In one way or another the nine chapters of this book all have to do with free logic. Most are updated revisions in and adaptations of previously published papers. The exceptions are Chapters 4 and 5, though Chapter 4 contains a revised segment from a recently published paper.

Chapter 1 began as an invited address to the Western Division of the American Philosophical Association Meetings in 1991, and at the request of the organizers of those meetings was subsequently published in slightly revised form in Philosophical Studies, 65 (1992), pp. 153–167. It is a critical analysis of Russell's famous theory of definite descriptions of which there are two quite distinct versions. The defining feature of either version is that definite descriptions are not singular terms. That the essence of Russell's theory has to do with logical grammar was stressed in my ‘Explaining away singular existence statements’, Dialogue, 1 (1963), pp. 381–389, and later, independently, by David Kaplan in ‘What is Russell's theory of descriptions?’ Physics, History and Logic (eds. W. Yourgrau and A. Breck), Plenum Press, New York (1970), pp. 277–288.

INTRODUCTION

Quine's concept of predication is intimately related to his notion of referential opacity. To treat the position of a singular term in a sentence as “purely referential”, and hence the sentence as “referentially transparent”, is to treat the very same sentence as a predication. “Predication”, he says, “joins a general term and a singular term to form a sentence that is true or false according as the general term is true or false of the object, if any, to which the singular term refers.” The conception of predication expressed in the quoted passage is not restricted to its author. Among others who hold it are many free logicians.

Quine has also written that “so long merely as the predicated general term is true of the object named by the singular term … the substitution of a new singular term that names the same object leaves the predication true.” So if a sentence is a predication, it satisfies the substitutivity of identity. Moreover, “in an opaque construction you … cannot in general supplant a general term by a coextensive term (one true of the same objects) … without disturbing the truth-value of the containing sentence. Such a failure is one of the failures of extensionality.”

The theory of predication under consideration is non-extensional precisely in the sense that it does not satisfy the principle that co-extensive general terms substitute for each other salva veritate. The proof of this fact is the first order of business.

INTRODUCTION

Consider the statement

(1) is believed by most philosophical logicians to be non-extensional according to each of two prominent conceptions of extensionality. First, it fails the salva veritate substitution test when ‘9’ is replace by the co-referential singular term ‘the number of planets’. Second, in possible world semantics, its truth is regarded as dependent on the reference-in-all-possible-worlds of ‘9’, or, alterntively, on the intension of ‘9’ conceived as a function from possible worlds to individuals.

More generally, a statement is SV-Extensional according to the salva veritate substitution conception just in case singular terms co-referential with a statement's constituent singular terms, predicates co extensive with the statement's constituent predicates, and statements co-valent with a statement's constituent statement(s), substitute in that statement salva veritate. A statement is not SV- Extensional if there is at least one failure of salva veritate substitution. (This is close to the characterization in Quine's Word and Object.)

A statement is TD-Extensional according to the truth-value dependence conception just in case its truth value depends only on the extensions simpliciter, if any at all, of its constituent singular terms, predicates and statements. A statement is not TD-Extensional if entities other than extensional entities areinvolved in the computation of its truth value.

INTRODUCTION

It is mildly ironic that the title of this chapter is an unfulfilled (or improper) definite description because Russell really had two versions of the theory of definite descriptions. The two versions differ in primary goals, character and philosophical strength.

The first version of Russell's theory of definite descriptions was developed in his famous essay of 1905, ‘On Denoting’. Its primary goal was to ascertain the logical form of natural language statements containing denoting phrases. The class of such statements included statements with definite descriptions, a species of denoting phrase, such as ‘The Prime Minister of England in 1904 favored retaliation’ and ‘The gold mountain is gold’. So the theory of definite descriptions contained in what Russell himself regarded as his finest philosophical essay is a theory about how to paraphrase natural language statements containing definite descriptions into an incompletely specified formal language about propositional functions. Russell used this version of his theory to disarm arguments such as Meinong's arguments for beingless objects. Such reasoning, he said, is the product of a mistaken view about the logical form of statements containing definite descriptions.

The second and later version is presented in that epic work of 1910, Principia Mathematica (hereafter usually Principia). Its primary goal. in contrast to the first version, was to provide a foundation for mathematics, indeed, to reduce all of mathematics to logic. In chapter *14 Russell introduces a special symbol, the inverted iota, and uses it to make singular term-like expressions out of quasi-statements.

LOGICAL TRUTH AND TRUTH-VALUE GAPS

Elementary microphysical statements can be neither true nor false without violating the classical codification of statement logic. The existence of such a possibility depends upon a revision in the standard explication of logical truth, a revision more harmonious with the idea of argument validity as merely truth-preserving. The revision in question, in turn, depends upon Bas van Fraassen's investigations into the semantical foundations of positive free logic, a species of logical system whose philosophical significance was first made plain in Henry Leonard's pioneering study of 1956, ‘The Logic of Existence’.

Students who steadfastly refuse to accept an argument as valid unless all of the component statements are in fact true frustrate teachers of logic. “Now look!” the teacher may heatedly emphasize, “the validity of an argument has to do with its form alone. So to say that an argument is valid is to say only that if its premises were true its conclusion would also be true!” But, then, not only is the argument from the pair of false statements

and

to the false statement

valid, but so is the argument from the pair of (allegedly) truth-valueless statements

and

to the (allegedly) truth-valueless statement

WHAT FREE LOGIC IS AND WHAT IT ISN'T

On page 149 of the second edition of their book, Deductive Logic, Hugues Leblanc and William Wisdom say the following about the origins of free logic.

This account is in one respect misleading, and in another inaccurate. It is misleading because Rolf Schock, independently either of Hintikka or of Leblanc and Hailperin, was developing a version of free logic in the early 1960s quite different in character from those mentioned by Leblanc and Wisdom. Moreover Schock's ideas were in certain respects more fully developed because he also supplied models for his own systematization. The writers mentioned by Leblanc and Wisdom had suggested – in print at any rate – only informally the semantical bases for their formulations. (Apparently Schock's ideas were known to some European scholars in the early 1960s but oddly were not disseminated. One can find an account of Schock's pioneering efforts in his 1968 book, Logics Without Existence Assumptions.

THE CONJECTURE

David Kaplan once suggested to me that the pair of self-contradictory statements:

and

“ought to have the same father”. But apparently they don't despite their family resemblance. Russell deduced (1) from a principle he presumed correctly to be a key ingredient of Meinong's theory of objects. That principle says:

On the other hand, Russell deduced (2) from a seemingly unrelated but no less fundamental principle in Frege's version of set theory, the principle of set abstraction. That principle, a version of the principle of comprehension, (in effect) says:

The lack of common ancestry between MP and FP, and hence between their respective consequences (1) and (2), enabled Russell to treat the theory of objects and the theory of sets (or classes) very differently. He thought (1) “demolished” the theory of objects, but he didn't think (2) destroyed the theory of classes. Russell's attitude was wrong, because Kaplan's suspicion of the common kinship of (1) and (2) is justified, and the proof of this fact is the next order of business.

The Nash solution of the bargaining problem whose frontier is the cubic curve with Equation (5.1.11) is the point on the frontier where [(1 + A)x − y − a] (y − b) takes its maximum value. We use the Lagrange multiplier method to find the maximum. Set the Lagrangian equal to

Then

It follows from the condition ∂Λ/∂y = 0 that

Set a − b = α. Then

However, if

then

Substitute the expression for y into Equation (5.1.11) and clear the denominator. The result is the following equation in x.

To find the optima in Example 5.2.1., we computed the partial derivative of P, the payoff function, with respect to y. The result is a quadratic in x with a single solution S(d, x) that is in the interval [0, 1] when x ∈ [0, 1] and d ∈ (0.5, 1.5). The solution S(d, x) substituted for y in P. When the resulting function of x and d is optimized, values to be in chosen in [0, 1], the result is the expression given in the second section of Chapter 5. One can then compute the y coordinate ŷ for the optimal point. The expression for is the following.

Each of the entries F1, …, F7 is an expression in d. The expressions are the following.

THE LEONTIEF THEOREM

In Chapter 1 we presented the concept of the (time) complexity of a function in an network model of computing. In this chapter we analyze the complexity of a function, or obtain bounds on its complexity. It is shown in Chapter 2 that when the class of elementary functions consists of Boolean functions the network model specializes to the finite-automaton model of computing. The network model is also related to an approach to computing and complexity called nomography (Bieberbach, 1922) that was pursued in mathematics from the late 19th into the mid-20th centuries. In this chapter we use some results from that literature to analyze the complexity of functions in the network model. Thus, the network model serves as a bridge connecting finite-automaton theory to classical mathematics – a connection we will emphasize below, when we consider the complexity of finite approximations of smooth functions (cf. Chapter 5).

A simple case to start with is that of networks. The class is the collection of real-valued differentiable functions of one real variable. The class contains no functions that can be used to reduce the n(>1) variables {x1, …, xn} to a single real variable. Whether a function G can be computed by an network can be decided by simply counting the number of variables of G. If n = 1, so that G is a function of one variable, and if g is some subset of, the time required to compute G by a G network can be very difficult to determine.