Overview
Studies in Semantics is the collective title for Rudolf Carnap’s interlocking treatises on formal semantics: Introduction to Semantics, Formalization of Logic, and Meaning and Necessity: A Study in Semantics and Modal Logic. Although each can be read on its own, the three are designed as a connected sequence. They move from a general exposition of semantical concepts, through a detailed examination of how far a formal calculus can capture the logic it is meant to formalize, to an application of the resulting apparatus to modal notions and intensional contexts. The common thread is Carnap’s attempt to construct an exact, systematic vocabulary for talking about meaning, reference, logical truth, and the interpretation of formal languages.
Introduction to Semantics
The first work is expository and programmatic. It begins by dividing the study of language into three branches: pragmatics, which treats language as used by speakers; semantics, which treats the relations between expressions and what they designate; and syntax, which treats the formal relations among expressions themselves. Carnap then distinguishes descriptive semantics, which is empirical and concerns actual languages, from pure semantics, which is a priori and concerns the construction and study of semantical systems. This distinction frames much of the book.
After a survey of the symbols and terminology of symbolic logic, the work introduces the central semantical concepts on which the rest depends. The relation of designation is analyzed, and a family of “L-concepts” is defined: logical truth, logical implication, logical equivalence, and their relatives, understood as truths or relations that hold solely in virtue of the meanings of the expressions involved. To make these notions precise, Carnap introduces the concept of an L-range—roughly, the set of possible state-descriptions or interpretations in which a sentence holds—and constructs a general semantics built upon it. The book also develops the complementary “F-concepts,” which concern factual rather than logical truth, and examines characteristic sentences, content, and range.
The later portions of Introduction to Semantics concern the relations between semantics and syntax. Carnap discusses true and false interpretations, L-true and L-false interpretations, and the contrast between exhaustive and L-exhaustive calculi. An appendix offers terminological remarks and an outline of further semantical problems, together with observations on the logical syntax of language. The overall aim is to show that semantical concepts can be defined with the same rigor as syntactical ones, and that a purely formal treatment of meaning is possible.
Formalization of Logic
The second work is more technical and narrower in focus. Its guiding question is what it means for a formal calculus to be a full formalization of a system of logic—that is, for the calculus to determine uniquely the intended meanings of its signs, rather than merely to generate the correct theorems. Carnap examines the propositional calculus in detail: its propositional connectives, elementary theorems concerning disjunction, negation, and the other connectives, extensionality, and the normal truth-table interpretation.
His central and influential argument is that the ordinary propositional calculus is not a full formalization of propositional logic. The reason is that the calculus admits “non-normal” interpretations of its signs—interpretations that satisfy all the theorems of the calculus but do not assign to negation and disjunction their intended truth-functional meanings. Because the calculus cannot exclude these deviant interpretations, it underdetermines the logic it is supposed to codify.
To remedy this, Carnap introduces “junctives”—new syntactic categories and rules of deduction designed to express the intended interpretation within the object language itself. He applies the L-concepts of the earlier volume to junctives, considers junctives in semantics and in syntax, and constructs a calculus that he argues is a full formalization of propositional logic. The discussion then extends to functional logic (the logic of quantification), where the functional calculus is examined in the same spirit, with attention to transfinite junctives and related constructions. The book is thus both a contribution to technical logic and a philosophical argument about the relationship between a formal system and its intended interpretation.
Meaning and Necessity
The third and best-known work applies the semantical framework to modality and intensional discourse. Its opening and most influential contribution is the method of extension and intension. The extension of a predicate is the class of objects to which it applies; the intension is the corresponding property. For sentences, the extension is a truth value and the intension is a proposition. Carnap uses this distinction to reconstruct, in a more precise and nominalistically acceptable form, Frege’s distinction between sense and reference. He discusses equivalence and L-equivalence, classes and properties, the extensions and intensions of sentences and individual expressions, individual descriptions, Frege’s treatment of descriptions, variables, and the difference between extensional and intensional contexts.
A substantial portion of the book is devoted to L-determinacy: the question of which designators have an intension fixed by the semantical rules of the language alone. Carnap treats the L-determinacy of individual expressions and of predicators, distinguishes logical from descriptive signs, and considers the reduction of extensions to intensions.
The method of the name relation is then examined at length. Carnap sets out the name relation, identifies an ambiguity in the method, discusses the unnecessary duplication of names, names of classes, and Frege’s distinction between nominatum and sense. He weighs the advantages and disadvantages of Frege’s approach, formulates the “antinomy of the name relation,” and proposes solutions to it.
The final major theme is the construction of metalanguages for semantics. Carnap considers the problem of reducing the entities to which a semantical metalanguage is committed, introduces a “neutral” metalanguage, and asks whether such a metalanguage is poorer in expressive power than its alternatives. He discusses neutral variables, the formulation of semantics within the neutral metalanguage, and the possibility of an extensional metalanguage for semantics. In the course of this, modal notions—necessity and possibility—are treated semantically, with necessity analyzed as a species of logical truth. The work as a whole is an extended defense of the viability of intensional and modal logic within a rigorously constructed semantical framework.
Character of the Series
Taken together, the three works display a consistent philosophical temperament: a preference for exact reconstruction over informal description, a concern with the interpretation of formal systems rather than merely their syntax, and a conviction that questions about meaning and modality can be answered by the careful construction of semantical metalanguages. Introduction to Semantics supplies the vocabulary; Formalization of Logic tests that vocabulary against the problem of how a calculus fixes its own interpretation; and Meaning and Necessity extends it to the intensional notions that had traditionally resisted formal treatment. The series is technical and presupposes familiarity with symbolic logic, but its central concerns—what it is for an expression to mean something, and how logical truth differs from factual truth—are stated in terms accessible to the careful philosophical reader.
State-Descriptions and the Semantics of Modality
Carnap gives the modal notions a precise semantical basis through state-descriptions—maximal descriptions of a possible total state of the world. A sentence is L-true when it holds in every state-description, L-false when it holds in none; L-implication and L-equivalence are defined by inclusion and identity of ranges. Necessity is then treated as L-truth, possibility as not L-false, and contingency as neither. This makes modal logic a branch of semantics: the modal operator is interpreted by the semantical rules of the language rather than left as an uninterpreted symbol.
Belief Contexts and Intensional Isomorphism
The book uses belief sentences as a central test for the extension/intension apparatus. In such contexts, substitution of materially equivalent sentences can fail, so Carnap treats belief as an intensional relation and introduces intensional isomorphism as a stricter condition for interchangeability. The analysis connects Frege’s problem of informative identity with the distinction between a sentence’s truth value and the proposition it expresses, and it shows how intensional contexts can be handled without abandoning a systematic semantics.
Meaning Postulates and Relative Analyticity
Carnap also considers how analyticity is fixed for a language that contains descriptive terms. Meaning postulates are introduced as semantical stipulations that determine which sentences are analytic by virtue of the meanings of nonlogical expressions. This extends L-determinacy beyond purely logical vocabulary and reinforces the book’s relativization of L-concepts to a semantical system: logical truth, analyticity, and necessity are not absolute metaphysical properties but are defined by the rules of the language under consideration.
