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.”
“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 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.
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...
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.
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.