11.1 INTRODUCTION
FRIEDRICH Ludwig Gottlob Frege was born on November 8, 1848 in the Hanseatic town of Wismar. He was educated in mathematics at the University of Jena and at the University of Göttingen, from which latter he received his doctorate in 1873. He defended his Habilitation the next year in Jena, and took up a position immediately at the University of Jena. Here he spent his entire academic career, lecturing in mathematics and logic, retiring in 1918. His death came on July 26, 1925 in the nearby town of Bad Kleinen.1
Frege is best known for three significant contributions to philosophy. The first is his development of modern quantified logic, a contribution as much to mathematics as to philosophy. The second is his pursuit of the thesis of logicism, the thesis that arithmetic (including the classical theory of the real numbers) is part of pure logic. The third is Frege’s account of the nature of language, including his noteworthy claim that there are two kinds of meaning possessed by virtually all significant pieces of language, commonly known in English as the sense and the reference of those pieces of language. The three contributions are closely connected, and can best be understood by following their parallel development throughout the course of Frege’s work. Following a brief overview of the three contributions, this chapter will proceed roughly chronologically to explore their interaction over the course of Frege’s major works.
11.1.1 Logicism
Frege’s logicism is the thesis that arithmetic (by which Frege means the usual theories of natural numbers, integers, and real numbers, but not including geometry) is part of logic. As Frege understands it, the truth of his version of logicism would imply that arithmetical truths are objective, in the sense of having no dependence on human mathematical (or other) activities, and that they are analytic, in a sense close enough to Kant’s that the truth of logicism so understood would entail that Kant is wrong about arithmetic.
Though the logicist project was later taken up by modern empiricists, Frege’s reasons for favoring it were not empiricist. His view was that arithmetical truth is clearly more deeply grounded than are any forms of synthetic knowledge, and that this depth, and an associated certainty and unrevisability, give good reason to take arithmetic to be grounded directly in logic. Motivation aside, Frege’s central view is that his logicist thesis is susceptible to direct demonstration, of a rigorous kind. This demonstration was to have consisted in (i) a thorough analysis of fundamental arithmetical truths and of their components, and (ii) a proof of those fundamental truths, so analyzed, from purely-logical premises. The attempt to carry out this demonstration occupies a substantial part of Frege’s research from 1879 through at least 1903, and involves as a crucial component the development of a formal system of quantified logic. That system embodies for the first time the principles that form the heart of modern logic today.
Frege’s attempt to demonstrate the truth of logicism was a failure. As he learned via a letter from Bertrand Russell in 1902, one of the principles that he, Frege, had taken to be a basic principle of logic was in fact paradoxical. This meant, for reasons that will be detailed, both that Frege’s analyses of arithmetical truths were flawed, and that his attempts to prove arithmetic from logic were unworkable. It also meant that the formal system of logic he presented in his mature work was inconsistent. The inconsistency in the formal system is easily remedied by removing a single problematic principle, but this remedy makes it impossible to use the system to prove the truths of arithmetic. That is to say: while the system salvaged from Frege’s own via the removal of the problematic parts is of considerable importance in its own right as a forerunner of modern logical systems, it is not sufficient for its original purpose, the defense of logicism.
11.1.2 Quantified Logic
Frege’s development of a formal system of logic was, as noted in section 11.2, motivated by the attempt to provide extremely rigorous proofs of the fundamental truths of arithmetic. By a “system” of logic we mean: (i) a language in which all of the statements appearing in proofs are to be expressed, together with (ii) a collection of fundamental truths to be taken as axioms in any proofs, and (iii) a collection of inference-rules, that is, rules by means of which truths can be inferred from other truths in order to generate proofs. To say that the system is “formal” is to say that components (i)–(iii) are specified entirely syntactically: what counts as a well-formed sentence, or an axiom, or an instance of an inference-rule, is determined entirely by the symbols and the order in which they appear; one need make no appeal to the meanings of the linguistic items in question in order to determine their status in the system. Frege’s own reason for insisting on the formal presentation of his deductive system was that of rigor: his idea was that a syntactic specification of proof would remove all ambiguity and unclarity, and would ensure that all steps taken in a proof were explicitly acknowledged. This is not to say that the formulas of Frege’s formal system were meaningless: on the contrary, each such formula expressed a determinate claim. Frege’s idea was that a syntactically-specified system of proof would give a rigorous way of demonstrating the logical grounding of the claims expressed by its sentences. This fundamental idea, that proofs can be carried out via principles that are specified entirely by means of their syntactic form, has since become a defining characteristic of modern logic.
Frege’s system is a system of quantified logic. The fundamental quantifiers are those notions expressed by the terms “all” and “there exist,” and it is the interaction between these notions, and those expressed by “and,” “not,” and “if…then,” that explains the validity of a huge swath of valid arguments, both in ordinary reasoning and in mathematics. What’s new in Frege’s logic, in addition to its explicit syntactic implementation, is the accurate treatment of the interaction between quantifiers and the other structural features of arguments. The inference, for example, from “Every even number is less than some odd number” and “2 is even” to “There is an odd number such that 2 is less than it” is a kind of inference handled smoothly by Frege’s and not by prior treatments of logic.
The axioms and inference-rules Frege introduces include all of those axioms and rules now familiar as the principles of classical first-order logic, plus two further kinds of principles in virtue of which Frege’s system is importantly richer. The first is the inclusion in Frege’s system of higher-order quantifiers. Though for various reasons the most popular systems of logic in use today do not include these quantifiers (but include only their first-order versions), logics including Frege-style higher-order quantifiers play an important role in both philosophical and mathematical arenas today. The second additional feature of Frege’s formal system in its mature version is that it includes notation for and principles governing what he called value-ranges, a version of the modern idea of sets. On this point Frege’s fundamental idea was flawed; this is the mistake, noted already and described in detail in section 11.2.3, that makes the formal system inconsistent. Formal systems of logic after Frege do not include this principle; his insights concerning value-ranges, subsequently cleaned up so as to avoid paradox, are now pursued via systems of set theory.
11.1.3 Philosophy of Language
Though the topic was arguably of secondary interest to Frege himself, he is perhaps best known for his views about language and meaning. Frege’s mature theory of language involves the view that words and sentences typically have two important semantic features, called by Frege the Sinn (“sense”) and the Bedeutung (“reference”) of those linguistic items. Consider an ordinary sentence, say:
(A)Alice likes geraniums.
One of the important things about this string of symbols is that it can be used, and indeed is typically used, to say something, to make a claim that’s either true or false. Frege’s view is that the claim expressed by this sentence, a claim that’s also expressed by other sentences (e.g. by sentences of German and of French) is the important thing to focus on when we’re talking about truth or falsehood. If Alice does in fact like geran
iums, then the claim—or, as Frege puts it, the Gedanke (“thought”) expressed by (A)—is true.
Thoughts, for Frege, are not mental entities (despite the terminology). The thought that happens to be expressed by (A) is, as he sees it, true; and this does not depend on any person’s having either entertained that idea, or produced that sentence. In Frege’s view, this independence between thoughts and people’s activities is immediately evident once we notice that, for example, Mount Aetna would have been covered in snow even if nobody had ever noticed that it was. That is to say: the thought Mount Aetna is covered in snow, a thought expressible by sentences in many different languages but not dependent on any particular language, is in fact true, and would have been true even if no person had ever seen or imagined that mountain.
Thoughts are, in this sense, the primary bearers of truth and falsehood. For similar reasons, they are also the primary bearers of such logical relations as entailment, consistency, inconsistency, and so on. When, having uttered (A), I go on to say “Therefore, someone likes geraniums,” I have noticed, according to Frege, that the thought expressed by
(B)Someone likes geraniums
follows logically from the thought expressed by (A). Similarly in the domain of arithmetic: when we set about to prove, for example, that 7 + 5 = 12, the thing we prove is not the sentence itself, but the thought it expresses. This forms a crucial part of Frege’s explanation of how researchers who speak different languages can nevertheless be engaged in the same pursuits: they can prove the same theorems, investigate the same questions, and agree or disagree about the same claims. In such cases, the items each investigator is concerned with are thoughts, each of which is expressible via various sentences in different languages.
This view of language-meaning extends to parts of sentences. Each part of a sentence (each word or phrase) makes a particular contribution to the thought expressed by that sentence. This contribution is called the “Sinn” (“sense”) of that part. Because the thought expressed by
(i)The morning star is bright
is a different thought from that expressed by
(ii)The evening star is bright.
Frege concludes that the phrases “the morning star” and “the evening star” have different senses, even though they happen to pick out the same object, the planet Venus. This object, the planet, is what Frege calls the “Bedeutung” (“reference”) of the phrase “the morning star.” (It is also, of course, the reference of “the evening star.”) The pattern established here—in which two singular terms can have the same reference while having different senses—forms a crucial part of Frege’s semantic theory. Both sense and reference are essential to the role of words and phrases in sentences: the sense of a word or phrase is what it contributes to the thought expressed, while the reference is what it contributes to determining the truth-value of that thought. We investigate more fully in section 11.2.2 the nature of senses and references, and their interaction.
11.2 THE HISTORICAL DEVELOPMENT OF FREGE’S WORK
11.2.1 Early Work: Begriffsschrift and Grundlagen
The work of Frege’s that has been most important to philosophers begins with his 1879 monograph entitled “Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens,” familiarly known as “Begriffsschrift.”2 In that work, Frege presents the first version of his formal system of logic. That system contains the formalism for first- and higher-order quantification, but does not contain the problematic notation or axioms for value-ranges. The system is elegant, powerful, and consistent.
Begriffsschrift announces the first step in Frege’s logicist program. It raises the question of whether the truths of arithmetic are provable by means of pure logic alone, and points out that the only way to settle this question is to provide clear analyses and rigorous proofs of those truths. The formal system is introduced as the means of presenting the proofs. Frege does not take a position in Begriffsschrift on the answer to the announced question, but makes some important steps towards its resolution. In addition to introducing the formal system for proofs, Frege provides here several crucial analyses of proto-arithmetical notions, including that of the ancestral of a binary relation, and of the notion of following in a series. Having given the analyses, Frege demonstrates that some surprisingly-rich claims regarding series, claims that will later prove pivotal in his analysis of arithmetic, can be proven by means of pure logic.
Frege’s views about the nature of logic begin to appear in this early work. Especially significant here is the early appearance of his anti-psychologism, a thesis that animates all of Frege’s further work. The view is that logic is a normative discipline, in the sense that it provides norms for valid reasoning, but that it has no grounding in psychological (or other) facts about human beings. It is, as Frege later puts it, the science of truth: its principles are principles governing the entailment relation that obtains between truths, whether or not people actually manage to reason in accordance with them. The principles of logic are, in short, conceived by Frege here and henceforth as principles against which one might measure the rationality of individual reasoning-processes, but not as descriptions of those processes.
Frege’s semantic theory is, in 1879, relatively rudimentary. He holds, as he will continue to hold, that logical entailment is a relation that obtains not between sentences, but between the things expressed by sentences. But in this early work, Frege has not yet introduced the two-tiered semantic theory in which sense is distinguished from reference, and does not yet employ the notion of thought. Here, the things expressed by sentences are known as contents of possible judgment, which contents are determined by the contents of the parts of the sentences in question. As to the nature of the contents of sub-sentential pieces of language, Frege does not have much to say in this period, but appears to take the contents of singular terms to be essentially what will later become their references, that is, the objects (e.g. geometric points or numbers) that they stand for.
Frege’s next major work is his Grundlagen der Arithmetik (Foundations of Arithmetic), published in 1884.3 The primary purpose of this work is to present, in ordinary language (i.e. in German, as opposed to his formal language), Frege’s own analyses of fundamental arithmetical truths, and to sketch the proofs of those truths from logical principles. Here Frege comes down clearly on the side of logicism, and takes this small book to provide the outline of the decisive demonstration of that thesis. En route to presenting his own account of the nature of arithmetic, Frege presents biting criticisms of a number of other accounts, and is particularly interested in undermining both empiricist (Millian) and what he takes to be overly “psychological” accounts of the subject. Frege’s own account of arithmetic turns on the idea that numbers are objects, and that their existence and fundamental properties are independent of our thoughts about them. They are, in that sense, objective: we do not create, but we discover, numbers and arithmetical truths. Frege also argues in Grundlagen that arithmetical facts are essentially facts about the sizes of collections: to say, for example, that 2 + 3 = 5 is to say something that logically entails various facts about the possibility of matching up unions of 2- and 3-membered collections with 5-membered collections.
The reconciliation of these two apparently-conflicting claims, that is, that arithmetical truths are about objects (the numbers), and that they are equivalent to facts about arbitrary collections, is at the heart of Frege’s view of the nature of arithmetic and of arithmetical discourse. The first central point in this account is Frege’s view of what we mean when we say something like “There are four pens on the desk.” His view is that in a case like this, we have attributed a property (roughly, the property of having four things falling under it) to the concept pen on the desk. (A concept (Begriff) here is not a mental entity, but is the kind of thing that a predicate stands for.) We have also, equivalently (in a way that needs explaining; see below) affirmed a relation between a particular object, the number four, and this conce
pt (pen on the desk). As Frege puts it, “the content of a statement of number is an assertion about a concept.”4
The connection between the object four and the property of having four things falling under it turns critically on the kind of object that a number is, for Frege. To explain this, we need first to explain the notion of an extension. What we mean when we talk about extensions of concepts is explained via the following equivalence: to say “the extension of the concept F = the extension of the concept G” is to say something equivalent to: “Everything that falls under F falls under G, and vice-versa.” This equivalence, the principle of extensionality, underwrites Frege’s novel view that we can obtain knowledge of objects without the aid of intuition: if we have analytic knowledge of some claim of the form “∀x(Fx iff Gx),” then (because the principle of extensionality is itself analytic, as Frege sees it), we can conclude via purely analytic principles the corresponding statement: “the extension of F = the extension of G.” To obtain the numbers themselves, we need the notion of a cardinality concept. A cardinality concept is a second-level concept (so, one under which concepts, rather than objects, fall), and it’s one under which a first-level concept falls if and only if that first-level concept has a specific number of objects falling under it. For example: the concept having exactly two objects falling under it, a concept under which fall all and only those first-level concepts under which fall exactly two objects, is a cardinality concept. Notice that for any first-level concept F, there is a cardinality-concept equinumerous with F, under which fall all and only those concepts equinumerous with F. Finally: numbers, for Frege, are the extensions of cardinality concepts. Specifically, where F is a first-level concept, the number of Fs is the extension of the cardinality concept equinumerous with F.
The Oxford Handbook of German Philosophy in the Nineteenth Century Page 41