Frege’s theory of meaning
Updated: Oct 18
An Introduction
Gottlob Frege’s theory of meaning distinguishes three elements: expression (a word, phrase, or sentence), sense (Sinn), and reference (Bedeutung). An expression’s sense presents its reference in a particular way: Frege describes sense as containing the “mode of presentation” (Frege, 1892). Thus, “the morning star” and “the evening star” designate the same object, Venus, but express different senses. This distinction explains why an identity statement can convey new knowledge even when its two names refer to one object. Sense is not a private mental image: it is something different speakers can grasp and share.
At sentence level, sense is a thought and reference is a truth-value: “We are therefore driven into accepting the truth value of a sentence as constituting its reference” (Frege, 1892).
Two principles organize Fregean semantics. The context principle directs us “never to ask for the meaning of a word in isolation, but only in the context of a proposition” (Frege, 1884/1953, Introduction). Rather than identifying word-meaning with an associated image, we explain an expression through its contribution to complete, truth-evaluable propositions. The context principle rejects semantic atomism: meanings are not first given as self-sufficient word-meanings and only subsequently assembled into propositions.
The principle of compositionality, in its modern formulation, states that a complex expression’s meaning depends on its constituents’ meanings and their arrangement. It is often called “Frege’s principle” (following Carnap), but that label is historically misleading if it implies that Frege himself formulated the doctrine in precisely this way. Carnap supplies the classic explicit formulation: “The extension of a compound expression is a function of the extensions of its components; and the intension of a compound expression is a function of the intensions of its components” (Carnap, 1947).
Frege’s context principle and the compositionality associated with his semantics also raise the question of semantic holism: the context principle gives explanatory priority to propositions, but does not by itself entail dependence of meaning on an entire language or theory.
Contextual explanation and compositional construction need not conflict. The meaning of context and the parts of the context are in dynamically interdependent: “It is enough if the proposition taken as a whole has a sense; it is this that confers on its parts also their content.” (Grundlagen, §60). Nor should the context principle automatically be equated with semantic holism. However, the meaning of an expression depends on its place within an interconnected system of expressions, inferences, or beliefs, rather than being independently determined in isolation.
Frege’s treatment of arithmetic illustrates this approach. He treats numbers as abstract objects and cardinal expressions, when used to designate those objects—as in “Two is prime”—as number-names or singular terms. These uses are nominal; their logical status should nevertheless be distinguished from the grammatical classification of cardinal expressions in counting constructions such as “two books.” Frege titles his discussion “Every individual number is a self-subsistent object” (Frege, 1884), while explaining numerical expressions through propositions, particularly statements of numerical identity.
The above example is an illustration of concept–object distinction: For Frege, objects are complete entities designated by singular terms, whereas concepts are unsaturated entities designated by predicates: “a concept is a function whose value is always a truth-value” (Frege, 1891; [1952, p. 30]). Thus, in “Two is prime”, two designates the object 2, while is prime designates a concept that yields the True for 2 and the False for 4.
Frege analyzes expressions into a function and an argument that completes it, explaining that “the function by itself must be called incomplete, in need of supplementation, or ‘unsaturated’” (Frege, 1891; [1952, p. 24]). His mathematical example is 2x3 + x: supplying 1 as argument gives the value 3; correspondingly, supplying an object to a conceptual function gives a truth-value rather than a number.
Frege gives this linguistic example: “Statements in general, just like equations or inequalities or expressions in Analysis, can be imagined to be split up into two parts; one complete in itself, and the other in need of supplementation, or ‘unsaturated.’ Thus, e.g., we split up the sentence ‘Caesar conquered Gaul’ into ‘Caesar’ and ‘conquered Gaul’. The second part is ‘unsaturated’—it contains an empty place; only when this place is filled up with a proper name, or with an expression that replaces a proper name, does a complete sense appear” (Frege, 1891, [1952, p. 31]).
The analysis of the sentence looks like this:
Component | Expression | Semantic role |
Argument-expression | Caesar | Designates the object Caesar |
Function-expression | ___ conquered Gaul | Designates a concept: a function from objects to truth-values |
Complete sentence | Caesar conquered Gaul | Expresses a thought and refers to a truth-value |
In modern notation, we can reconstruct the analysis as F(c), where F(x) means “x conquered Gaul” and c designates Caesar. Strictly, the words are expressions of the function and argument; the function takes the object Caesar—not the word Caesar—as its argument. Frege’s concept is a function whose value is a truth-value.
“I have kept to three fundamental principles: always to separate sharply the psychological from the logical, the subjective from the objective; never to ask for the meaning of a word in isolation, but only in the context of a proposition; never to lose sight of the distinction between concept and object” (Frege, 1884).
Frege also analyzes empty names and referential opacity in belief reports, where expressions can designate their customary senses rather than their ordinary referents. Together, these distinctions provide foundations for modern formal semantics.
References
Frege, G. (1891). Funktion und Begriff: Vortrag, gehalten in der Sitzung vom 9. Januar 1891 der Jenaischen Gesellschaft für Medizin und Naturwissenschaft [Function and concept: Lecture delivered at the meeting of the Jena Society for Medicine and Natural Science on January 9, 1891, (P. Geach, Trans.). In P. Geach & M. Black (Eds.), Translations from the philosophical writings of Gottlob Frege (pp. 21–41). Basil Blackwell]. Hermann Pohle.
Frege, G. (1892). Ueber Sinn und Bedeutung [Sense and reference (M. Black, Trans.). The Philosophical Review, 57(3), 209–230. 1948]. Zeitschrift für Philosophie und philosophische Kritik, 100(1), 25–50.
Frege, Gottlob. 1884. Die Grundlagen der Arithmetik: Eine logisch mathematische Untersuchung über den Begriff der Zahl[The foundations of arithmetic: A logico-mathematical enquiry into the concept of number (J. L. Austin, Trans.; 2nd ed.). Basil Blackwell. 1953]. Breslau: Wilhelm Koebner
Frege, G. (1923). Logische Untersuchungen. Dritter Teil: Gedankengefüge. Beiträge zur Philosophie des deutschen Idealismus, 3, 36–51. Compound thoughts (R. H. Stoothoff, Trans.). Mind, 72(285), 1–17. 1963
Carnap, Rudolf. 1947. Meaning and Necessity: A Study in Semantics and Modal Logic. Chicago: University of Chicago Press.
Comments