Selected Essays of Gottlob Frege

First published: “Funktion und Begriff,” 1891 (“Function and Concept,” 1952); “Über Begriff und Gegenstand,” 1892 (“On Concept and Object,” 1952); “Über Sinn und Bedeutung,” 1892 (“On Sense and Nominatum,” 1949; better known as “On Sense and Reference,” 1952); “Der Gedanke: Eine logische Untersuchung,” 1918 (“The Thought: A Logical Inquiry,” 1956)

Type of Philosophy: Epistemology, logic, philosophy of language

Context

Four essays express the essence of Gottlob Frege’s mature philosophy of language: “Function and Concept,” “Concept and Object,” the classic “On Sense and Reference,” and the later “The Thought.” (All are collected in The Frege Reader, 1997.) Although Frege conceived these papers as ancillary to the logicist project, they became the foundation of modern semantic theory. They contain significant revisions of Frege’s earlier views, yet they basically adhere to the three principles of Die Grundlagen der Arithmetik: Eine logische mathematische Untersuchung über den Begriff der Zahl (1884; The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of the Number, 1950): “There must be a sharp separation of the psychological from the logical, the subjective from the objective”; “the meaning of a word must be asked for in the context of a proposition, not in isolation”; and “the distinction of concept and object must be kept in mind.”

Functions and Concepts

“Function and Concept” was the first essay to clarify and systematize the semantic theses of Frege’s earlier works. The notion of function is taken from mathematics. In the expression “2x² + 2,” if x is replaced with 1, the result is 4. Frege says that “people call x the argument,” which is a useful way to think of it. However, for Frege an argument is defined as that which is not part of a function but combines with it to form a complete expression. The number signified by the whole expression is its value. For any argument there is only one value, and for mathematical functions it is always a number. The value-range of a function is thus a series of pairs of arguments and values. The value-ranges of two functions are identical if the functions always have the same value for the same argument. Thus, “x²-2x” and “x(x-2)” have the same value-ranges. When a function lacks an argument, Frege refers to it as incomplete, or “unsaturated.” When an argument is present, the function is complete, or “saturated.” A function has a value only for an argument.

Frege extended his definition of function from mathematical to universal use by considering functions such as x² = 4. He posited that the value of this function is a truth-value, either the True or the False. There are only two arguments,-2 and 2, for which the function’s value is the True; for all other arguments, it is the False. Therefore the expression “2² = 4” signifies the True just as “2²” signifies 4. He defines a function whose value is always a truth-value as a concept. Any predicate formed by omitting a proper name from a sentence will express a concept. However, a functional expression such as “the father of . . . ” will not, because for any appropriate argument (Mary), the value of the function is not a truth-value but a person (John).

The extension of a concept is a series of pairs, with one member being an object and the other a truth-value. This is an advance over the traditional conception, in which the extension of a concept consists of the objects that fall under it. According to that view, unicorn cannot signify a concept because no objects fall under it. This undesirable result is avoided by Frege’s theory, because for any argument there will be a truth-value—albeit for unicorn this is always the False.

As is perhaps now evident, Frege’s definitions of function and concept apply to statements of all kinds. For example, in the sentence “Caesar conquered Gaul,” the expression “conquered Gaul” contains an empty place; it is an incomplete expression. Only when this place is filled with a proper name (Caesar) or definite description does the sentence make sense. When used as such, however, arguments are not names but objects, and may, where appropriate, be persons, places, truth-values, value-ranges, extensions of concepts, or any other entity that is not a function. Moreover, a concept must be bounded: “It must be determinable, for any object, whether or not it falls under the concept.”

Objects and Concepts

The principal contention of “Concept and Object” is that concepts and objects are exclusive categories: Objects may fall under concepts but not the reverse. Frege anticipated a number of possible objections. One is that concepts can have properties, and to have a property must be to fall under a concept. Frege replied that a first-level concept can be subordinate to another first-level concept; for example, the concept mammal is subordinate to the concept animal. A concept may also fall within a second-level concept. We may say, “The concept unicorn is not instantiated (has no objects),” where instantiation is a property. Another possible objection stems from Frege’s criterion that any expression preceded by a definite article designates an object rather than a concept. How, then, does one construe sentences such as “The concept horse is a concept?” Because “the concept horse” begins with a definite article, it must refer to an object; yet because of its content, it must refer to a concept. Frege argued that “the concept horse” designates an object, even though it forced him to the awkward and controversial conclusion that “The concept horse is a concept” is false.

Sense and Reference

Frege’s seminal distinction between sense and reference was introduced in “Function and Concept” and developed in “On Sense and Reference,” one of the most influential works in analytic philosophy. In it Frege grappled with two language puzzles. The first concerns statements of identity such as “3 + 2 = 5,” “The morning star is identical to the evening star,” and “Regina is Davis’s mother.” These statements all take the form “a = b,” where “a” and “b” are either names or descriptions that designate individuals. The truth of “a = b” requires that the expressions flanking the identity sign refer to the same object. For example, “3 +2 = 5” is true only when “3 + 2” and “5” refer to the same number. Similarly, “The number of planets is nine” is true only when “The number of planets” is the same number as “nine.” This assumes that identity expresses a relation between signs, as indeed it did in Frege’s Begriffsschrift: Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (1879; “Begriffsschrift, a Formula Language, Modeled upon That of Arithmetic, for Pure Thought,” 1967; better known as Conceptual Notation, 1972).

The puzzle arises from the fact that the truth conditions for “a = b” are no different from those of “a = a.” Compare these two sentences: (1) The evening star is the evening star, and (2) The morning star is the evening star.

If, as astronomers have shown, the morning star and the evening star are just different aspects of the planet Venus, then (2) should be obtainable from (1) by substituting “the morning star” for “the evening star.” There is no difference between (1) and (2) at the level of reference because they refer to the same object. However, whereas (1) is almost trivially true (2) conveys genuine knowledge. Frege’s insight was that, because “the morning star” and “the evening star” express different thoughts, it follows that (1) and (2), though they share the same reference, also differ in the thoughts they express. Names are arbitrary signs; we can name an object anything we choose. Frege concluded that if “a = b” signifies a relation between signs, it expresses no knowledge about the extra-linguistic world.

The sense-reference distinction attempts to overcome this difficulty. Here, identity is a relation not between signs but between objects, or references; the informativeness of identity statements is explained by differences in the senses of the expressions flanking the identity sign. The sense of a linguistic item is what we grasp when we understand it; it is a common property—objective, immutable, and timeless. Senses are neither ideas nor objects. As Frege puts it, an idea is a subjective “mode of the individual mind” and may vary with person and place: It has an owner. In an ideal language, Frege asserts, every sign would have a unique sense. However, in natural languages, different signs may express the same sense. Moreover, not every sense has a reference; for example, “the least rapidly convergent series” has a sense but refers to no object. Fictional names also have a sense but no referent. If a supposedly fictional name is found to have a reference, then the thought remains the same but is “transposed from the realm of fiction to that of truth,” which further shows that sense is independent of reference. Ordinarily, though, the use of names presupposes that they have a reference.

Not only do proper names and definite descriptions have senses and references, but so do entire sentences. The sense of a sentence is the thought it expresses, and its reference is its truth-value, the object known as the True or the False. Every seriously propounded indicative sentence is the name of one or the other of these objects. It is by making a judgment that we take a sentence from the level of thought to the level of reference. Frege explicitly ignores sentences that do not make assertions because his aim is to describe, and where necessary to construct, a language that serves the needs of science. The reference of expressions within sentences is explained in terms of the contribution they make to the reference of the sentences in which they occur. For example, the reference of a predicate is a concept, and the reference of a quantifier, such as “all” or “some,” is a second-order concept, a function from concepts to truth-values.

Frege held that the references of complex expressions are determined by the references of their parts. This gives rise to the second linguistic puzzle, one concerning intensional contexts, in which expressions ascribe a “propositional attitude” such as “X knows that p,” “X believes that p,” and so on. Compare the following propositions: (3) Venus is the morning star, (4) Gottlob believes that the morning star is the morning star, and (5) Gottlob believes that Venus is the morning star.

In accordance with (3), the substitution of “Venus” for “the Morning Star” in (4) will yield (5). However, (4) is almost trivially true, whereas (5) is not, for it may be either true or false that Gottlob holds this belief. Frege interpreted this apparent failure of reference in intensional contexts as further evidence that a purely referential theory—one lacking senses—cannot account for the logical behavior of certain expressions. His reaction was not to deprive the expressions of referential status but to change the reference. In the resulting theory of indirect reference, belief is a relation between a believer and a thought. The reference is a not a truth-value but the sense of what is believed (or disbelieved).

Thought

The later essay “The Thought: A Logical Inquiry” explicates its subject and much more, for in it Frege develops his mature theories of truth and judgment. Here the True and the False are presented as indefinable, primitive objects. Frege’s three-tiered system is clearer than ever: At the level of signs is language, at the level of sense is thought, and at the level of reference are objects, which include truth-values as the references of thoughts. This translates into three realms of existent things: physical entities, mental entities, and a “third realm” of entities that are neither physical nor mental. Frege’s theory of judgment is marked by three levels: the grasping of a thought, the judgment that the thought is true (or false), and the assertion (or denial) of the thought. The first two are mental acts, whereas the third is an external manifestation of the judgment. The sense-reference distinction gives the notion of thought more clear-cut form because, for the most part, linguistic structures mirror the senses that they express. Although two or more expressions may share a reference yet differ in sense, no expression has more than one sense (in a given context). The greatest virtue of Fregean semantics is the clarity with which it can distinguish linguistic from other kinds of knowledge.

Relegating thoughts to a third realm results from Frege’s desire to preserve the objectivity of knowledge. The cost, however, was a weighty metaphysics that entails, for example, a causal relation between nonphysical entities and human cognition. Some argue that thoughts need not have this special status to fulfill their purpose. The question of whether all expressions have both a sense and a reference and what these are for each class of expressions has spawned much research. In some interpretations, the realm of sense is not independent of thinkers; rather, it contains humanity’s store of “accumulated knowledge.” Despite these disagreements, all philosophers of language owe a debt to Frege for providing the framework for such investigations.

Principal Ideas Advanced

•The True and the False are indefinable, primitive objects.

•Reality is three-tiered: At the level of signs is language; at the level of sense is thought; at the level of reference are objects.

•The significant expressions of a language have both a sense and a reference.

•Thoughts are unchanging and eternal; unlike ideas, they are common property and have no owners.

Bibliography

Beaney, Michael. Frege: Making Sense. London: Duckworth, 1996. An analysis of Frege’s logic and resulting philosophy.

Currie, Gregory. Frege: An Introduction to His Philosophy. Brighton, England: Harvester Press; Totowa, N.J.: Barnes and Noble, 1982. Offers an introduction to the central points of Gottlob Frege’s philosophical program and traces the historical development of his views. It provides a very clear explanation of Frege’s formal system, philosophy of mathematics, and philosophical logic. Recommended for advanced undergraduates.

Dummett, Michael. Frege: Philosophy of Language. 2d ed. Cambridge, Mass.: Harvard University Press, 1981. Although Dummett’s interpretations are hotly contested, this magisterial study is the definitive work on Frege. A very long work, it presupposes knowledge of the main currents of analytic philosophy and of symbolic logic in places, but the summaries impart key arguments very clearly. For advanced undergraduates.

Dummett, Michael. The Interpretation of Frege’s Philosophy. Cambridge, Mass.: Harvard University Press, 1987. This book is a reply to criticisms of Dummett’s Frege: Philosophy of Language. It is also shorter, less technical, and places Frege in a broader historical setting, thus making it suitable as an introduction.

Grossman, Reinhardt. Reflections on Frege’s Philosophy. Evanston, Ill.: Northwestern University Press, 1969. The author sees “On Sense and Reference” as the solution—though not a faultless one—to a number of problems emerging from Frege’s earlier work. This is a superb introductory text for the advanced undergraduate.

Kenny, Anthony. Frege. New York: Penguin Books, 1995. This marvelous survey briefly explains and assesses the full range of Frege’s work. Accurate but nontechnical, this is perhaps the most accessible introduction to Frege for the student or general reader.

Kneale, William, and Martha Kneale. The Development of Logic. Oxford: Clarendon Press, 1962. Though lacking the benefit of recent scholarship, this classic and authoritative text contains a profound yet accessible analysis of Frege’s career, his relation to his predecessors, and his theory of number. It devotes a chapter to his general logic, including expositions of his Conceptual Notation, the theory of sense and reference, and The Basic Laws of Arithmetic.

Schirn, Matthias, ed. Frege: Importance and Legacy. New York: Walter de Gruyter, 1996. This book examines Frege’s analytical philosophy and its lasting legacy.

Sluga, Hans D. Gottlob Frege. Boston: Routledge and Kegan Paul, 1980. A readable account of Frege’s theories set against the philosophical concerns of late nineteenth century Germany. Traces the influences of Gottfried Wilhelm Leibniz, Immanuel Kant, and later philosophers on Frege and his contemporaries.

Walker, Jeremy D. B. A Study of Frege. Ithaca, N.Y.: Cornell University Press, 1965. A thoughtful exposition, especially strong on the philosophy of language of Frege’s middle and late periods. Suitable for advanced undergraduates.

Weiner, Joan. Frege in Perspective. Ithaca, N.Y.: Cornell University Press, 1990. Though not expansive enough to serve as an introduction, this study is not addressed solely to specialists. It discusses Frege’s writings on number theory and the laws of thought and seeks to discover Frege’s motivation for attempting to prove the truths of arithmetic.