Introduction to Semantics by Rudolf Carnap
Introduction to Semantics is the first volume of Rudolf Carnap’s Studies in Semantics and a foundational work in the development of formal, or logical, semantics. It is not an introductory survey in the popular sense: it presupposes familiarity with modern symbolic logic and is addressed to logicians and philosophers of science. Its aim is to establish semantics as a rigorous discipline alongside logical syntax, and to show how the concepts of meaning, designation, truth, and logical modality can be explicated with precision.
The book opens by situating semantics within a general theory of signs. Carnap distinguishes three branches of semiotic: pragmatics, which studies language as used by speakers in concrete situations; semantics, which studies the relations between expressions and what they designate, or the truth-conditions of sentences, abstracting from language users; and syntax, which studies the formal structure of expressions. Semantics is then divided into pure and descriptive semantics. Pure semantics constructs and investigates artificial language systems with explicitly stated rules; descriptive semantics studies empirically the semantic properties of natural or historically given languages. Carnap’s principal concern is pure semantics, which he treats as a branch of logic rather than of empirical linguistics.
The central methodological tool is the concept of a semantical system, or language system. Such a system is specified by two kinds of rules: formation rules, which determine which strings of symbols count as well-formed sentences, and semantical rules, which supply the interpretation. Carnap distinguishes rules of designation, assigning objects to individual constants and extensions to predicates, from rules of truth, which determine the truth-conditions of sentences. Truth for atomic sentences is stipulated; truth for molecular sentences—negations, conjunctions, conditionals, and the like—is defined recursively by truth-functional rules. The concept of truth for a language is thus defined within a metalanguage, in the spirit of Tarski’s work, to which Carnap explicitly acknowledges his debt. Truth is relative to a language system: a sentence is true or false only under the rules of a given system.
To explicate logical truth and related notions, Carnap introduces the method of state-descriptions. A state-description is a class of sentences describing a possible state of the universe: for every atomic sentence of the language, the state-description contains either that sentence or its negation, but not both. A sentence is then L-true (logically true) if it holds in every state-description, L-false if it holds in none, and factual if it holds in some but not all. From these notions Carnap defines L-implication, L-equivalence, and the range of a sentence—the class of state-descriptions in which it is true. The L-concepts are offered as exact explicata for the traditional ideas of logical necessity, impossibility, and possibility: the necessary is the L-true, the possible is the not L-false, and so on. The apparatus thus provides a semantic reconstruction of modality and of analyticity without appeal to a special modal primitive.
A second major contribution is the treatment of extension and intension. Carnap distinguishes the extension of an expression—the object or objects to which it applies, or, for a sentence, its truth-value—from its intension, roughly the concept, property, or proposition it expresses. The intension of a predicate is determined by its range across state-descriptions; the intension of a sentence is the proposition it expresses, understood through the set of state-descriptions in which it is true; the intension of an individual expression is an individual concept, a function from state-descriptions to individuals. This framework is designed to handle intensional contexts and to provide a semantics for modal notions, and it is developed more fully in Carnap’s later writings on meaning and necessity.
Carnap also confronts the semantic paradoxes and the distinction between object language and metalanguage. Following Tarski, he insists that the semantic concepts of a language—truth, designation, satisfaction—cannot in general be defined within that same language without contradiction. A language may be syntactically universal, containing its own syntax, but it cannot be semantically universal; semantics requires a richer metalanguage. This hierarchical conception is integral to the book’s architecture and is used to diagnose the errors underlying the semantic antinomies.
An underlying philosophical theme is conventionalism about language forms. The construction of a language system is a matter of convention, to be judged by its utility, simplicity, and fruitfulness rather than by correspondence to a uniquely correct logic. Semantics, on this view, does not uncover metaphysical essences of meaning or reference; it supplies exact frameworks within which questions about truth, implication, and modality can be posed and answered. Philosophical disputes about meaning and necessity are thereby treated as amenable to logical reconstruction.
Throughout, the exposition is technical and systematic, making extensive use of symbolic notation and carefully stated definitions and theorems. Carnap works with simple model languages—small systems containing individual constants and predicates—before extending the treatment to richer languages with variables, quantifiers, and expressions of higher level. The discussion also touches on the relation between semantics and syntax, the status of semantical rules, the concept of logical truth, and the interpretation of scientific language, though the emphasis remains on logical foundations rather than applications.
The work is closely linked to its companion volume, Formalization of Logic, which concerns uninterpreted formal systems and their relation to semantic interpretation; together the two volumes present Carnap’s systematic account of the boundary between formal calculus and interpreted language. Introduction to Semantics remains a key document in the history of formal semantics, notable for its state-description method, its extension–intension distinction, and its attempt to make semantics a precise, conventionalist, and logically disciplined enterprise.
A further organizing distinction in the book is between pure and descriptive semantics. Pure semantics constructs semantical systems and defines their concepts; descriptive semantics applies those concepts to a historically given language. The formal apparatus of Introduction to Semantics belongs primarily to pure semantics, while descriptive semantics is treated as an application of definitions whose exactness derives from a constructed system. Correspondingly, semantic predicates are explicitly relativized: the book speaks of truth in a system \(S\), designation in \(S\), and L-truth in \(S\), rather than of truth or meaning simpliciter.
Within a semantical system, rules of designation connect descriptive constants with their designata, and rules of truth determine the truth-values of sentences recursively from atomic sentences. On this basis Carnap defines a family of L-concepts: a sentence is L-true if it is true in every state-description, L-false if false in every state-description, and L-indeterminate or factual if true in some and false in others; one sentence L-implies another when every state-description that makes the first true makes the second true, and L-equivalence is mutual L-implication. These definitions provide a formal reconstruction of logical truth and analyticity and show how implication and modal notions can be treated within a semantic framework.
The book also stresses that semantics concerns rules and truth-conditions rather than private mental meanings. Intensions are not psychological images but functions determined by the semantical rules of a system; to specify the meaning of an expression is to specify its range across state-descriptions. This anti-psychologistic, rule-based conception supports the treatment of alternative language systems as legitimate objects of formal study rather than as approximations to a single correct language.
