rule that was applied to turn T into T' may have been applied on a branch other than B. In this case B is still a branch of T' , and of course is still satisfiable, so T' is satisfiable. Now, for the rest of the proof assume a branch extension rule has been applied to the satisfiable branch B itself. And to be specific, say all the grounded formulas on B are true in the generalized Henkin model (M, A) with respect to the valuation v, where M = (1t,I,E). There are several cases, depending on which branch extension rule was applied. I consider only a few of these cases and leave the rest to you. Disjunction Suppose the grounded formula X VY occurred on Band a rule was applied to it . Then in T' the branch B has been replaced with two branches: B lengthened with X, and B lengthened with Y. All formulas on B are true in (M, A) with respect to valuation v, hence M H-v,A X V Y. Then either M H-v,A X or M l/-v,A Y. In the first case, all members of B lengthened with X, and in the second case, all members of B lengthened with Y, are true in (M, A) with respect to v . Either way, some branch of T' is satisfiable . Existential Quantifier Suppose the grounded formula (3a)(a) occurred on B and a rule was applied to it , so that in T' branch B has been lengthened with (p) where p is a parameter new to B, of the same type as a. Since all formulas on B are true in (M, A) with respect to v, M l/-v,A (3a) (a). Then, by definition of truth in a model, there must be some a -variant w of v such that M l/-w,A (a). Let a = {p/a}the substitution that replaces p by a -and consider the valuation ui" (Definition 2.26). I claim all formulas on B extended with (p) are true in (M, A) with respect to ui"; so the extended branch is satisfiable. First of all, v and tiJ agree on all variables except a. It is easy to see that wand ui" agree on all variables except p, so the only variables on which v and weT can differ are a and p. But a does not occur free in any formula on B, since these formulas are all grounded. And p does not occur either, since p was new to the branch. Conse-

45


quently all formulas on B are true in (M, A) with respect to Proposition 2.30.

ui";

by

Finally, note that since p did not occur in (3a)..a1, '" ,an.if!(a1 ,' " , a n ))(T1, ... ,Tn) E H , then ,if!(T1, '" , Tn) E H. This completes the definition of Hintikka sets. The task of relating them to models begins in the next subsection.

2.2

Pseudo-Models

The eventual goal is to construct a generalized Henkin model , starting with a Hintikka set. To do this a pseudo-model is first created, something that is much like a generalized Henkin model but with one significant difference: predicate abstracts are allowed to take on values that may lie outside the range of the quantifiers! This will pose no problems for the definition of truth in a pseudo-model since, for example, T1 (T2) can still be taken to be true if the value assigned to T2 is in the extension of the value assigned to T1 , whether or not these values are in quantifier ranges. Eventually it will be shown that we can dispose of the "pseudo" qualification on a pseudo-model. I begin by defining entities of each type. These are the things that can serve as values of predicate abstracts. In some ways the collection of entities is an analog of a Herbrand universe, familiar from treatments of first-order logic.

49


DEFINITION 4.7 (ENTITIES OF TYPE t) The notion of entity of type t is defined inductively, on the complexity of t. 1 Suppose t = O. If 7 is a grounded term of type t (thus a constant or parameter of type 0) , 7 is an entity of type t. 2 Suppose t = (tl, ... , tn) and the collection of entities of type ti has been defined for each i = 1, ... , n. Then (7, S) is an entity of type t provided 7 is a grounded term of type t, and S is a set whose members are of the form (E l, ... , En), where each E; is an entity of type ti·

I also define two mappings on entities. DEFINITION 4.8 (&, T) If the entity E is of a type other than 0, it is of the form (7, S); then T(E) = 7 and &(E) = S. I refer to &(E) as the extension of E. The definition of T (but not of E} is extended to entities of type 0 as well. If E is of type 0 it is, itself, a grounded term of £+(C); in this case T(E) = E. The idea is, if (7, S) is an entity of type t, it is something that could serve as a semantic value for the term 7 , with the extension explicitly coded in. One problem with entities is that Hintikka sets play no role-the collection of entities is the same no matter what Hintikka set we may have. Presumably, if we are trying to construct a model from a given Hintikka set , that should place some restrictions on what entities we want to consider. The next definition separates out those entities that will be in the range of quantifiers-it makes direct use of a Hintikka set. It is these entities that will make up the Henkin domains of a model. DEFINITION 4.9 (POSSIBLE VALUE) Let H be a Hintikka set. For each grounded term 7, define a collection of possible values of 7 relative to H. This is done inductively, on type complexity. 1 If 7 is a grounded term of type 0, the only possible value of 7 relative to H is 7 itself. 2 Suppose 7 is a grounded term of type (tl, ... , t n), and possible values relative to H have been specified for all grounded terms of types tl, ... ,tn' Then, an entity (7, S) is a possible value of 7 relative to H provided, for all grounded terms 71, ... , 7n of types tl , ... , t« respectively, and for all possible values E l, . . . , En of 71, ... , 7n: (a) If7(7l, (b) If-'7(7l ,

, 7n ) E H then (E l , ,7n) E H then (E l ,

,En) E S. , E n)

rf. S.

50


E is a possible value if it is a possible value for some grounded term.

Roughly the idea is, any possible value for r should have in its extension all those things the Hintikka set H requires, and should omit all the things H forbids. Any entity that meets these conditions will serve as a possible value. Clearly each possible value of a grounded term of type t, relative to a Hintikka set H, is an entity of type t. Item 1 of the definition of Hintikka set ensures that part 2 above is meaningful. Now that we have the notion of possible value, Henkin domains for our pseudo-models can be defined. 4.10 (RELATIVE HENKIN DOMAINS) Let H be a Hintikka set. A mapping, 'HH is defined, from types to entities, as follows. For each type t, 'HH(t) is the set of all entities of type t that are possible values relative to H.

DEFINITION

The languages L(C) and L +(C) are allowed to contain constant symbols. How to interpret these is rather arbitrary, within broad limits. 4.11 (ALLOWED INTERPRETATION) Let H be a Hintikka set. A mapping I is an allowed interpretation relative to H provided I assigns to each constant symbol A of type t some possible value for A, relative to H .

DEFINITION

We now have all the machinery needed to characterize an important class of generalized Henkin frames arising from Hintikka sets . 4.12 (RELATIVE GENERALIZED HENKIN FRAME) Let H be a Hintikka set. M = ('HH,I , £) is a generalized Henkin frame relative to H provided: DEFINITION

1 'HH is the relative Henkin domain function of Definition 4.10; 2 I is an allowed interpretation relative to H, Definition 4.11;

3 £ is the extension function of Definition

4.8.

To produce our pseudo-models we need some notion of an abstraction designation function . To define this we first need a little more machinery. DEFINITION 4.13 (W) Let v be a valuation in some generalized Henkin frame relative to a Hintikka set H . Define a substitution W as follows: a~

= T(v(a)).

Thus, if v(a) = (r, S) then aW = r. If v(a) = C, of type 0, then aW = C. Note that W substitutes grounded terms of L+(C) for variables, and

51


so if T is an arbitrary term, TlJ must be a grounded term. Similarly for formulas. Then, for any formula
, (v * I * AH)(rn)) E £((v * I * AH)(ro)) {:> ,(w *I * AH)(rn)) E £((w *I * AH)(ro)) {:> M If-w,AH ro(r1,' " ,rn ).

The various non-atomic cases are left to you. • Next we have several preliminary results, leading up to the proof of Proposition 4.16. Recall Definition 1.12: O"al,...,an is the substitution that is like 0" except that it is the identity on a1, .. . ,an' LEMMA 4.17 Let H be a Hintikka set, let M = (1iH'I , £) be a generalized frame relative to H , and let (M, AH) be a pseudo-model relative to H. Let (Aa1, ... , an. 0, and so T must be of the form (Aal ,'" , an.cI», where cI> is a formula whose degree is < k. And suppose (J is free for (Aal ,' .. , an.cI». Using the definition of substitution and Definition 4.14:

(v * I

* AH)( (Aal , '"

,an.cI»(J) = (v * I * AH)( (Aal , .. . , an.cI>(Jal,...,an)) =AH(V,(Aal, ... , D:n.cI>(Jal,oo. ,an)) = (a,S)

where

(Aal , ... , an' cI>(Jal '00 ' ,an )1J S = {(Ol ,'" , On) E 'HH(tl) x · .. X 'HH(tn) I M If-v,AH cI>(Jal,oo. ,an[aI/OI , ... , an/ On]}. a=


57

Similarly :

where

a, = (A0:1," " O:n . )~ V S' = {(0 1 , '" ,On) E 1tH(t1) X '" X 1tH(t n) I M Il-vu ,AH [0:1/01, ... ,O:n/On]}. So, we must show a = a' and S

= S'.

Part 1, a = a'. First of all, a = (A0:1 , .. . ,00n·aal ,..' ,an)W

= ((A0:1, .. ' ,00n.(TI,.. . ,Tn) k + 2. In this, the first k + 1 lines are members of S. Line k + 2 is from k + 1 by an abstract rule. Now continue this tableau to closure by copying over the steps of tableau T. This shows there is a closed tableau for a finite subset of S itself, so S must be inconsistent, which is a contradiction. _ Now, finally, we get"the completeness results. THEOREM 4 .30 Let cI> be a closed formula and let S be a set of closed formulas, all of L(C).

1 IfcI> is valid in generalized Henkin models, cI> has a basic tableau proof. 2 If cI> is a generalized Henkin consequence of S, cI> has a basic tableau derivation from S .

Proof Suppose there is no basic tableau derivation of cI> from S. Then there is no closed tableau for ocI>, allowing members of S to be added to the ends of open branches. It follows that S U { ocI>} is consistent. It can be extended to a maximal consistent, E-complete set H, by Proposition 4.28. The set H is a Hintikka set, by Proposition 4.29. Then by Corollary 4.24, Su {ocI>} is satisfiable in some generalized Henkin model , and consequently cI> is not a generalized Henkin consequence of S. This establishes part 2; part 1 has a simpler proof. _

66

3.


Miscellaneous Model Theory

Two of the main results about first-order logic are the Compactness and the Lowenheim-Skolem theorem. I already noted , in Section 3, that compactness does not hold for "true" higher-order logic. It is also easy to verify that the Lowenheim-Skolem theorem does not hold, since one can write a formula asserting an uncountable object exists. But things are very different if generalized Henkin models are used, instead of standard models. Then both theorems hold, just as in the first-order case. Compactness is easy to verify, now that completeness has been shown. Ldwenheim-Skolem takes more work. THEOREM 4.31 (COMPACTNESS) Let 8 be a set of closed formulas of L(C) . If every finite subset of 8 is satisfiable in some generalized Henkin model, so is 8 itself.

Proof Suppose 8 is not satisfiable in any generalized Henkin model-I show some finite subset of 8 is also not satisfiable. Let 1- abbreviate X A ,X, where X is some arbitrary closed formula of L( C). Since 8 is not satisfiable in any generalized Henkin model, 1- is true in every model in which the members of 8 are true (since there are none), so 1- is a generalized Henkin consequence of 8. By Completeness, 1- has a basic tableau derivation from 8 . A closed tableau, being a finite object, can use only a finite subset 8 0 of 8 . Now 1- has a basic tableau derivation from 8 0 , so by Soundness, 1- is a generalized Henkin consequence of 8 0 . If 8 0 were satisfiable in some generalized Henkin model, 1- would be true in it, which is not possible. Consequently 8 0 is unsatisfiable. _

The Lowenheim-Skolem theorem for first-order classical logic follows easily from the observation that models constructed in completeness proofs are countable. This does not apply directly to the generalized Henkin models constructed using tableaus. The reason is very simple. I showed how to construct a generalized Henkin frame M = ('HH' I, &) starting with a Hintikka set H . In this frame, the Henkin domains consisted of possible values for grounded terms, Definition 4.9. It is easy to see that 'HH(O) must be countable. But say T is a grounded term of type (0) such that no formulas of the form T(To) or 'T(TO) occur in H. (This can certainly happen-take the Hintikka set H to be the empty set!) Then (T,8) is a possible value for T for every subset 8 of 'HH(O), so 'HH((0)) is uncountable. We need some way around this difficulty. The main tool is contained in the following.


67

THEOREM 4.32 (CuT-ELIMINATION) Let S be a finite set of grounded formulas of L +(C). If there is a closed tableau beginning with S U { (T(a) = 7(,8))] is true in it , and hence 7(a) = 7(,8) is true in (M, A) with respect to any valuation V such that v(a) =r v(,8) . Set w to be a particular valuation such that w(a) = VI (,), w(,8) = V2 (,), and otherwise w is arbitrary. Since VI (,) =r V2 (,), we have w(a) =r w(,8) , so by the paragraph above, 7(a) = 7(,8) is true in (M ,A) with respect to w, in other words, A(w ,7(a)) < i A(w, 7(,8)) . Also

75

EQUALITY

v~o:h}(a) = vI (a{ ah } ) = VI(-Y) = w(a) . Likewise v~f1h}(,8) = w(,8) . Thus v~o:h} and w agree on the free variables of T(a), and v~{3h} and w agree on the free variables of T(,B). Now since A is proper, we can make use of the conditions of Definition 2.27, and we have the following.

A(VI,T(-y)) = A(vI,T(a){ah}) = A(v~o:h},T(a)) = A(w, T(a)) =I

A(w,T(,8))

A(v~{3h},T(,8)) = A(V2, T(,8){,8h }) = A(V2, T(-y)) =

• This Lemma justifies the following. Define an abstraction designation function by A(v, (Aal, . . . ,an.'X.X(x))(Q)] :> (>.X, X, Y.X(X) :> X(Y))(P, P,Q) ,{(V'x) [(>'XX(x))(P) == (>.XX(x))(Q)] :> (>.X, X ,Y.X(X) :> X(Y))(P, P, Q)} 1. (V'x) [(>'X.X(x))(P) == (>.XX(x))(Q)] 2. ,(>.X, X, Y.X(X) :> X(Y))(P, P,Q) 3. ,[P(P) :::> P(Q)] P(P) 5.

4.

,P(Q) 6.

»;

,[P(p) == Q(p)] 7. (>'X.X(p))(P) == (>.Y.Y(p))(Q) 10.

P=Q

8.

P(Q) 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.

EXTENSIONALITY

3.

79

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
.al ,'" ,an ..al, '" ,an..al, ... , an'.al, '" , an..al, '" , an. if it is not the case that M, r Il-v,A eI>. 3 M , r Il-v,A eI> /\ 'IT if M , r Il-v,A eI> and M, r Il-v,A 'IT.

4 For 0: of type t, M , r Il-v,A (Vo:)eI> if M , r Il-v,A eI>[o:IO] for every

o E 'H(t).

5 M , r Il-v,A DeI> if M , ~ Il-v,A eI> for all ~ E 9 such that rR~.

Finally, the following should be no surprise.

7.22 (HENKIN/KRIPKE MODEL) (M , A) is a Henkin/Kripke model provided that, for each predicate abstract (A0:1 , .. . , 00n.eI» ofL(C) of type it, A(V ,(A0:1 , '" , 00n.eI» ) is the function f given by the following: DEFINITION

The various theorems concerning uniqueness of an abstraction designation function, if one exists, and the good behavior of substitution (Section 6) all carryover to the modal setting. I leave this to you. The semantics just presented is extensional, in the sense of Part 1. A modal analog of generalized Henkin models can also be developed, along the lines of Section 5. Objects in the Henkin domains are no longer sets , and an explicit extension function must be added. The generalization is straightforward but complex , and I also leave this to you.

Chapter 8

MODAL TABLEAUS

1.

The Rules

There are several varieties of tableaus for modal logic. This book uses a version of prefixed tableaus. These incorporate a kind of naming device for possible worlds into the tableau mechanism, and do so in such a way that syntactic features of prefixes reflect semantic features of worlds. Prefixed tableau systems exist for most standard modal logics. Here I only give versions for K and 85 since these are the extreme cases. I refer you to the literature for modifications appropriate for other modal logics-see [FM9S] for instance.

1.1

Prefixes

There are two versions of what are called prefixes. The version for K is more complex, and variations on it also serve for many other modal logics. The version for 85 is simplicity itself. DEFINITION 8.1 (PREFIX) A K prefix is a finite sequence of positive integers, written with periods as separators (1.2.1.1 is an example). An 85 prefix is a single positive integer. Think of prefixes as naming worlds in some (unspecified) model. Prefix structure is intended to embody information about accessibility between worlds. For K , think of the prefixes 1.2.1.1, 1.2.1.2, 1.2.1.3, etc. as naming worlds accessible from the world that 1.2.1 names. For 85 one can take each world as being accessible from each world, so prefixes are simpler. Prefixes have two uses in tableau proofs, qualifying formulas and qualifying terms. I begin with terms.


106


As was done classically, a larger language allowing parameters is used for tableau proofs, with parameters for each type. But in addition, an intensional term T is allowed to have a prefix. If we think of (7 as designating a possible world, we should think of (7 T as representing the extensional object that T designates at (7. Formally, if T is of type it, then (7 T is of type t. But writing prefixes in front of terms makes formulas even more unreadable than they already are. Instead, in an abuse of language, I have chosen to write prefixes on terms as subscripts, T cn though of course the idea is the same, and I still often refer to them as prefixes. So, if one thinks of (7 as designating possible world r , and T as having the function f as its meaning , then T u should be thought of as designating the object f(r). By L + (C) is meant L( C) enlarged with parameters, and allowing prefixes (written as subscripts) on terms of intensional type (this includes parameters, but prefixes will not be needed on free variables that are not parameters). This extends the classical version of L +(C), since prefixes are permitted now. But just as classically, in proving a closed formula of L(C) it is formulas of L +(C) that will appear in proofs. I said prefixes had two roles. Qualifying formulas is the main one. DEFINITION 8.2 (PREFIXED FORMULA) A prefixed formula is an expression of the form (7 , where (7 is a prefix and is a formula of L+(C). Think of (7 as saying that formula is true at the world that (7 names. Note that this use of prefixes does not compound, that is, (7 is a prefixed formula if is a formula, and not something built up from prefixed formulas. DEFINITION 8.3 (GROUNDED) I call a term or a formula of L+(C) grounded if it contains no free variables, though it may contain parameters. As usual, tableau proofs are proofs of sentences--closed formulas--of L(C) . In the tableau, prefixed grounded formulas of L+(C) may appear. To construct a tableau proof of , begin with a tree that has 1 -, at its root, and nothing else. Think of 1 as an arbitrary world. This initial tableau intuitively asserts that is false at some world of some model, the world designated by 1. Next the tree is expanded according to branch extension rules to be given below. If we produce a tree that is closed, which means it embodies a contradiction, we have a proof of .

107

MODAL TABLEAUS

1.2

Propositional Rules

Since the modal tableau rules are rather complex, I've divided their presentation into categories, beginning here with the propositional ones. These are much as in the classical case, except that prefixes must be "carried along." In these, and throughout, I use a, a', and the like to stand for prefixes. DEFINITION 8.4 (CONJUNCTIVE RULES) For any prefix a,

a...,(X VY) a...,X a...,Y

a...,(X ~ Y) aX a...,Y

DEFINITION 8.5 (DOUBLE NEGATION RULE) For any prefix a,

DEFINITION 8.6 (DISJUNCTIVE RULES) For any prefix a ,

a...,(X 1\ Y)

s-ec la""Y a...,(X == Y) -a-...,"7"(X:;-;:--~--;-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. DEFINITION

8.7

(POSSIBILITY RULES FOR

K) If the prefix a .n is new

to the branch,

aOX a.n X

a...,OX a.n""X

108


DEFINITION 8.8 (POSSIBILITY RULES FOR 85) If the positive integer n is new to the branch, (5

OX

nX

(5

-,oX

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 FOR K) If the prefix a.ti already occurs on the branch,

DEFINITION

OX (5.nX (5

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

DEFINITION

OX nX

(5

(5

,OX

n,X

Many examples of the application of these propositional and modal rules can be found in [FM98] . 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) . DEFINITION 8.11 (EXISTENTIAL RULES) In the following , pt is a pa-

rameter of type t that is new to the tableau branch .

(5 (3a t )(at) (5 (pt)

(5

,(Vat) ((i) (5

, (pt)

Terms of the form 17 may vary their denotation from world to world of a model, because the extension of the intensional term 7 can change from world to world. Such terms should not be used when instantiating a universally quantified formula.

8.12 (RELATIVIZED TERM) If 7 is a grounded intensional term, 17 is a relativized term.

DEFINITION

MODAL TABLEA US

109

8.13 (UNIVERSAL RULES) In the following, grounded term of type t that is not relativized.

DEFINITION

(T ('v'at)~( at) . (T ~(Tt)

1.5

Tt

is any

(T ,(3at)~( at) (T ,~( Tt )

Abstraction Rules

The rules for predicate abstracts essentially correspond to Proposition 7.10. Note the presence of a subscript (prefix) on the predicate abstract. We must know at what world the abstract is to be evaluated before doing so. The next subsection provides machinery for the introduction of these subscripts. Note that the subscript on the abstract , and the prefix for the entire formula need not be the same. 8.14 (ABSTRACT RULES) In the following , TI, . . . non-relativized terms.

DEFINITION

(T' (Aal, .. .

,an .~(al , '" (T~(TI, ".

(T' '(Aal, . . .

1.6

,an .~(al ,'" (T'~(TI, '"

.r« are

,an))q(TI , . . . ,Tn) ,Tn) ,an))q(TI, . .. ,Tn) ,Tn)

Atomic Rules

Unlike classically, much can be done with atomic formulas in a modal tableau besides just using them to close branches . The first atomic rule says that, at a world, an intensional predicate applies to terms if those terms are in the extension of the predicate at that world. It corresponds to part 1a of Definition 7.9. 8.15 (INTENSIONAL PREDICATION RULES) Let T be a grounded intensional term, and TI, . . . , Tn be arbitrary grounded terms.

DEFINITION

(T (IT)(TI , '" ,Tn)

(T'T(TI , . .. ,Tn) (T ,(IT)(TI,''. ,Tn)

Relativized terms denote different objects in different worlds. In tableaus, their behavior depends on the prefix of the formula in which they appear. This leads us to the evaluation of relativized terms at prefixes. Think of T@(T as T evaluated at (T . On non-relativized terms, such evaluation has no effect-their meaning is world independent. DEFINITION

8 .16 (EVALUATION AT A PREFIX) Let (T be a prefix.

110


1 For a relativized term 1T, set (IT)@(J = Tu . 2 For a non-relativized term T, set T@(J

= T.

The next rule covers the case of an extensional predicate applying to terms. This corresponds to part 1b of Definition 7.9. DEFINITION 8.17 (EXTENSIONAL PREDICATION RULES) Let T be a grounded extensional term, and Tl, . . . , Tn be arbitrary grounded terms .

Here is a simple example of how these rules work. Suppose A is of intensional type i (0) and b is of type O. If (J A(b) occurs on a branch, we may add (J (1 A)(b) by an Intensional Predication Rule . Now the Extensional Predication Rule applies; (lA)@(J = Au and b@(J = b, so we may add (J Au(b). Think of this as saying, since A(b) is true at the world that (J designates, then b is in the extension of A at that world, an extension represented by Au . Finally, there are atomic formulas that must evaluate the same way no matter what world is involved. DEFINITION 8.18 (WORLD INDEPENDENT) We call an atomic formula T(Tl, '" ,Tn) world independent if none oj r, Tl, ... , Tn is relativized, and T is of extensional type. DEFINITION 8 .19 (WORLD SHIFT RULES) Let T(Tl,'" ,Tn) be world independent. If (J' already occurs on the branch,

(JT(T1 ,' " , Tn) (J'T(T1, ... ,Tn)

1. 7

(J ""T(T1 , (J'''''T(Tl,

, Tn) , Tn)

Proofs and Derivations

I'll begin with the easy part. DEFINITION 8 .20 (CLOSURE) A tableau branch is closed if it contains (J III and (J...,IlI, for some formula III of L +(C). A tableau is closed if each branch is closed. DEFINITION 8.21 (TABLEAU PROOF) For a sentence closed tableau beginning with 1...,(P is a proof of (P.

(p

of L(C) , a

A brief discussion of the complexities of modal consequence is in Section 6. More discussion can be found in [Fit83, Fit93, FM98]. Corresponding to the local/global semantic distinction of Definition 7.12, we have the following tableau version .

MODAL TABLEA US

111

DEFINITION 8.22 (LOCAL AND GLOBAL ASSUMPTIONS) Let Sand U be sets of sentences of L(C). A tableau uses S as global assumptions and U as local assumptions if the following two tableau rules are admitted.

Local Assumption Rule If Y is any member of U then 1 Y can be added to the end of any open branch.

Global Assumption Rule If Y is any member of S then a Y can be added to the end of any open branch on which a appears as a prefix.

DEFINITION 8.23 (TABLEAU DERIVATION) A sentence has a derivation from global assumptions S and local assumptions U if there is a closed tableau beginning with 1 -,, allowing the use of U and S as local and global assumptions respectively. This concludes the presentation of the basic tableau rules. It is a rather complex system. In Section 2 I give a few examples of proofs using the rules. I omit soundness and completeness proofs. The arguments are elaborations of those given earlier for classical logic. Complexity of presentation goes up , but no fundamentally new ideas arise . Consequently they are left as a huge exercise. There is one important consequence of the completeness proofs that we will need, however, and that is the fact that the system has the cutelimination property-see Theorem 4.32. Just as in the classical case (Corollary 4.34), it is a consequence of this that any previously proved result can simply be introduced into a tableau.

2.

Tableau Examples

Tableaus for classical logic are well-known, and even for propositional modal logics they are rather familiar. The abstraction and predication rules of the previous section are new, and I give two examples illustrating their uses. The examples use the K rules ; I do not give examples specifically for 85 here. EXAMPLE 8.24 This provides a proof for (7.3) which was verified valid in Example 7.13. The formula is

(AX.O(3x)X(x))(P) :::> O(AX.(3x)X(x))(P) in which x is a variable of type 0 and X is a variable and P a constant symbol, both of type j(O).

112


1 ,[(AX.O(3x)X(x))(P):) O(AX.(3x)X(x))(P)] (AX.O(3x)X(x))(P) 2. 1 1 'O(AX.(3x)X(x))(P) · 3. 1 l(AX.O(3x)X(x))(P) 4. 1 (AX.O(3x)X(x))r(P) 5. 1 O(3x)P(x) 6. 1.1 (3x)P(x) 7.

1.

1.1'(AX.(3x)X(x))(P) 8. 1.1, l(AX.(3x)X(x))(P) 9. 1.1,(AX.(3x)X(x))1.1(P) 10. 1.1,(3x)P(x) 11. In this, 2 and 3 are from 1 by a conjunctive rule; 4 is from 2 by intensional predication; 5 is from 4 by extensional predication; 6 is from 5 by predicate abstraction; 7 is from 6 by a possibility rule; 8 is from 3 by a necessitation rule; 9 is from 8 by intensional predication; 10 is from 9 by extensional predication; and 11 is from 10 by predicate abstraction. It should be obvious that useful derived rules could be introduced. For instance, the passage from 2 to 4 to 5 to 6 could be collapsed. Such rules are given in the next section. EXAMPLE

8.25 Here is a proof of (7.13), which was shown to be valid

earlier.

(,),X.O(3x)X(x))(jP)

:J

(,),X.(3x)X(x))(jP)

See Example 7.14 for a discussion of the significance of this formula.

1 ,[(,),X.O(3x)X(x))(jP):) (AX.(3x)X(x))(jP)] (AX.O(3x)X(x))(jP) 2. 1 1 ,(')'X.(3x)X(x))(jP) 3. 1 l(,),X.O(3x)X(x)) (lP) 4. 1 ,l(AX.(3x)X(x))(tp) 5. (AX.O(3x)X(x))r(PI) 6. 1 1 ,(AX.(3x)X(x))r(PI) 7. O(3x)PI(x) 8. 1 1 ,(3x)PI(x) 9. 1.1 (3x)PI(x) 10. 1.1 PI(p) 11. 1 PI(p) 12. 1 ,PI (p) 13.

1.

In this, 2 and 3 are from 1 by a conjunction rule; 4 is from 2 and 5 is from 3 by intensional predication; 6 is from 4 and 7 is from 5 by extensional

113

MODAL TABLEAUS

predication; 8 is from 6 and 9 is from 7 by predicate abstraction; 10 is from 8 by a possibility rule ; 11 is from 10 by an existential rule ; 12 is from 11 by a world shift rule ; and 13 is from 9 by a universal rule.

Exercises EXERCISE 2.1 Give a t ableau proof of the following

(AX.O(3x)X(x))(lP) :J (AX.(3x)X(x))(LP) where x is of type iO, X is of type (iO) and P is of type i(iO) . EXERCISE 2.2 Give a tableau proof of the following

O(3x)P(x) :J (3X)OP(LX) where x is of type 0, X is of type jO and P is of type i(O).

3.

A Few Derived Rules

The tableau examples in the previous section are short, but already quite complicated to read. In the interests of keeping things relatively simple, a few derived rules are introduced which serve to abbreviate routine steps. DEFINITION

8.26

Suppos e X is a world A branch closes if it contains a X and

(DERIVED CLOSURE RULE)

independ ent atomic formula. a' oX.

The justification for this is easy. Using the World Shift Rule, if a X is on a branch, we can add a' X, and then the branch closes according to the official closure rule . The official rule concerning intensional predication has a slightly more efficient version, in which we first apply intensional, then extensional predication rules . DEFINITION

8.27

Let be arbitrary grounded

(DERIVED INTENSIONAL PREDICATION R ULE)

be a grounded intensional term, and terms.

T

Tl , . .. , Tn

Also here are two derived rules for predicate abstracts, one in which the abstract has a prefix (subscript) , one in which it does not.

114


8.28 (DERIVED SUBSCRIPTED ABSTRACT RULE) In the following, Tl, ,Tn are arbitrary grounded terms.

DEFINITION

(>'0:1,

(5'

,00n .cI>(0:1,' " ,00n))q(Tl,'" ,Tn) cI>( Tl@(5', •• • ,Tn@(5')

(5

(5'

.(>'0:1,... ,00n.cI>( 0:1, .. . ,O:n))q(Tl, '" ,Tn) (5 .CI>(Tl @(5', • • • , Tn@(5')

This abbreviates the application of the extensional predication rule, followed by predicate abstraction. 8.29 (DERIVED UNSUBSCRIPTED ABSTRACT RULE) In the following , Tl, .. , ,Tn are grounded terms.

DEFINITION

(5

(5

(>'0:1, .. . , 00n.cI> (0:1, (5 cI>(Tl@(5,

,O:n))(T1, . .. ,Tn) ,Tn@(5)

.(>.0:1, .. . ,00n.cI>(0:1 , '" , 00n))(Tl ,' " ,Tn) (5 .cI>(Tl@(5, .. . ,Tn @(5 )

This rule abbreviates successive applications of intensional predication, extensional predication, and predicate abstraction.

Chapter 9

MISCELLANEOUS MATTERS

This chapter is something of a grab-bag. Some familiar topics, like equality, and some less familiar, like choice functions, are discussed.

1.

Equality

The tableau rules of the previous chapter do not mention equality or extensionality. These are treated exactly as in the classical setting, via axioms , though as we will see, extensionality requires some care .

1.1

Equality Axioms

If we want to take equality into account, we use the Equality Axioms, Definition 5.1, as global assumptions. Prom here on these will be assumed in this book. In Chapter 5 I presented some tableau rules that were derivable classically provided equality axioms were allowed. In the modal setting these rules (with prefixes added, of course) are also derived rules. They are stated again for reference.

Reflexivity Rule For a grounded, non-relativized term 7, and a prefix a that is already present on the branch,

Substitutivity Rule For grounded, non-relativized terms 71 and 72, a cI>( 71) a (71 = 72) a lP(72)


116


Here is an example that uses equality. To help understand what the example says, and see why it ought to be valid, I give an informal interpretation for it. Suppose we read modal operators temporally, so that OX means X will be the case no matter what the future brings, and OX means the future could turn out to be one in which X is true. Let p be a type iO constant symbol intended to be read, "the President of the United States." Thus p is an individual concept , and designates different people in different possible futures . Now, call a person Presidential material if the person could be President (say the person meets all the legal requirements, such as being at least 35, not having already served twice, and so on). Being Presidential material is a property of persons . If we assume we have a model whose domain is the population of the United States, being Presidential material is a type i (0) object and is expressed by the following abstract, where x is of type O.

(),x.O(lp = x)) Informally, this predicate applies to a person at a particular time if there is some possible future in which that person is the President of the United States. Next, call a property of persons statesmanlike if it will always apply to the President. Thus we are using statesmanlike as a property of properties of persons-being diplomatic is hopefully a statesmanlike property, for instance. As such, being statesmanlike is of type i ((0)). It is expressed by the following abstract, where X is of type (0), and applies to those properties that will always belong to the President, no matter who that will be.

(),X.oX(lp)) Now, the extension of the property of being Presidential material is a statesmanlike property since, no matter who turns out to be President, that person must have been of Presidential material. The following gives a tableau verification for this. EXAMPLE

9.1 Here is a proof of the formula:

(),X.oX(lp))(l(),x.O(lp = x)))


117

1 ,(AX.oX(lp))(!(Ax.O(lp = x))) 1 ,D(Ax.O(lp = x)h(lp) 2. 1.1 ,(Ax.O(lp = x)h(lp) 3. 1.1 ,(AX.O(.Jp = x)h(pu) 4. 1 ,O(lp = Pl.1) 5. 1.1 '(lP = 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 13 are of intensional type j(t), then the following is valid.

PROPOSITION

(Va)(Vf3)[(ta =113)

(a =

J

11)1

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. DEFINITION

and

9.3 (EXTENSIONALITY FOR INTENSIONAL TERMS) For a

13 of the same intensional type, (Va)(Vf3)[D(la =113)

J

(a = 13)]

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.

118


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 0: and /3 are of type (tl, . . . , t n ), II is of type tl, .. . , In is of type tn' DEFINITION

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 71 and 72, and for parameters PI, . . . , Pn that are new to the branch, (1"" [71(Pl, .. .

,Pn) =:72(Pl , .. . ,Pn)]

1(1(71 =72)

De Re and De Dicto Loosely speaking, asserting the necessary truth of a sentence is a de dicto 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 dicto 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.

2.

119


In the next few paragraphs, (3 is of some extensional type t, and T is of the corresponding intensional type it. Consider the expression (A(3.ocp((3))(l T), where cp(fJ) is some formula with only (3 free (for simplicity) . Say the expression is true at a world of a modal model. (I use a generalized HenkinjKripke model, but what is said applies to any version-extensional, standard-just as well.) Thus suppose M , I' If-v,A (A(3.oCP((3)) (IT). Let Or be the object that T designates at I', that is, (v*I*r*A)(T,r) = Or . Then we have M, r If-v,A Dcp(fJ) [(3jOr] . So at every alternative world, ~, we have M , ~ If-v,A cp((3)[(3jOrj, that is cp((3) is true, at ~, of the object that T designates at I' . The effect is that (A(3.ocp((3))(lT) asserts, of the object designated by T at I' that it has a necessary property. This is a de re use of necessity-ascribing a necessary property to a thing. Next consider the expression o (A(3.CP(fJ))(1 T) . This asserts the necessity of a sentence. It is a de dicto use of necessity-applying it to a sentence , a dictum. And in general the behavior is quite different from the de re version. If M, I' If-v,A D(A(3.CP((3))(l T) , then at each alternative world ~ we have M , ~ If-v,A (A(3.cp((3))(l T), and so M, ~ If-v,A cp((3)[wjOLl], where OLl is the designation of T at ~ , something that depends on~. We can thus think of the assertion D(A(3.cp((3))(l T) as being concerned with the sense of T and not just with the object it happens to denote in "our" world-we use the local designation of T, which can vary from world to world. One remarkable thing about de re and de dicto is that, if either happens to imply the other , for a particular term, then the two turn out to be equivalent for that term. The following makes this precise. In the next section the phenomena is linked to the notion of rigidity. 9.5 (De Rej De Dicto) Let T be a term of intensional type it , (3 be a variable of type t, and a be a variable of type i(t) . In a model:

DEFINITION

1 de re is equivalent to de dicto for

T

if the following is valid.

(Va)[(A(3.oa((3))(lT) == D(A(3.a((3))(lT)]

2 de re implies de dicto for

T


(vo) [(A(3.oa(fJ))(IT) :J D(A(3.a((3)) (IT)] 3 de dicto implies de re for

T


(Va)[D(>',8.a(,8»)(lT)

~

(>',8.Da(,8»)(lT)]

The formulas above are allowed to be open-free variables may be present. Equivalently, one can work with universal closures. In [FM98]

120


we used schemas instead of the formulas given above, because that was a first-order treatment and we did not have the higher-type quantifier (Vo) available . The interesting fact about the three notions above is: they all say the same thing. PROPOSITION

9.6 For any intensional term

7,

the following are equiv-

alent (in K).

1 de dicto is equivalent to de re for 2 de dicto implies de re for

7

3 de re implies de dicto for

7

7

Proof Obviously item 1 implies items 2 and 3. I give a tableau proof, in K, showing that item 2 implies item 3. A similar argument, which I leave to you, shows that item 3 implies item 2, and this is enough to complete the proof of the Proposition. To keep things simple, assume 7 has no free variables. Here is a closed tableau for .(Va)[(.>.,6.Da(,6))(! 7) :> 0(.>.,6.a(,6))(!7)], (negation of) de re implies de dicta. In it, at a certain point, use is made of an instance of the de dicta implies de re schema. The tableau begins as follows. 1 .(Va) [(.>.,6.Da(,6))(!7) :> 0(.>.,6.a(,6))(!7)] 1. 1 • [(>',6.0(,6))(17) :> 0(>.,6.(,6))(!7)J 2. 1 ('>',6.0(,6)) (!7) 3. 1 .0 (.>.,6. (,6))(!7) 4. 1 0(71) 5. 1.1.(.>.,6.(,6))(!7) 6. 1.1 .(71.1) 7. 1.1 (71) 8. 1 (Va)[0(.>.,6.a(,6))(!7):> (.>.,6.Da(,6))(!7)] 9. 1 0(.>.,6.(.>.,.(,):> (!7))(,6))(!7) ('>',6.0('>',.(,) :> (!7))(,6))(!7) 10.

Item 2 is from 1 by an existential rule, using as a new parameter of type i (t); items 3 and 4 are from 2 by a conjunctive rule; 5 is from 3 by an unsubscripted abstract rule ; 6 is from 4 by a possibility rule; 7 is from 6 by an unsubscripted abstract rule; 8 is from 5 by a necessity rule . Item 9 is a de dicta implies de re formula ; and item 10 is from 9 by a universal rule, using ('>',.(,) :> (!7)) to instantiate the quantifier. Using item 10, the tableau splits into two branches. I first present the left one, and afterwards the right .

121


1 .O('\,6.('\,.cI>(-y) ~ cI>(lT))(,6))(!T) 11. 1.2 .(,\,6. (ky.cI>(-y) ~ cI>(!T))(,6))(!T) 12. 1.2 .('\,.cI>(-y) ~ cI>(!T))(T1.2) 13. 1.2 .[cI>(T1.2) ~ cI>(!T)) 14. 1.2 cI>(T1.2) 15. 1.2.cI>(!T) 16. 1.2 cI>1.2(T1.2) 17. 1.2 .cI>1.2(T1.2) 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 ('\,6.o('\,.cI>(-y) ~ cI>(!T))(,6))(!T) 1 O('\,.cI>(-y) ~ cI>(!T))(Tl) 20. 1.1 ('\,.cI>(-y) ~ cI>(!T))(Tt} 21. 1.1 cI>(Tl) ~ cI>(lT) 22.

/

19.

-.

1.1.cI>(Tl) 23. 1.1 cI>(!T) 24. 1.1 cI>u (TU) 25. 1.1 ·cI>u (T1. i) 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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TYPES, TABLEAUS, AND CODEL 'S COD

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 the following is valid in it.

DEFINITION

7

is rigid in a normal model if

It is easy to see that the formula asserting rigidity of 7 is true at a world r of a normal model if and only if, at each world accessible from r, 7 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 rei 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 7 is rigid if and only if the de reide 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 (>',6.0(,6 =t7))(l7) and let B be the formula (Va)[D(>.,6.a(,6))(7) ~ (>',6.oa(,6)) (7)]. A says 7 is rigid, while B says de dicto implies de re for 7 . I first give a tableau proof of A ~ B.

123


1 --,(A:> B)

1. 1 (A,6.o(,6 =17))(17) 2. 1 --,(Va)[0(A,6.a(,6))(17):> (A,6.oa(,6))(17)] 3. 1 --,[0(A,6.(,6))(17):> (A,6.o(,6))(17)] 4. 1 o (A,6.(,6)) (17) 5. 1 --, (A,6.o (,6)) (17) 6. 1 --,0(71) 7. 1.1--,(7I) 8. 1.1 (A,6.(,6))(17) 9. 1.1 (71.1) 10. 1 0(71 =17) 11. 1.1 (71 =17) 12. 1.1 71 = ri.i 13. 1.1--, (71.1) 14.

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:> A)

1. 1 (Va)[0(A,6.a(,6))(17) :> (A,6.oa(,6))(17)] 2. 1 --,(A,6.o(,6 =17))(17) 3. 1 O(A,6.(A,. 17 = ,)(,6))(17) ~ (A,6.o(A,. 17 = ,)(,6))(17) 1 --,0(71 =17) 5. 1.1--'(71 =17) 6. 1.1--'(71 = 71.1) 7.

/

1 --,O(A,6.(A, . 17 = ,)(,6))(17) 8.

1.2--,(A,6.(A,. 17 = ,)(,6))(17) 1.2--,(A,.17 = ,)(71.2) 10. 1.2 --'(17 = 71.2) 11. 1.2--'(71.2 = 71.2) 12. 1.2 71.2 = 71.2 13.

~

4.

(A,6.o(A,. 17 = ,)(,6))(17) 14. 9. 1 O(A,. 17 = ,)(71) 15. 1.1 (A,. 17 = ,)(71) 16. 1.1 (17 = 71) 17. 1.1 71.1 = 71 18. 1.1--'(71 = 71) 19. 1.1 71 = 71 20. 1

124


• In this, 2 and 3 are from 1 by a conjunctive rule; 4 is from 2 by a universal rule, instantiating with the term (A,. 1T = ,); 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 G6del 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 Hj (0)). G6del stated his conditions more or less as follows, with quantifiers implied: P(X) ~ OP(X) and ,P(X) ~ O,P(X) . The second of these is equivalent to OP(X) ~ P(X) , and this form will be used in what follows. Positiveness is a second-order notion, but G6del's conditions can be extended to other orders as well. I call the resulting notion stability, which is not terminology that G6del used. 9.9 (STABILITY) Let T be a term of type jet). T satisfies the stability conditions in a model provided the following are valid in that model.

DEFINITION

(Va)[T(a) :) OT(a)] (Va)[OT(a) :) T(a)] The stability conditions come in pairs. In 85, however, these pairs collapse. PROPOSITION

9.10 In 85, (Va)[T(a) :) OT(a)] and (Va)[OT(a) :) T(a)]

are equivalent.

Proof Suppose (Va)[T(a) ~ OT(a)]. Contraposition gives (Va)[,OT(a) :) 'T(a)] . From necessitation and converse Barcan, (Va)O[,OT(a) :)

125


'7(a)], and so (\la)[0.07(a) ~ 0'7(a)], equivalently, (\la)[00'7(a) ~ 0'7(a)] . But in 85, X ~ OOX is valid, hence we have (\la)['7(a) ~ 0'7(a)]. By contraposition again, (\la)[.0'7(a) ~ "7(a)] , and hence (\la)[07(a) ~ 7(a)]. The converse direction is similar . _ In the stability conditions, 7 is being predicated of other things. On the other hand, to say 7 is rigid, or that the de re/ de dicto distinction vanishes for 7, involves other things being predicated of 7 . Here is the fundamental connection between stability and earlier items . THEOREM 9.11 An intensional term the stability conditions.

7

is rigid if and only if it satisfies

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 4>, and we form the expression 1a.4> , which is read as the a such that 4>, and is called a definite description. Syntactically it is treated like a term. Its free variables are those of 4>, except for a , and its type is the type of a. In a more formal presentation, all this would have been built into the definition of term and formula given earlier, but doing so adds much complexity at the start of the subject, so I am taking the easier route of explaining now what could have been done. DEFINITION 9.12 (DESCRIPTION DESIGNATION) The definite description 1a.4> designates , or is defined at the possible world r of M =

126

W,R, u.i,


if

M , r If-v (3,8) ('v'8)[ (Xo.)( 8) == (,8 = 8)J where ,8 and 8 are not free in .

Next, the behavior of definite descriptions in context is treated in Russell's style . As he so famously noted, scope issues are fundamental. There is a difference between "the King of France is non-bald," which is false since there is no King of France, and "it is not the case that the King of France is bald," which is true. Formally, it is the difference between (>.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 rei de dicto 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 ro(rd either, or both, of ro and rl 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 7a. be a definite description, and let ,8 and 8 be variables of the same type as a, that do not occur free in or in any of the terms ri below. DEFINITION

1 rO(1a..a.)(8) == (,8 = 8)J 1\ ro(,8)}. 2 ro(rl,'" , 7a. , ... ,rn) is an abbreviation for

(>',8.ro( rl , ... ,,8, . .. , rn)) (1a. 'Z3 .Z3(ZI)) (Z4)) }

(9.1)

On the other hand, we might choose to eliminate 7x.A(x) first, beginning with part 3 of the definition . If so, after a few steps we wind up with the following.

(3z2){('v'Z3)[(>.x .A(x))(Z3) == (Z2 = Z3)]/\ (>.zd3z4){('v'zs)[(>.y .B(y))(zs) == (Z4 = zs)]/\ ZI(Z4)})(Z2)}

(9.2)

Fortunately, (9.1) and (9.2) are equivalent . In general, the elimination procedure is confluent-different reduction sequences for the same atomic formula always lead to equivalent results . In a sense there are two kinds of definite descriptions, intensional and extensional, depending on the type of the variable a in ta .. Extensional definite descriptions are rather well-behaved, and I say little about them , but for intensional ones, some interesting issues can be raised. In Definition 9.7 I characterized a formal notion of rigidity. That definition can be extended to definite descriptions : call 7a . rigid at a world if the following is true at that world.

(>'13.0(13 =l( ux. DE(m.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 DE(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-(O) constant symbols. P(1a.DQ(a)) ::> DQ(1a.DQ(a))

From this it follows that (10.5) is valid in 85.

GODEL'S ARGUMENT, BACKGROUND

137

EXERCISE 3.2 Construct a model to show E(1O~.oE(a)) :J

OE(1a.oE(a)).

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

(-\,6.E(,6))(1Q.oE(a)) :J O(-\,6.E(,6))(1a.oE(a)) which, by the previous exercise, is not K valid. Show the following variant is valid (a K tableau proof is probably easiest) .

(-\,6.E(,6))(1a.oE(a)) :J (-\,6.oE(,6))(1Q.oE(a)) EXERCISE 3.4 Show why the valid K formula of Exercise 3.3 can not be used in a Descartes-style argument.

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-Gad's

138

TYPES, TABLEA US, AND GODEL 'S GOD

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 until 1978, 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 Codel'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

G6del ontological argument . Finally the publication of G6del'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. G6del's manuscript provides almost no explanation or motivation. It amounts to an invitation to others to elaborate. G6del'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 G6del wrote his notes , and the complexity of quantification in modal contexts was perhaps not well appreciated. Consequently, the exact logic G6del had in mind is unclear . Subsequently several people took up the challenge of putting the G6del 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 G6del'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 G6del argument, intended for a general audience, can be found at [Sma01] as well as at [Opp01]. 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 G6del.

6.

Godel's Argument, Informally

Before we get to precise details in the next Chapter, it would be good to run through G6del'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, G6del 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. G6del found it convenient to do this, and used the term positiveness for the resulting notion . Thus we should think of a positive property, in G6del's sense, as some conjunction of perfections in Leib-

140


niz's sense. At least, I am assuming this to be the case-e-Gcdel 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 Codel argument has the familiar two-part structure: Cod's existence is possible ; and Cod'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 som ething has property P. INFORMAL PROPOSITION

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 has P. • 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. Codel inst ead 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 Codel'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 9 had two essences, P and Q, each would be a property of 9 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 9 is a God, the essence of 9 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 9 has property G, then G is the essence of g. Suppose 9 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 9 has all positive properties, 9 then would have the property complementary to P. Since we are assuming 9 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 9 has the property of necessarily existing if the essence of 9 is necessarily instantiated. And here is the last of Godel'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 9 is a God. Then 9 has all positive properties, and these include necessary existence by Informal Axiom 5. Then the essence of 9 is necessarily instantiated, by Informal Definition 6. But the essence of 9 is being a God, by Informal Proposition 5. Thus the property of being a God is necessarily instantiated.

•

GODEL 'S ARGUMENT, BACKGROUND

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).

Proof In any modal logic at least as strong as K, if P :J Q is valid, so is OP :J OQ. Then by Informal Proposition 7, if it is possible that a God exists , it is possibly necessary that a God exists . In 85, OOP :J OP is valid, and the conclusion follows. • 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 gl and g2 possess this property. By Informal Proposition 5, G must be the essence of both gl and g2- Now, if P is any property of gl, G must entail P, by part 2 of Informal Definition 4. Since G is a property of sz, by part 1 of the same Informal Definition , P must also be a property of g2. Similarly, any property of 92 must be a property of 91. Since 91 and 92 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 Codel'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 O. We are interested in the intensional properties of this object, properties of type j(O). Among these properties are the ones GCidel 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 ofthe world). Only then [are] the axioms true. It may also mean pure 'at tribution' 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.

11.1 (POSITIVE) A constant symbol P of type i(i(O)) is designated to represent positiveness. It is an intensional property of intensional properties. Informally, P is positive if we have PCP).

DEFINITION

It is convenient to introduce the following abbreviation.

11.2 (NEGATIVE) If T is a term of type i(O), toke rrr as short for (>'X.'T(X)). Call T negative if'T is positive.

DEFINITION

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. Oodel 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. 11.3 (FORMALIZING INFORMAL AXIOM 1) (VX)[P(,X) ~ ,P(X)] (VX)[,P(X) ~ P(,X)]

AXIOM

A B

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 Codel gave it. [P(X) /\ D(Vx)(X(x) ~ Y(x))] ~ P(Y)

147


In this formula , x is a free variable of type O. For us, type-O quantification is possibilist, while for G6del 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 (Vx) quantifier to E. Since this relativization comes up frequently, it is best to make an official definition . DEFINITION 11.4 (EXISTENTIAL RELATIVIZATION)

(VEx) abbrevi-

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

AXIOM 11.5 (FORMALIZING INFORMAL AXIOM 2)

In the following, x is of type 0, X and Yare of type j(0). (VX)(VY){[P(X)

1\

D(VEx)(X(x) :) Y(x))] :) P(Y)}

At one point in his proof, G6del asserts that ('xx.x = x) must be positive if anything is, and (,Xx ....,x = x) must be negative. This is easy to see: P( ('xx.x = x)) is valid if anything is positive because anything strictly implies a validity, and we have Axiom 11.5. The assertion that (,Xx ....,x = x) is negative is equivalent to the assertion that (,Xx .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):) P( (,Xx .x

= x));

2 (3X)P(X) :) P(...,(,Xx....,x = x)). PROPOSITION 11.7

Assuming Axioms 11.3A and 11.5: (3X)P(X) :) ...,P((,Xx ....,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, (VX){P(X) :) O(3 Ex)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 p as 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, Codel 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 (>.x.X(x) 1\ Y(x)). AXIOM 11.9 (FORMALIZING INFORMAL AXIOM 3) (V'X)(V'Y){[P(X) 1\ P(Y)] :J 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) ~ (V'X)[Z(X) :J 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 j(j(O)), X is of type j(O) , and x is of type 0.) (X intersection of Z) ~ O(V'x){X(x) == (W)[Z(Y) :J Y(x)]} AXIOM 11.10 (ALSO FORMALIZING INFORMAL AXIOM 3) (V'Z){pos(Z) :J (V'X)[(Xintersection of Z) :J P(X)]}. Axiom 11.10 implies Axiom 11.9. I leave th e 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 t erse explanation, but here are the assumptions. (V'X)[P(X) :J OP(X)] (V'X)[--'P(X) :J O-'P(X)]

GODEL 'S ARGUMENT, FORMALLY

.....

..... .....

149


150

If the underlying logic is just K, equivalence of these two assumptions follows from Axioms l1.3A and l1.3B. 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) :) OP(X)]. P has been taken to be an intensional object, of type T(j (0)). Axiom 11.11 and Theorem 9.11 tells us that P is 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(0)) .

Exercises EXERCISE 2.1 Give a tableau proof that .(AX.• (X = x)) = (AX.X = x). More generally, show that for a type (O) term 7, .(.7) = 7. EXERCISE 2.2 Show that (V'X)[.P(X) :) O.P(X)] follows from Axiom 11.11 together with Axioms 11.3A and l1.3B. 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(0) term, where Y is type j(O).

(AX .(W)[P(Y) :) 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(0) term

(AX .(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)].

In K , with this assumption, (V'x)[G(x) == G*(x)]. Axiom l1.3B 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 l1.3B 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 In K, (3x)G*(x) :J (V'X)[-'P(X) :J P( -,X)]. Now we can show that God's existence is possible. G6del 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 In K 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 ofP) . 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. In K both of the following are consequences. O(3 Ex)G(x) and O(3x)G(x) .

Proof By Proposition 11.8,

(V'X){P(X)

:J

O(3Ex)X(x)},

hence trivially,

(V'X){P(X)

:J

O(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,

(V'X)(W){[P(X)

1\

o(V'x) (X(x) :J Y(x))]

:J

P(Y)}

152


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

(\fX){P(X) :::> O(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) :::> O(\fy)[G(y) :::> 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)] :::> P(X)} :::> 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 Codel 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 . Codel'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 rrX and a X => Y occur on a branch, a Y can be added. Schematically,

aX aX =>Y aY The justification for this is as follows.

aX 1. a X => Y

a,X 3.

2.

aY 4.

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) => Z

a,Y

aY => Z

a,X

a

X=> Y

aX

a,X

aX=Y aY

a X=Y a,Y

a (Val)" . (Van)(a} , .. . , an) a (T}, . . . ,Tn) for any grounded terms T}, .. . , Tn

a (3ad (3a n)( a}, ... ,an) a(P1 , , Pn ) for any new, distinct parameters p} , . . . , Pn

Now, here is the promised proof of equivalence. 11.18 Assume all the Axioms to this point, except for Axiom 11.10 and Axiom 11.9. The following are equivalent, using 85:

THEOREM

1 Axiom 11.10; 2 P(G) ;

3 O(3Ex)G(x);

154


4 O(3x)G(x). Proof We already know 1 implies 2, this is Proposition 11.16. Likewise 3 follows from 2, by Theorem 11.17. And the implication of 4 from 3 is trivial. Showing that 4 implies 2 is straightforward, using the fact that G and G* are equivalent, and the fact that positiveness is rigid. Here is a tableau derivation. 1 O(3x)G(x) 1. 1 -'P(G) 2. 1.1 (3x)G(x) 3. 1.1 G(g) 4. 1.1 (V'x)[G(x) == G*(x)] 5. 1.1 [G(g) == G*(g)] 6. 1.1 G*(g) 7. 1.1 (AX.(V'Y)[P(Y) == Y(x)])(g) 8. 1.1 (W)[1'(Y) == Y(g)] 9. 1.1 [P(G) == G(g)] 10. 1.1 P(G) 11. 1 (V'X)[-'P(X) :) O-,1'(X) 12. 1 -,1'(G) :) 0-,1'(G) 13. 1 D-,P( G) 14. 1.1-'P(G) 15. Item 3 is from 1 by a possibility rule; 4 is from 3 by an existential rule , with g as a new parameter; 5 is Proposition 11.14, and note that the modal version of Corollary 4.34 is being used here; 6 is from 5 by a universal rule ; 7 is from 4 and 6 by a derived rule; 8 is 7 unabbreviated; 9 is from 8 by an abstraction rule; 10 is from 9 by a universal rule ; 11 is from 10 and 4 by a derived rule ; 12 is an equivalent of Axiom 11.11; 13 is from 12 by a universal rule ; 14 is from 2 and 13 by a derived rule; 15 is from 14 by a necessity rule . Showing 2 implies 1 informally is also not hard. If C is any collection of positive properties, G entails every member of C by Exercise 3.1. It follows that G also entails the conjunction of C. Since 2 says G is positive, the conjunction of C is positive by Axiom 11.5. The informal argument just sketched can be turned into a proper tableau proof. In Figure 11.2 I give a proof that 2 implies Axiom 11.9, and I'll leave the argument for Axiom 11.10 as an exercise. In Figure 11.2, item 3 is from 2 by a (derived) existential rule ; 4 and 5 are from 3, and 6 and 7 are from 4 by conjunctive rules ; 8 is Axiom 11.5; 9 is from 8 by a derived universal rule; 10 is from 1 and 9 by a derived


1

P(G)

1.

1 -{'v'X)(W){[P(X) /\ P(Y)] ~ P(X /\

155

yn

2.

1 ""{[P(A) /\ P(B)] ~ P(A /\ Bn 3. 1 P(A) /\ P(B) 4. 1 ...,P(A /\ B) 5. 1 P(A) 6. 1 P(B) 7. 1 (VX)(VY){[P(X) /\ D(VEx)(X(x) ~ Y(x))] ~ p(yn 8. 1 [P(G) /\ D(VEx)(G(x) ~ (A /\ B)(x))] ~ P(A /\ B) 9. 1 D(VEx)(G(x) ~ (A /\ B)(x)) ~ P(A /\ B) 10. 1 ...,D(VEx)(G(x) ~ (A /\ B)(x)) 11. 1.1...,(VEx)(G(x) ~ (A /\ B))(x) 12. 1.1...,(Vx)[E(x) ~ (G(x) ~ (A /\ B)(x))] 13. 1.1...,[E(c) ~ (G(c) ~ (A /\ B)(c))] 14. 1.1 E(c) 15. 1.1...,(G(c) ~ (A /\ B)(c)) 16. 1.1 G(c) 17. 1.1 ...,(A /\ B)(c) 18. 1 (VX)[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 (AX.(VY)[P(Y) ~ Y(x)]}(c) 26. 1.1 (VY)[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",,(Ax.A(x) /\ B(x))(c) 32. 1.1...,[A(c) /\ B(c)] 33.

/~

1.1...,A(c)

34. 1.1...,B(c)

Figure 11.2.

35.

Proof that item 2 implies Axiom 11.9

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 O(3x)G(x) implies Axiom 11.10.

5.

Essence

Even though we ran into the old Descartes problem with half of the G6del 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, G6del defines a notion of essence that plays a central role, and is of interest in its own right. [Haz98] makes a case for calling G6del's notion character, reserving the term essence for something else. I follow G6del's terminology. The essence of something, x, is a property that entails every property that x possesses. G6del says it as follows. ip

Ess x

== (\f?jJ){?jJ(x)

~

D(\fy)[cp(y)

~

?jJ(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\ (\f?jJ){?jJ(x)

~ D(\fy)[cp(y) ~ ?jJ(y)]}

G6del states ip 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-O 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


j(0) and w is of type 0: (>'Y,x.Y(x) 1\ (YZ){Z(x) :) O(yEw)[Y(w) :) Z(w)]}) The property of being Godlike was defined earlier, Definition 11.12. A central fact about Godlikeness, from Codel'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. In K the following is provable. (Note that x is of type 0.)

THEOREM

(Yx)[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 In K the following is provable, assuming Axiom 11.11.

(Yx)[G*(x) :) £(G*, x)]. Proof Here is a closed K tableau to establish the theorem.

1 -,(Yx) [G* (x) :) £ (G* , x)] 1. l-,[G*(g) :) £(G*,g)] 2. 1 G*(g) 3. l-,£(G*,g) 4. 1-,{ G*(g) 1\ (YZ){ Z(g) :) D(yEw)[G* (w) :) Z(w)]}}

5.

/~

1,G*(g)

6. l-,(YZ){Z(g):) O(yEw)[G*(w) :) Z(w)]} 7.

Item 2 is from 1 by an existential rule, with 9 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):> O(VEw)[G*(w) :> Q(w)]} 1 Q(g) 9. 1 ...,O(VEw)[G*(w) j Q(w)] 10. 1.1...,(VEw)[G*(w) :> Q(w)] 11. 1.1...,{E(a) :> [G* (a) :> Q(a)]} 12. 1.1 E(a) 13. 1.1...,[G*(a) :> Q(a)] 14. 1.1 G*(a) 15. 1.1...,Q(a) 16. 1 (W)[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 (W)[P(Y) :> OP(Y)] 22. 1 P(Q) :> OP(Q) 23. 1 OP(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 rul e; 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){[£(X, z) 1\ £(Y, z)J :) O(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 &(P, a) 1. 1 &(Q,a) 2. l-,D(V'Ew)[p(w) == Q(w)] 3. 1 P(a) 4. 1 (V'Z)[Z(a) ~ D(V'Ew)[p(w) ~ Z(w)]] 5. 1 Q(a) 6. 1 (V'Z)[Z(a) ~ D(V'Ew)[Q(w) ~ Z(w)]] 7. 1 Q(a) ~ D(V'Ew)[p(w) ~ Q(w)] 8. 1 P(a) ~ D(V'Ew)[Q(w) ~ P(w)] 9.

l-,Q(a)

/~

10. 1 D(V'E w) [P(w) ~ Q(w)] 11.

l-,P(a)

/~

12. 1 D(V'E w) [Q(w) ~ 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) ~ [P(b) == Q(b)]} 15. 1.1 E(b) 16. 1.1-,[P(b) == Q(b)] 17.

/~

1.1 P(b) 18. 1.1-,P(b) 1.1-,Q(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 (\iEw)[P(w):) Q(w)] 22. 1.1 E(b):) [P(b) :) Q(b)] 23.

/~

1.1--,E(b) 24. 1.1 P(b) :) 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.

(\iX) (\iy){ E(X, y) :) D(\iEz)[X(z) :) (y = z)]} This can be proved using tableaus-I 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, (AX.X = c) and (AX.X c). Either property can be used in the proof.

t=

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 Codel calls necessary existence. DEFINITION 11.24 (NECESSARY EXISTENCE) (Formalizing Informal Definition 6) N abbreviates the following type j(0) term:

(Ax .(\iY)[E(Y,x) :) D(3 Ez)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. Note 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) :J D(3 Ex)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

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

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

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

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

O(3x)G(x) :J D(3 Ex)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)(Vy)[G(y) == (y = x)]. This Proposition has a curious Corollary. Since type-D 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 (Vx)[G(x):J E(x)]. 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 everyt hing else, is. But alternative versions of Godel's axioms have been proposed-I 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 (VX){P(X):J D(3 Ex)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,

(Vx)[G(x) :J t'(G ,x)].

(11.1)

Using the definition of t' , we have as a consequence

(Vx){G(x) :J (VZ){Z(x) :J D(VEw)[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 (VZ) 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 j(0). We get the following consequence. (Vx){G(x):J {(>.y .Q)(x):J D(VEw)[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 (Vx){G(x):J {Q:J D(VEw)[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) :J {Q :J o('v'Ew)(G(w) :J 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

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

Q :J 0[(3 Ew)G(w) :J Q).

(11.8)

Using the distributivity of 0 over implication, (11.8) gives us

Q:J [0(3 Ew)G(w) :J oQ).

(11.9)

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

Q:J oQ.

(11.10)

Most people have taken this as a counter to Oodel'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 j((0)) . Axiom 11.5 gets replaced with the following.

Revised Axiom 11.5 In the formula below, x is of type 0, and X and Y are of type (0). (V'X)(W){[P(X) /\ O(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 Yare 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 r of type (0), take or as short for !(Ax.or(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.

(V'X){P(X) :J O(3 Ex)X (x)} Analogs of G and G* are defined in the expected way. G is the following type j(0) term, where Y is type (0) and , as noted before, P is of type j((0)).

(AX.(W)[P(Y) :J Y(x)]) Likewise G* is the type j(0) term

(AX.(W)[P(Y) == Y(x)]). One can still prove (V'x)[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:

(AY,X.Y(x) /\ (V'Z){Z(x) :J O(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:

(V'x)[G*(x) :J £(lG*, 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):

(AX. (W)[£(Y, x) :J O(3 Ez)Y( z)).

166


Revised Axiom 11.25 is P(LN), where N is as just modified. With this est ablished, the rest of G6del's argument carries over directly, giving us the following.

D(3 Ez)(lG*)(z) The final step is the easy proof that this implies the desired D(3 Ez)G*(z), and hence D(3 Ez)G(z), and I leave this to you. So, we have the conclusion of G6del's argument. Finally, here is a model, adapted from [And90], that shows Sobel's continuation no longer applies. EXAMPLE 11.32 Construct a standard 85 model as follows. There are two possible worlds, call them r and zx. The accessibility relation always holds. The type-O 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~. Informally, all type-O objects exist at r , but only a exists at ~ . Call a type-(O) object positive if it applies to a. Interpret P so that at each world its extension is the collection of positive type-(O) 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 s 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 (3Ex)(3 Ey).(x = y). Since it asserts two objects actually exist, it is true at r, but not at ~ , and hence Q :J DQ 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

GODEL 'S ARGUMENT, FORMALLY

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

(V'x)[G(x) ::) £(!G ,x)]

(11.11)

which, unabbreviated, is

(V'x)[G(x) ::) (>'Y,x.Y(x) 1\ (V'Z){Z(x) ::) D(V'Ew)[Y(w) ::) Z(w)]}) (!G, x)]

(11.12)

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

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

(V'x)[G(x) ::) (>,y.Y(x)

1\

(V'Z){Z(x) ::) D(V'Ew)[Y(w) ::) Z(w)]})(!G)]

(11.13)

From this we trivially get the following.

(V'x)[G(x) ::) (>,y.(V'Z){Z(x) ::) D(V'Ew)[Y(w) ::) Z(w)]})(!G)]

(11.14)

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

(V'x)[G(x) ::) (>.Y, Z.Z(x) ::) D(V'Ew)[Y(w) ::) Z(w)])(!G, l(>.y.Q))]

(11.15)

Distributing the abstraction, this is equivalent to the following.

(V'x){G(x) ::) [(>.Y, Z.Z(x))(lG, l(>.y.Q)) ::) (>.Y, Z.D(V'Ew)(Y( w) ::) Z(w)))(lG, l(>.y.Q))]}

(11.16)


168

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

(Vx){ G(x) :J [Q:J (AY, z.o(VEw)(Y(w) :J Z(w)))(lG, l(Ay.Q))]}

(11.17)

From this we get

(3x)G(x) :J [Q:J (AY, Z.o(VEw) (Y(w) :J Z(w)))(lG, l(Ay.Q))]

(11.18)

and since we have (3x)G(x), we also have Q:J (AY, Z.o(VEw)(Y(w) :J Z(w)))(lG, l(Ay.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 (AY, Z.o((3 Ew)Y(w) :J Z(a)))(lG, l(Ay.Q)).

(11.20)

Using the distributivity of 0 over implication, (11.20) gives us Q:J (AY, Z.o(3 Ew)Y(w) :J DZ(a))(lG, l(Ay.Q)).

(11.21)

From (11.21) we get

Q :J[(AY,Z.o(3 Ew)Y(w))(lG,l(Ay .Q)):J (AY, Z.oZ(a))(lG, l(Ay.Q))].

(11.22)

Because Z has no free occurrences in D(3 Ew)Y(w) and Y has no free occurrences in Z(a), (11.22) can be simplified to

Q :J[(AY.o(3 Ew)Y(w))(lG) :J (AZ.DZ(a)) (l(Ay.Q))].

(11.23)

I don 't know the status of (AY.o(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:> (AZ.oZ(a))(l(Ay.Q)).

(11.24)

But (AZ.oZ(a))(l (Ay.Q)) is not equivalent to DQ, and that's an end of it. Expressing the essential idea of (AZ.oZ(a))(l (Ay.Q)) with somewhat informal notation, we might write it as (.xZ.oZ)(lQ), and so what has been established, assuming rigidity of G, is Q :> (AZ.oZ)(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 :> (.xZ.oZ)(lQ), and since Q is assumed to hold in the real world, we conclude (AZ.oZ)(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 G6del 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 i

(0) term ('xx.(W)[P(Y)

= DY(x)]).

There is a corresponding change in the definitions of essence and neeessaryexistence. 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

(AY, x.(V'Z){OZ(x) == O(V'Ew)[Y(w) :) 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 E, 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 G6del. DEFINITION 11.35 (NECESSARY EXISTENCE , ANDERSON VERSION) N A abbreviates the following type j(0) term: (AX.(W)[£A (Y, x) :) O(3 Ez)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 85 the following is provable.

(V'x)[CA(x):) £A(CA,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(N A ) . Now Theorem 11.26 turns into the following. THEOREM 11.38 Assume Axioms 11.11 and 11.37. In the logic 85,

(3x)C A(x) :) O(3 Ex)CA(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 85,

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)

~

(O(x) V O-,(x))]

~

(3Y)O(V'x)[Y(x) == (x)] .

Hajek shows that G6del's axioms do not lead to a proof of Q ~ OQ, 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 G6del'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 Codel '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

TYPES, TABLEA US, AND GODEL'S GOD

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.

References

Note to reader: At the end of each bibliography item is a list of the pages on which there is a reference to the item. [Ada95]

Robert Merrihew Adams. Introductory note to *1970. In Feferman et al. [FJWDG+95], pages 388-402. pages 138

[AG96]

C. Anthony Anderson and Michael Gettings. GOdel's ontological proof revisited. pages 167-172, 1996. In [Haj96a]. pages 139, 148

[And71]

Peter B. Andrews . Resolution in type theory. Journal of Symbolic Logic, 36(3):414-432, 1971. pages 42, 47

[And72]

Peter B. Andrews . General models and extensionality. Journal of Symbolic Logic, 37(2):395-397, 1972. pages 25

[And86]

Peter B. Andrews . An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof Academic Press, Orlando, Florida, 1986. pages xii

[And90]

C. Anthony Anderson. Some emendations of GOdel's ontological proof. Faith and philosophy, 7:291-303, 1990. pages 139, 166, 169, 171

[Bre72]

Aldo Bressan. A General Interpreted Modal Calculus. Yale University Press, 1972. pages 96, 102

[Car56]

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[Cha79]

M. J. Charlesworth. St . Anselm's Proslogion. University of Notre Dame Press, Notre Dame , IN, 1979. First edit ion, Oxford University Press, 1965. pages 134, 134

[Chu40]

Alonzo Church. A formulation of the simple theory of types . Journal of Symboli c Logic, 5:56-68, 1940. pages 4

173

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[Coc69]

Nino B. Cocchiarella. A completeness theorem in second order modal logic. Theoria, 35:81-103, 1969. pages 90, 96, 139

[Des51]

Rene Descartes. A Discourse on Method and Selected Writings. Everyman's Library. E. P. Dutton, New York, 1951. Translation by John Veitch. pages 135, 135

[Fit83]

Melvin C. Fitting. Proof Methods for Modal and Intu itionistic Logics. D. Reidel Publishing Co., Dordrecht, 1983. pages 95, 102, 110

[Fit93]

Melvin C. Fitting. Basic modal logic. pages 368-448. 1993. In [GHR94] Volume 1. pages 95, 110

[Fit96]

Melvin C. Fitting. First -Order Logic and Automated Theorem Proving. Springer-Verlag, 1996. First edition , 1990. pages 8, 10,33,37, 41, 47

[Fit98]

Melvin C. Fitting. Higher-order modal logic-a sketch. In Ricardo Caferra and Gernot Salzer, editors, Automated Deduction in Classical and Non-Classical Logics, pages 23-38. Springer Lecture Notes in Artificial Intelligence, 1761, 1998. pages 96

[FitOOa]

Melvin C. Fitting. Databases and higher types . In John Lloyd, Veronica Dahl, Ulrich Furbach , Manfred Kerber, Kung-Kiu Lau, Catuscia Palamidessi, Luis Moniz Pereira, Yehoshua Sagiv, and Peter Stuckey, editors, Computational Logic-CL 2000, pages 41-52. Springer Lecture Notes in Artificial Intelligence, 1861, 2000. pages xiii

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Melvin C. Fitting. Modality and databases. In Roy Dyckhoff, editor, Automated Reasoning with Analytic Tableaux and Related Methods, pages 19-39. Springer Lecture Notes in Artificial Intelligence, 1847, 2000. pages xiii, 96

[FJWDG+95]

Solomon Feferman , Jr. John W. Dawson, Warren Goldfarb, Charles Parsons, and Robert N. Solovay, editors. Kurt Gddel Collected Works, Volume III, Unpublished Essays and Lectures. Oxford University Press, New York, 1995. pages 173,174

[FM98]

Melvin C. Fitting and Richard Mendelsohn. First-Order Modal Logic. Kluwer, 1998. Paperback, 1999. pages xiv, 4, 89, 96, 101, 102, 102, 102, 105, 108, 110, 119, 122, 122, 125

[G70]

Kurt GOdel. Ontological proof. In Feferman et al. [FJWDG +95], pages 403-404. pages 139, 145

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Index

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

abstraction designation function, 21, 26, 103 proper, 22, 27 accessibility, 91 Anderson, C. A., 169-171 Anselm, 134 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 Godel, 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 int ensional object, 84, 91 interpretation , 11, 30, 51, 73, 92, 103 allowed ,50 K ,105

£( C),5 A abstraction, 3 Leibniz, G. , 137-140, 145, 148, 152 Lindstrom, P., 68 local assumption, 95, III Lcwenheim-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 possibl e value , 49 possible world , 91 predicate abstract, 5 predicate abstraction, 3 prefix , 105 pseudo-model, 47, 48, 51 quantification act ualist , 89, 91 possibilist, 89, 91 rigid , 121-124 rule a bst ract , 37, 109 branch extension, 35 conjunct ive, 35, 107 derived

TYPES, TABLEA US, AND CODEL'S COD closure, 113 extensionality, 77 int ensional predication, 113 reflexivity, 70 subs cripted abstract, 114 subs titutivity, 70 unsubscripted abstract, 114 disjunctive, 36, 107 double negation, 35, 107 existential, 36, 108 extensional, 118 extensional predication, 110 intensional pr edication, 109 necessity, 108 possibility, 107, 108 reflexivity, 115 substitutivity, 115 universal, 36, 109 world shift, 110 Russell, 8., 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 st a bilit y, 124-125 substitution, 8 free, 9 ta bleau, 33 basic , 35 derivation, 37, III 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 Zermel o-Fra enkel 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 ModalLogic. 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 andReasoning. Essays in Honour of Dov 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 of manyISBN 0-7923-6009-5

valuedReasoning. 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 Giidel's God. 2002

ISBN 0-7923-6940-8

ISBN 1-4020-0604-7

KLUWER ACADEMIC PUBLISHERS - DORDRECHT / BOSTON / LONDON

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