Mathematics] [computability Theory] - Types, Tableaus, And Godel's God. (melvin Fitting)

Types, Tableaus, and G¨odel’s God Melvin Fitting e-mail: [email protected] url: comet.lehman.cuny.edu/fitting Department of Mathematics and Computer Science Lehman College (CUNY) 250 Bedford Park Boulevard West Bronx, NY 10468-1589

Draft August 10, 2000

Preface What’s Here The term formal logic covers a broad range of concoctions. At one end are small, special-purpose systems; at the other are rich, expressive ones. Highertype modal 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 I’ve also applied it to provide a semantic foundation for some complex database systems, (Fitting 2000b, Fitting 2000a). Higher-type modal logic—also called intensional logic—is the subject of this book. There are two quite different aspects to a logic: the formal machinery for its own sake; and the formal machinery applied to problems. The mechanism of higher-type modal logic is complex and requires serious mathematics to develop properly, but if one is primarily interested in applications, much of this mathematics can be taken on faith. One of the interesting applications of intensional logic is the G¨odel ontological argument, which was formalized from the beginning. It provides the main example of applied higher-type modal logic that is considered here. On the other hand, if one is interested in the mathematical underpinnings of intensional logic, the details of this application can be omitted. It is a rare reader who will be equally interested in both the formal and the applied aspects. In a sense, then, this book has no audience. Instead there are (I hope) separate audiences for different parts of it. Philosophers interested in the G¨ odel ontological argument will find Part III pertinent. After a general survey of a few well-known ontological arguments, that of G¨ odel is analyzed in detail. While G¨ odel’s argument is formally correct, two 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 G¨ odel’s assumptions. Some ways out of this are explored. The other flaw is equally serious: G¨ odel assumes as an axiom something directly equivalent to his desired conclusion. The problematic axiom is a version of a principle Leibniz proposed as a way of dealing with a hole in an ontological proof of Descartes. The observation that Descartes, Leibniz, and G¨odel all have proofs that stick at the same point seems to be new in the literature. If you are interested in ontological arguments for their own sakes, start with Part III, and

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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. Now, here is a more detailed summary. Part I is devoted to higher-type classical logic. It begins with a discussion of syntax matters, Chapter 1. I present types in Sch¨ utte 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 the possibility of such models, but knowledge of them is not widespread. They are quite 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 (Takahashi 1967, Prawitz 1968, Toledo 1975, Andrews 1986, Shapiro 1991, Leivant 1994, Kohlhase 1995, Manzano 1996), 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 to the semantics. It is now that choices must be made, since quantified modal logic is not a thing, but a multitude. First of all, at ground level quantifiers could be actualist or possibilist— they 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. We 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 subsets, and so on. Going up intensionally, as Montague did, means we introduce

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functions from possible worlds to sets of ground-level objects, functions from possible worlds to sets of such things, and so on. What is presented here mixes both notions—both extensional and intensional objects are present. Classical tableau rules are adapted in Chapter 8, using prefixes, to produce modal systems. While the modal tableau rules are rather straightforward, they are new to the literature, and should be of interest. Since things are already quite complex, no completeness proof is given. If it were given, it would be a direct extension of the classical proof of Part I. Using modal semantics and tableaus, in Chapter 9 I discuss the relationships between rigidity, de re and de dicto usages, and what I call G¨ odel’s stability conditions, which arise in his proof of the existence of God. I also relate these 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 arguments. 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 G¨ odel argument itself. Formal methods are applied in Chapter 11, where G¨ odel’s argument is examined in great detail, flaws are found, and alternatives are discussed. This chapter brings together work from all parts of the book, but detailed knowledge of, say, the completeness proof is not needed. If this is what you are interested in, start here, and pick up earlier material as needed. Many of the uses of the formalism are relatively intuitive. Indeed, in G¨ odel’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, (Fitting & Mendelsohn 1998), a look at higher-order modal logic suggested itself. I thought I would use G¨ odel’s ontological argument as a paradigm, because it is one of the few examples I have run across that makes essential use of higher-order modal constructs. G¨ odel’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 G¨odel’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 G¨odel’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,

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there are essentially modal problems, some of which already arise at the firstorder 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—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 natural, 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 G¨ odel argument by others had generally used some version of an intensional logic, with origins in work of Carnap, (Carnap 1956), developed and applied by Montague, (Montague 1960, Montague 1968, Montague 1970), and formally elaborated in (Gallin 1975). 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 half 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 G¨ odel 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.

Contents Preface

I

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Classical Logic

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1 Classical Logic—Syntax 1.1 Terms and Formulas . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Classical Logic—Semantics 2.1 Classical Models . . . . . . 2.2 Truth in a Model . . . . . . 2.3 Problems . . . . . . . . . . 2.3.1 Compactness . . . . 2.3.2 Strong Completeness 2.3.3 Weak Completeness 2.3.4 And Worse . . . . . 2.4 Henkin Models . . . . . . . 2.5 Unrestricted Henkin Models 2.6 A Few Technical Results . .

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3 Classical Logic—Basic Tableaus 29 3.1 A Different Language . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Basic Tableaus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3 Tableau Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4 Soundness and Completeness 4.1 Soundness . . . . . . . . . . . . . . . . . 4.2 Completeness . . . . . . . . . . . . . . . 4.2.1 Hintikka Sets . . . . . . . . . . . 4.2.2 Pseudo-Models . . . . . . . . . . 4.2.3 Substitution and Pseudo-Models 4.2.4 Hintikka Sets and Pseudo-Models 4.2.5 Pseudo-Models and Models . . .

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4.2.6 Completeness At Last . . . . . . . . . . . . . . . . . . . . Miscellaneous Model Theory . . . . . . . . . . . . . . . . . . . . .

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5 Equality 62 5.1 Adding Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.2 Derived Rules and Tableau Examples . . . . . . . . . . . . . . . . 62 5.3 Soundness and Completeness . . . . . . . . . . . . . . . . . . . . 65 6 Extensionality 69 6.1 Adding Extensionality . . . . . . . . . . . . . . . . . . . . . . . . 69 6.2 A Derived Rule and an Example . . . . . . . . . . . . . . . . . . 69 6.3 Soundness and Completeness . . . . . . . . . . . . . . . . . . . . 70

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Modal Logic

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7 Modal Logic—Syntax and Semantics 7.1 Introduction . . . . . . . . . . . . . . . . . 7.2 Types and Syntax . . . . . . . . . . . . . 7.3 Constant Domains and Varying Domains 7.4 Standard Modal Models . . . . . . . . . . 7.5 Truth in a Model . . . . . . . . . . . . . . 7.6 Examples . . . . . . . . . . . . . . . . . . 7.7 Related Systems . . . . . . . . . . . . . . 7.8 Henkin/Kripke Models . . . . . . . . . . .

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8 Modal Tableaus 92 8.1 The Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 8.2 Tableau Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 97 8.3 A Few Derived Rules . . . . . . . . . . . . . . . . . . . . . . . . . 99 9 Miscellaneous Matters 9.1 Equality . . . . . . . . . 9.1.1 Equality Axioms 9.1.2 Extensionality . 9.2 De Re and De Dicto . . 9.3 Rigidity . . . . . . . . . 9.4 Stability Conditions . . 9.5 Definite Descriptions . . 9.6 Choice Functions . . . .

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Ontological Arguments

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10 Ontological Arguments, A Brief History 117 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 10.2 Anselm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

CONTENTS

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Descartes . . . . . . . . . . . Leibniz . . . . . . . . . . . . . G¨ odel . . . . . . . . . . . . . G¨ odel’s Argument, Informally

11 G¨ odel’s Argument, Formally 11.1 General Plan . . . . . . . . . 11.2 Positiveness . . . . . . . . . . 11.3 Possibly God Exists . . . . . 11.4 Objections . . . . . . . . . . . 11.5 Essence . . . . . . . . . . . . 11.6 Necessarily God Exists . . . . 11.7 Going Further . . . . . . . . . 11.7.1 Monotheism . . . . . . 11.7.2 Positive Properties are 11.8 More Objections . . . . . . . 11.9 A Solution . . . . . . . . . . . 11.10Anderson’s Alternative . . . . 11.11Conclusion . . . . . . . . . .

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Part I

Classical Logic

1

Chapter 1

Classical Logic—Syntax 1.1

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 only 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: (∃X)[X(x1 , . . . , xn ) ≡ ϕ(x1 , . . . , xn )]. where ϕ(x1 , . . . , xn ) is a formula with free variables as indicated. Such axioms ensure that to each formula corresponds an ‘object.’ The other approach is to elaborate the term-forming machinery, so that there is an explicit name for the object specified by a formula ϕ. This involves predicate abstraction, or λ-abstraction: hλx1 , . . . , xn .ϕ(x1 , . . . , xn )i. 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 (Fitting

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& Mendelsohn 1998), and makes discussion of major issues considerably easier here. Finally, one can characterize higher-order formulas in 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 (Sch¨ utte 1960). Alternatively, following (Church 1940), 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 a series of papers, (Gilmore 1998a, Gilmore 1998b, Gilmore 1999), 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 one-place, 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 1.1.1 [Type] 0 is a type. And if t1 , . . . , tn are types, ht1 , . . . , tn i is a type. t, t1 , t2 , t0 , etc. are generally used to represent types. An object of type 0 is intended to be a ground-level object—it corresponds to a constant symbol or variable in standard first-order logic. An object of type ht1 , . . . , tn i is a predicate that takes n arguments, of types t1 , . . . , tn respectively. Thus a constant symbol of type h0, 0, 0i, say, would be called a three-place relation symbol in standard first-order logic—it takes three ground-level arguments. But now we can have relation symbols of types such as hh0i, h0, 0i, 0i, to which nothing in first-order logic corresponds. Definition 1.1.2 [L(C)] Let C be a set of constant symbols, containing at least an equality symbol =ht,ti for each type t. I denote the classical higherorder language built up from C by L(C). The rest of this section amounts to the formal characterization of L(C).

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For each type t I assume there are infinitely many variable symbols of that type. (In addition to constant and variable symbols, certain other syntactic objects will be assigned types as well.) I generally use letters from the beginning of the Greek alphabet to represent variables, with the type written as a superscript: αt , β t , γ t , . . . . Likewise I generally use letters from the uppercase Latin alphabet as constant symbols, again with the type written as a superscript: At , B t , C t , Dt , . . . . As noted, equality is primitive, so for each type t there is a constant symbol =ht,ti of type ht, ti. 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—first-order, second-order, and so on. It is simplest to define this terminology first for types themselves. Definition 1.1.3 [Order] The type 0 is of order 0. The type ht1 , . . . , tn i has as its order the maximum of the orders of t1 , . . . , tn , plus one. Thus h0, 0i is of order 1, or first-order. Likewise h0, h0, 0ii is of order 2, or second-order. Types will play the fundamental role, but order provides a convenient way of referring to the maximum complexity of some construct. When I talk about the order of a constant or variable, I mean the order of its type. Likewise once formulas are defined, I may refer to the order of the formula, by which I mean the highest order of a typed part of it. Next I define the class of formulas, and their free variables. This definition is more complex than the corresponding first-order version because the notion of term cannot be defined first; both term and formula must be defined together. And to define both, I need the auxiliary notion of predicate abstract which is, itself, part of a mutual recursion involving Definitions 1.1.4, 1.1.5, and 1.1.6. Definition 1.1.4 [Predicate Abstract of L(C)] Suppose Φ is a formula of L(C) and α1 , . . . , αn is a sequence of distinct variables of types t1 , . . . , tn respectively. hλα1 , . . . , αn .Φi is a predicate abstract of L(C). Its type is ht1 , . . . , tn i, and its free variable occurrences are the free variable occurrences in the formula Φ, except for occurrences of the variables α1 , . . . , αn . Definition 1.1.5 [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). Its free variable occurrences were defined above. τ is used, with and without subscripts, to stand for terms. Definition 1.1.6 [Formula of L(C)] The notion of formula is given as follows.

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1. If τ is a term of type ht1 , . . . , tn i, and τ1 , . . . , τn is a sequence of terms of types t1 , . . . , tn respectively, then τ (τ1 , . . . , τn ) is a formula (atomic) of L(C). The free variable occurrences in it are the free variable occurrences of τ , τ1 , . . . , τn . 2. If Φ is a formula of L(C) so is ¬Φ. The free variable occurrences of ¬Φ are those of Φ. 3. If Φ and Ψ are formulas of L(C) so is (Φ∧Ψ). The free variable occurrences of (Φ ∧ Ψ) are those of Φ together with those of Ψ. 4. If Φ is a formula of L(C) and α is a variable then (∀α)Φ is a formula of L(C). The free variable occurrences of (∀α)Φ are those of Φ, except for occurrences of α. Example 1.1.7 Suppose αh0,0i is a variable of type h0, 0i (and so first-order), β 0 is a variable of type 0, and γ hh0,0i,0i is a variable of type hh0, 0i, 0i (secondorder). Both β 0 and γ hh0,0i,0i are terms. Then γ hh0,0i,0i (αh0,0i , β 0 ) is an atomic formula. Generally I will write the simpler looking γ(α, β), and give the information contained in the superscripts in a separate description. Since this atomic formula contains a variable γ of order 2, it is referred to as a second-order atomic formula. Definition 1.1.8 [Sentence] A formula with no free variables is a sentence. One can think of ∨, ⊃, ≡, and ∃ as defined symbols, with their usual definitions. But sometimes it is convenient to take them as primitive—I do whatever is most useful at the time. Also square and curly parentheses are used, as well as the official round ones, to aid readability. And finally, I write the equality symbol in infix position, following standard convention. Thus, for example, I write (αt =ht,ti β t ) in place of =ht,ti (αt , β t ). Several examples involving just first and second-order notions will be considered, so a few special alphabets are introduced informally, to make reading the examples a little easier. Order 0 1 2

Constants a, b, c, . . . A, B, C, . . . A, B, C, . . .

Variables x, y, z, . . . X, Y , Z, . . . X , Y, Z, . . .

Example 1.1.9 For this example I give explicit type information (in superscripts), until the end of the example. After this I omit the superscripts, and say in English what is needed to restore them. Suppose x0 , X h0i , and X hh0ii are variables (the first is of order 0, the second is of order 1, and the third is of order 2). Also suppose P hh0ii and g 0 are constant symbols of L(C) (the first is of order 2 and the second is of order 0). 1. Both X hh0ii (X h0i ) and X h0i (x0 ) are atomic formulas. All variables present have free occurrences.

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2. hλX hh0ii .X hh0ii (X h0i )i is a predicate abstract, of type hhh0iii. Only the occurrence of X h0i is free. 3. Since P hh0ii is of type hh0ii, hλX hh0ii .X hh0ii (X h0i )i(P hh0ii ) is a formula. Only X h0i is free. 4. [hλX hh0ii .X hh0ii (X h0i )i(P hh0ii ) ⊃ X h0i (x0 )] is a formula. The only free variable occurrences are those of X h0i and x0 . 5. (∀X h0i )[hλX hh0ii .X hh0ii (X h0i )i(P hh0ii ) ⊃ X h0i (x0 )] is a formula. The only free variable occurrence is that of x0 . 6. hλx0 .(∀X h0i )[hλX hh0ii .X hh0ii (X h0i )i(P hh0ii ) ⊃ X h0i (x0 )]i is a predicate abstract. It has no free variable occurrences, and is of type h0i. The type machinery is needed to guarantee that what is written is well-formed. Now that the exercise above has been gone through, I will display the predicate abstract without superscripts, as hλx.(∀X)[hλX .X (X)i(P) ⊃ X(x)]i, 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. Definition 1.1.10 [Degree] By the degree of a formula or term is meant the number of propositional connectives, quantifiers, and lambda-symbols it contains. 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.

1.2

Substitutions

Formulas can contain free variables, and terms that are much more 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 (Fitting 1996). Definition 1.2.1 [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. Composition of substitutions is the usual composition of functions.

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I generally denote substitutions by σ, with and without subscripts. Also I generally write xσ rather than σ(x). Most concern is with substitutions having finite 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 αi to τi and is the identity otherwise: {α1 /τ1 , . . . , αn /τn }. Clearly the composition of two substitutions with finite support is another substitution with finite support. Finally, the action of substitutions on variables is readily extended to terms and formulas generally. Definition 1.2.2 For a substitution σ, by σα1 ,... ,αn is meant the substitution that is like σ except that it is the identity on α1 , . . . , αn . Definition 1.2.3 Let σ be a substitution. The action of σ is extended recursively as follows. 1. Aσ = A for a constant symbol A. 2. hλα1 , . . . , αn .Φiσ = hλα1 , . . . , αn .Φσα1 ,... ,αn i. 3. [τ (τ1 , . . . , τn )]σ = τ σ(τ1 σ, . . . , τn σ). 4. [¬Φ]σ = ¬[Φσ]. 5. (Φ ∧ Ψ)σ = (Φσ ∧ Ψσ). 6. [(∀α)Φ]σ = (∀α)[Φσα ] Example 1.2.4 Let Φ be the formula (∃αhh0ii )[αhh0ii (hλβ 0 .γ h0i (β 0 )i)] and let σ be any substitution such that γ h0i σ = τ h0i . I compute Φσ. For convenience I omit type-indicating superscripts, but even so, the notion is a bit much. Sorry. (∃α)[α(hλβ.γ(β)i)]σ

= (∃α)[α(hλβ.γ(β)i)]σα = (∃α)[ασα (hλβ.γ(β)iσα )] = (∃α)[α(hλβ.γ(β)iσα )] = (∃α)[α(hλβ.[γ(β)]σα,β i)] = (∃α)[α(hλβ.(γσα,β )(βσα,β )i)] = (∃α)[α(hλβ.τ (β)i)]

The connection between substitutions and free variable occurrences is simple: it is only the free occurrences that can be changed by substitutions. I leave the proof to you as an exercise. Proposition 1.2.5 Let σ1 and σ2 be substitutions. 1. If σ1 and σ2 agree on the free variables of the term τ then τ σ1 = τ σ2 . 2. If σ1 and σ2 agree on the free variables of the formula Φ then Φσ1 = Φσ2 .

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Not all substitutions are appropriate; some do not properly respect the role of bound variables in the sense that they may replace a free occurrence of a variable in a formula with another variable that is “captured” by a quantifier or predicate abstract of the formula. The substitutions that are acceptable are called free substitutions. These play a significant role throughout what follows. Definition 1.2.6 [Free Substitution] The following characterizes when a substitution σ is free for a formula or term. 1. σ is free for a variable or constant. 2. σ is free for hλα1 , . . . , αn .Φi if σα1 ,... ,αn is free for Φ, and if β is any free variable of hλα1 , . . . , αn .Φi then βσ does not contain any of α1 , . . . , αn free. 3. σ is free for ¬Φ if σ is free for Φ. 4. σ is free for (Φ ∧ Ψ) if σ is free for Φ and σ is free for Ψ. 5. σ is free for (∀α)Φ if σα is free for Φ, and if β is a free variable of (∀α)Φ then βσ does not contain α free. It is not generally the case that Φ(α1 α2 ) = (Φα1 )α2 . But it is when appropriate freeness conditions are imposed. Theorem 1.2.7 Substitution is “compositional” under the following circumstances. 1. If σ1 is free for the formula Φ, and σ2 is free for the formula Φσ1 , then (Φσ1 )σ2 = Φ(σ1 σ2 ). 2. If σ1 is free for the term τ , and σ2 is free for the term τ σ1 , then (τ σ1 )σ2 = τ (σ1 σ2 ). 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 (Fitting 1996).

Exercises Exercise 1.2.1 Prove Proposition 1.2.5 by induction on degree. Conclude that if Φ is a sentence then Φσ = Φ for every substitution σ.

Chapter 2

Classical Logic—Semantics 2.1

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, after which 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 2.1.1 [Relation Types] Let S be a non-empty set. For each type t the collection [[t, S ]] is defined as follows. 1. [[0, S ]] = S. 2. [[ht1 , . . . , tn i, S ]] is the collection of all subsets of [[t1 , S ]] × · · · × [[tn , S ]]. O is an object of type t over S if O ∈ [[t, S ]]. O is systematically used, with or without subscripts, to stand for objects in this sense. For example, a member of [[h0, 0i, S ]] is a subset of S × S, and in standard first-order logic it would simply be called a two-place relation on S. But now relations of relations 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 denotation in the model to each constant symbol of the language. But that denotation must be consistent with type information.

9

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Definition 2.1.2 [Classical Model] A (higher-order) classical model for L(C) is a structure M = hD, Ii, where D 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) ∈ [[t, D]]. 2. If = is the equality constant symbol of type ht, ti then I(=) is the equality relation on [[t, D]].

2.2

Truth in a Model

Assume M = hD, Ii 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 °v Φ. Informally it can be read: the formula Φ is true in the model M, with respect to the valuation v which assigns meanings to free variables. But as will be seen, to properly define this one must also assign denotations to all terms. The denotation of a term of type t will be an object of type t over D. And this can not be done first, independently. The assignment of denotations to terms, and the determination of formula truth constitutes a mutually recursive pair of definitions, just as was the case for the syntactic notions of term and formula in Section 2.1. Still, it is all rather straightforward. Definition 2.2.1 [Valuation] The mapping v is a valuation in the classical model M = hD, Ii if v assigns to each variable αt of type t some object of type t, that is, v(αt ) ∈ [[t, D]]. Definition 2.2.2 [Variant] A valuation w is an α-variant of a valuation v if v and w agree on all variables except possibly α. More generally, w is an α1 , . . . , αn -variant if v and w agree on all variables except possibly α1 , . . . , αn . Now, the following two definitions constitute a single recursive characterization. Definition 2.2.3 [Denotation of a Term] Let M = hD, Ii be a classical model, and let v be a valuation in it. A mapping is defined, (v ∗ I), assigning to each term of L(C) a denotation for that term. 1. If A is a constant symbol of L(C) then (v ∗ I)(A) = I(A). 2. If α is a variable then (v ∗ I)(α) = v(α). 3. If hλα1 , . . . , αn .Φi is a predicate abstract of L(C) of type t, then (v ∗ I)(hλα1 , . . . , αn .Φi) is the following member of [[t, D]]: {hw(α1 ), . . . , w(αn )i | w is an α1 , . . . , αn variant of v and Γ °w Φ}

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Definition 2.2.4 [Truth of a Formula] Again let M = hD, Ii be a classical model, and let v be a valuation in it. The notion of formula Φ of L(C) being true in model M with respect to v, denoted M °v Φ, is characterized as follows. 1. For terms τ , τ1 , . . . , τn , M °v τ (τ1 , . . . , τn ) provided h(v ∗ I)(τ1 ), . . . , (v ∗ I)(τn )i ∈ (v ∗ I)(τ ). 2. M °v ¬Φ if it is not the case that M °v Φ. 3. M °v Φ ∧ Ψ if M °v Φ and M °v Ψ. 4. M °v (∀α)Φ if M °v0 Φ for every α-variant v 0 of v. There is an alternative notation that makes evaluating the truth of formulas in models somewhat easier. Definition 2.2.5 [Special Notation] Suppose v is a valuation, and w is the α1 , . . . , αn variant of v such that w(α1 ) = O1 , . . . , w(αn ) = On . Then, if M °w Φ this may be symbolized by M °v Φ[α1 /O1 , . . . , αn /On ]. Now part 3 of Definition 2.2.3 can be restated as follows. 3. (v ∗ I)(hλα1 , . . . , αn .Φi) = {hO1 , . . . , On i | M °v Φ[α1 /O1 , . . . , αn /On ]} Likewise part 4 of Definition 2.2.4 becomes 4. M °v (∀α)Φ if M °v Φ[α/O] for every object O of the same type as α. Defined symbols like ⊃ and ∃ have their expected behavior, which are explicitly stated below. Alternately, this can be considered an extension of the definition above. 5. M °v Φ ∨ Ψ if M °v Φ or M °v Ψ. 6. M °v Φ ⊃ Ψ if M °v Φ implies M °v Ψ. 7. M °v Φ ≡ Ψ if M °v Φ iff M °v Ψ. 8. M °v (∃α)Φ if M °v0 Φ for some α-variant v 0 of v; equivalently if M °v Φ[α/O] for some object O of the same type as α. As in first-order logic, if Φ has no free variables, M °v Φ 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 2.2.6 [Validity, Satisfiability, Consequence] Let Φ be a formula and S be a set of formulas. 1. Φ is valid if M °v Φ for every classical model M and valuation v.

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2. S is satisfiable if there is some model M and some valuation v such that M °v ϕ for every ϕ ∈ S. 3. Φ is a consequence of S provided, for every model M and every valuation v, if M °v ϕ for all ϕ ∈ S, then M °v Φ. The definitions above are of some complexity. Here is an example to help clarify their workings. Example 2.2.7 This example shows a formula that is valid and involves equality. In it, c is a constant symbol of type 0. The expression hλX.(∃x)X(x)i is a predicate abstract of type hh0ii, where X is of type h0i and x is of type 0. Intuitively it is the “being instantiated” predicate. Likewise the expression hλx.x = ci is a predicate abstract of type h0i, where x and c are of type 0. Intuitively this is the “being c” predicate. Since this predicate is, in fact, instantiated (by whatever c designates), the first predicate abstract correctly applies to it. That is, one should have the validity of the following. hλX.(∃x)X(x)i(hλx.x = ci)

(2.1)

I now verify this validity. Suppose there is a model M = hD, Ii. 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)(hλx.x = ci)

= {O | M °v (x = c)[x/O]} = {O | O = I(c)} = {I(c)}

We also have the following. (v ∗ I)(hλX.(∃x)X(x)i)

= {O | M °v (∃x)X(x)[X/O]} = {O | M °v X(x)[X/O, x/o] for some o} = {O | o ∈ O for some o} = {O | O 6= ∅}

Now we have (2.1) because M °v hλX.(∃x)X(x)i(hλx.x = ci) ⇔ (v ∗ I)(hλx.x = ci) ∈ (v ∗ I)(hλX.(∃x)X(x)i) ⇔

{I(c)} ∈ {O | O 6= ∅}.

You might try verifying, in a similar way, the validity of the following. ¬hλX.(∃x)X(x)i(hλx.¬(x = x)i)

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2.3

13

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 problems here.

2.3.1

Compactness

The compactness theorem for first-order logic says a set of formulas is satisfiable if every finite subset is. The higher-order analog does not hold, and counterexamples are easy to come by. 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—so the following formula is true in a model if and only if the domain of the model is finite. (∀X)[(function(X) ∧ one-one(X)) ⊃ onto(X)]

(2.2)

In (2.2) the following abbreviations are used. function(X) for (∀x)(∃y)(∀z)[X(x, z) ≡ (z = y)] one-one(X) for (∀x)(∀y)(∀z){[X(x, z) ∧ X(y, z)] ⊃ (x = y)} onto(X) for (∀y)(∃x)X(x, y) Also, define the following infinite list of formulas, where x 6= y abbreviates ¬(x = y). A2 A3 .. .

= (∃x1 )(∃x2 )[x1 6= x2 ] = (∃x1 )(∃x2 )(∃x3 )[(x1 6= x2 ) ∧ (x1 6= x3 ) ∧ (x2 6= 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 A2 , A3 , . . . , is certainly not satisfiable, but every finite subset is, so compactness fails. (In first-order classical logic this example turns around, and shows finiteness has no first-order characterization.)

2.3.2

Strong Completeness

A proof procedure is said to be (sound and) strongly complete if Φ has a derivation from a set S exactly when Φ is a logical consequence of S. Classical

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first-order logic has many proof procedures that are strongly complete for it, but there is no such proof procedure for higher-order logic. To see this, one doesn’t need an exact definition of proof procedure—it is enough that proofs be finite objects. Let S be the set of formulas defined in Section 2.3.1, a set which is not satisfiable though every finite subset is. The formula Φ ∧ ¬Φ is a logical consequence of S, since it is true in every model in which the members of S are true, namely none. If there were a strongly complete proof procedure, Φ ∧ ¬Φ would have a derivation from S. That derivation, being a finite object, could only use a finite subset of S, say S0 . Then Φ ∧ ¬Φ would be a logical consequence of S0 , and so S0 could not be satisfiable (otherwise there would be a model in which Φ ∧ ¬Φ were true). But every finite subset of S is satisfiable. Conclusion: no strongly complete proof procedure can exist for higher-order classical logic.

2.3.3

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. Higher-order classical logic does not even possess a weakly complete proof procedure. To show this, G¨ odel’s 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 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 h0, 0i. In addition to the abbreviations of Section 2.3.1, the following is needed. 0-exclude(S) inductive-set(P, S) induction(S)

for (∀x)¬S(x, 0) for P (0) ∧ (∀x)[P (x) ⊃ (∃y)(S(x, y) ∧ P (y))] for (∀P )[inductive-set(P, S) ⊃ (∀x)P (x)]

Now, let integer(S) be the formula function(S) ∧ one-one(S) ∧ 0-exclude(S) ∧ induction(S) It is not hard to show that integer(S) is true in a model hD, Ii if and only if the domain D is (isomorphic to) the natural numbers, using I(S) as successor. Consequently for any sentence Φ of arithmetic, Φ is true of the natural numbers if and only if integer(S) ⊃ Φ is valid. It is a standard requirement that the set of (G¨ odel 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 Φ such that

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integer(S) ⊃ Φ 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.

2.3.4

And Worse

I have been discussing higher-order classical logic, particularly its models, using conventional informal mathematics of the sort that every mathematician uses 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: 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-Fraenkel set theory. That is (assuming the axioms for set theory are consistent) there is a model of the Zermelo-Fraenkel 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 classical logic. 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 2.3.3, the following will do: (∃αh0,0i )integer(αh0,0i ). Next, one can identify a subset of the domain with a relation of type h0i. Then the collection of all subsets of the domain is a relation of type hh0ii, so the following says there is a powerset for the domain of a model: (∃β hh0ii )(∀γ h0i )β hh0ii (γ h0i ). 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 2.3.1. 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.

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Now, the real problem is: is the sentence CH valid or not? There are the following not very palatable options. 1. Assume the foundations for informal mathematics is Zermelo-Fraenkel set theory, formulated axiomatically. In this case, neither CH nor its negation is valid. 2. Assume that informal mathematics is being done in some particular model for the Zermelo-Fraenkel axioms. In this case, CH is definitely valid, or not, but it depends on which Zermelo-Fraenkel model is being considered. 3. Assume that higher-order classical logic itself supplies the theoretical foundations for mathematics. In this case CH either is valid or it is not, but which is it? I have reached perhaps the most basic difficulty of all with classical higherorder 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.

2.4

Henkin Models

As we saw in the previous section, higher-order classical logic is difficult to work with. Indeed, the 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 (Henkin 1950) 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 D, a universal quantifier whose variable is of type 0, (∀x), ranges over the members of D. If we have a universal quantifier, (∀X), whose variable is of type h0i, it ranges over the collection of properties of D, or equivalently, over the subsets of D. 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 D, we should work with enough of them, that is, we should take (∀X) as ranging over some collection of subsets of

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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, D, and this determines the range of quantification for each type. Specifically, we thought of a quantifier (∀αt ) as ranging over the members of [[t, D]]. 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 2.4.1 [Henkin Domain Function] H is a Henkin domain function if H is any function whose domain is the collection of types and, for each type ht1 , . . . , tn i, H(ht1 , . . . , tn i) is some non-empty collection of subsets of H(t1 ) × · · · × H(tn ). Sets of the form H(t) are called Henkin domains. The function H(t) = [[t, D]] is a Henkin domain function. In fact, if H is any Henkin domain function, and H(0) = D, then for every type t, H(t) ⊆ [[t, D]], with equality holding at t = 0. Definition 2.4.2 [Henkin Frame] The structure M = hH, Ii is a Henkin frame for a language L(C) if it meets the following conditions. 1. H is a Henkin domain function. 2. If A is a constant symbol of L(C) of type t, I(A) ∈ H(t). 3. I(=ht,ti ) is the equality relation on H(t) for each type t. The notion of valuation must be suitably restricted, of course. Definition 2.4.3 [Valuation] v is a valuation in a Henkin frame M = hH, Ii 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 2.2.3, for the meaning of a term, carries over word for word to a Henkin frame M. Also Definition 2.2.4, for truth in a model, carries over to M, with one restrictive change. Item 4, the universal quantifier condition, gets replaced with the following. 40 . Let M = hH, Ii be a Henkin frame and let αt be a variable of type t. M °v (∀αt )Φ if M °v Φ[αt /Ot ] for every Ot ∈ H(t), or equivalently, if M °v0 Φ for every αt -variant v 0 of v such that v 0 (αt ) ∈ H(t). The revised version of item 4 above says that quantifiers of type t range over just H(t) and not over all objects of type t. But there is a fundamental problem. Let M = hH, Ii be a Henkin frame, and suppose hλα1 , . . . , αn .Φi is a predicate abstract—to keep things simple for

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now, assume Φ itself contains no abstracts. Then for any valuation v, the characterization above determines whether or not M °v Φ. Now, according to Definition 2.2.3, the meaning (v ∗ I)(hλα1 , . . . , αn .Φi), to be assigned to the abstract, is {hO1 , . . . , On i | Γ °v Φ[α1 /O1 , . . . , αn /On ]}. The trouble is, we have no guarantee that this set will be a member of the appropriate Henkin domain. If it is not a member, quantifiers do not include it in their ranges. If this happens, we lose the validity of formulas like (∀α)Ψ(α) ⊃ Ψ(hλα1 , . . . , αn .Φi). The whole business becomes somewhat problematic since formulas like this clearly ought to be valid. What must be done is impose enough closure conditions on the Henkin domains of a Henkin frame to ensure that predicate abstracts always designate objects that are present in Henkin domains. There are several ways this can be done. Algebraic closure conditions can be formulated directly, though this takes some effort. I follow a different route that is somewhat easier. Essentially, I first allow predicate abstracts to designate members of Henkin domains in some arbitrary way, then I add the requirement that they be the “right” members. Definition 2.4.4 [Abstraction Designation Function] A function A is an abstraction designation function in the Henkin frame M = hH, Ii with respect to the language L(C) provided, for each valuation v in M, and for each predicate abstract hλα1 , . . . , αn .Φi of L(C) of type t, A(v, hλα1 , . . . , αn .Φi) is some member of H(t). Think of an abstraction designation function as providing a “meaning” for each predicate abstract. For the time being, such meanings can be quite arbitrary, except that they must be members of appropriate Henkin domains. Now earlier definitions get modified in straightforward ways (and these are the “official” versions). Definition 2.2.3 becomes the following. Definition 2.4.5 [Denotation of a Term in a Henkin Frame] Let M = hH, Ii be a Henkin frame, let v be a valuation, and let A be an abstraction designation function. A mapping, (v ∗ I ∗ A), is defined assigning to each term of L(C) a denotation for that term. 1. If A is a constant symbol of L(C) then (v ∗ I ∗ A)(A) = I(A). 2. If α is a variable then (v ∗ I ∗ A)(α) = v(α). 3. If hλα1 , . . . , αn .Φi is a predicate abstract of L(C) of type t, then (v ∗ I ∗ A)(hλα1 , . . . , αn .Φi) = A(v, hλα1 , . . . , αn .Φi). And Definition 2.2.4 becomes the following. Definition 2.4.6 [Truth of a Formula in a Henkin Frame] Let M = hH, Ii be a Henkin frame, let v be a valuation, and A be an abstraction designation function. A formula Φ of L(C) is true in model M with respect to v and A, denoted M °v,A Φ, if the following holds.

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1. For terms τ , τ1 , . . . , τn , M °v,A τ (τ1 , . . . , τn ) provided h(v ∗ I ∗ A)(τ1 ), . . . , (v ∗ I ∗ A)(τn )i ∈ (v ∗ I ∗ A)(τ ). 2. M °v,A ¬Φ if it is not the case that M °v,A Φ. 3. M °v,A Φ ∧ Ψ if M °v,A Φ and M °v,A Ψ. 4. M °v,A (∀αt )Φ if M °v,A Φ[αt /O] for every O ∈ H(t). Now we can impose a requirement that designations of predicate abstracts be “correct.” Definition 2.4.7 [Proper Abstraction Designation Function] Let M = hH, Ii be a Henkin frame and let A be an abstraction designation function in it, with respect to L(C). A is proper provided, for each predicate abstract hλα1 , . . . , αn .Φi and valuation v we have (v ∗ I ∗ A)(hλα1 , . . . , αn .Φi) = {hOn , . . . , On i | M °v,A Φ[α1 /O1 , . . . , αn /On }. Definition 2.4.8 [Henkin Model] Let M be a Henkin frame, and let A be an abstraction designation function in M. If A is proper, hM, Ai 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. Proposition 2.4.9 Let M = hH, Ii be a Henkin frame and let both A and A0 be proper abstraction designation functions, with respect to L(C). Then A = A0 . Proof The following two items are shown simultaneously, by induction on degree (Definition 1.1.10). From this the Proposition follows immediately. M °v,A Φ (v ∗ I ∗ A)(τ )

⇔ M °v,A0 Φ = (v ∗ I ∗ A0 )(τ )

(2.3) (2.4)

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 τ is a term of degree k. Since k could be 0, τ could be a constant symbol or a variable. If it is a constant symbol, (v ∗ I ∗ A)(τ ) = I(τ ) = (v ∗ I ∗ A0 )(τ ). Similarly if τ is a variable. Finally, τ could be a predicate abstract, hλα1 , . . . , αn .Φi, in which case Φ must be a formula of degree < k, so using the induction hypothesis with (2.3) we have (v ∗ I ∗ A)(hλα1 , . . . , αn .Φi) = {hO1 , . . . , On i | M °v,A Φ[α1 /O1 , . . . , αn /On ]} = {hO1 , . . . , On i | M °v,A0 Φ[α1 /O1 , . . . , αn /On ]} = (v ∗ I ∗ A0 )(hλα1 , . . . , αn .Φi)

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Thus (2.4) holds for terms of degree ≤ k. Now assume Φ is a formula of degree k. There are several cases, depending on the form of Φ. If Φ is atomic, it is τ (τ1 , . . . , τn ) where τ , τ1 , . . . , τn are all of degree ≤ k. Since (2.4) holds for terms of degree < k by assumption, and for terms of degree = k by the proof above, M °v,A τ (τ1 , . . . , τn ) ⇔ h(v ∗ I ∗ A)(τ1 ), . . . , (v ∗ I ∗ A)(τn )i ∈ (v ∗ I ∗ A)(τ ) ⇔ h(v ∗ I ∗ A0 )(τ1 ), . . . , (v ∗ I ∗ A0 )(τn )i ∈ (v ∗ I ∗ A0 )(τ ) ⇔ M °v,A0 τ (τ1 , . . . , τn ) If Φ is a negation, conjunction, or universally quantified formula, the result follows easily using the fact that (2.3) holds for its subformulas, by the induction hypothesis. We thus have (2.3) for formulas of degree k, and this concludes the induction. The pattern of the induction proof above will recur many times, with little variation of structure. We go from terms and formulas of degrees < k to terms of degrees ≤ k, and then to formulas of degrees ≤ k. The Proposition above allows us to give the following extension of Definition 2.4.8. Definition 2.4.10 [Henkin Model] If hM, Ai is a Henkin model, the proper abstraction designation function A is uniquely determined, so we may say the Henkin frame M itself is a Henkin model, and write M °v Φ for M °v,A Φ. Suppose D is some non-empty set, and we set H(t) = [[t, D]] for all types t. This gives us a Henkin domain function, as was noted earlier. And it is easy to see that hD, Hi will be a Henkin model. In fact, a sentence Φ is true in hD, Hi, as defined in this section, exactly when it is true in the higher-order model hD, Ii, as defined in Section 2.2. This says that “true” higher-order models are among the Henkin models. The real question is, are there any other Henkin models? The answer is, yes. The proof of the completeness theorem for tableaus will yield this as a byproduct. Definition 2.4.11 [Standard Model] The Henkin model M = hH, Ii is called a standard model if H(t) = [[t, D]] for all types t. Since standard models are among the Henkin models, any formula that is true in all Henkin models must be true in all standard models as well. But there is the possibility (a fact, as it happens) that there are formulas true in all standard models that are not true in all Henkin models. That is, the set of Henkin-valid formulas (Definition 2.5.9) is a subset of the set valid formulas (Definition 2.2.6), and in fact turns out to be a proper subset. By decreasing the set of validities, it opens up the possibility (again a fact, as it happens) that there may be a complete proof procedure with respect to this more restricted version of validity.

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2.5

21

Unrestricted Henkin Models

Unlike standard higher-order models, Henkin models are allowed to have some, but not necessarily all, of the relations permissible in principle at each type. This means there are more possibilities for Henkin models than for standard models. Even so, the objects in the domains of Henkin models are sets, and this imposes a restriction that we may want to avoid in certain circumstances. Sets are extensional objects—that is, a set is completely determined by its membership. Using the language of properties rather than sets, two extensional properties that apply to exactly the same things must be identical, and hence must have the same properties applying to them. Working with sets is sufficient for mathematics, but it is not always the right choice in every situation. Even if the terms “human being” and “featherless biped” happen to have the same extension, we might not wish to identify them. As another example, the properties of being the morning star and being the evening star have the same extension, but were thought of as distinct properties by the ancient Babylonians. Henkin himself (Henkin 1950) noted the possibility of a more general notion than what I am calling a Henkin model, “The axioms of extensionality can be dropped if we are willing to admit models whose domains contain functions which are regarded as distinct even though they have the same value for every argument.” Even so, extensionality has commonly been built into the treatment of Henkin models in the literature—(Andrews 1972) is one of the rare instances where a model without extensionality is constructed. As it happens, we will have need for a non-extensional version in carrying out the completeness proof for tableaus. Since such models are also of intrinsic interest, they are presented in some detail in this section. For Henkin frames, simply specifying the members of the Henkin domains tells us much. Since they are sets, there is a notion of membership, and it can be used in the definition of truth for atomic formulas. That is, sets come with their extensions fully determined. If we move away from sets this machinery becomes unavailable, and we must fill the gap with something else—I make use of an explicit extension function, denoted E. That is, for an arbitrary object O, E(O) gives us the extension of O. I also allow the possibility that equality may not behave as expected—I allow for non-normal frames and models. Definition 2.5.1 [Unrestricted Henkin Frame] M = hH, I, Ei is called an unrestricted Henkin frame for a language L(C) if it meets the following conditions. 1. H is a function whose domain is the collection of types. 2. For each type t, H(t) is some non-empty collection of objects (not necessarily sets). 3. If A is a constant symbol of L(C) of type t, I(A) ∈ H(t). 4. For each type t = ht1 , . . . , tn i, E maps H(t) to subsets of H(t1 ) × · · · × H(tn ).

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In addition, M is normal if E(I(=ht,ti )) is the equality relation on H(t) for each type t. Much of this definition is similar to that of Henkin frame. The members of H(t) are the objects of type t (which now need not be sets). The new item is the mapping E. Think of E(O) as the extension of the object O. Unrestricted Henkin models are built out of unrestricted Henkin frames. Much of the machinery is almost identical with that for Henkin models, but there are curious twists, so things are presented in detail, rather than just referring to earlier definitions. The definition of valuation is the same as before. Definition 2.5.2 [Valuation] v is a valuation in an unrestricted Henkin frame M = hH, I, Ei if v maps each variable of type t to some member of H(t). Next, just as with Henkin models, a function is needed that provides designations for predicate abstracts, then later we can require that it give us the “right” values. The wording is the same as before. Definition 2.5.3 [Abstraction Designation Function] A function A is an abstraction designation function in the unrestricted Henkin frame M = hH, I, Ei, with respect to the language L(C) provided, for each valuation v in M, and for each predicate abstract hλα1 , . . . , αn .Φi of L(C) of type t, A(v, hλα1 , . . . , αn .Φi) is some member of H(t). Term denotation is the same as before—terms designate objects in the various Henkin domains. Definition 2.5.4 [Denotation of a Term in an Unrestricted Henkin Frame] Let M = hH, I, Ei be an unrestricted Henkin frame, let v be a valuation, and let A be an abstraction designation function. A mapping, (v ∗ I ∗ A), is defined assigning to each term of L(C) a denotation for that term. 1. If A is a constant symbol of L(C) then (v ∗ I ∗ A)(A) = I(A). 2. If α is a variable then (v ∗ I ∗ A)(α) = v(α). 3. If hλα1 , . . . , αn .Φi is a predicate abstract of L(C) of type t, then (v ∗ I ∗ A)(hλα1 , . . . , αn .Φi) = A(v, hλα1 , . . . , αn .Φi). The following has a few changes from the earlier definition—to take the extension function into account the atomic case has been modified. Definition 2.5.5 [Truth of a Formula in an Unrestricted Henkin Frame] Again let M = hH, I, Ei be an unrestricted Henkin frame, let v be a valuation, and A be an abstraction designation function. A formula Φ of L(C) is true in model M with respect to v and A, denoted M °v,A Φ, provided the following. 1. For an atomic formula, M °v,A τ (τ1 , . . . , τn ) provided h(v ∗ I ∗ A)(τ1 ), . . . , (v ∗ I ∗ A)(τn )i ∈ E((v ∗ I ∗ A)(τ )).

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2. M °v,A ¬Φ if it is not the case that M °v,A Φ. 3. M °v,A Φ ∧ Ψ if M °v,A Φ and M °v,A Ψ. 4. M °v,A (∀αt )Φ if M °v,A Φ[αt /Ot ] for every Ot ∈ H(t). In item 1 above, τ (τ1 , . . . , τn ) is true if the designation of hτ1 , . . . , τn i is in the extension of the designation of τ . For Henkin frames, we were dealing with sets, and extensions were for free. Now we are dealing with arbitrary objects, and we must explicitly invoke the extension function E. I am about to impose a “correctness” requirement, analogous to Definition 2.4.7, but now there are three parts. The first part is similar to that for Henkin models, except that the extension function is invoked. The other parts need some comment. Suppose we have two predicate abstracts hλα1 , . . . , αn .Φi and hλα1 , . . . , αn .Ψi. In a Henkin model, if Φ and Ψ 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 2.5.6 Let M be an unrestricted Henkin frame (or a Henkin frame), and let A be an abstraction designation function in it. For each valuation v and substitution σ, define a new valuation v σ by: αv σ = (v ∗ I ∗ A)(ασ). Thus v σ assigns to a variable α the “meaning” of the term ασ. Definition 2.5.7 [Proper Abstraction Designation Function] Let M = hH, I, Ei be an unrestricted Henkin frame and let A be an abstraction designation function in it, with respect to L(C). A is proper provided, for each predicate abstract hλα1 , . . . , αn .Φi we have 1. E((v ∗ I ∗ A)(hλα1 , . . . , αn .Φi)) = {hO1 , . . . , On i | Γ °v,A Φ[α1 /O1 , . . . , αn /On ]} 2. If v and w agree on the free variables of hλα1 , . . . , αn .Φi then A(v, hλα1 , . . . , αn .Φi) = A(w, hλα1 , . . . , αn .Φi) 3. If σ is a substitution that is free for the term hλα1 , . . . , αn .Φi, then A(v, hλα1 , . . . , αn .Φiσ) = A(v σ , hλα1 , . . . , αn .Φi) The technical significance of items 2 and 3 above will be seen in the next section. When using Henkin frames, if a proper abstraction designation function

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exists, it is unique (Proposition 2.4.9). But with an unrestricted Henkin frame, it is entirely possible for there to be more than one proper abstraction designation function. Since there is this possibility, we must specify which one to use—the frame alone does not determine it. Definition 2.5.8 [Unrestricted Henkin Model] Let M be an unrestricted Henkin frame, and let A be an abstraction designation function in M. If A is proper, hM, Ai is an unrestricted Henkin model. Finally Definition 2.2.6 is broadened to the entire class of unrestricted Henkin models. Definition 2.5.9 [Validity, Satisfiability, Consequence] Let Φ be a formula and S be a set of formulas of L(C). 1. Φ is valid in unrestricted Henkin models if M °v,A Φ for every unrestricted Henkin model hM, Ai for L(C) and proper valuation v. 2. S is satisfiable in an unrestricted Henkin model hM, Ai for L(C) if there is some proper valuation v such that M °v,A ϕ for every ϕ ∈ S. 3. Φ is an unrestricted Henkin consequence of S provided, for every unrestricted Henkin model hM, Ai for L(C) and every proper valuation v, if M °v,A ϕ for all ϕ ∈ S, then M °v,A Φ. Similar terminology is used when confining things to unrestricted Henkin models that are normal, or to Henkin models themselves. We saw in Section 2.4 that the notion of Henkin model extended that of “true” higher-order model, since “true” models can be identified with standard Henkin models. In a similar way the notion of unrestricted Henkin model extends that of Henkin model, since Henkin models correspond to what will be called extensional unrestricted Henkin models (the definition is in the next section). Verifying this is postponed since it requires us to show that parts 2 and 3 of Definition 2.5.7 hold for Henkin models, and this involves some technical work. Assuming the result for the moment, it follows that there are unrestricted Henkin models because there are Henkin models; and we know there are Henkin models because there are standard models. The question is: have the various generalizations really generalized anything? In fact, they have. It is a consequence of the completeness proofs, which are given later, that there are Henkin models that are not standard, and there are unrestricted Henkin models that are not extensional Henkin models.

2.6

A Few Technical Results

There are several results of a rather technical nature that, nonetheless, are of fundamental importance. In fact, one of the technical propositions below

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25

concerns us immediately—it allows us to show that Henkin models are (isomorphically) among the unrestricted Henkin models. Since we do not yet know this, the first few items must treat Henkin models and unrestricted Henkin models separately. I omit proofs because they are very similar to proofs in Section 4.2.3 which are given fully. Proposition 2.6.1 Let hM, Ai be either a Henkin model or an unrestricted Henkin model, and let v and w be valuations. 1. If v and w agree on the free variables of the term τ (v ∗ I ∗ A)(τ ) = (w ∗ I ∗ A)(τ ). 2. If v and w agree on the free variables of the formula Φ M °v,A Φ ⇐⇒ M °w,A Φ. I leave the proof of the Proposition above as an exercise—see the proof of Proposition 4.2.8 as a guide. (For unrestricted Henkin models, part 2 of Definition 2.5.7 is needed.) Next I state a result that will be used in the next Chapter to establish the soundness of the tableau system. Proposition 2.6.2 Let hM, Ai be a Henkin model. For any substitution σ and valuation v: 1. If σ is free for the term τ then (v ∗ I ∗ A)(τ σ) = (v σ ∗ I ∗ A)(τ ). 2. If σ is free for the formula Φ then M °v,A Φσ ⇐⇒ M °vσ ,A Φ. Once again I omit the proof, and refer you to the proof of Proposition 4.2.10 for a similar argument. Among Henkin models, the standard ones correspond to “true” higher-order models. A similar phenomenon occurs here—among the unrestricted Henkin models certain ones correspond to Henkin models. Definition 2.6.3 [Extensional] An unrestricted Henkin frame hH, I, Ei is extensional provided that E(O) = E(O0 ) implies O = O0 for all objects O and O0 . An unrestricted Henkin model is extensional if its frame is. Suppose M = hH, Ii is a Henkin frame (Definition 2.4.2) and hM, Ai is a Henkin model. M can be converted into an unrestricted Henkin frame M0 = hH, I, Ei by setting E(O) = O for each object O of non-zero type. That is, we specify an extension function that gives us the usual set-theoretic notion of extension. It is easy to check that hM0 , Ai is an unrestricted Henkin frame— part 1 of Proposition 2.6.2 directly gives us part 3 of Definition 2.5.7, and likewise Proposition 2.6.1 gives us part 2. Obviously the unrestricted Henkin model that results is extensional. And equally obviously, evaluation of truth in the original Henkin model and in the unrestricted Henkin model just constructed is essentially the same.

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Conversely, suppose M = hH, I, Ei is an unrestricted Henkin frame that is extensional. Inductively define a mapping θ as follows. For objects O of type 0, θ(O) = O. And for an object O of type ht1 , . . . , tn i, set θ(O) = {hθ(O1 ), . . . , θ(On )i | hO1 , . . . , On i ∈ E(O)}. Define a new domain function H0 by setting H0 (t) = {θ(O) | O ∈ H(t)}. Using the fact that M is extensional, it is not hard to show that θ is 1-1 and onto between H(t) and H0 (t), for each type t. Finally, for each term τ , set I 0 (τ ) = θ(I(τ )). This gives us a Henkin frame hH0 , I 0 i. Thus, in effect, each unrestricted Henkin frame that is extensional is isomorphic to a Henkin frame as defined earlier. From now on I will treat Henkin models as being unrestricted Henkin models that are extensional, when it is convenient to do so. A result similar to Proposition 2.6.2, but for unrestricted Henkin models, is quite easy to establish, given the previous work. Proposition 2.6.4 Let hM, Ai be an unrestricted Henkin model. For any substitution σ and valuation v: 1. If σ is free for the term τ then (v ∗ I ∗ A)(τ σ) = (v σ ∗ I ∗ A)(τ ). 2. If σ is free for the formula Φ then M °v,A Φσ ⇐⇒ M °vσ ,A Φ. A proof of this is quite similar to that for Proposition 2.6.4 (which was not given), except for the induction step involving terms that are predicate abstracts, where a reduction to a simpler case is no longer possible. But for unrestricted Henkin models, we are given what we need for this step as part of the definition. (See part 3 of Definition 2.5.7). Part of the definition of (unrestricted) Henkin model is that each predicate abstract must have an interpretation that is an object with the “right” extension. But what predicate abstracts there are depends on what the language is. Given a language L(C), one would expect models to depend on the collection of constants—members of C—which the interpretation function, I, deals with. One would not expect the choice of free variables of L(C) to matter, but this is not entirely clear, since predicate abstracts can involve free variables. It is important to know that the choice of free variables, in fact, does not matter, since the machinery of tableau proofs will require the addition of new free variables to the language. In what follows, L(C) is the basic language, and L+ (C) is like L(C), with new variables added, but with the understanding that these new variables are never quantified or λ-bound. (This all takes on a significant role in the next chapter.) I note the fundamental problem: even with the restrictions imposed on the additional variables, the collection of predicate abstracts of L+ (C) properly extends that of L(C).

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Proposition 2.6.5 Every unrestricted Henkin model with respect to L(C) can be converted into an unrestricted Henkin model with respect to L+ (C) so that truth values for formulas of L(C) are preserved. There are two immediate consequences of this Proposition that I want to state, before I sketch its proof. First, any set S of sentences of L(C) that is satisfiable in some unrestricted Henkin model with respect to L(C) is also satisfiable in some unrestricted Henkin model with respect to L+ (C). And second, any sentence Φ of L(C) that is valid in all unrestricted Henkin models with respect to L+ (C) is also valid in all unrestricted Henkin models with respect to L(C) (because a L(C) countermodel can be converted into a L+ (C) countermodel). Proof The proof basically amounts to replacing the new variables of L+ (C) by some from L(C), to determine behavior of predicate abstracts. I only sketch the general outlines. Let M = hH, I, Ei be an unrestricted Henkin frame, and let hM, Ai be an unrestricted Henkin model with respect to L(C). Recall the following notational convention: {β1 /α1 , . . . , βn /αn } is the substitution that replaces each βi by the corresponding αi . Also, if v is a valuation with respect to L+ (C), by v 0 = v{β1 /α1 , . . . , βn /αn } I mean the valuation with respect to L(C) such that v 0 (αi ) = v(βi ), and on other free variables, v 0 and v agree. Now extend A to an abstraction designation function, A0 , suitable for L+ (C). For each predicate abstract hλγ1 , . . . , γk .Φi of L+ (C), and for each valuation v with respect to L+ (C), do the following. Let β1 , . . . , βn be all the free variables of Φ that are in the language L+ (C) but not in L(C), and let α1 , . . . , αn be a list of variables of the same corresponding types, that do not occur in Φ, free or bound. Now, set A0 (v, hλγ1 , . . . , γk .Φi) = A(v{β1 /α1 , . . . , βn /αn }, hλγ1 , . . . , γk .Φ{β1 /α1 , . . . , βn /αn }i) 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 βi . Now it is possible to show that hM, A0 i is an unrestricted Henkin model with respect to L+ (C), and truth values of sentences of L(C) evaluate the same with respect to A and A0 . One must show a more general result, involving formulas with free variables. The details are messy, and I omit them. Finally, Proposition 2.6.5 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 2.6.6 Every unrestricted Henkin model with respect to L+ (C) can be converted into an unrestricted Henkin model with respect to L(C) so that truth values for formulas of L(C) are preserved.

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Exercises Exercise 2.6.1 Give a proof of Proposition 2.6.1. Exercise 2.6.2 Give a proof of Proposition 2.6.2. Exercise 2.6.3 Give a proof of Proposition 2.6.4. Exercise 2.6.4 Give a proper proof that each unrestricted 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 first-order classical logic. Among them the semantic tableau procedure has a considerable attraction, (Smullyan 1968, Fitting 1996). It is intuitive, close to the intended semantics, and is automatable. For higher-order classical logic, semantic tableaus are rarely seen—most treatments in the literature are axiomatic. Among the notable exceptions are (Toledo 1975, Smith 1993, Kohlhase 1995, Gilmore 1998b). In fact, semantic tableaus retain much of their first-order ability to charm, and they are what I present here. Automatability becomes more problematic, however, for reasons that will become clear as we proceed. Consequently the presentation should be thought of as meant for human use, and intelligence in the construction of proofs is expected. This chapter examines what I call a basic tableau system; rules are lifted from those of first-order classical logic, and two straightforward rules for predicate abstracts are added. It is a higher-order version of the second-order system given in (Toledo 1975). I will show it corresponds to the unrestricted Henkin models from Section 2.5 of Chapter 2. In Chapters 5 and 6 I make additions to the system to narrow it to Henkin models.

3.1

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 which is called L+ (C). Before presenting the tableau rules, I describe the way in which the language is extended for proof purposes. 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 29

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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. Definition 3.1.1 [Parameters] In L(C), for each type t there is an infinite collection of free variables of that type. The language L+ (C) differs from L(C) in that, for each t there is also a second infinite list of free variables of type t, called parameters, a list disjoint from that of the free variables of L(C) itself. Parameters may appear in formulas in the same way as the original list of free variables but they are never quantified or λ bound. p, q, P , Q, . . . are used to represent 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 λ bound, we never need to worry about whether the introduction of a parameter will lead to its inadvertent capture by a quantifier or a λ—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 variable for the free variables of L(C) only, and when I want to include parameters in the discussion I will explicitly say so. The notion of truth in unrestricted 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 an unrestricted Henkin frame and an unrestricted Henkin model lies in the requirement that the extension of a formula appearing in a predicate abstract 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 an unrestricted Henkin model with respect to L+ (C) is apparently a stronger condition than requiring it be one with respect to L(C), though Section 2.6 establishes that this is not actually so. Definition 3.1.2 [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. 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.

3.2

Basic Tableaus

I now present the basic tableau system. It does not contain machinery for dealing with equality—that comes in the next chapter. The rules come from

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(Toledo 1975), where they were given for second-order logic. These rules, in turn, trace back to the sequent-style higher-order rules of (Prawitz 1968) and (Takahashi 1967). All tableau proofs are proofs of sentences—closed formulas—of L(C). A tableau proof of Φ is a tree that has ¬Φ 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 ¬Φ cannot happen, and so Φ 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 3.2.1 [Conjunctive Rules] X ∧Y X Y

¬(X ∨ Y ) ¬X ¬Y

¬(X ⊃ Y ) X ¬Y

X≡Y 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 3.2.2 [Double Negation Rule] ¬¬X X Next come the disjunctive rules. For these, if the formula above the line appears on a tableau branch, the end node can have two children added, labeled respectively with the two items shown below the line in the rule. In this case one says there is tableau branching. Definition 3.2.3 [Disjunctive Rules] X ∨Y X Y

¬(X ∧ Y ) ¬X ¬Y

X⊃Y ¬X Y

¬(X ≡ Y ) ¬(X ⊃ Y ) ¬(Y ⊃ X)

This completes the propositional connective rules. The motivation should be intuitively obvious. For instance, if X ∧ Y is true in a model, both X and Y are true there, and so a branch containing X ∧ Y can be extended with X and Y . If X ∨ Y is true in a model, one of them is true there. The corresponding

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tableau rule says if X ∨ 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 Φ(αt ) is a formula in which the variable αt , of type t, may have free occurrences. And suppose τ t is a term of type t. Then Φ(τ t ) is the result of carrying out the substitution {αt /τ t } in Φ(αt ), replacing all free occurrences of αt with occurrences of τ t . Now, here are the existential quantifier rules. Definition 3.2.4 [Existential Rules] In the following, pt is a parameter of type t that is new to the tableau branch. (∃αt )Φ(αt ) Φ(pt )

¬(∀αt )Φ(αt ) ¬Φ(pt )

The rules above embody the familiar notion of existential instantiation. Since the convention is that parameters are never quantified, we don’t have to worry about accidental variable capture. More precisely, in the rules above, the substitution {αt /pt } is free for the formula Φ(αt ). The universal rules are somewhat more straightforward. Once again, note that in them the substitution {αt /τ t } is free for the formula Φ(αt ). Definition 3.2.5 [Universal Rules] In the following, τ t is any grounded term of type t of L+ (C). (∀αt )Φ(αt ) Φ(τ t )

¬(∃αt )Φ(αt ) ¬Φ(τ t )

Finally we have the rules for predicate abstracts. Earlier notation is extended a bit so that, if Φ(α1 , . . . , αn ) is a formula, α1 , . . . , αn are distinct free variables, and τ1 , . . . , τn are grounded terms of the same respective types as α1 , . . . , αn , then Φ(τ1 , . . . , τn ) is the result of simultaneously substituting each τi for all free occurrences of αi in Φ. Definition 3.2.6 [Abstract Rules] hλα1 , . . . , αn .Φ(α1 , . . . , αn )i(τ1 , . . . , τn ) Φ(τ1 , . . . , τn ) ¬hλα1 , . . . , αn .Φ(α1 , . . . , αn )i(τ1 , . . . , τn ) ¬Φ(τ1 , . . . , τn ) Now what, exactly, constitutes a proof. Definition 3.2.7 [Closure] A tableau branch is closed if it contains Φ and ¬Φ, where Φ is a grounded formula. A tableau is closed if each branch is closed.

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Definition 3.2.8 [Tableau Proof] For a sentence Φ of L(C), a closed tableau beginning with ¬Φ is a proof of Φ. Definition 3.2.9 [Tableau Derivation] A tableau derivation of a sentence Φ from a set of sentences S, all of L(C), is a closed tableau beginning with ¬Φ, allowing the additional rule: at any point any member of S can be added to the end of any open branch. This concludes the presentation of the tableau rules. In the next section I give several examples of tableaus. Classical first-order tableau rules, as in (Smullyan 1968, Fitting 1996) are analytic—they only involve subformulas of the formula being proved. (It is not the case with the cut rule, but this is an eliminable rule.) Higher-order rules, for the most part, have an analytic nature as well. The important exception is the rule for the universal quantifier. It allows us to pass from (∀αt )Φ(αt ) to Φ(τ t ) where τ t is an arbitrary grounded term. Since terms can involve predicate abstracts, applications of this rule can introduce formulas that are not subformulas of the one being proved—indeed, they may be much more complicated. There is no way around this. In a sense, the introduction of predicate abstracts embodies the “creative element” of mathematics.

3.3

Tableau Examples

Tableaus for first-order classical logic are well-known, but the abstraction rules of the previous section are not widely familiar. I give a number of examples illustrating their uses. The first embodies the principle behind many diagonal arguments in mathematics. Example 3.3.1 Suppose there is a way of matching subsets of some set D with members of D. Let us call a member of D associated with a particular subset a code for that subset: every member of D must be a code, and nothing can be a code for more than one subset, though it is allowed that some subsets can have more than one code. Then, some subset of D must lack a code. (One consequence of this is Cantor’s Theorem: a set and its power set cannot be in a 1-1 correspondence.) To formulate this, let R(x, y) represent the relation: y is in the subset that has x as its code; so hλy.R(x, y)i represents the set coded by x. Then the following second-order sentence does the job. (∀R)(∃X)(∀x)¬[hλy.R(x, y)i = 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. (∀R)(∃X)(∀x)(∃y){[R(x, y) ∧ ¬X(y)] ∨ [¬R(x, y) ∧ X(y)]}

(3.2)

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I give a proof of (3.2). It is contained in Figure 3.3.1. In it, 2 is from 1 by an existential rule (P is a new parameter); 3 is from 2 by a universal rule (hλx.¬P (x, x)i 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 hλx.¬P (x, x)i 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. See (Hughes & Cresswell 1996a, pp 188-190) for a fuller treatment. Example 3.3.2 It is a well-known result of modal model theory that a relational frame is reflexive if and only if every instance of ¤P ⊃ 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, (∀y)[R(x, y) ⊃ P (y)] corresponds to P being true at every world accessible from x, and hence to ¤P being true at world x, where R plays the role of the accessibility relation. Then further, saying ¤P ⊃ P is true at x corresponds to (∀y)[R(x, y) ⊃ P (y)] ⊃ P (x). We want to say that if this happens at every world, and for all P , the relation R must be reflexive. Specifically, I give a tableau proof of the following. In it, take R to be a constant symbol. (∀x)R(x, x) ≡ (∀P )(∀x){(∀y)[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.

35 CHAPTER 3. CLASSICAL LOGIC—BASIC TABLEAUS

¬¬P (p, p)

¬(∀R)(∃X)(∀x)(∃y){[R(x, y) ∧ ¬X(y)] ∨ [¬R(x, y) ∧ X(y)]} 1. ¬(∃X)(∀x)(∃y){[P (x, y) ∧ ¬X(y)] ∨ [¬P (x, y) ∧ X(y)]} 2. ¬(∀x)(∃y){[P (x, y) ∧ ¬hλx.¬P (x, x)i(y)] ∨ [¬P (x, y) ∧ hλx.¬P (x, x)i(y)]} ¬(∃y){[P (p, y) ∧ ¬hλx.¬P (x, x)i(y)] ∨ [¬P (p, y) ∧ hλx.¬P (x, x)i(y)]} 4. ¬{[P (p, p) ∧ ¬hλx.¬P (x, x)i(p)] ∨ [¬P (p, p) ∧ hλx.¬P (x, x)i(p)]} 5. ¬[P (p, p) ∧ ¬hλx.¬P (x, x)i(p)] 6. ¬[¬P (p, p) ∧ hλx.¬P (x, x)i(p)] 7. @ @ @ @ ¬P (p, p) 8. ¬¬hλx.¬P (x, x)i(p) 9. hλx.¬P (x, x)i(p) 10. @ @ 12. ¬¬P (p, p) 13. ¬hλx.¬P (x, x)i(p) 14. ¬P (p, p) 16. @ @ 11. ¬hλx.¬P (x, x)i(p) ¬¬P (p, p) 15.

Figure 3.1: Tableau Proof of (∀R)(∃X)(∀x)(∃y){[R(x, y) ∧ ¬X(y)] ∨ [¬R(x, y) ∧ X(y)]}

3.

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¬{(∀P )(∀x){(∀y)[R(x, y) ⊃ P (y)] ⊃ P (x)} ⊃ (∀x)R(x, x)} 1. (∀P )(∀x){(∀y)[R(x, y) ⊃ P (y)] ⊃ P (x)} 2. ¬(∀x)R(x, x) 3. ¬R(p, p) 4. (∀x){(∀y)[R(x, y) ⊃ hλz.R(p, z)i(y)] ⊃ hλz.R(p, z)i(x)} 5. (∀y)[R(p, y) ⊃ hλz.R(p, z)i(y)] ⊃ hλz.R(p, z)i(p) 6. @ @ @ @ ¬(∀y)[R(p, y) ⊃ hλz.R(p, z)i(y)] 7. hλz.R(p, z)i(p) 8. ¬[R(p, q) ⊃ hλz.R(p, z)i(q) 9. R(p, p) 13. R(p, q) 10. ¬hλz.R(p, z)i(q) 11. ¬R(p, q) 12. In this, 2 and 3 are from 1 by a conjunctive rule; 4 is from 3 by an existential rule (p is a new parameter); 5 is from 2 by a universal rule (hλz.R(p, z)i 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. The last example is a version of the famous Knaster-Tarski theorem. Example 3.3.3 Let D be a set and let F be a function from its powerset to itself. F is called monotone provided, for each P, Q ⊆ D, if P ⊆ Q then F(P ) ⊆ F(Q). Theorem: any monotone function F on the powerset of D 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. Now, (∀x)(P (x) ⊃ 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 (∀x)(P (x) ⊃ Q(x)) ⊃ (∀x)(F(P, x) ⊃ F(Q, x)) says we have monotonicity. Then, the following embodies a version of the Knaster-Tarski theorem (F is a constant symbol). (∀P )(∀Q){[(∀x)(P (x) ⊃ Q(x)) ⊃ (∀x)(F(P, x) ⊃ F(Q, x))] ⊃

(3.4)

(∃S)(∀x)(F(S, x) ≡ S(x)) I leave the construction of a tableau proof of this to you as an exercise, but I give the following hint. Let Φ(P, x) abbreviate the formula (∀y)(F(P, y) ⊃ P (y)) ⊃ P (x). An appropriate term to consider during a universal rule application is: hλx.(∀P )Φ(P, x)i.

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A comment on the hint above. Rewriting (∀y)(F(P, y) ⊃ P (y)) using conventional function notation: it says F(P ) ⊆ P . Then Φ(P, x) says that x belongs to a set P if P meets the condition F(P ) ⊆ P . Then further, (∀P )Φ(P, x) says that x is in ∩{P | F(P ) ⊆ P }. So finally, hλx.(∀P )Φ(P, x)i represents the set ∩{P | 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 3.3.3 once again illustrates a fundamental point about higher-order 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 (Fitting 1996). 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 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 third-order terms in (Huet 1973), and improved to show unification for second-order terms is already undecidable, in (Goldfarb 1981). 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, (Huet 1975). Two freevariable tableau systems for higher-order classical logic, using unification, are presented in (Kohlhase 1995). The use of higher-order unification in this way traces back to resolution work of (Andrews 1971) and (Huet 1972). 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 3.3.1 Continuing the ideas of Example 3.3.2, give tableau proofs of the following. 1. (symmetry) (∀x)(∀y)[R(x, y) ⊃ R(y, x)] ≡ (∀P )(∀x){(∃y)[R(x, y) ∧ (∀z)(R(y, z) ⊃ P (z))] ⊃ P (x)}

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2. (transitivity) (∀x)(∀y)(∀z)[(R(x, y) ∧ R(y, z)) ⊃ R(x, z)] ≡ (∀P )(∀x){(∀y)[R(x, y) ⊃ P (y)] ⊃ (∀y)(∀z)[(R(x, y) ∧ R(y, z)) ⊃ P (z)]} Exercise 3.3.2 Give the tableau proof to complete Example 3.3.3. Exercise 3.3.3 Continuing with Example 3.3.3, the set ∩{P | F(P ) ⊆ P } is not only a fixed point of monotonic F, it is the smallest one. Dually, ∪{P | P ⊆ 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 complete with respect to unrestricted 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 (Takahashi 1967, Prawitz 1968), where they were applied to give a non-constructive proof of a cut elimination theorem.

4.1

Soundness

Soundness means that any sentence having a tableau proof must be valid. Tableau soundness arguments follow the same pattern for all logics: some notion 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 4.1.1 [Tableau Satisfiability] A tableau branch is satisfiable if the set of formulas on it is satisfiable in an unrestricted Henkin model for L+ (C) (see Definition 2.5.9). 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 4.1.2 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. 39

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Lemma 4.1.3 If a branch extension rule is applied to a satisfiable tableau, the result is another satisfiable tableau. 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 0 . It must be shown that T 0 is satisfiable. The rule that was applied to turn T into T 0 may have been applied on a branch other than B. In this case B is still a branch of T 0 , and of course is still satisfiable, so T 0 is satisfiable. Now, for the rest of the proof assume a branch extension rule has been applied to satisfiable branch B itself. And to be specific, say all the grounded formulas on B are true in the unrestricted Henkin model hM, Ai with respect to the valuation v, where M = hH, I, Ei. 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 ∨ Y occurred on B and a rule was applied to it. Then in T 0 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 hM, Ai with respect to valuation v, hence M °v,A X ∨Y . Then either M °v,A X or M °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 hM, Ai with respect to v. Either way, some branch of T 0 is satisfiable. Existential Quantifier Suppose the grounded formula (∃α)Φ(α) occurred on B and a rule was applied to it, so that in T 0 branch B has been lengthened with Φ(p) where p is a parameter new to B, of the same type as α. Since all formulas on B are true in hM, Ai with respect to v, M °v,A (∃α)Φ(α). Then, by definition of truth in a model, there must be some αvariant w of v such that M °w,A Φ(α). Let σ = {p/α}—the substitution that replaces p by α—and consider the valuation wσ (Definition 2.5.6). I claim all formulas on B extended with Φ(p) are true in hM, Ai with respect to wσ , so the extended branch is satisfiable. First of all, v and w agree on all variables except α. It is easy to see that w and wσ agree on all variables except p, so the only variables on which v and wσ can differ are α and p. But α 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. Consequently all formulas on B are true in hM, Ai with respect to wσ , by Proposition 2.6.1. Finally, note that since p did not occur in (∃α)Φ(α), then Φ(α) = Φ(p)σ. We have M °w,A Φ(α), and by Proposition 2.6.4 M °w,A Φ(α) ⇔ M °w,A Φ(p)σ ⇔ M °wσ ,A Φ(p). This completes the argument for the existential case.

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Abstraction Suppose the grounded formula hλα1 , . . . , αn .Φ(α1 , . . . , αn )i(τ1 , . . . , τn ) occurred on B, and a rule was applied to it, so that in T 0 branch B has been lengthened with Φ(τ1 , . . . , τn ). We are assuming that the formulas on B are all true in hM, Ai with respect to valuation v. I will show that this extends to include Φ(τ1 , . . . , τn ) as well. Let σ = {α1 /τ1 , . . . , αn /τn }. This substitution is free for Φ(α1 , . . . , αn ) because τ1 , . . . , τn must be grounded, and parameters are never quantified or lambda-bound. Now consider the valuation v σ . Note the following useful items. 1. v σ (αi ) = (v ∗ I ∗ A)(αi σ) = (v ∗ I ∗ A)(τi ) 2. If β is different from α1 , . . . , αn , v σ (β) = (v ∗ I ∗ A)(βσ) = (v ∗ I ∗ A)(β) = v(β). Since hλα1 , . . . , αn .Φ(α1 , . . . , αn )i(τ1 , . . . , τn ) is on B, we have M °v,A hλα1 , . . . , αn .Φ(α1 , . . . , αn )i(τ1 , . . . , τn ). For this to be the case h(v ∗ I ∗ A)(τ1 ), . . . , (v ∗ I ∗ A)(τn )i ∈ E((v ∗ I ∗ A)(hλα1 , . . . , αn .Φ(α1 , . . . , αn )i)). Since we have an unrestricted Henkin model, A is proper, so E((v ∗ I ∗ A)(hλα1 , . . . , αn .Φ(α1 , . . . , αn )i)) = {hw(α1 ), . . . , w(αn )i | w is an α1 , . . . , αn -variant of v and M °w,A Φ(α1 , . . . , αn )} and consequently M °w,A Φ(α1 , . . . , αn ) where w is the α1 , . . . , αn variant of v such that w(α1 ) = (v ∗I ∗A)(τ1 ), . . . , w(αn ) = (v ∗I ∗A)(τn ). But, by items 1 and 2 above, v σ itself is this α1 , . . . , αn -variant of v. We thus have M °vσ ,A Φ(α1 , . . . , αn ). Now, by Proposition 2.6.4, M °v,A Φ(α1 , . . . , αn )σ, that is, M °v,A Φ(τ1 , . . . , τn ). There are other cases—I leave them to you.

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Theorem 4.1.4 (Soundness) If a sentence Φ of L(C) has a tableau proof, Φ must be true in all unrestricted Henkin models with respect to L(C). Proof Suppose Φ has a tableau proof, but is not true in all unrestricted Henkin models with respect to L(C)—I derive a contradiction. Since Φ is not true in all unrestricted Henkin models with respect to L(C), {¬Φ} is satisfiable, and by Proposition 2.6.5, is so in an unrestricted Henkin model with respect to L+ (C). A tableau proof of Φ begins with a tableau consisting of a single branch, containing the single formula ¬Φ, so this must be a satisfiable tableau. As we apply branch extension rules, we continue to get satisfiable tableaus, by Lemma 4.1.3. Since Φ is provable, we must get a closed tableau. Hence there must be a closed, satisfiable tableau, which is impossible according to Lemma 4.1.2. Essentially the same argument also establishes the following. Theorem 4.1.5 Let S be a set of sentences and Φ be a single sentence of L(C). If Φ has a tableau derivation from S, then Φ is an unrestricted Henkin consequence of S.

4.2

Completeness

The proof of completeness, for basic tableaus, with respect to unrestricted Henkin models, is of considerable intricacy. It is spread over several subsections, each devoted to a single aspect of it. All the basic ideas go back to (Takahashi 1967, Prawitz 1968), where they were used to non-constructively establish a cut-elimination theorem for higher-order Gentzen systems. I also use aspects of the (second-order) presentation of (Toledo 1975), in particular the central goal, for us, is to prove that something called a Hintikka set is satisfiable. This contains the essence of the proofs of (Takahashi 1967, Prawitz 1968). (Andrews 1971) abstracted the Takahashi, Prawitz ideas to prove a higher-order Model Existence Theorem which could have been appealed to, but the ideas of the completeness proof are pretty and deserve to be better known, hence the full presentation. In outline, the completeness proof is as follows. In Section 4.2.1 the notion of a Hintikka set is defined: a set of grounded formulas of L+ (C) meeting certain closure conditions. These closure conditions bear an obvious relationship to the tableau rules. In Section 4.2.2 pseudo-models are introduced. These are the closest we come, in higher-order logic, to the Herbrand models familiar in the first-order setting. Unfortunately, they are not true models in the higher-order sense, because objects assigned as meanings for predicate abstracts might lie outside the range allowed for quantifiers. In Section 4.2.3 some rather technical (but important) results about the behavior of substitution in pseudo-models are shown. In Section 4.2.4 it is established that each Hintikka set is satisfiable in some pseudo-model. Section 4.2.5 shows that pseudo-models can be converted into true unrestricted Henkin models, and so each Hintikka set is satisfiable in

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such a model. Finally in Section 4.2.6 it is shown how to extract a Hintikka set from a failed tableau proof attempt, and this puts the last step in place for the completeness proof.

4.2.1

Hintikka Sets

Hintikka sets are fairly familiar from propositional and first-order logics—see (Fitting 1996) and (Smullyan 1968) for instance. They play a similar role in the higher-order case, though arguments about them are much more complex. You should note that the basic tableau rules all correspond directly to Hintikka set conditions (I omit the connective ≡ as a small convenience). Definition 4.2.1 [Hintikka Set] A non-empty set H of grounded formulas of L+ (C) is a Hintikka set if it meets the following conditions. 1. Atomic Case. If Φ is atomic, not both Φ ∈ H and ¬Φ ∈ H. 2. Conjunctive Cases. (a) If (Φ ∧ Ψ) ∈ H then Φ ∈ H and Ψ ∈ H. (b) If ¬(Φ ∨ Ψ) ∈ H then ¬Φ ∈ H and ¬Ψ ∈ H. (c) If ¬(Φ ⊃ Ψ) ∈ H then Φ ∈ H and ¬Ψ ∈ H. 3. Disjunctive Cases. (a) If (Φ ∨ Ψ) ∈ H then either Φ ∈ H or Ψ ∈ H. (b) If ¬(Φ ∧ Ψ) ∈ H then either ¬Φ ∈ H or ¬Ψ ∈ H. (c) If (Φ ⊃ Ψ) ∈ H then either ¬Φ ∈ H or Ψ ∈ H. 4. Double Negation Case. If ¬¬Φ ∈ H then Φ ∈ H. 5. Universal Cases. (a) If (∀αt )Φ(αt ) ∈ H then Φ(τ t ) ∈ H for every grounded term τ t . (b) If ¬(∃αt )Φ(αt ) ∈ H then ¬Φ(τ t ) ∈ H for every grounded term τ t . 6. Existential Cases. (a) If (∃αt )Φ(αt ) ∈ H then Φ(pt ) ∈ H for at least one parameter pt . (b) If ¬(∀αt )Φ(αt ) ∈ H then ¬Φ(pt ) ∈ H for at least one parameter pt . 7. Abstraction Cases. (a) If hλα1 , . . . , αn .Φ(α1 , . . . , αn )i(τ1 , . . . , τn ) ∈ H, then Φ(τ1 , . . . , τn ) ∈ H. (b) If ¬hλα1 , . . . , αn .Φ(α1 , . . . , αn )i(τ1 , . . . , τn ) ∈ H, then ¬Φ(τ1 , . . . , τn ) ∈ H. This completes the definition of Hintikka sets. The task of relating them to models begins in the next subsection.

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4.2.2

44

Pseudo-Models

The eventual goal is to construct an unrestricted Henkin model, starting with a Hintikka set. To do this a pseudo-model is first created, something that is much like an unrestricted Henkin model but with some significant differences. One clear difference is in the treatment of predicate abstracts; they will be 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, τ1 (τ2 ) can still be taken to be true if the value assigned to τ2 is in the extension of the value assigned to τ1 , whether or not the value of τ1 is in quantifier range. Eventually, of course, it will be shown that we can dispose of the “pseudo” part of a pseudo-model. We 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. Definition 4.2.2 [Entities of type t, Extension] The notion of entity of type t is defined inductively, on the complexity of t. 1. Suppose t = 0. If τ is a grounded term of type t (thus a constant or parameter of type 0), τ is an entity of type t. 2. Suppose t = ht1 , . . . , tn i and the collection of entities of type ti has been defined for each i = 1, . . . , n. Then hτ, Si is an entity of type t, provided τ is a grounded term of type t, and S is a set whose members are of the form hE1 , . . . , En i, where each Ei is an entity of type ti . Define an extension mapping on entities of types other than 0 by setting E(hτ, Si) = S, and refer to S as the extension of entity hτ, Si. The idea is, if hτ, Si is an entity of type t, it is something that could serve as a semantic value for the term τ , with the extension explicitly coded in. One problem with entities as just defined 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. Eventually it is these entities that will make up the Henkin domains of a model. Definition 4.2.3 [Possible Value] Let H be a Hintikka set. For each grounded term τ , define a collection of possible values relative to H. This is done inductively, on type complexity. 1. If τ is a grounded term of type 0, the only possible value of τ relative to H is τ itself. 2. Suppose τ is a grounded term of type ht1 , . . . , tn i, and possible values relative to H have been specified for all grounded terms of types t1 , . . . ,

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tn . Then, an entity hτ, Si is a possible value of τ relative to H provided, for all grounded terms τ1 , . . . , τn of types t1 , . . . , tn respectively, and for all possible values E1 , . . . , En of τ1 , . . . , τn : (a) If τ (τ1 , . . . , τn ) ∈ H then hE1 , . . . , En i ∈ S. (b) If ¬τ (τ1 , . . . , τn ) ∈ H then hE1 , . . . , En i 6∈ S. E is a possible value if it is a possible value for some grounded term. Roughly the idea is, any possible value for τ 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 is needed for part 2 above to be meaningful. Valuations were defined earlier, in unrestricted Henkin models. That terminology is extended to pseudo-models as well. Note that arbitrary entities are allowed and not just possible values. Definition 4.2.4 [Valuation] A mapping v from variables and parameters to entities is a valuation provided it assigns to each variable or parameter of type t some entity of type t. The languages L(C) and L+ (C) are allowed to contain constant symbols. How to interpret these is essentially arbitrary, within broad limits. Definition 4.2.5 [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. Grounded terms have entities associated with them. We also need some machinery for going the other way, from entities back to grounded terms. Definition 4.2.6 [T and ← v−] A mapping T from entities E to grounded terms is defined as follows. If E is of type 0 it is, itself, a grounded term of L+ (C); in this case T (E) = E. If E is of type ht1 , . . . , tn i it is of the form hτ, Si; in this case T (E) = τ . Next, let v be a valuation. Define a substitution ← v− as follows: α← v− = T (v(α)). Note that ← v− substitutes grounded terms of L+ (C) for variables, and so if τ is an arbitrary term, τ ← v− must be a grounded term, and similarly for formulas. Then, for any formula Φ and any valuation v, Φ← v− is something that could be a member of a Hintikka set. The next bit of business is to assign truth values to formulas, and entities to predicate abstracts—this is the fundamental construction. It is done via an inductive definition on formulas and terms.

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Definition 4.2.7 [Pseudo-Model] Let H be a Hintikka set and let I be an allowed interpretation relative to H. For each valuation v, define another mapping denoted by vH,I —and call vH,I a pseudo-model. The mapping vH,I maps formulas of L+ (C) to truth values true and false, and maps terms to entities. Atomic vH,I (τ (τ1 , . . . , τn )) = true if hvH,I (τ1 ), . . . , vH,I (τn )i ∈ E(vH,I (τ )), and otherwise vH,I (τ (τ1 , . . . , τn )) = false. Negation vH,I (¬X) = false if vH,I (X) = true and vH,I (¬X) = true if vH,I (X) = false. Conjunction vH,I (X ∧ Y ) = true if vH,I (X) = true and vH,I (Y ) = true; otherwise vH,I (X ∧ Y ) = false. (The other propositional cases are similar, and are omitted.) Universal Quantification vH,I ((∀α)Φ) = true if wH,I (Φ) = true for every valuation w that is an α-variant of v, and that assigns to α some possible value relative to H. Otherwise vH,I ((∀α)Φ) = false. (The existential case is similar, and is omitted.) Constant Symbols, Variables, Parameters vH,I (τ ) = I(τ ) if τ is a constant symbol. If τ is a a variable or a parameter, vH,I (τ ) = v(τ ). Predicate Abstract If τ is the term hλα1 , . . . , αn .Φi, set vH,I (τ ) = hτ ← v−, Si where S

= {hw(α1 ), . . . , w(αn )i | wH,I (Φ) = true where w is an α1 , . . . , αn variant of v}.

In the definition above, the quantification clause, in effect, says quantifiers are thought of as ranging over possible values, and not over arbitrary entities. On the other hand, values assigned to predicate abstracts can be entities that might not be possible values. In the next subsection some significant technical facts about pseudo-models are shown. Once these are out of the way, it will be possible to extract a proper unrestricted Henkin model from a pseudo-model.

Exercises Exercise 4.2.1 Show that if entity E is a possible value, then E must be a possible value of T (E).

4.2.3

Substitution and Pseudo-Models

Pseudo-models are strange, hybrid, things, because of the mix of arbitrary entities and allowed values in interpreting quantifiers and predicate abstracts. In this subsection valuations and substitutions are shown to be well-behaved with

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respect to pseudo-models. The proofs are rather technical, so I begin with the statements of the two Propositions to be established, after which their proofs are given, broken into a number of Lemmas. On a first reading you might want to just read the Propositions and skip over the proofs. The first item should be compared with Proposition 2.6.1. Proposition 4.2.8 Let H be a Hintikka set and let I be an allowed interpretation relative to H. Also, let v and v 0 be two valuations, and let X be either a term or a formula of L+ (C). If v and v 0 agree on the free variables and 0 (X). parameters of X then vH,I (X) = vH,I The second item is an analog to Propositions 2.6.2 and 2.6.4. Of course Definition 2.5.6 must be modified to adapt it to present circumstances, since there is no explicit abstraction designation function in a pseudo-model. Definition 4.2.9 Let v be a valuation, and vH,I be a pseudo-model. For a substitution σ, by v σ now is meant the valuation given by v σ (α) = vH,I (ασ), where α is a variable or parameter. Thus v σ (α) gives us the denotation of the term ασ in the pseudo-model vH,I . Proposition 4.2.10 Let v be a valuation, and vH,I be a pseudo-model. For a σ (X) for every formula substitution σ, if σ is free for X, then vH,I (Xσ) = vH,I and term X. Now I turn to the proofs, which are given in considerable detail since these results are critical to the completeness argument, and I want the reasoning on record. On a first reading, skip the proofs and move on to the next section. Proof of 4.2.8 Suppose the result is known for terms and formulas whose degree is < k. I show the result is also true for those of degree k itself, beginning with terms. Assume τ is a term of degree k, and v and v 0 agree on the free variables and parameters of τ . If k happens to be 0, τ is a constant symbol, variable, or parameter. In these cases the result is immediate. Now suppose k 6= 0, and so τ = hλα1 , . . . , αn .Φi, where Φ is of degree < k. 0 (hλα1 , . . . , αn .Φi) = ha0 , S 0 i. Let us say vH,I (hλα1 , . . . , αn .Φi) = ha, Si and vH,I 0 0 We must show a = a and S = S . Suppose α is a variable or parameter that occurs free in τ . Then α← v− = ← −0 0 0 T (v(α)) = T (v (α)) = α v , using the assumption that v and v must agree on ← − ← − v− and v 0 agree on the free α. By definition, a = τ ← v− and a0 = τ v 0 , and ← variables of τ , so a = a0 by Proposition 1.2.5. Next, suppose hE1 , . . . , En i ∈ S. We must have wH,I (Φ) = true where w is the α1 , . . . , αn variant of v such that w(α1 ) = E1 , . . . , w(αn ) = En . Let w0 be the same as v 0 except that w0 (α1 ) = E1 , . . . , w0 (αn ) = En . Since v and w agree on the free variables and parameters of hλα1 , . . . , αn .Φi, then v 0 and w0 must agree on the free variables and parameters of Φ. Then by the

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0 (Φ) = true, and it follows that hE1 , . . . , En i ∈ S 0 . induction hypothesis, wH,I 0 Thus S ⊆ S . A similar argument shows S 0 ⊆ S. This completes the induction step for terms, and I turn next to formulas. Suppose Φ is of degree k and v and v 0 agree on the free variables and parameters of Φ. By the induction hypothesis, we have the Proposition for terms and formulas of degree < k, and by what was just shown, we also have it for terms of degree k itself. Now we have several cases. Suppose Φ is atomic, τ0 (τ1 , . . . , τn ), where each τi must be of degree ≤ k. Then, using the induction hypothesis,

vH,I (τ0 (τ1 , . . . , τn )) = true ⇔ hvH,I (τ1 ), . . . , vH,I (τn )i ∈ E(vH,I (τ0 )) 0 0 0 (τ1 ), . . . , vH,I (τn )i ∈ E(vH,I (τ0 )) ⇔ hvH,I 0 ⇔ vH,I (τ0 (τ1 , . . . , τn )) = true. The various non-atomic cases are left to you. Next we have several preliminary results, leading up to the proof of Proposition 4.2.10. Lemma 4.2.11 Let H be a Hintikka set, let I be an allowed interpretation relative to H, and let v be a valuation. Also, suppose the substitution σ is free for hλα1 , . . . , αn .Φi, and further that σ is the identity map on parameters and variables that do not occur free in hλα1 , . . . , αn .Φi. 1. If w is an α1 , . . . , αn variant of v then wσα1 ,... ,αn is an α1 , . . . , αn variant of v σ . 2. Conversely, if u is an α1 , . . . , αn variant of v σ then u = wσα1 ,... ,αn for some α1 , . . . , αn variant w of v. 3. v σα1 ,... ,αn (αi ) = v(αi ), for i = 1, . . . , n. Proof Part 1. Suppose w is an α1 , . . . , αn variant of v. Let β be a variable or parameter other than α1 , . . . , αn . It must be shown that wσα1 ,... ,αn (β) = v σ (β). Here are the steps; the reasons follow. wσα1 ,... ,αn (β)

= = = =

wH,I (βσα1 ,... ,αn ) wH,I (βσ) vH,I (βσ) v σ (β)

(4.1) (4.2) (4.3) (4.4)

Above, (4.1) is by Definition 4.2.9 for wσα1 ,... ,αn , and (4.2) is because β is different from α1 , . . . , αn . Also (4.4) follows from (4.3) by Definition 4.2.9 again, for v σ . The key item is getting (4.3) from (4.2), and for this it is enough to show v and w agree on the free variables and parameters of βσ, and then appeal to Proposition 4.2.8. The argument for this follows.

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If β does not occur free in hλα1 , . . . , αn .Φi, βσ = β by hypothesis, we are assuming β is different from α1 , . . . , αn , and v and w agree on all variables except α1 , . . . , αn , so v and w trivially agree on the free variables and parameters of βσ in this case. Now suppose β does occur free in hλα1 , . . . , αn .Φi. Since σ is free for hλα1 , . . . , αn .Φi, βσ cannot contain any of α1 , . . . , αn free. Once again v and w must agree on the free variables and parameters of βσ, since v and w can only differ on α1 , . . . , αn . Part 2. Suppose u is an α1 , . . . , αn variant of v σ . Define a valuation w as follows. w(αi ) = u(αi ) i = 1, . . . , n w(β) = v(β) β 6= α1 , . . . , αn By definition, w is an α1 , . . . , αn variant of v. I will show wσα1 ,... ,αn = u. The argument is in two parts. wσα1 ,... ,αn (αi )

= = = =

wH,I (αi σα1 ,... ,αn ) wH,I (αi ) w(αi ) u(αi )

(4.5) (4.6) (4.7) (4.8)

In this, (4.5) is by definition of wσα1 ,... ,αn . Then (4.6) is because σα1 ,... ,αn is the identity on αi . Next, (4.7) is because αi is a variable, and finally (4.8) is by definition of w. Now suppose β 6= α1 , . . . , αn . wσα1 ,... ,αn (β)

= = = = =

wH,I (βσα1 ,... ,αn ) wH,I (βσ) vH,I (βσ) v σ (β) u(β)

(4.9) (4.10) (4.11) (4.12) (4.13)

Here (4.9) is by definition of wσα1 ,... ,αn . Then (4.10) is by definition of σα1 ,... ,αn . Next, (4.11) follows exactly as (4.3) did above. Finally (4.12) is by definition of v σ , and (4.13) is because u and v σ are α1 , . . . , αn variants. Part 3. v σα1 ,... ,αn (αi ) = vH,I (αi σα1 ,... ,αn ) = vH,I (αi ) = v(αi ). Lemma 4.2.12 Let vH,I be a pseudo-model. For any term τ of L+ (C), τ ← v− = T (vH,I (τ )). Proof Suppose first that τ is a predicate abstract. Then by Definition 4.2.7, v−, Si for a particular set S, and so T (vH,I (τ )) = τ ← v−. If τ is a vH,I (τ ) = hτ ← ← − ← − variable or parameter, τ v = T (v(τ )) by definition of v , and v(τ ) = vH,I (τ ) by definition of vH,I again, for variables. Finally, the case of τ being a constant symbol is straightforward.

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The next two Lemmas will give us the induction steps of the main item we σ in place of are after. In the interests of simple notation I have written vH,I σ (v )H,I . Lemma 4.2.13 Let H be a Hintikka set and let I be an allowed interpretation relative to H. Assume that if σ is free for X, then σ (X) vH,I (Xσ) = vH,I

(4.14)

for each formula X of degree < k. Then (4.14) also holds for each term X of degree k itself. Proof Assume the hypothesis, and suppose τ is a term of degree k. If k is 0, τ must be a constant symbol, a variable, or a parameter. If it is a variable or σ (α) = v σ (α); and this, in turn, is vH,I (ασ), using parameter, say α, then vH,I σ the definition of v . The case of a constant symbol is trivial. Now suppose k > 0, and so τ must be of the form hλα1 , . . . , αn .Φi, where Φ is a formula whose degree is < k. And suppose σ is free for hλα1 , . . . , αn .Φi. Using the definition of substitution and Definition 4.2.7: vH,I (hλα1 , . . . , αn .Φiσ)

= vH,I (hλα1 , . . . , αn .Φσα1 ,... ,αn i) = ha, Si

where v− a = hλα1 , . . . , αn .Φσα1 ,... ,αn i← S = {hw(α1 ), . . . , w(αn )i | wH,I (Φσα1 ,... ,αn ) = true where w is an α1 , . . . , αn variant of v}. Similarly: σ (hλα1 , . . . , αn .Φi) = ha0 , S 0 i vH,I

where a0 S0

← − = hλα1 , . . . , αn .Φiv σ = {hu(α1 ), . . . , u(αn )i | uH,I (Φ) = true where u is an α1 , . . . , αn variant of v σ }.

So, we must show a = a0 and S = S 0 . Part 1, a = a0 .

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First of all, v− a = hλα1 , . . . , αn .Φσα1 ,... ,αn i← v− = (hλα1 , . . . , αn .Φi)σ)← ← − = hλα1 , . . . , αn .Φi(σ v )

(4.15) (4.16)

In this, (4.15) is by definition of substitution. (Recall we are assuming that σ is v− replaces variables by grounded terms, free for hλα1 , . . . , αn .Φi.) Also, since ← and parameters are never bound, substitution ← v− is free for hλα1 , . . . , αn .Φiσ. Then (4.16) follows by Theorem 1.2.7. ← − v− and v σ are Then to show a = a0 it is enough to show the substitutions σ ← the same. Let β be a variable or parameter. β(σ ← v−) = (βσ)← v− by definition ← − of composition for substitutions. And, using Definitions 4.2.6 and 4.2.9, β v σ = v− and T (vH,I (βσ)) are the same, by T (v σ (β)) = T (vH,I (βσ)). Finally, (βσ)← Lemma 4.2.12. We thus have shown that a = a0 . Part 2, S = S 0 . Using Proposition 1.2.5 it can be assumed that σ is the identity on variables and parameters that do not occur free in hλα1 , . . . , αn .Φi. S

= {hw(α1 ), . . . , w(αn )i | wH,I (Φσα1 ,... ,αn ) = true where w is an α1 , . . . , αn variant of v} σα1 ,... ,αn (Φ) = true = {hw(α1 ), . . . , w(αn )i | wH,I where w is an α1 , . . . , αn variant of v} = {hwσα1 ,... ,αn (α1 ), . . . , wσα1 ,... ,αn (αn )i | σα1 ,... ,αn (Φ) = true wH,I where w is an α1 , . . . , αn variant of v} = {hu(α1 ), . . . , u(αn )i | uH,I (Φ) = true where u is an α1 , . . . , αn variant of v σ }.

(4.17) (4.18)

(4.19)

Since σ is free for hλα1 , . . . , αn .Φi, by Definition 1.2.6, σα1 ,... ,αn must be free for Φ. And since Φ must be of degree < k, we have (4.17) by the hypothesis of the Lemma. Then (4.18) is by part 3 of Lemma 4.2.11. Finally, (4.19) follows by parts 1 and 2 of Lemma 4.2.11 Lemma 4.2.14 Let H be a Hintikka set and let I be an allowed interpretation relative to H. Assume that if σ is free for X, then σ (X) vH,I (Xσ) = vH,I

(4.20)

for each formula X of degree < k and each term X of degree ≤ k. Then (4.20) also holds for each formula X of degree k itself.

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Proof Assume the hypothesis. Suppose Φ is a formula of degree k. There are several cases depending on the form of Φ. If Φ is of the form τ (τ1 , . . . , τn ), each of τ , τ1 , . . . , τn must be of degree ≤ n. Then, using the hypothesis about terms, vH,I (Φσ) = true ⇔ ⇔ ⇔ ⇔ ⇔ ⇔

vH,I ((τ (τ1 , . . . , τn ))σ) = true vH,I (τ σ(τ1 σ, . . . , τn σ)) = true hvH,I (τ1 σ), . . . , vH,I (τn σ)i ∈ E(vH,I (τ σ)) σ σ σ (τ1 ), . . . , vH,I (τn )i ∈ E(vH,I (τ )) hvH,I σ (τ (τ1 , . . . , τn )) = true vH,I σ (Φ) = true vH,I

Next, if Φ is a propositional combination of simpler formulas, the argument is straightforward using the hypothesis about formulas, and is left to you. Finally, if Φ is of the form (∀α)Ψ the argument, in outline, is as follows. vH,I ([(∀α)Ψ]σ) = true ⇔ vH,I ((∀α)[Ψσα ]) = true ⇔ wH,I (Ψσα ) = true for every valuation w that is an α-variant of v where w(α) is a possible value relative to H σα (Ψ) = true for every ⇔ wH,I valuation w that is an α-variant of v where w(α) is a possible value relative to H ⇔ uH,I (Ψ) = true for every valuation u that is an α-variant of v σ where u(α) is a possible value relative to H σ ((∀α)Ψ) = true ⇔ vH,I

(4.21)

(4.22)

(4.23)

In this, the equivalence of (4.21) and (4.22) is by the hypothesis about formulas, and the equivalence of (4.22) and (4.23) is by a result similar to that stated in Lemma 4.2.11, but for quantified formulas rather than for predicate abstracts. Finally, the central item we have been aiming at. Proof of 4.2.10 The proof, of course, is by induction on the degree of X. Suppose the result is known for formulas and terms of degree < k. Then by Lemma 4.2.13 the result holds for terms of degree k, and then by Lemma 4.2.14 it holds for formulas of degree k as well.

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4.2.4

53

Hintikka Sets and Pseudo-Models

Given a Hintikka set, we know how to create a pseudo-model from it. It would be nice if the various formulas in the Hintikka set turned out to map to true in that pseudo-model. This turns out to be the case, and will be shown below. But there is a troublesome feature of the definition of pseudo-model: quantifiers range over possible values, but predicate abstracts can have general entities as values. It would also be nice if the values assigned to predicate abstracts turned out to be possible values after all. This too happens to be the case, and will also be shown below. In fact, both of the things we desire will be shown simultaneously, in one big result. Then we can conclude that each Hintikka set is satisfiable in a well behaved pseudo-model. Definition 4.2.15 [Allowed Valuation] Let H be a Hintikka set. Call a valuation v allowed relative to H if, for each variable or parameter α, v(α) is some possible value, relative to H. Theorem 4.2.16 Let H be a Hintikka set, let I be an allowed interpretation, and let v be an allowed valuation, relative to H. v−, relative to 1. For each term τ of L+ (C), vH,I (τ ) is a possible value for τ ← H. v− ∈ H then vH,I (Φ) = true. 2. For each formula Φ of L+ (C), if Φ← Proof Both parts of the theorem are shown by a simultaneous induction on degree. Assume they hold for formulas and terms of degree < k. It will first be shown that item 1 holds for terms of degree k; then it will be shown that item 2 holds for formulas of degree k. Part 1. Let τ be a term of degree k. If k happens to be 0, τ is a constant symbol, variable, or parameter. If τ is a constant symbol A, A← v− = A, and vH,I (A) = I(A), which is a possible value of A because I is an allowed interpretation. If τ is a variable or parameter, α, vH,I (α) = v(α) is some possible value E because v is an allowed valuation. But then α← v− = T (v(α)) = T (E), and E is a possible value of T (E) by Exercise 4.2.1. v−, Si Now suppose τ = hλα1 , . . . , αn .Φ(α1 , . . . , αn )i. Then vH,I (τ ) = hτ ← where S = {hw(α1 ), . . . , w(αn )i | wH,I (Φ) = true where w is an α1 , . . . , αn variant of v}. I show hτ ← v−, Si is a possible value of τ ← v− relative to H. To do this I must show that if E1 is a possible value for τ1 , . . . , En is a possible value for τn , then 1. (τ ← v−)(τ1 , . . . , τn ) ∈ H implies hE1 , . . . , En i ∈ S; 2. ¬(τ ← v−)(τ1 , . . . , τn ) ∈ H implies hE1 , . . . , En i 6∈ S. I show the first of these; the second is similar. Assume (τ ← v−)(τ1 , . . . , τn ) ∈ H. That is, v−](τ1 , . . . , τn ) ∈ H. [hλα1 , . . . , αn .Φ(α1 , . . . , αn )i←

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By definition of substitution we have v−α1 ,... ,αn ](α1 , . . . , αn )i(τ1 , . . . , τn ) ∈ H. hλα1 , . . . , αn .[Φ← Since H is a Hintikka set, we also have [Φ← v−α1 ,... ,αn ](τ1 , . . . , τn ) ∈ H and since τ1 , . . . , τn are grounded terms and so have no free occurrences of α1 , . . . , αn [Φ(τ1 , . . . , τn )]← v− ∈ H. Now, let w be the α1 , . . . , αn -variant of v such that w(α1 ) = E1 , . . . , w(αn ) = −= w En . Since Ei is a possible value for the grounded term τi it follows that αi ← ← − ← − τi . And if β 6= α1 , . . . , αn then β w = β v . Then it follows that − ∈ H. w [Φ(α1 , . . . , αn )]← Since Φ(α1 , . . . , αn ) must be of lower degree than k the induction hypothesis applies, and wH,I (Φ(α1 , . . . , αn )) = true. Then hw(α1 ), . . . , w(αn )i ∈ S, so hE1 , . . . , En i ∈ S, which is what we wanted. This concludes the induction step for terms. Part 2. Let Φ be a formula of degree k. By the induction hypothesis, the result holds for formulas and terms of degree < k, and by part 1 it also holds for terms of degree k. Now we have several cases, depending on the form of Φ. I only present a few of them. v− ∈ H. That is, Suppose Φ is τ0 (τ1 , . . . , τn ) and [τ0 (τ1 , . . . , τn )]← v−(τ1 ← v−, . . . , τn ← v−) ∈ H. τ0 ← Each τi is of degree ≤ k so by the induction hypothesis, each vH,I (τi ) is a possible value for τi ← v−. It follows immediately from the definition of possible value (Definition 4.2.3) that hvH,I (τ1 ), . . . , vH,I (τn )i ∈ E(vH,I (τ0 )) and so vH,I (τ0 (τ1 , . . . , τn ) = true. Suppose Φ is X ∧ Y , and (X ∧ Y )← v− ∈ H. By definition of substitution, ← − ← − (X v ∧ Y v ) ∈ H. Since H is a Hintikka set, X ← v− ∈ H and Y ← v− ∈ H. But

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each of X and Y is of lower degree than Φ, so by the induction hypothesis, vH,I (X) = true and vH,I (Y ) = true. It follows that vH,I (X ∧ Y ) = true. Suppose Φ is (∀α)Ψ(α) and [(∀α)Ψ(α)]← v− ∈ H. By definition of substitution, (∀α)[Ψ← v−α ](α) ∈ H. Let w be an arbitrary α-variant of v that assigns to α some possible value, say E, of the same type as α. Since E is a possible value, it is the possible value of some grounded term, say τ . Now by definition of Hintikka v− ∈ H, since τ does not contain α free. We set, [Ψ← v−α ](τ ) ∈ H, and so [Ψ(τ )]← − = τ , and if β 6= α, β ← − = β← − ∈ H. But Ψ(α) is of have α← w w v−, so [Ψ(α)]← w lower degree than Φ, so by the induction hypothesis, wH,I (Ψ(α)) = true. Since w was arbitrary, vH,I ((∀α)Ψ(α)) = true. The other cases are similar and are omitted. Corollary 4.2.17 Let H be a Hintikka set, let I be an allowed interpretation, and let v be an allowed valuation, relative to H. Suppose further that v assigns to each parameter p some possible value for p. Then H is satisfied in the pseudomodel vH,I . That is, if Φ ∈ H then vH,I (Φ) = true. Proof If v assigns to each parameter p some possible value for p, then p← v− = T (v(p)) = p. Consequently for each grounded formula Φ we have Φ← v− = Φ. The result then follows from the previous Theorem.

4.2.5

Pseudo-Models and Models

So far, a satisfiability result has been shown using pseudo-models. Now it is shown that a pseudo-model can be converted to an actual unrestricted Henkin model, and we get a much better version of the satisfiability result as a consequence. It starts with the construction of a candidate for an unrestricted Henkin model. Definition 4.2.18 [Constructed Structure] Let H be a Hintikka set and let I be an allowed interpretation relative to H. For each type t, set H(t) to be the collection of all type t possible values relative to H. An extension function, E, was defined for entities in Definition 4.2.2. Then M = hH, I, Ei is an unrestricted Henkin frame. If v is a valuation in M then v is also an allowed valuation in the sense of Definition 4.2.15. Set A(v, hλα1 , . . . , αn .Φi) = vH,I (hλα1 , . . . , αn .Φi). The structure hM, Ai is said to be constructed from H and I. Theorem 4.2.19 Let H be a Hintikka set and I be an allowed interpretation relative to H. Let hM, Ai be constructed from H and I. 1. For each term τ of L+ (C), (v ∗ I ∗ A)(τ ) = vH,I (τ ), for any valuation v in M. 2. For each formula Φ of L+ (C), M °v,A Φ ⇐⇒ vH,I (Φ) = true, for any valuation v in M. 3. The abstraction designation function A is proper, and hence hM, Ai is an unrestricted Henkin model.

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Proof Finally we have a simple proof. Part 1 has three cases, in each of which (v ∗ I ∗ A)(τ ) and vH,I (τ ) agree. If τ is a constant symbol, both functions are defined to give I(τ ). If τ is a variable or parameter, both functions are defined to give v(τ ). And finally, if τ is a predicate abstract, (v ∗ I ∗ A)(τ ) = A(v, τ ) and this is vH,I (τ ) by definition. Part 2 also has several parts. If Φ = τ0 (τ1 , . . . , τn ) is atomic, then using the first part, M °v,A τ0 (τ1 , . . . , τn ) ⇔ h(v ∗ I ∗ A)(τ1 ), . . . , (v ∗ I ∗ A)(τn )i ∈ E((v ∗ I ∗ A)(τ0 )) ⇔ hvH,I (τ1 ), . . . , vH,I (τn )i ∈ E(vH,I (τ0 )) ⇔ vH,I (τ0 (τ1 , . . . , τn )) = true. Beyond the atomic level, the argument is by a straightforward induction on degree, using Definitions 2.5.5 and 4.2.7. For part 3, there are three items that must be shown, according to Definition 2.5.7. The first is that E((v ∗ I ∗ A)(hλα1 , . . . , αn .Φi)) = {hw(α1 ), . . . , w(αn )i | w is an α1 , . . . , αn variant of v and Γ °w,A Φ}. This is immediate from parts 1 and 2 and the definition of vH,I on predicate abstracts. The second is, if v and w agree on the free variables of hλα1 , . . . , αn .Φi then A(v, hλα1 , . . . , αn .Φi) = A(w, hλα1 , . . . , αn .Φi). This follows from the definition given for A and Proposition 4.2.8. The third, and last, item is that if σ is a substitution that is free for the term hλα1 , . . . , αn .Φi, then A(v, hλα1 , . . . , αn .Φiσ) = A(v σ , hλα1 , . . . , αn .Φi), and this follows using Proposition 4.2.10. Corollary 4.2.20 Every Hintikka set is satisfiable in an unrestricted Henkin model. Proof By Corollary 4.2.17 and the Theorem above. (Recall, it was shown in Section 2.6 that a choice between L(C) and L+ (C) was not significant when considering models for formulas from the language L(C).)

4.2.6

Completeness At Last

Most of the work of showing completeness is over. All that is left is to connect Hintikka sets with tableaus. This can be done in either of two ways. One could give a systematic tableau construction procedure, designed to ensure everything that can be done is eventually done in fact. Then one would show that the set of formulas on an unclosed branch of such a tableau is a Hintikka set. This approach involves considerable attention to detail, and is not what I have chosen to do here. The other technique involves maximal consistent sets, much like in the standard axiomatic approach. Things must be adapted to tableaus, of course, but this is the direction I have chosen because it is considerably simpler.

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Definition 4.2.21 [Consistency] Call a set S of grounded formulas of L+ (C) consistent if no basic tableau beginning with any finite subset of S closes. If S is not consistent, call it inconsistent. Thus a set S is inconsistent if there is a closed tableau beginning with some finite subset. Definition 4.2.22 [Maximal Consistency] A set S is maximally consistent if it is consistent but no proper extension of it is consistent. For propositional logic, working with maximal consistent sets is sufficient to prove completeness, but with quantifiers involved, more is needed. Definition 4.2.23 [E-Complete] A set S of grounded formulas of L+ (C) is E-complete if: 1. ¬(∀α)Φ(α) ∈ S implies ¬Φ(p) ∈ S for some parameter p. 2. (∃α)Φ(α) ∈ S implies Φ(p) ∈ S for some parameter p. It will be shown that lots of maximal consistent, E-complete sets exist, and they are Hintikka sets. From this, completeness follows easily. The primary difference between a tableau completeness proof and an axiomatic one is that with tableaus, maximal consistency and E-completeness give us the implications that make up the definition of a Hintikka set, while in an axiomatic version, these implications become equivalences. The stronger version, in fact, is more than is needed. But now, to work. Proposition 4.2.24 If S is a consistent set of closed formulas of L(C), S can be extended to a maximal consistent, E-complete set of grounded formulas of L+ (C). Proof The set of formulas of L+ (C) is countable; let Ψ1 , Ψ2 , Ψ3 , . . . be an enumeration of all of them. Also, let p1 , p2 , p3 , . . . be an enumeration of all parameters of L+ (C) of all types. Now we construct a sequence of sets of formulas. Each set in the sequence will meet two conditions: it is consistent, and infinitely many parameters of each type do not appear in it. Here is the construction. Let S0 = S. This is consistent by hypothesis, and contains no parameters at all, so both of the conditions are met. Suppose Sn has been defined, and the conditions are met. 1. If Sn ∪ {Ψn+1 } is not consistent, let Sn+1 = Sn . 2. If Sn ∪ {Ψn+1 } is consistent, and Ψn+1 is not an existentially quantified formula or the negation of a universally quantified formula, let Sn+1 = Sn ∪ {Ψn+1 }.

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3. Finally, if Sn ∪ {Ψn+1 } is consistent, and Ψn+1 is (∃α)Φ(α), choose the first parameter p in the enumeration of parameters, of the same type as α, that does not appear in Sn or in (∃α)Φ(α), and set Sn+1 = Sn ∪ {(∃α)Φ(α), Φ(p)}. And similarly if Ψn+1 is ¬(∀α)Φ(α). Note that Sn+1 meets the conditions again. In case 3, consistency needs a small argument, which I leave to you. Finally, let S∞ be S0 ∪ S1 ∪ S2 ∪ . . . . I leave to you the easy verification that S∞ will be consistent, E-complete, and maximal. Proposition 4.2.25 If S is a set of grounded formulas of L+ (C) that is maximal consistent and E-complete, S is a Hintikka set. Proof Let S satisfy the hypothesis of the Proposition. It is a simple matter to verify that S meets each of the Hintikka set conditions. One is presented as an example. Suppose hλα1 , . . . , αn .Φ(α1 , . . . , αn )i(τ1 , . . . , τn ) ∈ S, but Φ(τ1 , . . . , τn ) 6∈ S; we derive a contradiction. If S ∪ {Φ(τ1 , . . . , τn )} were consistent, Φ(τ1 , . . . , τn ) would be in S, since S is maximally consistent. Consequently S ∪ {Φ(τ1 , . . . , τn )} is not consistent, so there is a closed tableau for some finite subset, which must include Φ(τ1 , . . . , τn ), since S itself is consistent. Thus there are formulas X1 , . . . , Xk ∈ S such that there is a closed tableau, call it T , beginning with X1 , . . . , Xk , Φ(τ1 , . . . , τn ). Now, hλα1 , . . . , αn .Φ(α1 , . . . , αn )i(τ1 , . . . , τn ) ∈ S. Construct a tableau as follows. Begin with X1 .. .

1.

Xk k. hλα1 , . . . , αn .Φ(α1 , . . . , αn )i(τ1 , . . . , τn ) k + 1. Φ(τ1 , . . . , τn ) 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 4.2.26 Let Φ be a closed formula and let S be a set of closed formulas, all of L(C). 1. If Φ is valid in unrestricted Henkin models, Φ has a basic tableau proof. 2. If Φ is an unrestricted Henkin consequence of S, Φ has a basic tableau derivation from S.

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Proof Suppose there is no basic tableau derivation of Φ from S. Then there is no closed tableau for ¬Φ, allowing members of S to be added to the ends of open branches. It follows that S ∪ {¬Φ} is consistent. It can be extended to a maximal consistent, E-complete set H, by Proposition 4.2.24. The set H is a Hintikka set, by Proposition 4.2.25. Then by Corollary 4.2.20, S ∪ {¬Φ} is satisfiable in some unrestricted Henkin model, and consequently Φ is not an unrestricted Henkin consequence of S. This establishes part 2; part 1 has a simpler proof.

4.3

Miscellaneous Model Theory

Two of the main results about first-order logic are the Compactness and the L¨ owenheim-Skolem theorem. I already noted, in Section 2.3, that compactness does not hold for higher-order logic. It is also easy to verify that the L¨owenheimSkolem theorem does not hold, since one can write a formula asserting an uncountable object exists. But things are very different if unrestricted 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. L¨owenheim-Skolem takes more work. Theorem 4.3.1 (Compactness) Let S be a set of closed formulas of L(C). If every finite subset of S is satisfiable in some unrestricted Henkin model, so is S itself. Proof Suppose S is not satisfiable in any unrestricted Henkin model—I show some finite subset of S is also not satisfiable. Let ⊥ abbreviate X ∧ ¬X, where X is some arbitrary closed formula of L(C). Since S is not satisfiable in any unrestricted Henkin model, ⊥ is true in every model in which the members of S are true (since there are none), so ⊥ is an unrestricted Henkin consequence of S. By Completeness, ⊥ has a basic tableau derivation from S. A closed tableau, being a finite object, can use only a finite subset S0 of S. Now ⊥ has a basic tableau derivation from S0 , so by Soundness, ⊥ is an unrestricted Henkin consequence of S0 . If S0 were satisfiable in some unrestricted Henkin model, ⊥ would be true in it, which is not possible. Consequently S0 is unsatisfiable. The L¨ owenheim-Skolem theorem for first-order classical logic follows easily from the observation that models constructed in completeness proofs are countable. Unfortunately, this does not apply directly to the unrestricted Henkin models constructed using tableaus. The reason is very simple. I showed how to construct an unrestricted Henkin frame M = hH, I, Ei starting with a Hintikka set H. In this frame, the Henkin domains consisted of possible values for grounded terms, Definition 4.2.3. It is easy to see that H(0) must be countable. But say τ is a grounded term of type h0i such that no formulas of the form τ (τ0 ) or ¬τ (τ0 ) occur in H. (This can certainly happen—take the Hintikka set H to

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be the empty set!) Then hτ, Si is a possible value for τ for every subset S of H(0), so H(h0i) is uncountable. We need some way around this difficulty. The main tool is contained in the following. Theorem 4.3.2 (Cut-Elimination) Let S be a finite set of grounded formulas of L+ (C). If there is a closed tableau beginning with S ∪ {Φ}, and a closed tableau beginning with S ∪ {¬Φ}, then there is a closed tableau beginning with S. This Theorem is a version of Gentzen’s famous Haputsatz, or cut elimination theorem, for higher-order logic. It is an important result about classical firstorder logic that closed tableaus for S∪{Φ} and for S∪{¬Φ} can be constructively converted into one for S. There is no constructive proof for the higher-order case, but the result can be obtained provided we are willing to drop constructivity. Such a proof was given in (Prawitz 1968) and in (Takahashi 1967), and their argument has appeared here, in disguise, as a completeness proof. To finish things off I sketch the remaining ideas involved in a proof of the Theorem. Proof Suppose there are closed tableaus for S ∪ {Φ} and for S ∪ {¬Φ}. Then neither set is satisfiable. It follows that S itself is not satisfiable, for if there were an unrestricted Henkin model in which its members were true, one of Φ or ¬Φ would be true there. It remains to show that the unsatisfiability of S implies there must be a closed tableau beginning with S. Suppose the contrary: there is no closed tableau beginning with S, so that S is a consistent set. Proposition 4.2.24 says a consistent set of L(C) sentences can be extended to a maximal consistent, E-complete set—the same proof can easily be made to work even if the set contains parameters, provided it omits infinitely many of them. Since S is finite, it certainly omits infinitely many parameters, so we can extend it to a maximal consistent, E-complete set, which must be a Hintikka set. Corollary 4.2.20 says Hintikka sets are satisfiable. Since S is a subset of a satisfiable set, it too must be satisfiable, but it is not. This contradiction concludes the proof. This immediately gives us the following important result. Corollary 4.3.3 (Cut Rule) The addition of the following Cut Rule does not change the class of provable formulas: at any point, split a branch, and add ¬Φ to one side, and Φ to the other, where Φ is any grounded formula. The way this result is most often used is embodied in the following. Corollary 4.3.4 If Φ has a tableau proof, Φ can be added as a line to any tableau, without expanding the class of provable formulas. Proof Suppose Φ has a tableau proof, and so there is a closed tableau for ¬Φ. And now suppose we are constructing another tableau, and we wish to use Φ in

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that construction. We can proceed as follows. .. .

¬Φ

@ @ Φ

That is, we have used an application of a cut. Now, on the left branch, introduce the steps appropriate to close it, which exist because we are assuming there is a closed tableau for ¬Φ. This leaves the right branch, and the effect was to add Φ to the tableau. Now, go back through the proof of completeness given earlier. Proposition 4.2.24 said we could extend a consistent set to a maximal consistent, Ecomplete one. Using the Lemma above, it follows that a maximal consistent set must contain either Φ or ¬Φ for every grounded formula Φ. Since this is the case, each grounded term can, in fact, have only one possible value associated with it. Thus the particular model constructed in the completeness argument must have countable Henkin domains, since the family of grounded terms for each type is countable. We thus have the following. Theorem 4.3.5 (L¨ owenheim-Skolem) Let S be a set of closed formulas of L(C). If S is satisfiable in some unrestricted Henkin model, S is satisfiable in an unrestricted Henkin model whose domain function H meets the condition that H(t) is countable for every type t. The results above have both good and bad points. It is obviously good to be able to prove such powerful model-theoretic facts about a logic. The bad side is that Lindstr¨ om’s Theorem says, since the version of higher-order logic based on unrestricted Henkin models satisfies the theorems above, it is simply an equivalent to first-order logic. This does not mean nothing has been gained. The higher-order formalism is natural for the expression of things whose translation into first-order versions would be unnatural. And finally, if a sentence is not provable, it must have an unrestricted Henkin counter-model, but if it is provable, it must be true in all unrestricted Henkin models, and among these are the standard higher-order models! Thus we have a means of getting at higher-order validities—we just can’t get at all of them this way.

Chapter 5

Equality The basic tableau rules of Chapter 3 do not give any special role to equality. It is time to bring it into the picture. This is done by adding axioms to the tableau system, which has the effect of narrowing things to normal unrestricted Henkin models. In addition, some useful derived tableau rules will be presented.

5.1

Adding Equality

Leibniz’s principle is that objects are equal just in case they have the same properties. This principle is most easily embodied in axioms, rather than in tableau-style rules. Definition 5.1.1 [Equality Axioms] For each type t, the following sentence is an equality axiom: (∀αt )(∀β t )[(αt =ht,ti β t ) ≡ (∀γ hti )(γ hti (αt ) ⊃ γ hti (β t ))] EQ denotes the set of equality axioms. I will show that a closed formula Φ of L(C) is valid in normal unrestricted Henkin models if and only if Φ has a tableau derivation from EQ. But before that is done I give some handy derived tableau rules, and examples of their use.

5.2

Derived Rules and Tableau Examples

There are two derived rules involving equality that are more “tableau-like” in flavor, and are what I primarily use in constructing tableau proofs and derivations. I do not know if they can serve as full replacements for the official Equality Axioms, since I have been unable to prove a completeness theorem using them. Nonetheless, the derived rules below are the ones I generally use in practice.

62

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Definition 5.2.1 [Derived Reflexivity Rule] For a grounded term τ of L+ (C), at any point in a proof (τ = τ ) may be added to the end of a tableau branch. Schematically, (τ = τ ) Justification of Derived Reflexivity Rule Let τ be a grounded term of type t. (τ = τ ) can be added to the end of a branch via the following sequence of steps. (I omit type designations for variables, which are easy to restore if you feel the need). (∀α)(∀β)[(α = β) ≡ (∀γ)(γ(α) ⊃ γ(β))] (∀β)[(τ = β) ≡ (∀γ)(γ(τ ) ⊃ γ(β))] 2. [(τ = τ ) ≡ (∀γ)(γ(τ ) ⊃ γ(τ ))] 3. [(τ = τ ) ⊃ (∀γ)(γ(τ ) ⊃ γ(τ ))] 4. [(∀γ)(γ(τ ) ⊃ γ(τ )) ⊃ (τ = τ )] 5.

¬(∀γ)(γ(τ ) ⊃ γ(τ ))

@ @ 6. (τ = τ )

1.

7.

In this, 1 is an equality axiom; 2 is from 1 and 3 is from 2 by universal rules; 4 and 5 are from 3 by a conjunction rule; 6 and 7 are from 5 by a disjunction rule. Clearly the left branch continues to closure. The remaining open branch, the right one, indeed has (τ = τ ) on it. The next rule embodies the familiar notion of substitutivity of equals for equals. Definition 5.2.2 [Derived Substitutivity Rule] Suppose Φ(α) is a formula of L+ (C) in which the variable α may have free occurrences, but no other variables occur free. Also suppose τ1 and τ2 are grounded terms of the same type as α. As usual, let Φ(τ1 ) denote the result of replacing free occurrences of α in Φ(α) with occurrences of τ1 ; and similarly for Φ(τ2 ). Then, if both Φ(τ1 ) and (τ1 = τ2 ) occur on a tableau branch, Φ(τ2 ) can be added to the branch end. Schematically, Φ(τ1 ) (τ1 = τ2 ) Φ(τ2 ) Justification of Derived Substitutivity Rule Assume τ1 and τ2 are grounded terms of type t, and Φ(τ1 ) and (τ1 = τ2 ) occur on a tableau branch. I show Φ(τ2 ) can be added to the end of the branch. Again I omit type desig-

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nations. Φ(τ1 ) (τ1 = τ2 ) .. . (∀α)(∀β)[(α = β) ≡ (∀γ)(γ(α) ⊃ γ(β))] 1. (∀β)[(τ1 = β) ≡ (∀γ)(γ(τ1 ) ⊃ γ(β))] 2. [(τ1 = τ2 ) ≡ (∀γ)(γ(τ1 ) ⊃ γ(τ2 ))] 3. [(τ1 = τ2 ) ⊃ (∀γ)(γ(τ1 ) ⊃ γ(τ2 ))] 4. [(∀γ)(γ(τ1 ) ⊃ γ(τ2 )) ⊃ (τ1 = τ2 )] 5. @

¬(τ1 = τ2 )

@ 6. (∀γ)(γ(τ1 ) ⊃ γ(τ2 )) 7. hλα.Φ(α)i(τ1 ) ⊃ hλα.Φ(α)i(τ2 )

¬hλα.Φ(α)i(τ1 ) ¬Φ(τ1 ) 11.

8.

@ @ 9. hλα.Φ(α)i(τ2 ) Φ(τ2 ) 12.

10.

In this, 1 is an equality axiom; 2 is from 1 and 3 is from 2 by universal rules; 4 and 5 are from 3 by a conjunction rule; 6 and 7 are from 4 by a disjunction rule; 8 is from 7 by a universal rule, using the term hλα.Φ(α)i; 9 and 10 are from 8 by a disjunction rule; 11 is from 9 and 12 is from 10 by a predicate abstract rule. The two left branches are closed, leaving the right one which contains Φ(τ2 ). Now I give several examples of tableau derivations using the derived rules. The first example is (intentionally) a simple one. It appeared earlier as Example 2.2.7, where an informal reading was given, and validity was shown directly. Example 5.2.3 Here is a proof of hλX.(∃x)X(x)i(hλx.x = ci). ¬hλX.(∃x)X(x)i(hλx.x = ci) ¬(∃x)hλx.x = ci(x) 2. ¬hλx.x = ci(c) 3. ¬(c = c) 4. (c = c) 5.

1.

In this, 2 is from 1 by an abstract rule; 3 is from 2 by a universal rule; 4 is from 3 by an abstract rule, and 5 is by the derived reflexivity rule. The next example shows how, by using the derived rules, we can prove a version of the equality axioms. Example I give a tableau proof (using derived rules, not axioms) of (∀α)(∀β)[(∀γ)(γ(α) ⊃ γ(β)) ⊃ (α = β)].

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¬(∀α)(∀β)[(∀γ)(γ(α) ⊃ γ(β)) ⊃ (α = β)] 1. ¬(∀β)[(∀γ)(γ(P ) ⊃ γ(β)) ⊃ (P = β)] 2. ¬[(∀γ)(γ(P ) ⊃ γ(Q)) ⊃ (P = Q)] 3. (∀γ)(γ(P ) ⊃ γ(Q)) 4. ¬(P = Q) 5. hλX.¬(X = Q)i(P ) ⊃ hλX.¬(X = Q)i(Q) 6.

¬hλX.¬(X = Q)i(P ) ¬¬(P = Q) 9.

@ @ 7. hλX.¬(X = Q)i(Q) ¬(Q = Q) 10. (Q = Q) 11.

8.

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

Exercises Exercise 5.2.1 Prove the following characterization of equality—it says it is the smallest reflexive relation. (∀x)(∀y){(x = y) ≡ (∀R)[(∀z)R(z, z) ⊃ R(x, y)]} Exercise 5.2.2 Give a tableau derivation of the following from EQ. (∀αhti )(∀β hti )[(αhti = β hti ) ⊃ (∀γ t )(αhti (γ t ) ≡ β hti (γ t ))] More generally, one can do the same with the following. (∀αht1 ,... ,tn i )(∀β ht1 ,... ,tn i )[(αht1 ,... ,tn i = β ht1 ,... ,tn i ) ⊃ (∀γ1t1 ) · · · (∀γntn )(αht1 ,... ,tn i (γ1t1 , . . . , γntn ) ≡ β ht1 ,... ,tn i (γ1t1 , . . . , γntn ))]

5.3

Soundness and Completeness

The results of this section combine to prove the following.

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Theorem 5.3.1 Let Φ be a closed formula and let S be a set of closed formulas of L(C). 1. Φ is valid in all normal unrestricted Henkin models if and only if Φ has a tableau derivation from EQ. 2. Φ is a consequence of S with respect to normal unrestricted Henkin models if and only if Φ has a tableau derivation from S ∪ EQ. The theorem above combines soundness and completeness. One direction, soundness, is almost immediate. Every equality axiom is true in every normal unrestricted Henkin model, so the implications from right to left in Theorem 5.3.1 follow immediately from Theorems 4.1.4 and 4.1.5. As usual, the completeness direction is more work. The key item is to prove the following Proposition. Once we have it, completeness follows immediately using part 2 of Theorem 4.2.26. Proposition 5.3.2 Given an unrestricted Henkin model in which all members of EQ are true, there is a normal unrestricted Henkin model in which exactly the same closed formulas of L(C) are true. The rest of this section is given over to a proof of Proposition 5.3.2—it is broken up into constructions and Lemmas. The ideas are the same as in G¨ odel’s original completeness proof for first-order logic with equality—bring equivalence classes into the picture. For the rest of this section, assume hM, Ai is an unrestricted Henkin model, M = hH, I, Ei, and all members of EQ are true in this model. For O1 , O2 ∈ H(t), let us write O1 =I O2 as a more readable alternative notation for hO1 , O2 i ∈ I(=ht,ti ). Thus =I is the interpretation of the equality constant symbol (of a particular type, which will be indicated only if needed). Since all equality axioms are true in hM, Ai, it is an easy consequence that =I is an equivalence relation. For each O ∈ H(t), let O be the equivalence class determined by O, that is, O = {O0 | O =I O0 }. Define a new Henkin domain mapping by setting H(t) = {O | O ∈ H(t)}. Also, define a new interpretation by setting I(A) to be the equivalence class containing I(A), that is, I(A) = I(A). Lemma 5.3.3 If O1 = O2 then E(O1 ) = E(O2 ). Proof Suppose O1 = O2 , that is, O1 =I O2 , and say O1 and O2 are of type hti. In Exercise 5.2.2 you were asked to give a tableau derivation of (∀α)(∀β)[(α = β) ⊃ (∀γ)(α(γ) ≡ β(γ))] from EQ. Then by soundness, this sentence is valid in hM, Ai. It follows that (∀γ)(α(γ) ≡ β(γ)) is also true with respect to any valuation assigning O1 to α and O2 to β. From this we immediately get that the sets E(O1 ) and E(O2 ) must be the same. A similar argument applies if O1 and O2 are of type ht1 , . . . , tn i.

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The Lemma above justifies the following. For O ∈ H(ht1 , . . . , tn i), set E(O) = {hO1 , . . . , On i | hO1 , . . . , On i ∈ E(O)}. We have now created a new unrestricted Henkin frame M = hH, I, Ei. Lemma 5.3.4 The unrestricted Henkin frame M = hH, I, Ei is normal. Proof The following calculation establishes normality. hO1 , O2 i ∈ E(I(=)) ⇔ ⇔ ⇔ ⇔

hO1 , O2 i ∈ E(I(=)) hO1 , O2 i ∈ E(I(=)) O1 =I O2 O1 = O2

For each valuation v in M, let v be the corresponding valuation in M given by v(α) = v(α). It is easy to see that each valuation in M is v for some valuation v in M. Lemma 5.3.5 Let τ be a predicate abstract. If v1 = v2 then A(v1 , τ ) =I A(v2 , τ ). Proof For convenience, say τ has a single free variable, γ. The more general case is treated similarly. From now on I’ll write τ as τ (γ). Let α and β be variables of the same type as γ, that do not occur in τ (γ) (free or bound). Since hM, Ai is normal, (∀α)(∀β)[(α = β) ⊃ (τ (α) = τ (β))] is valid in it, and hence τ (α) = τ (β) is true in hM, Ai with respect to any valuation w such that w(α) =I w(β). Let σ1 be the substitution {α/γ}. Then [τ (α)]σ1 = τ (γ), so A(v1 , τ (γ)) = A(v1 , τ (α)σ1 ) = A(v1σ1 , τ (α)), since A is proper, Definition 2.5.7. Likewise, if σ2 is the substitution {β/γ}, then A(v2 , τ (γ)) = A(v2 , τ (β)σ2 ) = A(v2σ2 , τ (β)). Now let w be a valuation such that w(α) = v1σ1 (α), w(β) = v2σ2 (β), and otherwise w is arbitrary. The only free variable of τ (α) is α, on which w and v1σ1 agree, hence since A is proper, A(v1σ1 , τ (α)) = A(w, τ (α)). Similarly A(v2σ2 , τ (β)) = A(w, τ (β)). Finally, w(α) = v1σ1 (α) = v1 (ασ1 ) = v1 (γ). Similarly w(β) = v2 (γ). Since v1 = v2 , v1 (γ) =I v2 (γ) so we have w(α) =I w(β). It now follows from the observation at the start of the proof, that τ (α) = τ (β) is true in hM, Ai with respect to valuation w, and hence A(w, τ (α)) =I A(w, τ (β)). Combining this with items above, we have A(v1 , τ (γ)) =I A(v2 , τ (γ)). This Lemma justifies the following. Define an abstraction designation function by A(v, hλα1 , . . . , αn .Φi) = A(v, hλα1 , . . . , αn .Φi). Now the final step. Lemma 5.3.6 For each valuation v in M: 1. (v ∗ I ∗ A)(τ ) = (v ∗ I ∗ A)(τ ) for each term τ of L(C);

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2. M °v,A Φ ⇐⇒ M °v,A Φ for each formula Φ of L(C); 3. A is proper, and hence hM, Ai is an unrestricted Henkin model. Proof Part 1 follows for variables, constant symbols, and predicate abstracts by definition of v, I, and A respectively. Part 2 is by an induction on the degree of Φ, which I leave to you. Finally, for part 3 it is necessary to verify the three parts of Definition 2.5.7. I check part 3 and leave the other parts to you. Let σ be a substitution that is free for hλα1 , . . . , αn .Φi, a term which I abbreviate as τ . It must be shown that A(v, τ σ) = A(v σ , τ ). We have the following. A(v, τ σ)

= A(v, τ σ) = A(v σ , τ ) = A(v σ , τ )

But also we have the following, for each variable α. v σ (α)

= = = =

v σ (α) v(ασ) v(ασ) v σ (α)

Exercises Exercise 5.3.1 Give the details of the proof that =I is an equivalence relation. Exercise 5.3.2 Supply the proof of part 2 of Lemma 5.3.6.

Chapter 6

Extensionality Extensionality says that properties applying to the same objects are identical. Just as was done with equality in Chapter 5, extensionality is added via axioms. Throughout this chapter it is, of course, assumed that equality is available.

6.1

Adding Extensionality

The extensionality axioms simply assert the equality of co-extensional properties. Definition 6.1.1 [Extensionality Axioms] Each sentence of the following form is an extensionality axiom, where α and β are of type ht1 , . . . , tn i, γ1 is of type t1 , . . . , γn is of type tn . (∀α)(∀β){(∀γ1 ) · · · (∀γn )[α(γ1 , . . . , γn ) ≡ β(γ1 , . . . , γn )] ⊃ [α = β]} EXT denotes the set of extensionality axioms. I will show that a closed formula Φ of L(C) is valid in normal Henkin models (note that I’ve dropped the qualifier “unrestricted”) if and only if Φ has a tableau derivation from EQ ∪ EXT. But first some examples.

6.2

A Derived Rule and an Example

Extensionality was embodied in a set of axioms. There is a derived tableau rule that expresses the same idea, in a rather more useful form. Definition 6.2.1 [Derived Extensionality Rule] Suppose τ1 and τ2 are two grounded terms, both of type ht1 , . . . , tn i. At any point in a tableau construction the end of a branch can be split, with one fork labeled (τ1 = τ2 ), and the other fork labeled ¬(τ1 (pt11 , . . . , ptnn ) ≡ τ2 (pt11 , . . . , ptnn )) where pt11 , . . . , ptnn are parameters new to the branch. Schematically, for new parameters: ¬ [τ1 (p1 , . . . , pn ) ≡ τ2 (p1 , . . . , pn )] (τ1 = τ2 ) 69

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The justification of this rule is quite straightforward, and I leave it as an exercise. Here is an example that illustrates the use of this Derived Extensionality Rule. Example 6.2.2 I give a proof of the following formula. (∀x) [hλX.X(x)i(P ) ≡ hλX.X(x)i(Q)] ⊃ hλX , X, Y.X (X) ⊃ X (Y )i(P, P, Q)

¬{(∀x) [hλX.X(x)i(P ) ≡ hλX.X(x)i(Q)] ⊃ hλX , X, Y.X (X) ⊃ X (Y )i(P, P, Q)} 1. (∀x) [hλX.X(x)i(P ) ≡ hλX.X(x)i(Q)] 2. ¬hλX , X, Y.X (X) ⊃ X (Y )i(P, P, Q) 3. ¬ [P(P ) ⊃ P(Q)] 4. P(P ) 5. ¬P(Q) 6. ! aa !! aa ! ! aa ! ! a P = Q 8. ¬ [P (p) ≡ Q(p)] 7. P(Q) 9. hλX.X(p)i(P ) ≡ hλY.Y (p)i(Q) 10. 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 τ1 to be P and τ2 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 6.2.1 Give a proof of (3.1) from Example 3.3.1. Exercise 6.2.2 Show that the rule contained in Definition 6.2.1 is, in fact, a derived rule, using EXT.

6.3

Soundness and Completeness

I sketch a proof that the sentences having tableau proofs using EQ ∪ 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 ∪ EXT are valid in normal Henkin models.

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Completeness also takes very little work. Using results of Chapter 4, if a sentence Φ does not have a tableau proof using EQ ∪ EXT as axioms, there is an unrestricted Henkin model hM, Ai (where M = hH, I, Ei) in which Φ is false, but in which all of EQ ∪ EXT are true. Since the members of EQ are true, by results of Chapter 5, we can take hM, Ai to be normal. I claim it is also extensional in the sense of Definition 2.6.3, that is, if E(O) = E(O0 ) then O = O0 , where O and O0 are objects in the model domain. I now show this. Suppose E(O) = E(O0 ), where O and O0 are of type hti for simplicity (the general case is similar). The following is a member of EXT (in it, α and β are of type hti, and γ is of type t) (∀α)(∀β){(∀γ)[α(γ) ≡ β(γ)] ⊃ [α = β]} and so this sentence true in hM, Ai. Let v be a valuation such that v(α) = O and v(β) = O0 . Then M °v,A {(∀γ)[α(γ) ≡ β(γ)] ⊃ [α = β]}. But since E(O) = E(O0 ) it is easy to see we also have M °v,A (∀γ)[α(γ) ≡ β(γ)] and so M °v,A α = β. Since hM, Ai is normal, it follows that v(α) = v(β), that is, O = O0 . Since hM, Ai is extensional it is isomorphic to a Henkin model, as was shown in Section 2.6. And trivially, isomorphism preserves sentence truth.

Part II

Modal Logic

72

Chapter 7

Modal Logic—Syntax and Semantics 7.1

Introduction

The second part of this book investigates a logic of intensions and extensions, using a possible world semantics. I will assume you have some familiarity with such semantics—technicalities can be postponed. First, a point about terminology. The intensional/extensional distinction is an old one. Unfortunately, the word “extensionality” has already been given a technical meaning in Part I, where Henkin models that did or did not satisfy the extensionality axioms were considered. The use of “extension” in this part, while related, is not the same. I briefly tried using the word “denotation” here, but finally it seemed unnatural, and I resigned myself to using the word “extension” after all. As a matter of fact, the Axioms of Extensionality will be assumed throughout Part II for those terms that will be called extensional, so any confusion of meanings between the classical and the modal settings should be minimal. The machinery in Part I had no place for intensions—meanings. In a normal Henkin model, if terms intended to denote the morning star and the evening star have the same extension, as they do in the real world, they are equal, and so share all properties. They cannot be distinguished. Montague and his students, notably Gallin, developed a purely intensional logic. In this, extensions could only be handled indirectly—in some sense an extension could be an intension that did not vary with circumstances. While this can be made to work, it treats extensions as second class objects, and leads to a rather complicated development. What is presented here is a modification of the Montague/Gallin approach, in which both extensions and intensions are first class objects. What are the underlying intuitions? An extensional object will be much as it was in Part I: a set or relation in the usual sense. The added construct is that of intensional object, or concept, and this is treated in the Carnap

73

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tradition. A phrase like, say, “the royal family of England,” has a meaning, an intension. At any particular moment, that meaning can be used to determine a particular set of people, constituting its extension. But that extension will vary with time. For different phrases, there may be different mechanisms for determining extensions as circumstances vary. The one thing common to all such intensional phrases is that they, somehow, induce mappings from circumstances to extensions. Abstracting to the minimum useful structure, in a possible world model an intensional object will be a function from possible worlds to extensional objects. Here is an example using the terminology just introduced. Suppose we take possible worlds as people, with an S5 accessibility relation—every person is accessible to every other person. And suppose the ground-level domain is a bunch of real-world objects. Any one person will classify some of those objects as being red. Because of differences in vision, and perhaps culture, this classification may vary from person to person. Nonetheless, there is a common concept of red, or else communication would not be possible. We can identify it with the function that maps each person to the set of objects that person classifies as red. And similarly for other colors. In addition, each person has a notion of color, though this too may vary from person to person. One person may think of ultra-violet as a color, another not. We can think of the color concept as a mapping from persons to the set of colors for that person. If we assert that red is a color for a particular person, we mean the red concept is in the extension of the color concept for that person. The extension of the red concept for that person plays no role here. Sometimes extensions are needed too. Certainly if we ask someone whether or not some object is red, the extension of the red concept, for that person, is needed to answer the question. Here is another example in this direction. Assume the word “tall” has a definite, non-fuzzy, meaning. Say everybody gets together and votes on which people are tall, or say there is a tallness czar who decides to whom the adjective applies. The key point is that the meaning of “tall,” even though precise, drifts with time. Average height of the general population has increased over the last several generations, so someone who once was considered tall might not be considered so today. Now suppose I say, “Someday everybody will be tall.” There is more than one ambiguity here. On the one hand I might mean that at some point in the future, everybody then alive will be a tall person. On the other hand I might mean that everybody now alive will grow, and so at some point everybody now alive will be a tall person. Let us read modal operators temporally, so that ¤X informally means that X is true and will remain true, and ♦X means that X either is true or will be true at some point in the future. Also, let us use T (x) as a tallness predicate. (The examples that follow assume an actualist reading of the quantifiers, and eventually I will adopt a version of a possibilist reading. For present purposes, this is a point of no fundamental importance. For now, think in terms of varying domain models, with quantifiers ranging over different domains at different worlds.) The two readings of the sentence are easily expressed as follows.

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(∀x)♦T (x)

(7.1)

♦(∀x)T (x)

(7.2)

Formula (7.1) refers to those alive now, and says at some point they will all be tall. Formula (7.2) refers to those alive at some point in the future and asserts, of them, that they will be tall. All this is standard; the problem is with the adjective “tall.” Do we mean that at some point in the future everybody (read either way) will be tall as they use the word in the future, or as we use the word now? If we interpret things intensionally, T (x) at a possible world would be understood according to that world’s meaning of tall. There is no way, using purely intensional machinery, to formalize the assertion that, at some point in the future, everybody will be tall as we understand the term. But this is what is most likely meant if someone says, “Someday everybody will be tall.” Here is another example, one that goes the other way. Suppose a member of the Republican Party, call him R, says, “necessarily the proposed tax cut is a good thing.” Suppose we take as the possible worlds of a model the collection of all Republicans, and assume a sentence is true at a world if that Republican thinks the sentence true. (We assume Republicans are entirely rational, so we don’t have to worry about contradictory beliefs.) Let us assume ¤X means that every Republican thinks X is the case, which means X is necessary for Republicans. (Technically, this gives us an S5 modality.) Now, how do we formalize the sentence above? Let c be a constant symbol whose intended meaning is, “the proposed tax cut,” and let G be a “goodness” predicate. Then ¤G(c) seems reasonable as a formalization. What should it mean to say it is true for R? One possibility is that R means every Republican thinks the tax cut is good, as R understands the word good. This may not be what was meant. After all, R might consider something good only if it personally benefitted him. Another Republican might think something good if it eventually benefitted the poor. Such a Republican probably would not think a tax cut good simply because it benefitted R but he might believe it would eventually benefit the poor, and so would be good in his sense. Probably R is saying every Republican thinks a tax cut is good, for his own personal reasons. The notion of what is good can vary from Republican to Republican, provided they all agree that the proposed tax cut is a good thing. But the mere fact that we can consider more than one reading tells us that a simple formalization like ¤G(c) is not sufficient. Here will be presented a logic of both intension and extension, of both sense and reference. In one of the examples above, color is an intensional object. It is a function from persons to sets of concepts like red, blue, and so on. As such, it is the same function for each person. The extension of color for a particular person is the color function evaluated at that person, and thus it is a particular set of concepts, such as red but not infra-red, and so on, quite possibly different from person to person. We need a logic in which both intensions and extensions are first-class objects. The machinery for doing this makes for complicated

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looking formulas. But I point out, in everyday discourse all the machinery is present but hidden—we infer it from our knowledge of what we think must have been meant. Formalization naturally requires complex machinery—it is making explicit what our minds do automatically.

7.2

Types and Syntax

Now begins the formal treatment. And I begin by defining a notion of type. I want it to include the types of classical logic, as defined in Section 1.1. I also want it to include the purely intensional types of the Montague tradition, as given in (Gallin 1975). Definition 7.2.1 [Type] The notion of a type, extensional and intensional, is given by the following. 1. 0 is an extensional type. 2. If t1 , . . . , tn are types, extensional or intensional, ht1 , . . . , tn i is an extensional type. 3. If t is an extensional type, ↑t is an intensional type. A type is an intensional or an extensional type. The ideas behind the definition above are these. As usual, 0 is to be the type of ground-level objects, unanalysed “things.” The type ht1 , . . . , tn i is intended to be analogous to types in part I. The type ↑t is the new piece of machinery—an object of such a type will be a function on possible worlds. Recall the example involving colors from Section 7.1; it can be used to give a sense of how these types are intended to be applied. In that example, real-world objects are those of type 0. A set of real-world objects is of type h0i so, for instance, the set of objects some particular person considers red is of this type; this is the extension of red for that person. The intensional object red, mapping each person to that person’s set of red objects, is of type ↑h0i. A set of such intensional objects is of type h↑h0ii, so for a particular person, that person’s set of colors is of this type—the extension of color for that person. Finally, the intensional object color, mapping each person to that person’s set of colors, is an object of type ↑h↑h0ii. For another example, assume possible worlds are possible situations, and the ground-level objects include people. In each particular situation, there is a tallest person in the world. The tallest person, in each situation, is an object of type 0. The tallest person concept is an object of type ↑0—it associates with each possible world the tallest person in that possible world. As a final example example, suppose t is an extensional type, so that ↑t is intensional. The two-place relation: the intensional object X of type ↑t has the extensional object y of type t as its extension, is a relation of type h↑t, ti.

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The language of Part I must be expanded to allow for modality. Just as classically, C is a set of constant symbols containing at least an equality symbol =ht,ti for each type t, though the set of types is now larger. Note that the equality symbols are all of extensional type. Using them we can form the intensional terms hλx, y.x = yi and hλx, y.¤(x = y)i, as needed. We also have variables of each type. There is one new piece of machinery, an operator ↓, which plays a role in term formation. As usual, terms and formulas must be defined together in a mutual recursion. Definition 7.2.2 [Term of L(C)] Terms are characterized as follows. 1. A constant symbol or variable of L(C) of type t is a term of L(C) of type t. If it is a constant symbol, it has no free variable occurrences. If it is a variable, it has one free variable occurrence, itself. 2. If Φ is a formula of L(C) and α1 , . . . , αn is a sequence of distinct variables of types t1 , . . . , tn respectively, then hλα1 , . . . , αn .Φi is a term of L(C) of the intensional type ↑ht1 , . . . , tn i. Its free variable occurrences are the free variable occurrences of Φ, except for occurrences of the variables α1 , . . . , αn . 3. If τ is a term of L(C) of type ↑t then ↓τ is a term of type t. It has the same free variable occurrences that τ has. The predicate abstract hλα1 , . . . , αn .Φi is of type ↑ht1 , . . . , tn i above, and not of type ht1 , . . . , tn i, essentially because Φ can vary its meaning from world to world, and so hλα1 , . . . , αn .Φi itself is world dependent. Case 3 above makes use of what may be called an extension-of operator, converting a term of an intensional type to a term of the corresponding extensional one. Continuing with the color example, suppose r is the intensional notion of red, of type ↑h0i, mapping each person to that person’s set of red objects. Then for a particular person, ↓r would be that person’s set of red objects—the extension of r for that person, and an extensional object of type h0i. Definition 7.2.3 [Modal Formula of L(C)] The definition of formula of L(C) is as follows: 1. If τ is a term of either type ht1 , . . . , tn i or type ↑ht1 , . . . , tn i, and τ1 , . . . , τn is a sequence of terms of types t1 , . . . , tn respectively, then τ (τ1 , . . . , τn ) is a formula (atomic) of L(C). The free variable occurrences in it are the free variable occurrences of τ , τ1 , . . . , τn . 2. If Φ is a formula of L(C) so is ¬Φ. The free variable occurrences of ¬Φ are those of Φ. 3. If Φ and Ψ are formulas of L(C) so is (Φ∧Ψ). The free variable occurrences of (Φ ∧ Ψ) are those of Φ together with those of Ψ.

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4. If Φ is a formula of L(C) and α is a variable then (∀α)Φ is a formula of L(C). The free variable occurrences of (∀α)Φ are those of Φ, except for occurrences of α. 5. If Φ is a formula of L(C) so is ¤Φ. The free variable occurrences of ¤Φ are those of Φ. Item 1 above needs some comment, and again the example concerning colors should help make things clear. Suppose r is the intensional notion of red, of type ↑h0i. And suppose c is an extensional notion of color, the set of colors for a particular person—call the person George. Also let C be the intensional version of color, mapping each person to that person’s extension of color. c is of type h↑h0ii, and C is of type ↑h↑h0ii I take both C(r) and c(r) to be atomic formulas. If we ask whether they are true for George, no matter which formula we use, we are asking if r is a color for George. But if we ask whether they are true for Natasha, we are asking different questions. C(r) is true for Natasha if r is a color for Natasha, while c(r) is true for Natasha if r is a color for George. No matter which, both c(r) and C(r) make sense, and are considered well-formed. I use ♦ to abbreviate ¬¤¬ in the usual way, or I tacitly treat it as primitive, as is convenient at the time.

7.3

Constant Domains and Varying Domains

Should quantifiers range over what does exist, or over what might exist? That is, should they be actualist or possibilist? This is really a first-order question. A flying horse may or may not exist. In the world of mythology, such a being does exist. In the present world, it does not. But the property of being a flying horse does not exist in some worlds and lack existence in others. In the present world nothing has the flying-horse property, but that does not mean the property itself is non-existent. Thus actual/possible existence issues really concern type 0 objects, so the discussion that follows assumes a first-order setting. As presented in (Fitting & Mendelsohn 1998) and also in (Hughes & Cresswell 1996b), the distinction between actualist and possibilist quantification can be seen to be that between varying domain modal models and constant domain ones. In a varying domain modal model, one can think of the domain associated with a world as what actually exists at that world, and it is this domain that a quantifier ranges over when interpreted at that world. In a constant domain model one can think of the common domain as representing what does or could exist, and this is the same from world to world. Of course a choice between constant and varying domain models makes a substantial difference. Both the Barcan formula and its converse are valid in a constant domain setting, but neither is in a varying domain one. As it happens, while a choice between constant and varying domain models makes a difference technically, at a deeper level such a choice is essentially an arbitrary one. If we choose varying domains as basic, we can restrict attention to constant domain models by requiring the Barcan formula and its converse

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to hold. (Technically this requirement involves an infinite set of formulas but if equality is available, a single formula will do.) Thus when using actualist quantification we can still determine constant domain validity. The other direction is even easier—if we have possibilist quantification we can also determine varying domain validity. And on this topic I present a somewhat more detailed discussion. Suppose quantification is taken in a possibilist sense—domains are constant. Nonetheless, at each world we can intuitively divide the (common) domain into what ‘actually’ exists at that world and what does not. Introduce a predicate symbol E of type ↑h0i for this purpose. At a world, E(x) is true if x has as its value an object one thinks of as existing at that world, and is false otherwise. Then the effect of varying domain quantification can be had by relativising all quantifiers to E. That is, replace (∀x)ϕ by (∀x)(E(x) ⊃ ϕ) and replace (∃x)ϕ by (∃x)(E(x) ∧ ϕ). What we get, at least intuitively, simulates an actualist version of quantification. All this can be turned into a formal result. Suppose we denote the relativization of a first-order formula ϕ, as described above, by ϕE . It can be shown that ϕ is valid in all varying domain models if and only if ϕE is valid in all constant domain models. Possibilist quantification can simulate actualist quantification. I note in passing that (Cocchiarella 1969) actually has two kinds of quantifiers, corresponding to actualist and possibilist, though it is observed that a quantifier relativization of the sort described above could be used instead. The discussion above was in a first-order setting. As observed earlier, when higher types are present the actualist/possibilist distinction is only an issue for type 0 objects. I have made the choice to use possibilist type 0 quantifiers. The justification is that, first, such quantifiers are easier to work with, and second, they can simulate actualist quantifiers, so nothing is lost. When I say they are easier to work with, I mean that both the semantics and the tableau rules are simpler. So there is considerable gain, and no loss. Officially, from now on the formal language will be assumed to contain a special constant symbol, E, of type ↑h0i, which will be understood informally as an existence predicate.

7.4

Standard Modal Models

I begin the formal presentation of semantics for higher-order modal logic with the modal analog of standard models. The new piece of semantical machinery added to that for classical logic is the possible world structure. Definition 7.4.1 [Kripke Frame] A Kripke frame is a structure hG, Ri. In it, G is a non-empty set (of possible worlds), and R is a binary relation on G (called accessibility). An augmented frame is a structure hG, R, Di where hG, Ri is a frame, and D is a non-empty set, the (ground-level) domain. The notion of a Kripke frame should be familiar from propositional modal logic treatments, and I do not elaborate on it. As usual, different restrictions on

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R give rise to different modal logics. The only two I will be interested in are K, for which there are no restrictions on R, and S5, for which R is an equivalence relation. Note that the ground-level domain, D, is not world dependent, since the choice was to take type-0 quantification as possibilist and not actualist. Next I say what the objects of each type are, relative to a choice of groundlevel domain. This is analogous to what was done in Part I, in Definition 2.1.1. To make the definition easier to state, I use some standard notation from set theory. The first item is something that was used before, but I include it here for completeness sake. 1. For sets A1 , . . . , An , A1 × · · · × An is the collection of all n-tuples of the form ha1 , . . . , an i, where a1 ∈ A1 , . . . , an ∈ An . The 1-tuple hai is generally identified with a. 2. For a set A, P(A) is the power set of A, the collection of all subsets of A. 3. For sets A and B, AB is a function space, the set of all functions from B to A. Definition 7.4.2 [Objects, Extensional and Intensional] Let G be a non-empty set (of possible worlds) and let D be a non-empty set (the ground-level domain). For each type t, I define the collection [[t, D, G ]], of objects of type t with respect to D and G, as follows. 1. [[0, D, G ]] = D. 2. [[ht1 , . . . , tn i, D, G ]] = P([[t1 , D, G ]] × · · · × [[tn , D, G ]]). G

3. [[↑t, D, G ]] = [[t, D, G ]] . O is an object of type t if O ∈ [[t, D, G ]]. O is an intensional or extensional object according to whether its type is intensional or extensional. As before, O is used, with or without subscripts, to stand for objects. Now the final notion of the section. Definition 7.4.3 [Modal Model] A (higher-order) modal model for L(C) is a structure M = hG, R, D, Ii, where hG, R, Di is an augmented frame and I is an interpretation. The interpretation I must meet the following conditions. 1. If At is a constant symbol of type t, I(At ) is an object of type t in hG, R, Di. 2. If =ht,ti is an equality constant symbol, I(=ht,ti ) is the equality relation on [[t, D, G ]].

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7.5

81

Truth in a Model

In this section I say how truth is to be assigned to formulas, at worlds, in models, and how values should be assigned to terms. I lead up to a proper definition after a few preliminary notions. Definition 7.5.1 [(Modal) Valuation] The mapping v is a modal valuation in the modal model M = hG, R, D, Ii if v assigns to each variable αt of type t some object of type t, that is, v(αt ) ∈ [[t, D, G ]]. A variant of a valuation is defined exactly as classically. The definition of term designation needs modification, of course. A term like ↓τ is intended to designate the extension of the intensional object designated by τ . To determine this a context is needed—the designation of τ where, under what circumstances? Consequently the designation of a term is defined with respect to a valuation, an interpretation, and a context—a possible world. Definition 7.5.2 [Designation of a Term] Let M = hG, R, D, Ii be a modal model, let v be a valuation in it, and let Γ ∈ G be a possible world. Define a mapping (v ∗ I ∗ Γ), assigning to each term an object that is the designation of that term. 1. If A is a constant symbol of L(C) then (v ∗ I ∗ Γ)(A) = I(A). 2. If α is a variable then (v ∗ I ∗ Γ)(α) = v(α). 3. If τ is a term of type ↑t then (v ∗ I ∗ Γ)(↓τ ) = (v ∗ I ∗ Γ)(τ )(Γ) 4. If hλα1 , . . . , αn .Φi is a predicate abstract of L(C) of type ↑ht1 , . . . , tn i, then (v∗I ∗Γ)(hλα1 , . . . , αn .Φi) is the function that assigns to an arbitrary world ∆ the following member of [[ht1 , . . . , tn i, D, G ]]: {hw(α1 ), . . . , w(αn )i | w is an α1 , . . . , αn variant of v and M, ∆ °w Φ} Item 4 tells us this definition is part of a mutual recursion—Definition 7.5.4 below is the other part. Item 3 is a little awkward to read. (v ∗ I ∗ Γ)(τ )(Γ) means: evaluate τ using (v ∗ I ∗ Γ), getting a function, an intension, then evaluate that function at Γ. Generally the simpler notation (v ∗ I ∗ Γ)(τ, Γ) will be used for this. Similarly for v(α, Γ) and I(A, Γ), when α and A are of intensional type. Classically, some alternative notation was introduced to make working with models easier, Definition 2.2.5. That carries over to a modal context in a natural way Definition 7.5.3 [Special Notation] Suppose v is a valuation, and w is the α1 , . . . , αn variant of v such that w(α1 ) = O1 , . . . , w(αn ) = On . Then, if M, Γ °w Φ this will generally be symbolized as follows. M, Γ °v Φ[α1 /O1 , . . . , αn /On ].

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Part 4 of Definition 7.5.2 can be restated using the new notation: 4. If hλα1 , . . . , αn .Φi is a predicate abstract of L(C) of type ↑ht1 , . . . , tn i, then (v ∗ I ∗ Γ)(hλα1 , . . . , αn .Φi) is the function f given by the following. f (Γ) = {hO1 , . . . , On i | M, Γ °v Φ[α1 /O1 , . . . , αn /On ]} The next item should be compared with Definition 2.2.4: worlds must now be taken into account. And of course Definitions 7.5.2 and 7.5.4 involve a mutual recursion, just as was the case classically. Definition 7.5.4 [Truth of a Formula] Let M = hG, R, D, Ii be a modal model, and let v be a valuation in it. The notion of formula Φ being true at world Γ of G in model M with respect to v, denoted M, Γ °v Φ, is characterized as follows. 1. For an atomic formula τ (τ1 , . . . , τn ), (a) If τ is of an extensional type, M, Γ °v τ (τ1 , . . . , τn ) provided h(v ∗ I ∗ Γ)(τ1 ), . . . , (v ∗ I ∗ Γ)(τn )i ∈ (v ∗ I ∗ Γ)(τ ). (b) If τ is of an intensional type, M, Γ °v τ (τ1 , . . . , τn ) provided M, Γ °v (↓τ )(τ1 , . . . , τn ). This reduces things to the previous case. 2. M, Γ °v ¬Φ if it is not the case that M, Γ °v Φ. 3. M, Γ °v Φ ∧ Ψ if M, Γ °v Φ and M, Γ °v Ψ. 4. M, Γ °v (∀α)Φ if M, Γ °v0 Φ for every α-variant v 0 of v. 5. M, Γ °v ¤Φ if M, ∆ °v Φ for all ∆ ∈ G such that ΓR∆. As usual, other connectives, quantifiers, and modal operators can be introduced via definitions, with the expected behavior. For instance: M, Γ °v ♦Φ if M, ∆ °v Φ for some ∆ ∈ G such that ΓR∆. Note that part 4, the quantifier case, can be given in a different form using the alternative notation. It is often much handier to work with. 4. M, Γ °v (∀α)Φ if M, Γ °v Φ[α/O] for all objects O of the same type as α. It follows from the definitions, that M, Γ °v ¤Φ[α1 /O1 , . . . , αn /On ] if and only if M, ∆ °v Φ[α1 /O1 , . . . , αn /On ] for all ∆ such that ΓR∆. It also follows from the definitions that hλα.Φ(α)i(τ ) and Φ(τ ) are equivalent under certain circumstances. For instance, this is the case if τ is a constant symbol of either intensional or extensional type. It is not the case if τ involves the extension-of operator, ↓. Rather than give exact conditions, I give the following, which is more useful for present purposes. Proposition 7.5.5 Suppose that (v ∗ I ∗ Γ)(τ1 ) = O1 , . . . , (v ∗ I ∗ Γ)(τn ) = On in model M. Then M, Γ °v hλα1 , . . . , αn .Φi(τ1 , . . . , τn ) ⇔ M, Γ °v Φ[α1 /O1 , . . . , αn /On ].

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7.6

83

Examples

Here is a simple informal example to start with. Suppose we take possible worlds to be points in time (within a reasonable range from near past to near future). Also take the accessibility relation to always hold, so that ¤Φ means Φ holds at all times. Does there exist, now, somebody whose parents are necessarily not alive? Certainly—the oldest person in the world. After all, the oldest person can never have living parents. But on the other hand, there was a time when the oldest person had living parents. There seems to be a discrepancy here. Say ϕ(x) is read as “x has no living parents. We are asking about the truth of (∃x)¤ϕ(x). The key question is, what type of variable is x? If we think of the quantifier as ranging over objects—so x is of type 0—then when we say the oldest person in the world instantiates the existential quantifier we are saying a particular person does so. If we designate the oldest person now as the value of x, instantiating the existential quantifier, while ϕ(x) is certainly true now, for this value of x, there are earlier worlds in which the person who is the oldest now had living parents. Thus we do not have ¤ϕ(x), where x has as value the oldest person in the present world. The proposed instantiation for the existential quantifier does not work. More generally, it is easy to see that (∃x)¤ϕ(x) can never be true, now or at any other point of time, provided we think of quantifiers as ranging over objects or individuals. On the other hand if quantifiers range over individual concepts—so that x is of type ↑0—we would certainly have the truth of (∃x)¤ϕ(x) since taking the value of x to be the oldest-person concept would serve as a correct instantiation of ¤ϕ(x). The type theory of (Bressan 1972) makes intensional objects basic. The second-order logic of (Cocchiarella 1969) quantifies over extensional objects at the first-order level, and over intensional objects at the second-order level. The higher-order modal logic of (Fitting 1998), which is a forerunner to this book, had quantification only over extensional objects. Finally, the first-order treatment of (Fitting & Mendelsohn 1998) involves a kind of mixed system, and more will be said about it shortly. Now for some further examples, which will be treated more formally. Example 7.6.1 Suppose x is a variable of type 0 and P is a constant symbol of type ↑h0i. The following formula is valid, where X is of type ↑h0i. hλX.♦(∃x)X(x)i(P ) ⊃ ♦hλX.(∃x)X(x)i(P )

(7.3)

I leave it to you to verify the validity of this—one way is to show both the antecedent and the consequent are equivalent to ♦(∃x)P (x). On the other hand, the following formula is not valid, where X is of type h0i. hλX.♦(∃x)X(x)i(↓P ) ⊃ ♦hλX.(∃x)X(x)i(↓P )

(7.4)

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Here is an informal illustration to help you understand intuitively why this formula is invalid. Suppose that on an island there are two societies, optimists and pessimists, separated by a volcano. The optimists, being nonetheless realistic, accept something as really possible only if the pessimists believe it. Now, suppose the optimists think the volcano is beautiful, while the pessimists think nothing is beautiful. (I know, it follows that the optimists, while thinking the volcano is beautiful, also don’t think that is possible. That’s the set-up.) Take for P the concept of beauty—it maps each society to the set of things that society accepts as beautiful. For the optimists, hλX.♦(∃x)X(x)i(↓P ) is true, because in the optimist society the extension of P is the set consisting of the volcano, so the formula asserts that ♦(∃x)X(x) is the case, when X is understood to be that set, and indeed this is possible, since even the pessimists would agree that something is in the set consisting of the volcano. Essentially, for the optimists the antecedent asserts that the pessimists believe the optimists think something is beautiful. On the other hand, ♦hλX.(∃x)X(x)i(↓P ) is not true for the optimists, because hλX.(∃x)X(x)i(↓P ) is not the case for the pessimists, and this happens because the pessimists do not think anything is beautiful. This informal example can be turned into a formal argument. Here is a model, M = hG, R, D, Ii, in which (7.4) is not valid. The collection of worlds, G, contains two members, Γ and ∆, with ΓR∆. Think of Γ as the optimists and ∆ as the pessimists. The domain, D is the set {7} (think of the number 7 as the volcano). I show 7 available at both worlds as a reminder that domains are constant and quantifiers are possibilist. The constant symbol P is interpreted to be a type ↑h0i object: the function that is {7} at Γ and ∅ at ∆. Thus I(P ) is true of 7 at Γ, and of nothing at ∆. This gives us the model—it is presented schematically below.

Γ

7

I(P, Γ) = {7}

? ∆

7

I(P, ∆) = ∅

The first claim is that, for an arbitrary valuation v, we have M, Γ °v hλX.♦(∃x)X(x)i(↓P ).

(7.5)

Since (v ∗ I ∗ Γ)(↓P ) = (v ∗ I ∗ Γ)(P, Γ) = I(P, Γ) = {7}, by Proposition 7.5.5 we will have (7.5) provided we have M, Γ °v ♦(∃x)X(x)[X/{7}]

(7.6)

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which will be the case provided we have M, ∆ °v (∃x)X(x)[X/{7}].

(7.7)

But, since 7 ∈ {7}, we have M, ∆ °v X(x)[X/{7}, x/7]

(7.8)

and hence we have (7.7). We thus have established (7.5). Next it is shown that M, Γ 6°v ♦hλX.(∃x)X(x)i(↓P )

(7.9)

which, together with (7.5), gives us the invalidity of (7.4). Well, suppose otherwise, that is, suppose we had M, Γ °v ♦hλX.(∃x)X(x)i(↓P ).

(7.10)

M, ∆ °v hλX.(∃x)X(x)i(↓P ),

(7.11)

Then we must have

and so, since (v ∗ I ∗ ∆)(↓P ) = ∅, M, ∆ °v (∃x)X(x)[X/∅].

(7.12)

It is easy to see we can not have this, and thus we have (7.9). Example 7.6.2 This example is one that is unexpected on superficial consideration, although deeper thought says it should not be. The following formula is valid, with types of variables and constants as in (7.4). hλX.♦(∃x)X(x)i(↓P ) ⊃ hλX.(∃x)X(x)i(↓P )

(7.13)

To show validity, suppose M = hG, R, D, Ii is an arbitrary model, Γ ∈ G is an arbitrary world, and v is an arbitrary valuation. Suppose M, Γ °v hλX.♦(∃x)X(x)i(↓P ).

(7.14)

M, Γ °v ♦(∃x)X(x)[X/O]

(7.15)

Then

where O = (v ∗ I ∗ Γ)(↓P ) = I(P, Γ). Then, for some ∆ ∈ G such that ΓR∆, M, ∆ °v (∃x)X(x)[X/O]

(7.16)

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and so, for some object o we have M, ∆ °v X(x)[X/O, x/o].

(7.17)

For (7.17) to be the case, we must have o ∈ O. Now, M, Γ °v X(x)[X/O, x/o]

(7.18)

M, Γ °v (∃x)X(x)[X/O]

(7.19)

M, Γ °v hλX.(∃x)X(X)i(↓P )

(7.20)

since o ∈ O. Consequently

and finally,

since O = (v ∗ I ∗ Γ)(↓P ). Since we went from (7.14) to (7.20), the validity of (7.13) has been established. Some comments on the example above. The point is, the term ↓P is given broad scope in both the antecedent and the consequent of the implication. This essentially says its meaning in alternative worlds will be the same as in the present world. Under these circumstances, existence of something falling under ↓P in an alternate world is equivalent to existence of something falling under ↓P in the present world. This is just a formal variation on the old observation that, in Kripke models, if relation symbols could not vary their interpretation from world to world, modal operators would have no visible effect. The distinction between intensional and extensional types is complex. The following two examples should help make clear the role of the ↓ operator. Example 7.6.3 Let x and c be of type ↑0, and P be of type ↑h↑0i. The following formula is valid. ¤P (c) ⊃ (∃x)¤P (x)

(7.21)

To show (7.21) is valid let M = hG, R, D, Ii be an arbitrary model, Γ be an arbitrary world in G, and v be an arbitrary valuation. Suppose we had the following. M, Γ °v ¤P (c)

(7.22)

Let ∆ be an arbitrary world such that ΓR∆. We must have M, ∆ °v P (c) from which it follows that

(7.23)

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M, ∆ °v P (x)[x/I(c)].

87

(7.24)

Since ∆ was arbitrary, we have M, Γ °v ¤P (x)[x/I(c)]

(7.25)

M, Γ °v (∃x)¤P (x)

(7.26)

and hence

Since we went from (7.22) to (7.26), the validity of (7.21) has been established. Example 7.6.4 This continues the previous example. Let c be of type ↑0, but now let x be of type 0 and P be of type ↑h0i. The following formula is not valid. ¤P (↓c) ⊃ (∃x)¤P (x)

(7.27)

To show the non-validity of (7.27) a specific model, M = hG, R, D, Ii, is constructed. In this model, G consists of three possible worlds: Γ, ∆, Ω. We have ΓR∆, ΓRΩ, and R holds in no other cases. The domain D is {1, 2}. I interprets c by a function that is 1 at ∆, 2 at Ω, and either 1 or 2 at Γ (it won’t matter). Likewise I interprets P by the function that is {1} at ∆, {2} at Ω, and some arbitrary value at Γ. Here is the model schematically.

1, 2

Γ @

@ @

∆

1, 2

I(c, ∆) = 1 I(P, ∆) = {1}

@ R @ Ω

1, 2

I(c, Ω) = 2 I(P, Ω) = {2}

I leave it to you to check that (7.27) is not valid in this model. Example 7.6.5 The last example is in three parts. To make the type structure work out, let x, y, and z be of type 0, and X, Y , and Z be of type ↑0. Consider the following three formulas.

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(∀Z)hλx.¤hλy.x = yi(↓Z)i(↓Z) (∀z)hλx.¤hλy.x = yi(z)i(z)

(7.28) (7.29)

(∀Z)hλX.¤hλY.X = Y i(Z)i(Z)

(7.30)

Of the formulas above, (7.28) is not valid, but (7.29) and (7.30) are both valid. I leave the work to you. I note that in (Fitting & Mendelsohn 1998) it was shown that, in a first-order setting, the constructions used above relate directly to rigidity. Both extensional and intensional objects, as such, are the same from world to world, but the extensional object designated by an intensional object can vary. This is what the example illustrates.

Exercises Exercise 7.6.1 Show the formula (7.3) is valid. Exercise 7.6.2 This is a variation on formula (7.13); the formula looks the same, but the types are different. Show the validity of hλX.♦(∃x)X(x)i(↓P ) ⊃ hλX.(∃x)X(x)i(↓P ) where x is of type ↑ 0 and P is of type ↑ h↑ 0i. The fact that ground level quantification is possibilist—constant domain—will be needed. Exercise 7.6.3 Show the validity of the following, which looks a little like a version of the Barcan formula: ♦(∃x)P (x) ⊃ (∃X)♦P (↓X). x is of type 0, X is of type ↑0 and P is of type ↑h0i. Exercise 7.6.4 Show the non-validity of the following, where x is of type 0, X is of type h0i, and P is of type ↑h0i. ♦(∃x)hλX.X(x)i(↓P ) ⊃ (∃x)hλX.♦X(x)i(↓P ) Exercise 7.6.5 Verify the claims made in Example 7.6.5.

7.7

Related Systems

There have been many other versions of quantified modal logics in the literature. Here I briefly say how a few of them relate to the one presented here. First-order modal logic, as given in (Hughes & Cresswell 1996a) or (Fitting 1983) say, has variables and constant symbols of type 0, and predicate symbols of types ↑h0, 0, . . . , 0i. Thus quantification is over ground-level objects; constant symbols designate such objects and hence are rigid. Predicates, of course, vary in meaning from world to world—they are intensional. Treating them extensionally would force modal logic to collapse to classical.

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In (Fitting & Mendelsohn 1998), conventional first-order modal logic is extended by allowing non-rigid terms, and an abstraction mechanism. Relating things to the present system, variables are still of type 0, but constant symbols are of type ↑0: they are individual concepts. Allowing intensional constant symbols greatly enhances the expressibility of the language. Predicate symbols are still of types ↑h0, 0, . . . , 0i. The fit between intension and extension is achieved by treating hλx.Φi(c), where c is a constant symbol, as if it were hλx.Φi(↓c) in the present system. In effect, this means the logic of (Fitting & Mendelsohn 1998) can be embedded in the higher-type version given here. (Actually, this is not quite correct, since the logic of (Fitting & Mendelsohn 1998) allows function symbols, and partial designation, neither of which is the case here. But with these exceptions, the embedding claim is correct.) Montague proposed a higher-order modal logic specifically as a logic of intensions, in (Montague 1960, Montague 1968, Montague 1970). It is presented most fully in (Gallin 1975). Essentially, it is the present system with only intensional types (except at the lowest level). To be more specific, define a Gallin/Montague type, as follows. 1. 0 is a Gallin/Montague type. 2. If t1 , . . . , tn are Gallin/Montague types, so is ↑ht1 , . . . , tn i. Then the logic of (Gallin 1975) can be identified with the sublogic of the system given here, in which all constant symbols and variables are restricted to be of some Gallin/Montague type. Indeed, the present system was created by adding extensional types to the Gallin logic. Bressan is a pioneer in the study of higher-order modal logics (Bressan 1972). I must confess that I do not fully understand his presentation. It is an S5 system rather like that of Gallin, though Gallin’s is for a broader variety of logics. In it extensional objects are not explicitly present, but rather are identified with constant intensional objects. Also abstractions are not taken as primitive, but rather are defined in terms of definite descriptions.

7.8

Henkin/Kripke Models

In the classical case there were good reasons for introducing non-standard higher-order models, and those same reasons apply in the modal case as well. Since modal versions of Henkin and unrestricted Henkin models are relatively straightforward extensions of the classical versions, I confine things to a brief sketch, and refer to Part I and your intelligence for the details. Definition 7.4.1 specified Kripke frames and augmented Kripke frames. What takes the place of augmented Kripke frames is the following. Definition 7.8.1 [Henkin/Kripke Frame] Let hG, Ri be a Kripke frame. H is a Henkin domain function in this frame if it is a function on the collection of types and:

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1. H(0) = D 2. H(ht1 , . . . , tn i) ⊆ P(H(t1 ) × · · · × H(tn )). 3. H(↑t) ⊆ [H(t)]G . I is an interpretation if it maps each constant symbol of L(C) of type t to a member of H(t). Finally, M = hG, R, H, Ii is a Henkin/Kripke frame for L(C). If items 2 and 3 above hold with =, and not just ⊆, the Henkin/Kripke model is standard. Standard models correspond exactly to the models defined in Section 7.4. Definition 7.8.2 [Abstraction Designation Function] A function A is an abstraction designation function in the Henkin/Kripke frame M = hG, R, H, Ii, with respect to the language L(C), provided A(v, hλα1 , . . . , αn .Φi) is some object of type t in M, for each valuation v in M and for each predicate abstract hλα1 , . . . , αn .Φi of L(C) of type t. Term designation gets the obvious modification. Definition 7.8.3 [Designation of a Term] Let M = hG, R, H, Ii be a Henkin/Kripke frame with A an abstraction designation function in it. For each valuation v in it, define a mapping (v ∗ I ∗ Γ ∗ A) assigning to each term a designation for that term, in the context (possible world) Γ. 1. If A is a constant symbol of L(C) then (v ∗ I ∗ Γ ∗ A)(A) = I(A). 2. If α is a variable then (v ∗ I ∗ Γ ∗ A)(α) = v(α). 3. If τ is a term of type ↑t then (v ∗ I ∗ Γ ∗ A)(↓τ ) = (v ∗ I ∗ Γ ∗ A)(τ )(Γ). 4. If hλα1 , . . . , αn .Φi is a predicate abstract of L(C) of type ↑ht1 , . . . , tn i, then (v ∗ I ∗ Γ ∗ A)(hλα1 , . . . , αn .Φi) = A(v, hλα1 , . . . , αn .Φi). As usual, (v ∗ I ∗ Γ ∗ A)(τ, Γ) is written for (v ∗ I ∗ Γ ∗ A)(τ )(Γ). Now truth, at a world, also has the expected characterization. Definition 7.8.4 [Truth of a Formula] Let M = hG, R, H, Ii be a Henkin/ Kripke frame, let A be an abstraction designation function, and let v be a valuation. 1. For an atomic formula τ (τ1 , . . . , τn ), (a) If τ is of an intensional type, M, Γ °v τ (τ1 , . . . , τn ) provided h(v ∗ I ∗ Γ ∗ A)(τ1 ), . . . , (v ∗ I ∗ Γ ∗ A)(τn )i ∈ (v ∗ I ∗ Γ ∗ A)(τ, Γ). (b) If τ is of an extensional type, M, Γ °v τ (τ1 , . . . , τn ) provided h(v ∗ I ∗ Γ ∗ A)(τ1 ), . . . , (v ∗ I ∗ Γ ∗ A)(τn )i ∈ (v ∗ I ∗ Γ ∗ A)(τ ).

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2. M, Γ °v,A ¬Φ if it is not the case that M, Γ °v,A Φ. 3. M, Γ °v,A Φ ∧ Ψ if M, Γ °v,A Φ and M, Γ °v,A Ψ. 4. For α of type t, M, Γ °v,A (∀α)Φ if M, Γ °v0 ,A Φ for every α-variant v 0 of v such that v 0 (α) ∈ H(t). 5. M, Γ °v,A ¤Φ if M, ∆ °v,A Φ for all ∆ ∈ G such that ΓR∆. Finally, the following should be no surprise. Definition 7.8.5 [Henkin/Kripke Model] hM, Ai is a Henkin/Kripke model provided that, for each predicate abstract hλα1 , . . . , αn .Φi of L(C) of type ↑t, (v ∗ I ∗ Γ ∗ A)(hλα1 , . . . , αn .Φi) is the function f given by the following: f (∆) = {hO1 , . . . , On i ∈ H(t) | M, ∆ °v Φ[α1 /O1 , . . . , αn /On ]}. The various theorems concerning uniqueness of an abstraction designation function, if one exists, and the good behavior of substitution (Section 2.6) all carry over to the modal setting. I leave this to you. The semantics just presented is extensional, in the sense of Part I. A modal analog of unrestricted Henkin models can also be developed, along the lines of Section 2.5. 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 8.1

The Rules

There are several varieties of tableau rules for modal logic. This book uses a version of prefixed tableaus. These incorporate a kind of naming mechanism 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 S5 since these are the extreme cases. I refer you to the literature for modifications appropriate for other modal logics—see (Fitting & Mendelsohn 1998) for instance. 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 S5 is simplicity itself. Definition 8.1.1 [Prefix] A K prefix is a finite sequence of positive integers, written with periods as separators (1.2.1.1 is an example). An S5 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 S5 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. 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 τ is allowed to have a prefix. If we think of σ as designating a possible world, we should think of σ τ as representing the extensional object that τ designates at σ. Formally, if τ is of type ↑t, then σ τ 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 92

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terms as subscripts, τσ , though of course the idea is the same, and I still often refer to them as prefixes. So, if one thinks of σ as designating possible world Γ, and τ as having the function f as its meaning, then τσ should be thought of as designating the object f (Γ). By L+ (C) is meant L(C) enlarged with parameters, and allowing prefixes (written as subscripts) on intensional terms including parameters of intensional type (they will not be needed on free variables that are not parameters). 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. I now turn to the one that gave them their name. Definition 8.1.2 [Prefixed Formula] A prefixed formula is an expression of the form σ Φ, where σ is a prefix and Φ is a formula of L+ (C). Think of σ Φ as saying that formula Φ is true at the world that σ names. Note that this use of prefixes does not compound, that is, σ Φ is a prefixed formula if Φ is a formula, and not something built up from prefixed formulas. Definition 8.1.3 [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 Φ, begin with a tree that has 1 ¬Φ at its root, and nothing else. Think of 1 as an arbitrary world. This initial tableau intuitively asserts that Φ 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 Φ. Propositional and quantifier branch extension rules are just as in the classical case, except that prefixes must be “carried along.” I give the rules explicitly, to make sure this is understood. In these, and throughout, I use σ, σ 0 , and the like to stand for prefixes. Definition 8.1.4 [Conjunctive Rules] For any prefix σ, σX ∧Y σX σY

σ ¬(X ∨ Y ) σ ¬X σ ¬Y

σ ¬(X ⊃ Y ) σX σ ¬Y

σX ≡ Y σX ⊃ Y σY ⊃ X

Definition 8.1.5 [Double Negation Rule] For any prefix σ, σ ¬¬X σX

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Definition 8.1.6 [Disjunctive Rules] For any prefix σ, σX ∨Y σX σY

σ ¬(X ∧ Y ) σ ¬X σ ¬Y

σX ⊃ Y σ ¬X σ Y

σ ¬(X ≡ Y ) σ ¬(X ⊃ Y ) σ ¬(Y ⊃ X)

This completes the classical connective rules. The motivation should be intuitively obvious. For instance, if X ∧ Y is true at a world named by σ, both X and Y are true there, and so a branch containing σ X ∧ Y can be extended with σ X and σ Y . Next come the modal rules. Naturally these differ between the two logics being considered. It is here that the structure of prefixes plays a role. The idea is, if ♦X 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 ♦X is true. Definition 8.1.7 [Possibility Rules for K] If the prefix σ.n is new to the branch, σ ♦X σ.n X

σ ¬¤X σ.n ¬X

Definition 8.1.8 [Possibility Rules for S5] If the positive integer n is new to the branch, σ ♦X nX

σ ¬¤X n ¬X

Notice that for both logics there is a newness condition. This implicitly treats ♦ as a kind of existential quantifier. Correspondingly, the following rules treat ¤ as a version of the universal quantifier. Definition 8.1.9 [Necessity Rules for K] If the prefix σ.n already occurs on the branch, σ ¤X σ.n X

σ ¬♦X σ.n ¬X

Definition 8.1.10 [Necessity Rules for S5] For any positive integer n that already occurs on the branch, σ ¤X nX

σ ¬♦X n ¬X

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Many examples of the application of these propositional rules can be found in (Fitting & Mendelsohn 1998). I do not give any here. For the existential quantifier rules, just as in the classical case, parameters must be introduced. Thus proofs of sentences of L(C) are forced to be in the larger language L+ (C). Definition 8.1.11 [Existential Rules] In the following, pt is a parameter of type t that is new to the tableau branch. σ (∃αt )Φ(αt ) σ Φ(pt )

σ ¬(∀αt )Φ(αt ) σ ¬Φ(pt )

Terms of the form ↓τ may vary their denotation from world to world of a model, because the extension of the intensional term τ can change from world to world. Such terms should not be used when instantiating a universally quantified formula. Definition 8.1.12 [Relativized Term] If τ is a grounded intensional term, ↓τ is a relativized term. Definition 8.1.13 [Universal Rules] In the following, τ t is any grounded term of type t that is not relativized. σ (∀αt )Φ(αt ) σ Φ(τ t )

σ ¬(∃αt )Φ(αt ) σ ¬Φ(τ t )

Now I give the rules for atomic formulas. The first rule essentially says that, at a world, an intensional predicate applies to terms if those terms are in the extension of the predicate at that world. Definition 8.1.14 [Intensional Predication Rules] Let τ be a grounded intensional term, and τ1 , . . . , τn be arbitrary grounded terms. σ τ (τ1 , . . . , τn ) σ (↓τ )(τ1 , . . . , τn )

σ ¬τ (τ1 , . . . , τn ) σ ¬(↓τ )(τ1 , . . . , τn )

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 τ @σ as τ evaluated at σ. On non-relativized terms, such evaluation has no effect. Definition 8.1.15 [Evaluation At a Prefix] Let σ be a prefix. 1. For a relativized term ↓τ , take (↓τ )@σ = τσ . 2. For a non-relativized term τ , take τ @σ = τ . The next rule covers the case of an extensional predicate applying to terms. Essentially, it says we eliminate the relativized terms by evaluation.

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Definition 8.1.16 [Extensional Predication Rules] Let τ be a grounded extensional term, and τ1 , . . . , τn be arbitrary grounded terms. σ τ (τ1 , . . . , τn ) σ (τ @σ)(τ1 @σ, . . . , τn @σ)

σ ¬τ (τ1 , . . . , τn ) σ ¬(τ @σ)(τ1 @σ, . . . , τn @σ)

Here is a simple example of how these rules work. Suppose A is of intensional type ↑h0i and b is of type 0. If σ A(b) occurs on a branch, we may add σ (↓A)(b) by an Intensional Predication Rule. Now the Extensional Predication Rule applies; (↓A)@σ = Aσ and b@σ = b, so we may add σ Aσ (b). Think of this as saying, since A(b) is true at the world that σ designates, then b is in the extension of A at that world. There are atomic formulas that must evaluate the same way no matter what world is involved. Definition 8.1.17 [World Independent] An atomic formula τ (τ1 , . . . , τn ) is called world independent if none of τ , τ1 , . . . , τn is relativized, and τ is extensional. Next is a rule that says sometimes the specific prefix does not really matter. Definition 8.1.18 [World Shift Rules] Let τ (τ1 , . . . , τn ) be world independent. If σ 0 already occurs on the branch, σ τ (τ1 , . . . , τn ) σ 0 τ (τ1 , . . . , τn )

σ ¬τ (τ1 , . . . , τn ) σ 0 ¬τ (τ1 , . . . , τn )

Finally, rules intended to capture the meaning of predicate abstracts. They correspond to Proposition 7.5.5. Note the presence of a prefix on the predicate abstract. We must know at what world the abstract is to be evaluated before doing so. Thus a previous application of an Extensional Predication Rule must be involved. And since this needs an extensional term, while predicate abstracts are intensional, a still prior application of Intensional Predication must be involved. Finally, σ and σ 0 need not be the same—a World Shift Rule may also have been applied at some point. Definition 8.1.19 [Predicate Abstract Rules] In the following, τ1 , . . . , τn are non-relativized terms. σ 0 hλα1 , . . . , αn .Φ(α1 , . . . , αn )iσ (τ1 , . . . , τn ) σ Φ(τ1 , . . . , τn ) σ 0 ¬hλα1 , . . . , αn .Φ(α1 , . . . , αn )iσ (τ1 , . . . , τn ) σ ¬Φ(τ1 , . . . , τn ) Finally what, exactly, constitutes a proof or a derivation. Definition 8.1.20 [Closure] A tableau branch is closed if it contains σ Ψ and σ ¬Ψ, for some formula Ψ of L+ (C).

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Definition 8.1.21 [Tableau Proof] For a sentence Φ of L(C), a closed tableau beginning with 1 ¬Φ is a proof of Φ. In classical logic, when one says Φ follows semantically from a set S of sentences, one means that Φ is true in all the models in which the members of S are true. Modally things are more complex, since we not only have models, but possible worlds within them to deal with. Still, the semantical treatment of local and global assumptions is sufficiently intuitive. Φ is said to follow from a set S of global assumptions and a set U of local assumptions provided, for every modal model in which the members of S are true at every world, Φ is true at each world at which the members of U are true. An analysis of this notion, with special consideration to deduction theorems, can be found in (Fitting 1983, Fitting 1993, Fitting & Mendelsohn 1998). There is a notion of derivation corresponding to this. Definition 8.1.22 [Local and Global Assumptions] Let S and 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 σ Y can be added to the end of any open branch on which σ appears as a prefix. Definition 8.1.23 [Tableau Derivation] A sentence Φ has a derivation from global assumptions S and local assumptions U if there is a closed tableau beginning with 1 ¬Φ, 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 8.2 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 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 cut-elimination property—see Theorem4.3.2. It is a consequence of this that any previously proved result can simply be introduced into a tableau. The argument is simple. Suppose Φ has been given a tableau proof, and now, in a later tableau, we wish to introduce Φ onto a tableau branch (with

8.2

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 S5.

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Example 8.2.1 This provides a proof for (7.3) which was verified valid in Example 7.6.1. The formula is hλX.♦(∃x)X(x)i(P ) ⊃ ♦hλX.(∃x)X(x)i(P ) in which x is a variable of type 0 and X is a variable and P a constant symbol, both of type ↑h0i. 1 ¬[hλX.♦(∃x)X(x)i(P ) ⊃ ♦hλX.(∃x)X(x)i(P )] 1 hλX.♦(∃x)X(x)i(P ) 2. 1 ¬♦hλX.(∃x)X(x)i(P ) 3. 1 ↓hλX.♦(∃x)X(x)i(P ) 4. 1 hλX.♦(∃x)X(x)i1 (P ) 5. 1 ♦(∃x)P (x) 6. 1.1 (∃x)P (x) 7. 1.1 ¬hλX.(∃x)X(x)i(P ) 8. 1.1 ¬ ↓hλX.(∃x)X(x)i(P ) 9. 1.1 ¬hλX.(∃x)X(x)i1.1 (P ) 10. 1.1 ¬(∃x)P (x) 11.

1.

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 8.2.2 Here is a proof of (7.13), which was shown to be valid earlier. hλX.♦(∃x)X(x)i(↓P ) ⊃ hλX.(∃x)X(x)i(↓P ) See Example 7.6.2 for a discussion of the significance of this formula. 1 ¬[hλX.♦(∃x)X(x)i(↓P ) ⊃ hλX.(∃x)X(x)i(↓P )] 1 hλX.♦(∃x)X(x)i(↓P ) 2. 1 ¬hλX.(∃x)X(x)i(↓P ) 3. 1 ↓hλX.♦(∃x)X(x)i(↓P ) 4. 1 ¬ ↓hλX.(∃x)X(x)i(↓P ) 5. 1 hλX.♦(∃x)X(x)i1 (P1 ) 6. 1 ¬hλX.(∃x)X(x)i1 (P1 ) 7. 1 ♦(∃x)P1 (x) 8. 1 ¬(∃x)P1 (x) 9. 1.1 (∃x)P1 (x) 10. 1.1 P1 (p) 11. 1 P1 (p) 12. 1 ¬P1 (p) 13.

1.

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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 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 8.2.1 Give a tableau proof of the following hλX.♦(∃x)X(x)i(↓P ) ⊃ hλX.(∃x)X(x)i(↓P ) where x is of type ↑0, X is of type h↑0i and P is of type ↑h↑0i. Exercise 8.2.2 Give a tableau proof of the following ♦(∃x)P (x) ⊃ (∃X)♦P (↓X) where x is of type 0, X is of type ↑0 and P is of type ↑h0i.

8.3

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 8.3.1 [Derived Closure Rule] Suppose X is a world independent atomic formula. A branch closes if it contains σ X and σ 0 ¬X. The justification for this is easy. Using the World Shift Rule, if σ X is on a branch, we can add σ 0 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 8.3.2 [Derived Intensional Predication Rule] Let τ be a grounded intensional term, and τ1 , . . . , τn be arbitrary grounded terms. σ τ (τ1 , . . . , τn ) σ τσ (τ1 @σ, . . . , τn @σ)

σ ¬τ (τ1 , . . . , τn ) σ ¬τσ (τ1 @σ, . . . , τn @σ)

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

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Definition 8.3.3 [Derived Subscripted Abstract Rule] In the following, τ1 , . . . , τn are arbitrary grounded terms. σ 0 hλα1 , . . . , αn .Φ(α1 , . . . , αn )iσ (τ1 , . . . , τn ) σ Φ(τ1 @σ 0 , . . . , τn @σ 0 ) σ 0 ¬hλα1 , . . . , αn .Φ(α1 , . . . , αn )iσ (τ1 , . . . , τn ) σ ¬Φ(τ1 @σ 0 , . . . , τn @σ 0 ) This abbreviates the application of the extensional predication rule, followed by predicate abstraction. Definition 8.3.4 [Derived Unsubscripted Abstract Rule] In the following, τ1 , . . . , τn are grounded terms. σ hλα1 , . . . , αn .Φ(α1 , . . . , αn )i(τ1 , . . . , τn ) σ Φ(τ1 @σ, . . . , τn @σ) σ ¬hλα1 , . . . , αn .Φ(α1 , . . . , αn )i(τ1 , . . . , τn ) σ ¬Φ(τ1 @σ, . . . , τn @σ) 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.

9.1

Equality

The tableau rules of the previous section 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.

9.1.1

Equality Axioms

If we want to take equality into account, we use the Equality Axioms, Definition 5.1.1, as global assumptions. From here on these will be assumed without further comment. In Chapter 5 I presented some tableau rules that were derivable 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 τ , and a prefix σ that is already present on the branch, σ (τ = τ ) Substitutivity Rule For grounded, non-relativized terms τ1 and τ2 , σ Φ(τ1 ) σ (τ1 = τ2 ) σ Φ(τ2 ) 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. 101

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Suppose we read modal operators temporally, so that ¤X means X will be the case no matter what the future brings, and ♦X means the future could turn out to be one in which X is true. Let p be a type ↑0 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 been elected 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 ↑h0i object and is expressed by the following abstract, where x is of type 0. hλx.♦(↓p = x)i 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 ↑hh0ii. It is expressed by the following abstract, where X is of type h0i. hλX.¤X(↓p)i Now, it is clear that the extension of the property of being Presidential material is a statesmanlike property since, no matter who turns out to be President, that person was of Presidential material. The following gives a tableau verification for this. Example 9.1.1 Here is a proof of the formula: hλX.¤X(↓p)i(↓hλx.♦(↓p = x)i)

1 ¬hλX.¤X(↓p)i(↓hλx.♦(↓p = x)i) 1 ¬¤hλx.♦(↓p = x)i1 (↓p) 2. 1.1 ¬hλx.♦(↓p = x)i1 (↓p) 3. 1.1 ¬hλx.♦(↓p = x)i1 (p1.1 ) 4. 1 ¬♦(↓p = p1.1 ) 5. 1.1 ¬(↓p = p1.1 ) 6. 1.1 ¬(p1.1 = p1.1 ) 7. 1.1 (p1.1 = p1.1 ) 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.

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9.1.2

103

Extensionality

Extensionality can, of course, be imposed by assuming the Extensionality Axioms of Chapter 6, Definition 6.1.1, as global assumptions. The trouble is, doing so for intensional terms yields undesirable results, as the following shows. Proposition 9.1.2 Assume the Extensionality Axioms apply to intensional terms. If α and β are of intensional type ↑hti, then (∀α)(∀β)[(↓α =↓β) ⊃ (α = β)] The proof of this is left to you. It is almost immediate, using the Intensional Predication Rules. The problem with this result is that it tells us that if two intensional objects happen to coincide 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 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 9.1.3 [Extensionality for Intensional Terms] For α and β of the same intensional type, (∀α)(∀β)[¤(↓α =↓β) ⊃ (α = β)] 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. I restate them here for convenience. Definition 9.1.4 [Extensionality for Extensional Terms] Each sentence of the following form is an extensionality axiom, where α and β are of type ht1 , . . . , tn i, γ1 is of type t1 , . . . , γn is of type tn . (∀α)(∀β){(∀γ1 ) · · · (∀γn )[α(γ1 , . . . , γn ) ≡ β(γ1 , . . . , γn )] ⊃ [α = β]} 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 τ1 and τ2 , and for parameters p1 , . . . , pn that are new to the branch, σ ¬ [τ1 (p1 , . . . , pn ) ≡ τ2 (p1 , . . . , pn )] σ (τ1 = τ2 )

9.2

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

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States is a citizen of the United States,” is a de dicto application of necessity. It 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,” is a de re application of necessity. It asserts, of the President, that he is and always will be at least 50 years old. Since the President, at the time of writing this, 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, with 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 relationships 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. In what follows, β is of some extensional type t, and τ is of the corresponding intensional type ↑t. Consider the expression hλβ.¤Φ(β)i(↓τ ), where Φ(β) is some formula with only β free (for simplicity). Say the expression is true at a world of a modal model. (I use an unrestricted Henkin/Kripke model, but what is said applies to any version—extensional, standard—just as well.) Thus suppose M, Γ °v,A hλβ.¤Φ(β)i(↓τ ). Let O be the object that τ designates at Γ, that is, (v ∗ I ∗ Γ ∗ A)(τ, Γ) = O. Then we have M, Γ °v,A ¤Φ(β)[β/O]. So at every alternative world, Φ(β) is true of the object O that τ designates at Γ. The effect is that hλβ.¤Φ(β)i(↓τ ) asserts, of the object designated by τ at Γ that it has a necessary property. This is a de re use of necessity—applying a necessary property to a thing. Next consider the expression ¤hλβ.Φ(β)i(↓τ ). 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, Γ °v,A ¤hλβ.Φ(β)i(↓τ ), then in each alternative world ∆ we have M, ∆ °v,A hλβ.Φ(β)i(↓τ ), and so M, ∆ °v,A Φ(β)[w/O∆ ], where O∆ is the designation of τ at ∆, something that depends on ∆. We can thus think of the assertion ¤hλβ.Φ(β)i(↓τ ) as being concerned with the sense of τ and not just with the object it happens to denote in “our” world—we use the local designation of τ , 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. Definition 9.2.1 [De Re/De Dicto] Let τ be a term of intensional type ↑t, β be a variable of type t, and α be a variable of type hti. In a model:

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1. de re is equivalent to de dicto for τ if the following valid. (∀α)[hλβ.¤α(β)i(↓τ ) ≡ ¤hλβ.α(β)i(↓τ )] 2. de re implies de dicto for τ if the following is valid. (∀α)[hλβ.¤α(β)i(↓τ ) ⊃ ¤hλβ.α(β)i(↓τ )] 3. de dicto implies de re for τ if the following is valid. (∀α)[¤hλβ.α(β)i(↓τ ) ⊃ hλβ.¤α(β)i(↓τ )] The formulas above are allowed to be open—free variables may be present. Equivalently, one can work with universal closures. In (Fitting & Mendelsohn 1998) 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 (∀α) available. The interesting fact about the three notions above is: they all say the same thing. Proposition 9.2.2 For any intensional term τ , the following are equivalent (in K). 1. de dicto is equivalent to de re for τ 2. de dicto implies de re for τ 3. de re implies de dicto for τ 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 τ has no free variables. Here is a closed tableau for ¬(∀α)[hλβ.¤α(β)i(↓τ ) ⊃ ¤hλβ.α(β)i(↓τ )], de re implies de dicto. In it, at a certain point, use is made of an instance of the de dicto implies de re schema. The tableau begins as follows. 1 ¬(∀α) [hλβ.¤α(β)i(↓τ ) ⊃ ¤hλβ.α(β)i(↓τ )] 1. 1 ¬ [hλβ.¤Φ(β)i(↓τ ) ⊃ ¤hλβ.Φ(β)i(↓τ )] 2. 1 hλβ.¤Φ(β)i(↓τ ) 3. 1 ¬¤hλβ.Φ(β)i(↓τ ) 4. 1 ¤Φ(τ1 ) 5. 1.1 ¬hλβ.Φ(β)i(↓τ ) 6. 1.1 ¬Φ(τ1.1 ) 7. 1.1 Φ(τ1 ) 8. 1 (∀α)[¤hλβ.α(β)i(↓τ ) ⊃ hλβ.¤α(β)i(↓τ )] 9. 1 ¤hλβ.hλγ.Φ(γ) ⊃ Φ(↓τ )i(β)i(↓τ ) hλβ.¤hλγ.Φ(γ) ⊃ Φ(↓τ )i(β)i(↓τ ) 10.

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Item 2 is from 1 by an existential rule, using Φ as a new parameter; 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 the de dicto implies de re formula; and item 10 is from 9 by a universal rule, using hλγ.Φ(γ) ⊃ Φ(↓τ )i to instantiate the quantifier. Using item 10, the tableau splits into two branches. I first present the left one, and afterward the right. 1 ¬¤hλβ.hλγ.Φ(γ) ⊃ Φ(↓τ )i(β)i(τ ) 11. 1.2 ¬hλβ.hλγ.Φ(γ) ⊃ Φ(τ )i(β)i(↓τ ) 12. 1.2 ¬hλγ.Φ(γ) ⊃ Φ(↓τ )i(τ1.2 ) 13. 1.2 ¬[Φ(τ1.2 ) ⊃ Φ(↓τ )) 14. 1.2 Φ(τ1.2 ) 15. 1.2 ¬Φ(↓τ ) 16. 1.2 ¬Φ(τ1.2 ) 17. 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 16 by an extensional predication rule. The branch is closed because of 15 and 17. Now I show the right branch, below item 10. 1 hλβ.¤hλγ.Φ(γ) ⊃ Φ(↓τ )i(β)i(↓τ ) 1 ¤hλγ.Φ(γ) ⊃ Φ(↓τ )i(τ1 ) 19. 1.1 hλγ.Φ(γ) ⊃ Φ(↓τ )i(τ1 ) 20. 1.1 Φ(τ1 ) ⊃ Φ(↓τ ) 21.

1.1 ¬Φ(τ1 )

18.

@ @ 22. 1.1 Φ(↓τ ) 23. 1.1 Φ(τ1.1 ) 24.

In this part, 18 is from 10 by a disjunctive rule; 19 is from 18 by an unsubscripted abstract rule; 20 is from 19 by a necessity rule; 21 is from 20 by an unsubscripted abstract rule; 22 and 23 are from 21 by a disjunctive rule; 24 is from 23 by an extensional predication rule. Closure is by 8 and 22, and by 7 and 24.

Exercises Exercise 9.2.1 Give the tableau proof needed to complete the argument for Proposition 9.2.2.

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9.3

107

Rigidity

Kripke, in (Kripke 1980), discussed the philosophical ramifications of the notion of rigidity at some length, with a key claim being that names are rigid. His setting was first-order modal logic, treated informally. A term is taken to be rigid if it designates the same thing in all possible worlds. In (Fitting & Mendelsohn 1998) 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, that is, if we use normal models. (Whether they are standard, Henkin, or unrestricted 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. Definition 9.3.1 The intensional term τ is rigid in a normal model if the following is valid. hλβ.¤(β =↓τ )i(↓τ ) It is easy to see that the formula asserting rigidity of τ is valid at a world Γ of a normal model if and only if, at each world accessible from Γ, τ designates the same object that it designates at Γ itself. Thus asserting validity for the rigidity formula indeed captures the notion of rigidity for terms that we have in mind. If an intensional term is rigid, it does not matter in which possible world we determine its designation. But then, if both necessitation and designation by a rigid intensional term are involved in the same formula, it should not matter whether we determine what the term designates before or after we move to alternative worlds when taking necessitation into account. In other words, for rigid intensional terms the de re/de dicto distinction should vanish. In fact it does, and as it happens, the converse is also the case. The following is a higher order version of a first order argument from (Fitting & Mendelsohn 1998). Proposition 9.3.2 In K, the intensional term τ is rigid if and only if the de re/de dicto distinction vanishes, that is, if and only if any (and hence all) parts of Proposition 9.2.2 hold. Proof This is shown by proving two implications, using tableau rules for K including rules for equality. Let A be the formula hλβ.¤(β =↓τ )i(↓τ ) and let B be the formula (∀α)[¤hλβ.α(β)i(τ ) ⊃ hλβ.¤α(β)i(τ )]. A says τ is rigid, while B says de dicto implies de re for τ . I first give a tableau proof of A ⊃ B.

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1 ¬(A ⊃ B) 1. 1 hλβ.¤(β =↓τ )i(↓τ ) 2. 1 ¬(∀α)[¤hλβ.α(β)i(↓τ ) ⊃ hλβ.¤α(β)i(↓τ )] 1 ¬[¤hλβ.Φ(β)i(↓τ ) ⊃ hλβ.¤Φ(β)i(↓τ )] 4. 1 ¤hλβ.Φ(β)i(↓τ ) 5. 1 ¬hλβ.¤Φ(β)i(↓τ ) 6. 1 ¬¤Φ(τ1 ) 7. 1.1 ¬Φ(τ1 ) 8. 1.1 hλβ.Φ(β)i(↓τ ) 9. 1.1 Φ(τ1.1 ) 10. 1 ¤(τ1 =↓τ ) 11. 1.1 (τ1 =↓τ ) 12. 1.1 τ1 = τ1.1 13. 1.1 ¬Φ(τ1.1 ) 14.

108

3.

In this tableau, 2 and 3 are from 1 by a conjunctive rule; 4 is from 3 by an existential rule, with Φ as a new 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 again 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 (∀α)[¤hλβ.α(β)i(↓τ ) ⊃ hλβ.¤α(β)i(↓τ )] 2. 1 ¬hλβ.¤(β =↓τ )i(↓τ ) 3. 1 ¤hλβ.hλγ. ↓τ = γi(β)i(↓τ ) ⊃ hλβ.¤hλγ. ↓τ = γi(β)i(↓τ ) 1 ¬¤(τ1 =↓τ ) 5. 1.1 ¬(τ1 =↓τ ) 6. 1.1 ¬(τ1 = τ1.1 ) 7.

4.

@ @ 1 ¬¤hλβ.hλγ. ↓τ = γi(β)i(↓τ ) 8. 1 hλβ.¤hλγ. ↓τ = γi(β)i(↓τ ) 1.2 ¬hλβ.hλγ. ↓τ = γi(β)i(↓τ ) 9. 1 ¤hλγ. ↓τ = γi(τ1 ) 15. 1.1 hλγ. ↓τ = γi(τ1 ) 16. 1.2 ¬hλγ. ↓τ = γi(τ1.2 ) 10. 1.2 ¬(↓τ = τ1.2 ) 11. 1.1 (↓τ = τ1 ) 17. 1.1 τ1.1 = τ1 18. 1.2 ¬(τ1.2 = τ1.2 ) 12. 1.1 ¬(τ1 = τ1 ) 19. 1.2 τ1.2 = τ1.2 13. 1.1 τ1 = τ1 20.

14.

In this, 2 and 3 are from 1 by a conjunctive rule; 4 is from 2 by a universal rule, instantiating with the term hλγ. ↓τ = γi; 5 is from 3 by an unsubscripted

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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.

9.4

Stability Conditions

In his ontological argument G¨ odel makes essential use of what he called “positiveness,” which is a property of properties of things. He does 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 second-order constant symbol P to represent positiveness, and take it to be of type ↑h↑h0ii. G¨ odel stated his conditions more or less as follows, with quantifiers implied: P(X) ⊃ ¤P(X) and ¬P(X) ⊃ ¤¬P(X). The second of these is equivalent to ♦P(X) ⊃ P(X), and this form will be used in what follows. Positiveness is a second-order notion, but G¨odel’s conditions can be extended to other orders as well. I call the resulting notion stability, which is not terminology that G¨ odel used. Definition 9.4.1 [Stability] Let τ be a term of type ↑hti. τ satisfies the stability conditions in a model provided the following are valid in that model. (∀α)[τ (α) ⊃ ¤τ (α)] (∀α)[♦τ (α) ⊃ τ (α)] The stability conditions come in pairs. In S5, however, these pairs collapse. Proposition 9.4.2 In S5, (∀α)[τ (α) ⊃ ¤τ (α)] and (∀α)[♦τ (α) ⊃ τ (α)] are equivalent. Proof Suppose (∀α)[τ (α) ⊃ ¤τ (α)]. Contraposition gives (∀α)[¬¤τ (α) ⊃ ¬τ (α)]. By necessitation and the converse Barcan formula, (∀α)¤[¬¤τ (α) ⊃ ¬τ (α)], and so (∀α)[¤¬¤τ (α) ⊃ ¤¬τ (α)], or equivalently, (∀α)[¤♦¬τ (α) ⊃ ¤¬τ (α)]. But in S5, X ⊃ ¤♦X is valid, hence we have (∀α)[¬τ (α) ⊃ ¤¬τ (α)]. By contraposition again, (∀α)[¬¤¬τ (α) ⊃ ¬¬τ (α)], and hence (∀α)[♦τ (α) ⊃ τ (α)]. The converse direction is similar. In the stability conditions, τ is being predicated of other things. On the other hand, to say τ is rigid, or that the de re/de dicto distinction vanishes for τ , involves other things being predicated of τ . Here is the fundamental connection between stability and earlier items.

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Theorem 9.4.3 An intensional term τ is rigid if and only if it satisfies the stability conditions. Proof This is most easily established using tableaus. And it is a good workout. I leave it to you to supply the details.

Exercises Exercise 9.4.1 Complete the proof of Theorem 9.4.3 by giving appropriate closed tableaus. Recall the derived extensionality rule given in Definition 6.2.1.

9.5

Definite Descriptions

As is well-known, Russell handled definite descriptions by translating them away, (Russell 1905). 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 (Fitting & Mendelsohn 1998) we showed how both of these approaches extended to first-order modal logic. Further extending this to higher-order modal logic adds greatly to the complexity, so I confine things to a Russell-style treatment here. Suppose we have a formula Φ, and we form the expression α.Φ, which is read as the α such that Φ, and is called a definite description. Syntactically it is treated like a term. Its free variables are those of Φ, except for α, and its type is the type of α. 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. ι

Definition 9.5.1 [Description Designation] The definite description α.Φ designates, or is defined at possible world Γ of M = hG, R, H, Ii if ι

M, Γ °v (∃β)(∀δ)[hλα.Φi(δ) ≡ (β = δ)] where β and δ are not free in Φ. 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 hλx.¬B(x)i( y.K(y)) and ¬hλx.B(x)i( y.K(y)). There is a similar distinction to be made between hλx.¤B(x)i( y.K(y)) and ¤hλx.B(x)i( y.K(y)) since definite descriptions generally act non-rigidly, and so the de re/de dicto issue arises. ι

ι

ι

ι

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Note that in all the examples above, scope of a definite description was indicated by the use of a predicate abstract. Now hλx.¤B(x)i( y.K(y)) is an atomic formula, as are hλx.B(x)i( y.K(y)) and hλx.¬B(x)i( y.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 τ0 (τ1 ) either, or both, of τ0 and τ1 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. ι

ι

ι

Definition 9.5.2 [Descriptions In Atomic Context] Let α.Φ be a definite description, and let β and δ be variables of the same type as α, that do not occur free in Φ or in any of the terms τi below. ι

1. τ0 ( α.Φ) is an abbreviation for ι

(∃β){(∀δ)[hλα.Φi(δ) ≡ (β = δ)] ∧ τ0 (β)}. 2. τ0 (τ1 , . . . , α.Φ, . . . , τn ) is an abbreviation for ι

hλβ.τ0 (τ1 , . . . , β, . . . , τn )i( α.Φ). ι

3. ( α.Φ)(τ1 , . . . , τn ) is an abbreviation for ι

hλβ.β(τ1 , . . . , τn )i( α.Φ). ι

4. τ0 (τ1 , . . . , ↓( α.Φ), . . . , τn ) is an abbreviation for ι

hλβ.τ0 (τ1 , . . . , ↓β, . . . , τn )i( α.Φ). ι

5. (↓ α.Φ)(τ1 , . . . , τn ) is an abbreviation for ι

hλβ.(↓β)(τ1 , . . . , τn )i( α.Φ). ι

The definition above provides a routine for the elimination of definite descriptions. The problem is, there may be more than one way of following the routine. For instance, consider the atomic formula ( x.A(x))( y.B(y)), which contains two definite descriptions. If we eliminate ( y.B(y)) first, beginning with an application of part 1 of the definition, and then eliminate ( x.A(x)), we wind up with the following. ι

ι ι

ι

(∃z1 ){(∀z2 )[hλy.B(y)i(z2 ) ≡ (z1 = z2 )] ∧ (∃z4 ){(∀z5 )[hλx.A(x)i(z5 ) ≡ (z4 = z5 )] ∧ hλz3 .z3 (z1 )i(z4 )}}

(9.1)

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

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(∃z2 ){(∀z3 )[hλx.A(x)i(z3 ) ≡ (z2 = z3 )] ∧ hλz1 .(∃z4 ){(∀z5 )[hλy.B(y)i(z5 ) ≡ (z4 = z5 )] ∧ z1 (z4 )}i(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 α in α.Φ. 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 9.3.1 I defined a formal notion of rigidity. That definition can be extended to definite descriptions: call α.Φ rigid at a world if the following is true at that world. ι

ι

hλβ.¤(β =↓( α.Φ))i(↓( α.Φ)). ι

ι

Semantically speaking, to say this is true at a world Γ amounts to saying: α.Φ designates at world Γ, α.Φ designates at all worlds accessible from Γ, and at Γ and every world accessible from it, α.Φ designates the same thing. The following Proposition makes this precise. ι

ι

ι

Proposition 9.5.3 The formula hλβ.¤(β =↓( α.Φ))i(↓( α.Φ)) is equivalent in K to the conjunction of the following three formulas. ι

ι

1. (∃β)(∀δ)[hλα.Φi(δ) ≡ (β = δ)] 2. (∀β)[hλα.Φi(β) ⊃ ¤hλα.Φi(β)] 3. (∀β)[♦hλα.Φi(β) ⊃ hλα.Φi(β)]. In other words, this Proposition says ( α.Φ) is rigid if and only if ( α.Φ) designates and hλα.Φi satisfies the stability conditions. ι

ι

Exercises Exercise 9.5.1 Show the equivalence of (9.1) and (9.2). (Classical tableaus could be used, since modal operators do not explicitly appear.) Exercise 9.5.2 Use K tableaus to prove Proposition 9.5.3. (This is a long exercise.)

9.6

Choice Functions

In a Henkin/Kripke model, not all objects of a standard model need be present. We need some mechanism to ensure that many are, so non-standard models have a sufficiently rich universe. Abstraction provides one way of doing this. If

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Φ is a formula, there must be an intensional object in a Henkin/Kripke model to serve as the designation for hλα.Φi, and so there must be extensional objects to supply the designations for ↓hλα.Φi at each particular world. But for some purposes this is still not enough. In effect, the example just given starts with an intensional object, and moves to extensional objects derivatively. We need some machinery for moving in the other direction as well. Suppose, in a Henkin/Kripke model, we have somehow picked out an extensional object of the same type at each world—say we call the object we choose at world Γ, OΓ . It seems plausible that there should be an intensional object: the chosen object. That is, there should be an intensional object f whose value, at each world Γ, is the object OΓ . More generally, suppose at each world we have selected a non-empty set of extensional objects, all of the same type. Say at world Γ we select the set SΓ . Again it seems plausible that there should be an intensional object—a selected object—an mapping f whose value at each world Γ is some member of SΓ . Given the formal machinery up to this point, the existence of the intensional objects posited above cannot be guaranteed. (At least, I believe this to be the case. I do not have a proof.) To postulate existence of such intensional objects using some sort of axiom requires quantification over possible worlds, which we cannot do, but we can approximate to it by use of the ¤ operator. What we wind up with is the following postulate, which I call a choice axiom because, in effect, it posits the existence of choice functions in the standard set-theoretic sense. Definition 9.6.1 [Choice Axiom] Let t be an extensional type, and let α be of type hti, β be of type t, and γ be of type ↑t. The following is the choice axiom of type t. (∀α)[¤(∃β)α(β) ⊃ (∃γ)¤α(↓γ) Informally, the axiom says that if, at each world the set of things such that α is non-empty—¤(∃β)α(β)—then there is a choice function γ that picks out something such that α at each world—(∃γ)¤α(↓ γ). I give one example of a Choice Axiom application. Suppose α is an extensional variable, and α.Φ designates in every possible world. That is, in each possible world, the Φ is meaningful. Then, plausibly, there should be an intensional object that, in each world, designates the thing that is the Φ of that world—that is, the term ζ.¤hλα.Φi(↓ζ) should also designate. More loosely, the Φ concept should also designate. Recall, Definition 9.5.1 says what it means for a definite description to designate, and since hλζ.¤hλα.Φi(↓ ζ)i(η) ≡ ¤hλα.Φi(↓ η), things can be simplified a little. ι

ι

Proposition 9.6.2 Assume the Choice Axiom (Definition 9.6.1) and Extensionality for Intensional Terms (Definition 9.1.3). Assume α, β, and δ are of extensional type t, and γ and η are of type ↑t. The following is valid in all K models. ¤(∃β)(∀δ)[hλα.Φi(δ) ≡ (β = δ)] ⊃ (∃γ)(∀η)[¤hλα.Φi(↓η) ≡ (γ = η)]

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Proof Assume ¤(∃β)(∀δ)[hλα.Φi(δ) ≡ (β = δ)] is true at a possible world. I show that (∃γ)(∀η)[¤hλα.Φi(↓η) ≡ (γ = η)] must also be true there. Start with ¤(∃β)(∀δ)[hλα.Φi(δ) ≡ (β = δ)]

(9.3)

¤(∃β){hλα.Φi(β) ∧ (∀δ)[hλα.Φi(δ) ⊃ (β = δ)]}.

(9.4)

which is equivalent to

Instantiating the universal quantifier in the choice axiom with hλη.hλα.Φi(η) ∧ (∀δ)[hλα.Φi(δ) ⊃ (η = δ)]i (9.4) implies (∃γ)¤{hλα.Φi(↓γ) ∧ (∀δ)[hλα.Φi(δ) ⊃ (↓γ = δ)]}

(9.5)

which is equivalent to (∃γ){¤hλα.Φi(↓γ) ∧ ¤(∀δ)[hλα.Φi(δ) ⊃ (↓γ = δ)]}.

(9.6)

Since the Barcan and converse Barcan formulas are valid in the semantics, this is equivalent to (∃γ){¤hλα.Φi(↓γ) ∧ (∀δ)¤[hλα.Φi(δ) ⊃ (↓γ = δ)]}.

(9.7)

This, in turn, implies the following formula. I leave the justification to you. (∃γ){¤hλα.Φi(↓γ) ∧ (∀η)¤[hλα.Φi(↓η) ⊃ (↓γ =↓η)]}.

(9.8)

Using distributivity of necessity over implication, this implies (∃γ){¤hλα.Φi(↓γ) ∧ (∀η)[¤hλα.Φi(↓η) ⊃ ¤(↓γ =↓η)]}

(9.9)

and using Extensionality for Intensional Terms, this implies (∃γ){¤hλα.Φi(↓γ) ∧ (∀η)[¤hλα.Φi(↓η) ⊃ (γ = η)]}

(9.10)

which is equivalent to (∃γ)(∀η)[¤hλα.Φi(↓η) ≡ (γ = η)] and we are done.

(9.11)

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Exercises Exercise 9.6.1 Give tableau proofs of the Barcan formula, and of the converse Barcan formula. Exercise 9.6.2 Give a tableau proof to show (9.7) implies (9.8).

Part III

Ontological Arguments

116

Chapter 10

Ontological Arguments, A Brief History 10.1

Introduction

There are many directions from which people have tried to prove the existence of God. There have been arguments based on design: a complex universe must have had a designer. There have been attempts to show the existence of an ethical sense implies the existence of God. There have been arguments based on causality: trace the chain of effect and cause backward and one must reach a first cause. Ontological arguments seek to establish the existence of God based on pure logic: the principles of reasoning require that God be part of ones ontology. It does not matter if such arguments have holes. Religious belief, like much that is fundamentally human, is not really the product of reason. We are emotional animals, and one of the uses of proof, in the various senses above, is to sway emotion. Proof is often just a rhetorical device, one among many. But this takes us too far afield. Here we are interested in ontological arguments only. Independently of whether one believes their conclusion to be true, the logical machinery used in such arguments is often ingenious, and merits serious study. It is generally accepted that such arguments contain flaws, but saying exactly where the flaw lies is not easy, and is subject to controversy. It happens that different analyses of the same argument will locate an error at different points. Often this happens because the notions involved in a particular ontological argument are vague and subject to interpretation. G¨ odel’s ontological argument is rather unique in that it is entirely precise—the premises are clearly set forth, and the reasoning can be formalized. But we will see that here too there is room for interpretation, and things are not as clear as they first seem.

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118

Anselm

Historically, the first ontological argument is that of St. Anselm (1033 – 1109), given in his book Proslogion. A detailed examination of his argument can be found in (Hartshorne 1965). Here is the argument itself, in a somewhat technical translation from (Charlesworth 1979). Even the Fool, then, is forced to agree that something-thanwhich-nothing-greater-can-be-thought exists in the mind, since he understands this when he hears it, and whatever is understood is in the mind. And surely that-than-which-a-greater-cannot-be-thought cannot exist in the mind alone. For if it exists solely in the mind even, it can be thought to exist in reality also, which is greater. If then that-than-which-a-greater-cannot-be-thought exists in the mind alone, this same that-than-which-a-greater-cannot-be-thought is that-than-which-a-greater-can-be-thought. But this is obviously impossible. Therefore there is absolutely no doubt that somethingthan-which-a-greater-cannot-be-thought exists both in mind and in reality. Put into more modern terms, Anselm defined God to be a maximally conceivable being. This term—maximally conceivable being—must denote something, since “whatever is understood is in the mind.” But a maximally conceivable being must have the property of existence, because if it did not, we could conceive of a greater being, namely one that also had the existence property. My understanding of this is that, read with some charity, it shows the phrase “maximally conceivable being,” if it designates anything, must designate something that exists. The flaw lies in the failure to properly verify that the phrase designates at all—to show it is not in the same category as “the round square.” Indeed, Anselm’s way of justifying this, by claiming that it exists in the mind, is exactly what was attacked by his contemporary Gaunilo, in his counterargument, A Reply on Behalf of the Fool. A modern translation of this can also be found in (Charlesworth 1979). Anselm’s argument was the ancestor of various later versions, all of which involve some notion of maximality. An easily accessible discussion of the family of ontological arguments in general is in the on-line Stanford Encyclopedia of Philosophy (Oppy 1996), and (Oppy 1995, Plantinga 1965) are recommended as more detailed studies.

10.3

Descartes

Descartes (1598 – 1650) gave several different ontological arguments. Here is one version, in which he defines God to be a being whose necessary existence is part of the definition. It is from the Appendix to The Principles of Philosophy, (Descartes 1951). I omit the Definitions and Axioms to which the quote refers.

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Proposition I The existence of God is known from the consideration of his nature alone. Demonstration To say that an attribute is contained in the nature or in the concept of a thing is the same as to say that this attribute is true of this thing, and that it may be affirmed to be in it (Definition IX). But necessary existence is contained in the nature or in the concept of God (by Axiom X). Hence it may with truth be said that necessary existence is in God, or that God exists. Here is another version of the same argument, this time from The Meditations, book V, (Descartes 1951). . . . because I cannot conceive God unless as existing, it follows that existence is inseparable from him, and therefore that he really exists; not that this is brought about by my thought, or that it imposes any necessity on things, but, on the contrary, the necessity which lies in the thing itself, that is, the necessity of the existence of God, determines me to think in this way, for it is not in my power to conceive a God without existence, that is a being supremely perfect, and yet devoid of an absolute perfection, as I am free to imagine a horse with or without wings. The underlying idea in this argument is starkly simple: God is the most perfect being, the being having all perfections, and among these is necessary existence. Put a little differently, necessary existence is part of the essence of God. And here we have reached an ontological argument that can be easily formalized. Recall the discussion in Chapter 7, Section 7.3. The type-0 objects are possibilist—they are the same from world to world, and represent what might exist, not what does. If we want to relativize things to what actually exists, we need a type-h0i “existence” predicate, E, about which nothing special need be postulated at this point. Now, suppose we define God to be the necessarily existent being, that is, the being g such that ¤E(g). If such a being exists, it must satisfy its defining property, and hence we have E(g) ⊃ ¤E(g).

(10.1)

Given (10.1), using the rule of necessitation, we have the following. ¤[E(g) ⊃ ¤E(g)]

(10.2)

From (10.2), using the K principle ¤(P ⊃ Q) ⊃ (♦P ⊃ ♦Q) we have the next implication. ♦E(g) ⊃ ♦¤E(g)

(10.3)

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Finally we use something peculiar to S5 (and some slightly weaker logics, a point of no importance here). The principle needed is ♦¤P ⊃ ¤P , and so from (10.3) we have the following. ♦E(g) ⊃ ¤E(g)

(10.4)

We thus have a proof that God’s existence is necessary, if possible. And for Descartes, God’s existence is possible because possibility is identified with conceivability, and Descartes simply takes it for granted that God is conceivable. Russell’s treatment of definite descriptions applies quite well in a modal setting—Chapter 9, Section 9.5. The use of g above was an informal way of avoiding a formal definite description—note that I gave no real proof for (10.1). Let us recast the argument using definite descriptions—the necessarily existent being is α.¤E(α) and I assume g is an abbreviation for this type-0 term. Now (10.1) unabbreviates to the following. ι

E( α.¤E(α)) ⊃ ¤E( α.¤E(α)).

(10.5)

ι

ι

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 S5, a fact I leave to you as an exercise. In fact, using S5, the entire argument above is entirely correct! The real problem with the Descartes argument lies in the assumption that God’s existence is possible. In S5 both ¤E(g) ⊃ E(g) and E(g) ⊃ ♦E(g) are trivially valid. Since ♦E(g) ⊃ ¤E(g) has been shown to be valid, we have the equivalence of E(g), ♦E(g), and ¤E(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 10.3.1 Give an S5 tableau proof of the following, where P and Q are type-h0i constant symbols. P ( α.¤Q(α)) ⊃ ¤Q( α.¤Q(α)) ι

ι

From this it follows that (10.5) is valid in S5. Exercise 10.3.2 Construct a model to show E( α.¤E(α)) ⊃ ¤E( α.¤E(α)). ι

ι

is not valid in K. Exercise 10.3.3 Formula 10.5 can also be written as hλβ.E(β)i( α.¤E(α)) ⊃ ¤hλβ.E(β)i( α.¤E(α)) ι

ι

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which, by the previous exercise, is not K valid. Show the following variant is valid (a K tableau proof is probably easiest). hλβ.E(β)i( α.¤E(α)) ⊃ hλβ.¤E(β)i( α.¤E(α)) ι

ι

Exercise 10.3.4 Show why the valid K formula of Exercise 10.3.3 can not be used in a Descartes-style argument.

10.4

Leibniz

Leibniz (1646 – 1716) accepted the Descartes argument, discussed at length in the previous section, as being correct as far as it went. 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, (Leibniz 1956). Descartes’ reasoning about the existence of a most perfect being assumed that such a being can be conceived or is possible. If it is granted that there is such a concept, it follows at once that this being exists, because we set up this very concept in such a way that it at once contains existence. But it is asked whether it is in our power to set up such a being, or whether such a concept has reality and can be conceived clearly and distinctly, without contradiction. For opponents will say that such a concept of a most perfect being, or a being which exists through its essence, is a chimera. Nor does it suffice for Descartes to appeal to experience and allege that he experiences this very concept in himself, clearly and distinctly. This is not to complete the demonstration but to break it off, unless he shows a way in which others can also arrive at an experience of this kind. For whenever we inject experience into our demonstrations, we ought to show how others can produce the same experience, unless we are trying to convince them solely through our own authority. Leibniz’s remedy amounted to an attempt to prove that God’s existence is possible, where God is defined to be the being having all perfections—again a maximality notion. Intuitively, a perfection is an atomic property that is, in some sense, good to have, positive. Leibniz based his proof on the compatibility of all perfections, from which he took it to follow that all perfections could reside in a being—God’s existence is possible. Here is another quote from Two Notations for Discussion with Spinoza, (Leibniz 1956). 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.

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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 G¨odel’s argument as well.

10.5

G¨ odel

G¨ odel (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, G¨ odel 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. G¨ odel’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. G¨ odel worked on his ontological argument over many years. According to (Adams 1995), there is a partial version in his papers dated about 1941. In 1970, believing he would die soon, G¨ odel showed his proof to Dana Scott. In fact G¨ odel 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. Scott’s version of the proof was published in (Scott 1987). G¨ odel’s original version appeared in (Sobel 1987), based on a few pages of G¨ odel’s handwritten notes. Scott also wrote some brief notes, based on his conversation with G¨odel, and (Sobel 1987) provides these as well. In fact, (Sobel 1987) has served as something of a Bible (pun intended) for the G¨ odel ontological argument. Finally the publication of G¨ odel’s collected works has brought a definitive version before the public, (G¨ odel 1970). Still, the notion of a definitive version is rather elusive in this case. G¨ odel’s manuscript provides almost no explanation or motivation. It amounts to an invitation to others to elaborate. G¨ odel’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 G¨odel wrote his notes, and the complexity of quantification in modal contexts was perhaps not well appreciated. Consequently, the exact logic G¨odel had in mind is unclear. Subsequently several people took up the challenge of putting the G¨ odel argu-

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ment on a firm foundation and exposing any hidden assumptions. (Sobel 1987), playing Gaunilo to G¨ odel’s Anselm, showed the argument could be applied to prove more than one would want. I’ll discuss this point in the next Chapter. (Anderson & Gettings 1996) 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 (Anderson 1990, H´ ajek 1996a). The present Chapter can be thought of as part of the continuing tradition of explicating G¨ odel. People have generally used the second-order modal logic of (Cocchiarella 1969), sometimes rather informally.

10.6

G¨ odel’s Argument, Informally

Before we get to precise details in the next Chapter, it would be good to run through G¨ odel’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, G¨ odel 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. G¨ odel found it convenient to do this, and called the resulting notion positiveness. Thus we should think of a positive property, in G¨ odel’s sense, as some conjunction of perfections in Leibniz’s sense. At least, I am assuming this to be the case—G¨ odel says nothing explicit about the matter. The most notable difference between G¨odel and Leibniz is that, where Leibniz tried to use what are essentially informal notions in a rigorous way, G¨ odel introduces formal axioms concerning them. Here are G¨odel’s axioms, and his argument, set forth in everyday English. A formalized version will be found in the next Chapter. The G¨ odel argument has the familiar two-part structure: God’s existence is possible; and God’s existence is necessary, if possible. I’ll take these in order. I’ll begin with the axioms for positiveness. The first is rather strong. (Axiom numbering is not that of G¨ odel.) Informal Axiom 1 Exactly one of a property or its complement is positive. It follows that there must be positive properties. If we call a property negative if it is not positive, 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 positive.

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This brings us to our first interesting result. Informal Proposition 1 Any positive property is possibly instantiated. That is, if P is positive, it is possible that something has property P . Proof Suppose P is positive. Let N be some negative property (the complement of P will do). It cannot be that P entails N , or else N would be positive. So it is not necessary that everything having P has N , that is, it is possible that something has P without having N . So it is possible that something has P . Next, G¨ odel simply takes the compatibility of perfections, which Leibniz attempted to prove, as an axiom. We will see later on that this is a problematic assumption. Informal Axiom 3 The conjunction of any collection of positive properties is positive. Now G¨ odel defines God, or rather, defines the property of being Godlike, essentially the same way Leibniz did. Informal Definition 1 A God is any being that has every positive property. This gives us part one of the argument rather easily. Informal Proposition 2 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 1, 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. They are as follows. 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 G¨odel’s proof is quite different from that of Descartes, and rather ingenious. To carry out the argument, G¨ odel introduces a pair of notions that are of interest in their own right. Informal Definition 2 A property G is the essence of an object g if: 1. g has property G; 2. G entails every property of g.

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Strictly speaking, in the definition above I should have said an essence rather than the essence, but it is an easy argument that essences are unique, if they exist at all. Very simply, if an object g had two essences, P and Q, each would be a property of g by part 1, and then each would entail the other by part 2. G¨ odel 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 3 If g is a God, the essence of g is being a God. Proof Let’s state what we must show a little more precisely. Suppose G is the conjunction of all positive properties, so having property G is what it means to be a God. It must be shown that if an object g has property G, then G is the essence of g. Suppose g has property G. Then automatically we have part 1 of Informal Definition 2. Suppose also that P is some property of g. By Informal Axiom 1, if P were not positive its complement would be. Since g has all positive properties, g then would have the property complementary to P . Since we are assuming g has P itself, we would have a contradiction. It follows that P must be positive. Since G is the conjunction of all positive properties, clearly G entails P . Since P was arbitrary, G entails every property of g, and we have part 2 of Informal Definition 2. Here is the second of G¨odel’s two new notions. Informal Definition 3 An object g has the property of necessarily existing if the essence of g is necessarily instantiated. And here is the last of G¨odel’s axioms. Informal Axiom 5 Necessary existence, itself, is a positive property. Informal Proposition 4 If a God exists, a God exists necessarily. Proof Suppose a God exists, say object g is a God. Then g has all positive properties, and these include necessary existence by Informal Axiom 5. Then the essence of g is necessarily instantiated, by Informal Definition 3. But the essence of g is being a God, by Informal Proposition 3. Thus the property of being a God is necessarily instantiated. Now we present the second part of the ontological proof. Informal Proposition 5 If it is possible that a God exists, it is necessary that a God exists (assuming the logic is S5). Proof In any modal logic at least as strong as K, if P ⊃ Q is valid, so is ♦P ⊃ ♦Q. Then by Informal Proposition 4, if it is possible that a God exists, it is possibly necessary that a God exists. In S5, ♦¤P ⊃ ¤P is valid, and the conclusion follows.

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Finally, by Informal Propositions 2 and 5, we have our conclusion. Informal Theorem 6 Assuming all the axioms, and assuming the underlying logic is S5, 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. Let G be the property of being Godlike—the maximal positive property—and suppose both g1 and g2 possess this property. By Informal Proposition 3, G must be the essence of both g1 and g2 . Now, if P is any property of g1 , G must entail P , by part 2 of Informal Definition 2. Since G is a property of g2 , by part 1 of the same Informal Definition, P must also be a property of g2 . Similarly, any property of g2 must be a property of g1 . Since g1 and g2 have the same properties, they are identical. This concludes the informal presentation of G¨odel’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 10.6.1 Show that only God can have a positive essence. (This exercise is due to Ioachim Teodora Adelaida of Bucharest.)

Chapter 11

G¨ odel’s Argument, Formally 11.1

General Plan

The last Chapter ended with an informal presentation of G¨ odel’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 G¨ odel 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 the desired conclusion. The second kind of objection is that G¨ odel’s axioms are too strong, and lead to a collapse of the modal system involved. Various extensions and modifications of G¨ odel’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.

11.2

Positiveness

God, if one exists, will be taken to be an object of type 0. We are interested in the intentional properties of this object, properties of type ↑h0i. Among these properties are those G¨odel calls positive, and which we can think of as conjunctive combinations of Leibniz’s perfections. At least that is how I understand positiveness. G¨ odel’s ideas on the subject are given almost no explanation in his manuscript—here is what is said, using the translation of (G¨ odel 1970). Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world). Only then [are]

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the axioms true. It may also mean pure ‘attribution’ as opposed to ‘privation’ (or containing privation). This is not something I profess to understand. But what is significant is that, rather than attempting to define positiveness, G¨ odel characterized it axiomatically. In this section I present his basic axioms concerning the notion, and I explore some of their consequences. Definition 11.2.1 [Positive] A constant symbol P of type ↑h↑h0ii is designated to represent positiveness. It is an intentional property of intentional properties. Informally, P is positive if we have P(P ). G¨ odel assumes that each item must be exactly one of positive or negative. G¨ odel’s axiom (which he actually stated using exclusive-or) can be broken into two implications. Here they have been formulated as two separate axioms, since they play different roles. Axiom 11.2.2 (Corresponding to Informal Axiom 1) A (∀X)[P(¬X) ⊃ ¬P(X)] B (∀X)[¬P(X) ⊃ P(¬X)] Of these, Axiom 11.2.2A is certainly plausible: if a property is negative, the property should not also be positive. But Axiom 11.2.2B is more problematic: it says one of a property or its complement must be positive. Perhaps G¨odel had in mind something like the notion of a maximal consistent set of formulas, familiar from the Lindenbaum/Henkin approach to proving classical completeness. At any rate, these are the 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 G¨odel gave it. [P(X) ∧ ¤(∀x)(X(x) ⊃ Y (x))] ⊃ P(Y ) In this formula, x is a free variable of type 0. For us, type-0 quantification is possibilist, while for G¨ odel 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 (∀x) quantifier to E. Since this relativization comes up frequently, it is best to make an official definition. Definition 11.2.3 [Existential Relativization] (∀E x)Φ abbreviates (∀x)[E(x) ⊃ Φ], and (∃E x)Φ abbreviates (∃x)[E(x) ∧ Φ]. Axiom 11.2.4 (Corresponding to Informal Axiom 2) In the following, x is of type 0, X and Y are of type ↑h0i. (∀X)(∀Y ){[P(X) ∧ ¤(∀E x)(X(x) ⊃ Y (x))] ⊃ P(Y )}

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At this point it is convenient to introduce the following abbreviation. Definition 11.2.5 [Negative] If τ is a term of type ↑h0i, take ¬τ as short for hλx.¬τ (x)i. Call τ negative if ¬τ is positive. Loosely, at a world in a model, ¬τ denotes the complement of whatever τ denotes. It is easy to check formally that τ = ¬(¬τ ), given extensionality for intensional terms, Definition 9.1.3. At one point in his proof, G¨ odel asserts that hλx.x = xi must be positive if anything is, and hλx.¬x = xi must be negative. This is easy to see: P(hλx.x = xi) is valid if anything is positive because anything strictly implies a validity, and we have Axiom 11.2.4. The assertion that hλx.¬x = xi is negative is equivalent to the assertion that hλx.x = xi is positive. We thus have the following consequences of Axiom 11.2.4. Proposition 11.2.6 Assuming Axiom 11.2.4: 1. (∃X)P(X) ⊃ P(hλx.x = xi); 2. (∃X)P(X) ⊃ P(¬hλx.¬x = xi). Proposition 11.2.7 Assuming Axioms 11.2.2A and 11.2.4: (∃X)P(X) ⊃ ¬P(hλx.¬x = xi). Now we have a result from which the possible existence of God will follow immediately, given a key assumption about positiveness. Proposition 11.2.8 (Corresponding to Informal Proposition 1) Assuming Axioms 11.2.2A and 11.2.4, (∀X){P(X) ⊃ ♦(∃E x)X(x)}. Proof The idea has already been explained, in the proof of Informal Proposition 1 in Section 10.6. 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 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 11.2.7; 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 G¨ odel, rather than proving any two perfections could apply to the same object, G¨ odel assumes the positive properties are closed under conjunction. This turns out to be a critical assumption. In stating the assumption, read X ∧ Y as abbreviating hλx.X(x) ∧ Y (x)i.

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1 ¬(∃X)P(X) 1 ¬P(P ) 21.

1 ¬(∀X){P(X) ⊃ ♦(∃E x)X(x)} 1. 1 ¬{P(P ) ⊃ ♦(∃E x)P (x)} 2. 1 P(P ) 3. 1 ¬♦(∃E x)P (x) 4. 1 (∀X)(∀Y ){[P(X) ∧ ¤(∀E x)(X(x) ⊃ Y (x))] ⊃ P(Y )} 5. 1 (∀Y ){[P(P ) ∧ ¤(∀E x)(P (x) ⊃ Y (x))] ⊃ P(Y )} 6. 1 [P(P ) ∧ ¤(∀E x)(P (x) ⊃ hλx.¬x = xi(x))] ⊃ P(hλx.¬x = xi) 7. @ @ @ @ 1 P(hλx.¬x = xi) 9. 8. 1 (∃X)P(X) ⊃ ¬P(hλx.¬x = xi) 18. @ @ @ @ 19. 1 ¬P(hλx.¬x = xi) 1 ¬[P(P ) ∧ ¤(∀E x)(P (x) ⊃ hλx.¬x = xi(x))] @ @ @ @ 1 ¬P(P ) 10. 1 ¬¤(∀E x)(P (x) ⊃ hλx.¬x = xi(x)) 11. 1.1 ¬(∀E x)(P (x) ⊃ hλx.¬x = xi(x)) 12. 1.1 ¬((E(p) ∧ P (p)) ⊃ hλx.¬x = xi(p)) 13. 1.1 E(p) ∧ P (p) 14. 1.1 ¬hλx.¬x = xi(p) 15. 1.1 ¬(∃E x)P (x) 16. 1.1 ¬(E(p) ∧ P (p)) 17.

Figure 11.1: Tableau Proof of (∀X)[P(X) ⊃ ♦(∃x)X(x)]

20.

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Axiom 11.2.9 (Corresponding to Informal Axiom 3) (∀X)(∀Y ){[P(X) ∧ P(Y )] ⊃ P(X ∧ Y )} G¨ odel 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 11.2.9 as stated—the serious issue is that of an infinite number, which G¨ odel needs. (Anderson & Gettings 1996) 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 ↑h↑h0ii): pos(Z) ⇔ (∀X)[Z(X) ⊃ P(X)] 2. X applies to those objects which possess exactly the properties falling under Z—roughly, X is the (necessary) intersection of Z. (In this, Z is of type ↑h↑h0ii, X is of type ↑h0i, and x is of type 0.) (Xintersection of Z) ⇔ ¤(∀x){X(x) ≡ (∀Y )[Z(Y ) ⊃ Y (x)]} Axiom 11.2.10 (Also Corresponding to Informal Axiom 3) (∀Z){pos(Z) ⊃ (∀X)[(Xintersection of Z) ⊃ P(X)]}. Axiom 11.2.10 implies Axiom 11.2.9. I leave the verification to you. I’ll finish this section with two technical assumptions that G¨ odel makes “because it follows from the nature of the property.” I don’t understand this terse explanation, but here are the assumptions. P(X) ⊃ ¤P(X) ¬P(X) ⊃ ¤¬P(X) If the underlying logic is just K, equivalence of these two assumptions follows from Axioms 11.2.2A and 11.2.2B. And if the underlying logic is S5, as it must be for part of G¨ odel’s argument, equivalence also follows by Proposition 9.4.2. Consequently the version used here can be simplified. Axiom 11.2.11 (Corresponding to Informal Axiom 4) (∀X)[P(X) ⊃ ¤P(X)]. P has been taken to be an intentional object, of type ↑h↑h0ii. Axiom 11.2.11 and Theorem 9.4.3 tells us that P is rigid. In effect the intentionality of P is illusory—since it is rigid it could just as well have been an extensional object of type h↑h0ii.

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Exercises Exercise 11.2.1 Give a tableau proof that ¬hλx.¬(x = x)i = hλx.x = xi. More generally, show that for a type h0i term τ , ¬(¬τ ) = τ . Exercise 11.2.2 Show (∀X)[¬P(X) ⊃ ¤¬P(X)] follows from Axiom 11.2.11 together with Axioms 11.2.2A and 11.2.2B. Exercise 11.2.3 Axiom 11.2.10 implies Axiom 11.2.9. Hint: use equality.

11.3

Possibly God Exists

G¨ odel defines something to be Godlike if it possesses all positive properties. Definition 11.3.1 [Corresponding to Informal Definition 1] G is the following type ↑h0i term, where Y is type ↑h0i. hλx.(∀Y )[P(Y ) ⊃ Y (x)]i. 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 11.3.2 [Also Corresponding to Informal Definition 1] G∗ is the type ↑h0i term hλx.(∀Y )[P(Y ) ≡ Y (x)]i. The following result is easily proved; I leave it to you as an exercise. Proposition 11.3.3 Assume Axiom 11.2.2B, (∀X)[¬P(X) ⊃ P(¬X)]. In K, with this assumption, (∀x)[G(x) ≡ G∗ (x)]. Axiom 11.2.2B 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. And from this Axiom 11.2.2B follows, even if the existence in question is possibilist. Here is a formal statement of this. Once again I leave the proof to you. Proposition 11.3.4 In K, (∃x)G∗ (x) ⊃ (∀X)[¬P(X) ⊃ P(¬X)]. Now we can show that God’s existence is possible. G¨odel assumes that positive properties are compatible. Since G∗ is, in effect, the conjunction of all positive properties, it must be positive, and hence so must G be. Proposition 11.3.5 In K Axiom 11.2.10 implies P(G).

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Once again I leave the formal verification to you. What must be shown is the following. (∀Z)(∀X){[pos(Z) ∧ (X intersection of Z)] ⊃ P(X)} ⊃ P(G) Essentially, this is the case because, as is easy to verify, we have each of pos(P) and (G intersection of P). Now the possibility of God’s existence is easy. In fact, it can be proved with an actualist quantifier, though only the weaker possibilist version is really needed for the rest of the argument. Theorem 11.3.6 Assume Axioms 11.2.2A, 11.2.4, and 11.2.10. In K both of the following are consequences. ♦(∃E x)G(x) and ♦(∃x)G(x). Proof By Proposition 11.2.8, (∀X){P(X) ⊃ ♦(∃E x)X(x)}, hence trivially, (∀X){P(X) ⊃ ♦(∃x)X(x)}. By the Proposition above, P(G). The result is immediate. Note that the full strength of Proposition 11.2.8 was not really needed for the possibilist conclusion. In fact, if we modify Axiom 11.2.4 so that quantification is possibilist, (∀X)(∀Y ){[P(X) ∧ ¤(∀x)(X(x) ⊃ Y (x))] ⊃ P(Y )} we would still be able to prove Proposition 11.2.8 in the weaker form (∀X){P(X) ⊃ ♦(∃x)X(x)} and the G¨ odel proof would still go through.

Exercises Exercise 11.3.1 Give a tableau proof that G entails any positive property: (∀X){P(X) ⊃ ¤(∀y)[G(y) ⊃ X(y)]}. You will need Axiom 11.2.11. Exercise 11.3.2 Give a tableau proof for Proposition 11.3.3. Exercise 11.3.3 Give a tableau proof for Proposition 11.3.4. Exercise 11.3.4 Give a tableau proof for Proposition 11.3.5. Exercise 11.3.5 Give a tableau proof of (∀Z)(∀X){[pos(Z) ∧ (X intersection of Z)] ⊃ P(X)} ⊃ P(G).

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11.4

134

Objections

G¨ odel replaced Leibniz’s attempted proof of the compatibility of perfections by an outright assumption, given here as Axiom 11.2.10. Dana Scott noted that the only use G¨ odel makes of this Axiom is to show being Godlike is positive, and proposed taking P(G) itself as an axiom. Indeed, Scott has always maintained that the G¨ odel 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. G¨odel’s assumption concerning the compatibility of positive properties turns out to be equivalent to the possibility of God’s existence. We will see later on that G¨ odel’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 made above. 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. If σ X and σ X ⊃ Y occur on a branch, σ Y can be added. Schematically, σX σX ⊃ Y σY The justification for this is as follows. σ X 1. σX ⊃ Y

σ ¬X

3.

2.

@ @ σ Y 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. σX σ (X ∧ Y ) ⊃ Z σY ⊃ Z σX σX ≡ Y σY

σ ¬Y σ X⊃Y σ ¬X σ ¬X σ X≡Y σ ¬Y

¨ CHAPTER 11. GODEL’S ARGUMENT, FORMALLY

σ (∀α1 ) · · · (∀αn )Φ(α1 , . . . , αn ) σ Φ(τ1 , . . . , τn ) for any grounded terms τ1 , . . . , τn

135

σ (∃α1 ) · · · (∃αn )Φ(α1 , . . . , αn ) σ Φ(P1 , . . . , Pn ) for any new, distinct parameters P1 , . . . , Pn

Now, here is the promised proof of equivalence. Theorem 11.4.1 Assume all the Axioms to this point, except Axiom 11.2.10. The following are equivalent in S5: 1. Axiom 11.2.10; 2. P(G); 3. ♦(∃E x)G(x); 4. ♦(∃x)G(x). Proof We already know 1 implies 2, this is Proposition 11.3.5. Likewise 3 follows from 2, by Theorem 11.3.6. 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 ♦(∃x)G(x) 1. 1 ¬P(G) 2. 1.1 (∃x)G(x) 3. 1.1 G(g) 4. 1.1 (∀x)[G(x) ≡ G∗ (x)] 5. 1.1 [G(g) ≡ G∗ (g)] 6. 1.1 G∗ (g)] 7. 1.1 hλx.(∀Y )[P(Y ) ≡ Y (x)]i(g) 8. 1.1 (∀Y )[P(Y ) ≡ Y (g)] 9. 1.1 [P(G) ≡ G(g)] 10. 1.1 P(G) 11. 1 (∀X)[¬P(X) ⊃ ¤¬P(X) 12. 1 [¬P(G) ⊃ ¤¬P(G) 13. 1 ¤¬P(G) 14. 1.1 ¬P(G) 15. Item 3 is from 2 by a possibility rule; 4 is from 3 by an existential rule, with g as a new parameter; 5 is Proposition 11.3.3, and note that the modal version of Corollary 4.3.4 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 by a derived rule; 12 is an equivalent of Axiom 11.2.11; 13 is from 12 by a universal rule; 14 is from 2 and 13 by a derived rule; 15 is from 14 by a universal rule.

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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 11.3.1. It follows that G also entails the conjunction of C. Since 2 says G is positive, the conjunction of C is positive by Axiom 11.2.4. 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 11.2.9, and I’ll leave the argument for Axiom 11.2.10 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 11.2.4; 9 is from 8 by a derived rule; 10 is from 9 by a derived 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 11.2.11; 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 universal 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 11.4.1 Give a tableau proof that ♦(∃x)G(x) implies Axiom 11.2.10.

11.5

Essence

Even though we ran into the old Descartes problem with half of the G¨ odel 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, G¨odel defines a notion of essence that plays a central role, and is of interest in its own right. (Hazen 1998) makes a case for calling G¨odel’s notion character, reserving the term essence for something else. I follow G¨odel’s terminology. The essence of something, x, is a property that entails every property that x possesses. G¨odel says it as follows. ϕ Ess x ≡ (∀ψ){ψ(x) ⊃ ¤(∀y)[ϕ(y) ⊃ ψ(y)]} As 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 G¨ odel, and inserted a conjunct ϕ(x) into the definition. I will follow him in this. ϕ Ess x ≡ ϕ(x) ∧ (∀ψ){ψ(x) ⊃ ¤(∀y)[ϕ(y) ⊃ ψ(y)]} G¨ odel states ϕ Ess x as a formula rather than a term—in the version in this book an explicit predicate abstract is used. Also, I assume the type-0 quantifier that appears is actualist, and so in my version the existence predicate, E, must appear. E(P, q) is intended to assert that P is the essence of q.

¨ CHAPTER 11. GODEL’S ARGUMENT, FORMALLY

1 P(G) 1. 1 ¬(∀X)(∀Y ){[P(X) ∧ P(Y )] ⊃ P(X ∧ Y )} 2. 1 ¬{[P(A) ∧ P(B)] ⊃ P(A ∧ B)} 3. 1 P(A) ∧ P(B) 4. 1 ¬P(A ∧ B) 5. 1 P(A) 6. 1 P(B) 7. 1 (∀X)(∀Y ){[P(X) ∧ ¤(∀E x)(X(x) ⊃ Y (x))] ⊃ P(Y )} 8. 1 [P(G) ∧ ¤(∀E x)(G(x) ⊃ (A ∧ B)(x))] ⊃ P(A ∧ B) 9. 1 ¤(∀E x)(G(x) ⊃ (A ∧ B)(x)) ⊃ P(A ∧ B) 9. 1 ¬¤(∀E x)(G(x) ⊃ (A ∧ B)(x)) 11. 1.1 ¬(∀E x)(G(x) ⊃ (A ∧ B))(x) 12. 1.1 ¬(∀x)[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 (∀X)[P(X) ⊃ ¤P(X)] 19. 1 P(A) ⊃ ¤P(A) 20. 1 P(B) ⊃ ¤P(B) 21. 1 ¤P(A) 22. 1 ¤P(B) 23. 1.1 P(A) 24. 1.1 P(B) 25. 1.1 hλx.(∀Y )[P(Y ) ⊃ Y (x)]i(c) 26. 1.1 (∀Y )[P(Y ) ⊃ Y (c)] 27. 1.1 P(A) ⊃ A(c) 28. 1.1 P(B) ⊃ B(c) 29. 1.1 A(c) 30. 1.1 B(c) 31. 1.1 ¬hλx.A(x) ∧ B(x)i(c) 32. 1.1 ¬[A(c) ∧ B(c)] 33. @

1.1 ¬A(c)

@ 34. 1.1 ¬B(c)

35.

Figure 11.2: Proof that item 2 implies Axiom 11.2.9

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Definition 11.5.1 [Essence, Corresponding to Informal Definition 2] E abbreviates the following type ↑h↑h0i, 0i term, in which Z is of type ↑h0i and w is of type 0: hλY, x.Y (x) ∧ (∀Z){Z(x) ⊃ ¤(∀E w)[Y (w) ⊃ Z(w)]}i The property of being Godlike was defined earlier, Definition 11.3.1. A central fact about Godlikeness, from G¨ odel’s notes, is that it is the essence of any being that is Godlike. Theorem 11.5.2 (Corresponding to Informal Proposition 3) Assume Axioms 11.2.2B and 11.2.11. In K the following is provable. (Note that x is of type 0.) (∀x)[G(x) ⊃ E(G, x)]. Rather than giving a direct proof, if we use Proposition 11.3.3 it follows from a similar result concerning G∗ , provided Axiom 11.2.2B is assumed. Since such a result has a somewhat simpler proof, that is what is actually shown. Theorem 11.5.3 In K the following is provable, assuming Axiom 11.2.11. (∀x)[G∗ (x) ⊃ E(G∗ , x)]. Proof Here is a closed K tableau to establish the theorem. 1 ¬(∀x)[G∗ (x) ⊃ E(G∗ , x)] 1. 1 ¬[G∗ (g) ⊃ E(G∗ , g)] 2. 1 G∗ (g) 3. 1 ¬E(G∗ , g) 4. 1 ¬{G∗ (g) ∧ (∀Z){Z(g) ⊃ ¤(∀E w)[G∗ (w) ⊃ Z(w)]}} 1 ¬G∗ (g)

5.

@ @ 6. 1 ¬(∀Z){Z(g) ⊃ ¤(∀E w)[G∗ (w) ⊃ Z(w)]}

7.

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

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1 ¬{Q(g) ⊃ ¤(∀E w)[G∗ (w) ⊃ Q(w)]} 1 Q(g) 9. 1 ¬¤(∀E w)[G∗ (w) ⊃ Q(w)] 10. 1.1 ¬(∀E w)[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 (∀Y )[P(Y ) ≡ Y (g)] 17. 1 P(Q) ≡ Q(g) 18. 1 P(Q) 19. 1.1 (∀Y )[P(Y ) ≡ Y (a)] 20. 1.1 P(Q) ≡ Q(a) 21. 1 (∀Y )[P(Y ) ⊃ ¤P(Y )] 22. 1 P(Q) ⊃ ¤P(Q) 23. 1 ¤P(Q) 24. 1.1 P(Q) 25. 1.1 Q(a) 26.

139

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 3; 23 is from 22 by a universal rule; 24 is from 19 and 23 by a derived rule; 25 is from 24 by a necessity rule; 26 is from 21 and 25 by a derived rule. The branch is closed by 16 and 26. In the notes Dana Scott made when G¨ odel 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 11.5.4 Assume the modal logic is K. The following is provable. (∀X)(∀Y )(∀z){[E(X, z) ∧ E(Y, z)] ⊃ ¤(∀E w)[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 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.

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1 E(P, a) 1. 1 E(Q, a) 2. 1 ¬¤(∀E w)[P (w) ≡ Q(w)] 3. 1 P (a) 4. 1 (∀Z)[Z(a) ⊃ ¤(∀E w)[P (w) ⊃ Z(w)]] 5. 1 Q(a) 6. 1 (∀Z)[Z(a) ⊃ ¤(∀E w)[Q(w) ⊃ Z(w)]] 7. 1 Q(a) ⊃ ¤(∀E w)[P (w) ⊃ Q(w)] 8. 1 P (a) ⊃ ¤(∀E w)[Q(w) ⊃ P (w)] 9. @

1 ¬Q(a)

@ 10. 1 ¤(∀E w)[P (w) ⊃ Q(w)]

11.

@

@ 12. 1 ¤(∀E w)[Q(w) ⊃ P (w)]

1 ¬P (a)

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 ¬(∀E w)[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) 20. 1.1 ¬Q(b) 19. 1.1 Q(b) 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. 1.1 (∀E w)[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)

1.1 ¬P (b)

25.

@ @ 26. 1.1 Q(b)

27.

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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 11.5.5 Assume the modal logic is K, including equality. The following is valid. (∀X)(∀y){E(X, y) ⊃ ¤(∀E z)[X(z) ⊃ (y = z)]} This can be proved using tableaus—I leave it to you as an exercise.

Exercises Exercise 11.5.1 Give a tableau proof for Theorem 11.5.5. Hint: for a parameter c, one can consider the property of being, or not being, c, that is, hλx.x = ci and hλx.x 6= ci. Either can be used. Exercise 11.5.2 Give a tableau proof to establish Theorem 11.5.2 directly, without using G∗ .

11.6

Necessarily God Exists

In this section I present a version of G¨odel’s argument that God’s possible existence implies His existence necessarily. It begins with the introduction of an auxiliary notion that G¨ odel calls necessary existence. Definition 11.6.1 [Necessary Existence, Corresponding to Informal Definition 3] N abbreviates the following type ↑h0i term: hλx.(∀Y )[E(Y, x) ⊃ ¤(∃E z)Y (z)]i. The idea is, something has the property N of necessary existence provided any essence of it is necessarily instantiated. G¨odel now makes a crucial assumption: necessary existence is positive. Axiom 11.6.2 (Corresponding to Informal Axiom 5) P(N ). Given this final axiom, G¨ odel shows that if (some) God exists, that existence cannot be contingent. An informal sketch of the proof was given in Section 10.6 of Chapter 10, and it can be turned into a formal proof—see Informal Propositions 4 and 5. I will leave the details as exercises, since you have seen lots of worked out tableaus now. Here is a proper statement of G¨ odel’s result, with all the assumptions explicitly stated. Note that the necessary actualist existence of God follows from His possibilist existence.

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Theorem 11.6.3 (Corresponding to Informal Proposition 4) Assume Axioms 11.2.2B, 11.2.11, and 11.6.2. In the logic K, (∃x)G(x) ⊃ ¤(∃E x)G(x). I leave it to you to prove this, using the informal sketch as a guide. Now G¨ odel’s argument can be completed. Theorem 11.6.4 (Corresponding to Informal Proposition 5) Assume Axioms 11.2.2B, 11.2.11, and 11.6.2. In the logic S5, ♦(∃x)G(x) ⊃ ¤(∃E x)G(x). Proof From Theorem 11.6.3, (∃x)G(x) ⊃ ¤(∃E x)G(x). By necessitation, ¤[(∃x)G(x) ⊃ ¤(∃E x)G(x)]. By the K validity ¤(A ⊃ B) ⊃ (♦A ⊃ ♦B), ♦(∃x)G(x) ⊃ ♦¤(∃E x)G(x). Finally, in S5, ♦¤A ⊃ ¤A, so we conclude ♦(∃x)G(x) ⊃ ¤(∃E x)G(x).

Now we are at the end of the argument. Corollary 11.6.5 Assume all the Axioms. In the logic S5, ¤(∃E x)G(x). Proof By Theorems 11.6.4 and 11.3.6.

Exercises Exercise 11.6.1 Give a tableau proof to show Theorem 11.6.3. Use various earlier results as assumptions in the tableau.

11.7

Going Further

G¨ odel’s axioms admit more consequences than just those of the ontological argument. In this section a few of them are presented.

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143

Monotheism

Does there exist exactly one God? The following says “yes.” You are asked to prove it, as Exercise 11.7.1. Proposition 11.7.1 (∃x)(∀y)[G(y) ≡ (y = x)]. This Proposition has a curious Corollary. Since type-0 quantification is possibilist, it makes sense to ask if there are gods that happen to be non-existent. But Corollary 11.6.5 tells us there is an existent God, and the Proposition above tells us it is the only one, existent or not. Consequently we have the following. Corollary 11.7.2 (∀x)[G(x) ⊃ E(x)]. Proposition 11.7.1 tells us we can apply the machinery of definite descriptions. By Definition 9.5.1, x.(∀Y )[P(Y ) ⊃ Y (x)] always designates, and consequently so does x.G(x). Proposition 9.5.3 tells us this will be a rigid designator provided G(x) is stable. It follows from Sobel’s argument in Section 11.8 that it, and everything else, is. But alternative versions of G¨ odel’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 G¨ odel (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. ι

ι

11.7.2

Positive Properties are Necessarily Instantiated

Proposition 11.2.8 says that positive properties are possibly instantiated. In (Sobel 1987), it is observed that a consequence of Corollary 11.6.5 is that every positive property is necessarily instantiated. Proposition 11.7.3 (∀X){P(X) ⊃ ¤(∃E x)X(x)}. I leave the easy proof of this to you.

Exercises Exercise 11.7.1 Give a tableau proof for Proposition 11.7.1. Hint: you will need Corollary 11.6.5, Theorem 11.5.2, and Theorem 11.5.5. Exercise 11.7.2 Provide a tableau proof for Proposition 11.7.3. Hint: by Corollary 11.6.5, a Godlike being necessarily exists. Such a being has all positive properties, so every positive property is instantiated. Now, build this into a tableau.

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11.8

144

More Objections

In Section 11.4 we saw that one of G¨odel’s Axioms was equivalent to the possible existence of God. Other objections have been raised that are equally as serious, and one of these is taken up now. (Sobel 1987) showed that G¨ odel’s axiom system is so strong it implies that whatever is the case is so of necessity, Q ⊃ ¤Q. 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 have 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 slightly informal version of the argument given by Sobel. For simplicity, assume Q is a formula that contains no free variables. By Theorem 11.5.2, (∀x)[G(x) ⊃ E(G, x)].

(11.1)

Using the definition of E, we have as a consequence (∀x){G(x) ⊃ (∀Z){Z(x) ⊃ ¤(∀E w)[G(w) ⊃ Z(w)]}}.

(11.2)

There is a minor nuisance to deal with. In the formula (11.2) I would like to instantiate the quantifier (∀Z) with Q, but this is not a ‘legal’ term, so instead I use the term hλy.Qi to instantiate. In it, y is of type 0, and so hλy.Qi is of type ↑h0i. We get the following consequence. (∀x){G(x) ⊃ {hλy.Qi(x) ⊃ ¤(∀E w)[G(w) ⊃ hλy.Qi(w)]}}.

(11.3)

Now to undo the technicality just introduced, note that since y does not occur free in Q, hλy.Qi(x) ≡ hλy.Qi(w) ≡ Q, and so we have (∀x){G(x) ⊃ {Q ⊃ ¤(∀E w)[G(w) ⊃ Q]}}.

(11.4)

Since x does not occur free in the consequent, (11.4) is equivalent to the following: (∃x)G(x) ⊃ {Q ⊃ ¤(∀E w)(G(w) ⊃ Q)}.

(11.5)

We have Corollary 11.6.5, from which (∃x)G(x) follows. Then from (11.5) and (11.6) we have

(11.6)

¨ CHAPTER 11. GODEL’S ARGUMENT, FORMALLY

Q ⊃ ¤(∀E w)(G(w) ⊃ Q).

145

(11.7)

Since Q has no free variables, (11.7) is equivalent to the following: Q ⊃ ¤[(∃E w)G(w) ⊃ Q].

(11.8)

Using the distributivity of ¤ over implication, (11.8) gives us Q ⊃ [¤(∃E w)G(w) ⊃ ¤Q].

(11.9)

Finally (11.9), and Corollary 11.6.5 again, give the intended result, Q ⊃ ¤Q.

(11.10)

Most people have taken this as a counter to G¨odel’s argument—if the axioms are strong enough to admit this consequence, something is wrong. In the next two sections I explore some ways out of the difficulty.

11.9

A Solution

Sobel’s demonstration that the G¨ odel 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 G¨ odel had in mind intensional properties when talking about positiveness and essence. But, suppose not—suppose extensional properties were intended. In this section I reformulate G¨odel’s argument under this alternative interpretation. It is one way of solving the problem Sobel raised. In this section only I will take P to be a constant symbol of type ↑hh0ii. Axiom 11.2.4 gets replaced with the following. Revised Axiom 11.2.4 In the following, x is of type 0, X and Y are of type h0i, and (∀E x)Φ abbreviates (∀x)[E(x) ⊃ Φ]. (∀X)(∀Y ){[P(X) ∧ ¤(∀E x)(X(x) ⊃ Y (x))] ⊃ P(Y )} Note that this has the same form as Axiom 11.2.4, but the types of variables X and Y are now extensional rather than intensional. This will be the general pattern for changes. The definition of negative, for instance, is modified as follows. For a term τ of type h0i, take ¬τ as short for ↓hλx.¬τ (x)i. Then Axioms 11.2.2A and 11.2.2B, 11.2.10, and 11.2.11, all have their original form, but with variables changed from intensional to extensional type.

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The analog of Proposition 11.2.8 still holds, but with extensional variables involved. (∀X){P(X) ⊃ ♦(∃E x)X(x)} Analogs of G and G∗ are defined in the expected way. G is the following type ↑h0i term, where Y is type h0i. hλx.(∀Y )[P(Y ) ⊃ Y (x)]i Likewise G∗ is the type ↑h0i term hλx.(∀Y )[P(Y ) ≡ Y (x)]i. One can still prove (∀x)[G(x) ≡ G∗ (x)]. Essence must be redefined, but again it is only variable types that are changed. E now abbreviates the following type ↑hh0i, 0i term, in which Z is of type h0i and w is of type 0: hλY, x.Y (x) ∧ (∀Z){Z(x) ⊃ ¤(∀E w)[Y (w) ⊃ Z(w)]}i Theorem 11.5.3 plays an essential role in the G¨ odel proof, and it too continues to hold, in a slightly modified form: (∀x)[G∗ (x) ⊃ E(↓G∗ , 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 11.6.2 as well, to use the modified definition of Necessary Existence. For this section, N abbreviates the following type ↑h0i term, in which Y is of type h0i: hλx.(∀Y )[E(Y, x) ⊃ ¤(∃E z)Y (z)i. Revised Axiom 11.6.2 is P(N ), where N is as just modified. With this established, the rest of G¨ odel’s argument carries over directly, giving us the following. ¤(∃E z)(↓G∗ )(z) The final step is the easy proof that this implies the desired ¤(∃E z)G∗ (z), and hence ¤(∃E z)G(z), and I leave this to you. So, we have the conclusion of G¨ odel’s argument. Finally, here is a model, adapted from (Anderson 1990), that shows Sobel’s continuation no longer applies. Example 11.9.1 Construct a standard S5 model as follows. There are two possible worlds, call them Γ and ∆. The accessibility relation always holds. The

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type-0 domain is the set {a, b}. Since this is a standard model, the remaining types are fully determined. The existence predicate, E, is interpreted to have extension {a, b} at Γ and {a} at ∆. Informally, all type-0 objects exist at Γ, but only a exists at ∆. Call a type-h0i object positive if it applies to a. Interpret P so that at each world its extension is the collection of positive type-h0i 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 E is also rigid, with extension {h{a}, ai, h{b}, bi} 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 (∃E x)(∃E y)¬(x = y). Since it asserts two objects actually exist, it is true at Γ, but not at ∆, and hence Q ⊃ ¤Q is not true at Γ. 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 inferences now are more complex. I leave it to you to verify their correctness, using tableaus. Once again, assume Q contains no free variables. We try to prove Q ⊃ ¤Q, starting more or less as we did before. (∀x)[G(x) ⊃ E(↓G, x)]

(11.11)

which, unabbreviated, is (∀x)[G(x) ⊃

(11.12)

hλY, x.Y (x) ∧ (∀Z){Z(x) ⊃ ¤(∀ w)[Y (w) ⊃ Z(w)]}i(↓G, x)] E

where Y and Z are of type h0i, unlike in (11.2) where they were of type ↑h0i. The variable x is of type 0, and it is easy to show the following simpler formula is a consequence of (11.12). (∀x)[G(x) ⊃ hλY.Y (x) ∧ (∀Z){Z(x) ⊃ ¤(∀E w)[Y (w) ⊃ Z(w)]}i(↓G)] (11.13) From this we trivially get the following. (∀x)[G(x) ⊃ hλY.(∀Z){Z(x) ⊃ ¤(∀E w)[Y (w) ⊃ Z(w)]}i(↓G)]

(11.14)

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Next, in the argument of Section 11.8, we instantiated the quantifier (∀Z) with the term hλy.Qi. Of course we cannot do that now, since hλy.Qi is an intensional term, while the present quantifier is extensional. Apply the extension-of operator, getting ↓hλy.Qi, and use this instead. But universal instantiation involving relativized terms is a little tricky. If ↓τ is a relativized term of the same type as Z, (∀Z)ϕ(Z) ⊃ ϕ(↓τ ) is not generally valid. What is valid is (∀Z)ϕ(Z) ⊃ hλZ.ϕ(Z)i(↓τ ). So what we get from formula (11.14) when we instantiate the quantifier is the following consequence. (∀x)[G(x) ⊃ hλY, Z.Z(x) ⊃ ¤(∀E w)[Y (w) ⊃ Z(w)]i(↓G, ↓hλy.Qi)]

(11.15)

Distributing the abstraction, this is equivalent to the following. (∀x){G(x) ⊃ (11.16) [hλY, Z.Z(x)i(↓G, ↓hλy.Qi) ⊃ hλY, Z.¤(∀E w)(Y (w) ⊃ Z(w))i(↓G, ↓hλy.Qi)]} The variable x does not occur free in hλy.Qi and Y does not occur in Z(x), so hλY, Z.Z(x)i(↓G, ↓hλy.Qi) is simply equivalent to Q, so (11.16) reduces to the following. (∀x){G(x) ⊃ [Q ⊃ hλY, Z.¤(∀E w)(Y (w) ⊃ Z(w))i(↓G, ↓hλy.Qi)]}

(11.17)

From this we get (∃x)G(x) ⊃ [Q ⊃ hλY, Z.¤(∀E w)(Y (w) ⊃ Z(w))i(↓G, ↓hλy.Qi)]

(11.18)

and since we have (∃x)G(x), we also have Q ⊃ hλY, Z.¤(∀E w)(Y (w) ⊃ Z(w))i(↓G, ↓hλy.Qi).

(11.19)

Since Q has no free variables, (11.19) can be shown to be equivalent to the following, where a is a new constant symbol introduced to keep formula formation correct. Q ⊃ hλY, Z.¤((∃E w)Y (w) ⊃ Z(a))i(↓G, ↓hλy.Qi).

(11.20)

Using the distributivity of ¤ over implication, (11.20) gives us Q ⊃ hλY, Z.¤(∃E w)Y (w) ⊃ ¤Z(a)i(↓G, ↓hλy.Qi).

(11.21)

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From (11.21) we get Q⊃

(11.22) [hλY, Z.¤(∃ w)Y (w)i(↓G, ↓hλy.Qi) ⊃ hλY, Z.¤Z(a)i(↓G, ↓hλy.Qi)]. E

Because Z has no free occurrences in ¤(∃E w)Y (w) and Y has no free occurrences in Z(a), (11.22) can be simplified to Q⊃ [hλY.¤(∃E w)Y (w)i(↓G) ⊃ hλZ.¤Z(a)i(↓hλy.Qi)].

(11.23)

I don’t know the status of hλY.¤(∃E w)Y (w)i(↓G), that is, whether or not it follows from the axioms used in this section. It does hold provided G is rigid, so in particular, it holds in the model of Example 11.9.1. Consequently, in settings like that model (11.23) reduces to the following. Q ⊃ hλZ.¤Z(a)i(hλy.Qi).

(11.24)

But hλZ.¤Z(a)i(hλy.Qi) is not equivalent to ¤Q, and that’s an end of it. Expressing the essential idea of hλZ.¤Z(a)i(hλy.Qi) with somewhat informal notation, we might write it as hλZ.¤Zi(Q), and so what has been established, assuming rigidity of G, is Q ⊃ hλZ.¤Zi(Q)

(11.25)

and this is quite different from Q ⊃ ¤Q. In ¤Z, the variable Z is given 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 ⊃ ¤Q, it would necessarily be raining—¤Q—and so in every alternative world, it would be raining. But what we have is Q ⊃ hλZ.¤Zi(Q), and since Q is assumed to hold in the real world, we conclude hλZ.¤Zi(Q). This 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.

11.10

Anderson’s Alternative

One solution to the objection Sobel raised has been presented. In (Anderson 1990) some different, quite reasonable, modifications to the G¨ odel axioms are proposed that manage to avoid Sobel’s conclusion. For this section I return to the use of intensional variables. Axiom 11.2.2B 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 11.2.2B.

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The most fundamental change, however, is elsewhere. Definition 11.3.1 and its alternative, Definition 11.3.2, are discarded. Instead there is a requirement that a Godlike being have positive properties essentially. Definition 11.10.1 [Godlike, Anderson Version] GA is the type ↑h0i term hλx.(∀Y )[P(Y ) ≡ ¤Y (x)]i. There is a corresponding change in the definitions of essence and necessary existence. Definition 11.5.1 gets replaced by the following Definition 11.10.2 [Essence, Anderson Version] E A abbreviates the following type ↑h↑h0i, 0i term: hλY, x.(∀Z){¤Z(x) ≡ ¤(∀E w)[Y (w) ⊃ Z(w)]}i 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 E has been replaced by an equivalence. The definition of necessary existence, Definition 11.6.1, is replaced by a version of the same form, except that Anderson’s definition of essence is used in place of that of G¨ odel. Definition 11.10.3 [Necessary Existence, Anderson Version] N A abbreviates the following type ↑h0i term: hλx.(∀Y )[E A (Y, x) ⊃ ¤(∃E z)Y (z)i. Now, what happens to earlier reasoning? Of course Proposition 11.2.8 still holds, since Axioms 11.2.2A and 11.2.4 remain unaffected. Theorem 11.5.2 turns into the following. Theorem 11.10.4 In S5 the following is provable. (∀x)[GA (x) ⊃ E A (GA , x)]. I leave it to you to verify the theorem, using tableaus say. Next, Anderson replaces Axiom 11.6.2 with a corresponding version asserting that his modification of necessary existence is positive. Axiom 11.10.5 (Anderson’s Version of 11.6.2) P(N A ). Now Theorem 11.6.3 turns into the following. Theorem 11.10.6 Assume Axioms 11.2.11 and 11.10.5. In the logic S5, (∃x)GA (x) ⊃ ¤(∃E x)GA (x).

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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 11.10.7 Assume all the Axioms 11.2.2A, 11.2.4, 11.2.10, 11.2.11, and 11.10.5. In the logic S5, ¤(∃E x)GA (x). 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 (Anderson 1990) for details.

Exercises Exercise 11.10.1 Supply a tableau argument for Theorem 11.10.4. Do the same for Theorem 11.10.6.

11.11

Conclusion

G¨ odel’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. (H´ ajek 1995) studies the role of the comprehension axioms—work that is summarized in (H´ ajek 1996a). Completely general comprehension axioms are implicit in my presentation, they are present as the assumption that every abstract has a meaning. H´ ajek confines things to a second-order intensional logic, augmented with one third-order constant to handle positiveness. In this setting H´ ajek introduces what he calls a cautious comprehension schema: (∀x)[G(x) ⊃ (¤Φ(x) ∨ ¤¬Φ(x))] ⊃ (∃Y )¤(∀x)[Y (x) ≡ Φ(x)]. H´ ajek shows that G¨odel’s axioms do not lead to a proof of Q ⊃ ¤Q, provided cautious comprehension replaces full comprehension, but the necessary existence of God still can be concluded. H´ ajek refutes a claim by Magari, (Magari 1988), that a subset of G¨ odel’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 G¨ odel’s axioms, with cautious comprehension, can be interpreted in Anderson’s system, with full comprehension. The results of H´ ajek 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 11.7, some further consequences of G¨odel’s axioms were discussed. I don’t know what happens to these when the axioms are modified in the various

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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 puzzles, in (Smullyan 1983), Chapter 10. You should find this fun, and a bit of a relief after the rather heavy going of the book you just finished.

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