Mathematical Knowledge and Sentential vs. Subsentential Priority

In this chapter, as in Chapter 7, an example is given that shows how Frege’s lessons can be put to work on contemporary issues. The focus here is on two papers, written by Paul Benacerraf in 1965 and 1973, that are still of concern to many philosophers today. In the first, Benacerraf argues that, although it seems obvious that numbers are objects, in fact numbers cannot be objects. In the second, Benacerraf presents an epistemological puzzle that seems to undermine our claims to have mathematical knowledge—even knowledge of elementary facts about numbers. These puzzles challenge our everyday understanding both of the nature of numbers and of our knowledge about them. In this chapter, it is argued that both puzzles depend on our presupposing the subsentential priority view. And both puzzles, it is argued, vanish once we accept Frege’s sentential priority view.

Why Frege’s Apparently Absurd View Is not Absurd at All

This chapter is largely an examination of the significance one of Frege’s views has for contemporary thought. The view, which was labeled the “apparently absurd view” in Chapter 6, is that (1) it is appropriate to give definitions of terms already in use that are, in part, stipulative and (2) it is appropriate to take sentences in which such terms appear as, already (pre-definition), having truth-values. It is argued that, although this view may seem absurd, it is perfectly in line with some scientific practices, in particular unexceptionable practices routinely used in epidemiology. If we follow Hilary Putnam’s view about the significance of our deference to experts, we should accept Frege’s apparently absurd view as not absurd in the least. Moreover, what we see on examination of this view are reasons for rejecting a number of contemporary views about vague language, including those of Field, Fine, Fodor and Lepore, and Williamson.

Soundness, Epistemology, and Frege’s Project

Insofar as the use of natural language to introduce, discuss, and argue about features of a formal system is metatheoretic, there can be no doubt: Frege has a metatheory. But what kind of metatheory? Although the model theoretic semantics with which we are familiar today is a post-Fregean development, most believe that Frege offers a proto-soundness proof for his logic that intrinsically exploits a truth predicate and metalinguistic variables. In this chapter it is argued that he neither uses, nor has any need to use, a truth predicate or metalinguistic variables in justifications of his basic laws and rules. The purpose of the discussions that are typically understood as constituting Frege’s metatheory is, rather, elucidatory. And once we see what the aim of these particular elucidations is, we can explain Frege’s otherwise puzzling eschewal of the truth predicate in his discussions of the justification of the laws and rules of inference.

Reference, the Context Principle, and the Significance of Sentential Priority

Several sections of Basic Laws (§§10, 28–31) appear to offer intrinsically metatheoretic and, indeed, proto-model theoretic proofs. These sections are also notoriously puzzling: the proofs seem obviously incorrect, and it is difficult to understand how Frege could have thought that they worked. In this chapter it is argued that the puzzles in question are artifacts of the Standard Interpretation. They result, in particular, from the assumption that a subsentential expression’s having Bedeutung amounts to its referring to an extra-linguistic entity. The solution to the apparent difficulties is to see that a version of Frege’s context principle—a sentential priority view—is operative even in his later works.

Frege’s New Logic and the Function/Argument Regimentation

Frege says his new logical language is not designed to play the role of natural language. It is, rather, a tool designed for specific scientific purposes. The role he assigns to his logical language forces him to abandon traditional subject–predicate analyses of statements in favor of a new kind of analysis in terms of function and argument. But the function-argument analysis, in its original version, gives rise to several problems, among them the problem about identity with which Frege begins “On Sinn and Bedeutung.” This chapter traces through the problems with the original version of the new logical language and Frege’s later solutions. A key part of these solutions is to take sentences to be object names. And, while this may be problematic in an account of the workings of natural languages, it is unproblematic in a language that plays the role of Frege’s logical language.

The Context Principle, Sentential Priority, and the Pursuit of Truth

Frege’s attempt to show that arithmetic is a part of logic requires definitions of the numbers. What criteria determine whether his definitions are acceptable? It seems to stand to reason that the definition of, say, the number one must pick out the object to which we have been referring, all along, when we use the numeral “1.” However, Frege does not assume that there are objects to which we have been referring all along when we use numerals. Definitions of the numbers must be, at least in part, stipulative. But how, then, can a science based on Frege’s definitions be our science of arithmetic? The key to answering this question is Frege’s sentential priority view. To be accurate to our arithmetic, Frege’s definition does not need to preserve reference; what they need to preserve is, rather, the truth of sentences expressing the “well known properties of the numbers.”

Metaphysics and the Standard Interpretation

Frege is celebrated as an arch-Platonist and an arch-realist. He is renowned for claiming that truths of arithmetic are eternally true and independent of us, our judgments, and our thought; that there is a “third realm” containing nonphysical objects that are not ideas. In this chapter it is argued that, to sustain the view that Frege means these renowned claims as statements of metaphysical doctrines, it is necessary to reject most of his claims about the unsaturatedness of functions. It is widely believed that these claims should be rejected, since they give rise to apparently insuperable problems. In this chapter, however, it is argued that the problems result, not from Frege’s claims about unsaturatedness but, rather, the view that they are meant to form part of a metaphysical theory. Properly understood, Frege has no problem with the concept horse.

Language and the Standard Interpretation

On the Standard Interpretation, Frege means to be giving some sort of theory of the workings of language. The interpretation is easily justified, many think, by mere reportage of the words on his pages. This chapter begins with a closer look at this reportage and shows that a great deal of the widely accepted reportage is inaccurate. The chapter goes on to exhibit problems with the Standard Interpretation. Frege’s aim is to develop, not a perfect language, but a logically perfect language—a language adequate for the expression and evaluation of inference. And, as it turns out, a language that does the work of a natural language cannot satisfy the demands that must be satisfied by a logically perfect language.