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## Context

(Student Guide to World Philosophy)

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

(Student Guide to World Philosophy)

“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 “2*x*2 + 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, “*x*2-2*x*” 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 *x*2 = 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...

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