Logical Semantics

a branch of logic that deals with the study of the meaning and sense (in Russian, znachenie and smysl) of concepts and propositions and of their formal analogues—the interpretations of expressions (terms and formulas) of different calculi (formal systems). The first and foremost task of logical semantics is to define precisely the concepts of “meaning,” “sense,” and “interpretation,” and, accordingly, the concepts of “truth,” “definability,” “expressibility,” “consequence,” and “model” (including such general and primary concepts as “set,” “object,” and “correspondence”).

Important semantic problems arise as a result of the difference between the content and extension of concepts and between the meaning and truth value of propositions. Properties such as equivalence and consequence that are related to the content of concepts and the signification of propositions are called intensional; properties related to the extension of concepts and the truth value of judgments are called extensional. Propositions and concepts that are intensionally equivalent are also extensionally equivalent, although the opposite is generally false. For example, the statements “The Volga flows into the Caspian Sea” and “2 × 2 = 4” are equivalent extensionally but not intensionally. Any pair of propositions that are equivalent in the ordinary sense of the term illustrate the preceding assertion (see below the discussion of analytic and synthetic truth).

In logical semantics the fundamental relation between an expression and its interpretation has, after detailed analysis, proved to be not a two-place but a three-place relation. That is, the concept of an interpretation is “stratified” into an extensional and an intensional level. Following the tradition established by G. Frege, the author of the first fundamental studies on logical semantics, by the Austrian logician R. Carnap, and by the contemporary American logician A. Church, each proper noun (including, in the broad sense of the term, cardinal numbers and any nouns with definite articles or demonstrative pronouns) is contrasted with, on the one hand, its denotative object (denotation, or nominatum) and, on the other, the sense, or concept expressed by the noun. The terms of this “semantic triangle” are first defined for natural languages and only then, with certain restrictions, are they transferred to formalized languages.

The relations between noun, denotation, and concept are generally not univocal. Thus, nouns that are homonyms have several different concepts; on the other hand, different nouns that are synonyms may correspond to one and the same concept. The nominative relation between a noun and its denotation is also not univocal (Frege’s example was that of the nouns “morning star” and “evening star,” which have the same common denotation, the planet Venus, but different concepts). The concept however completely defines the denotation, that is, of course, if such exists; for example the noun “Pegasus” has a sense but lacks a denotation.

Unlike natural languages, formalized languages are, as a rule, constructed such that every noun has exactly one sense. On the other hand, in most formalized languages synonymy is retained, synonyms being connected, by definition, through relations of equality (equivalence, identity). The elimination of synonymy is in a number of cases theoretically impossible since there is no algorithm for determining identity between arbitrary expressions (“words”) in a sufficiently broad class of formalized languages.

The foundations for a systematic construction of contemporary logical semantics were set forth in the works of A. Tarski, who was chiefly drawn to the analysis of the notions “truth,” “realizability,” “definability,” and “denotation” and to the possibilities of their exact definition. It was found that all these notions are defined for formalized languages by means of richer languages that serve as metalanguages for formalized languages (“object” languages). To define the corresponding notions for nonformalized languages it is first necessary to formalize them, after which they obey the same scheme. A metalanguage, in turn, may be formalized, but the semantic notions (“truth,” etc.) of a formalized metalanguage must be defined by means of yet another metalanguage. Mixing language and metalanguage at any level inevitably leads to semantic paradoxes.

Following the American logician W. V. O. Quine, logicians distinguish the properties of linguistic expressions that can be characterized in terms of arbitrary interpretations (models) of a given language and that are invariant from one interpretation to another and linguistic properties that can be defined in terms of any one interpretation. The first range of questions relates to the theory of sense while the second to the theory of reference (theory of denotation).

The notions of “sense,” “synonymy,” “intelligibility,” and “semantic consequence” are dealt with in a theory of sense. This field of logical semantics is still at a very elementary stage in its development. The theory of reference, which is based on concepts of “truth,” “denotation,” and “nomination,” has already proved quite productive. One of the results of the theory of reference was Tarski’s theorem that the truth predicate of any non-contradictory language system cannot be determined using its own resources. The meaning of Tarski’s theorem, which establishes specific limitations on the expressible means of formal languages, is in many ways similar to the role in mathematics of Gödel’s famous theorem in metamathematics that states that sufficiently rich calculi of mathematical logic are in principle deductively incomplete. The very structure of the demonstrations of these two remarkable clauses reveal profound analogies. They yield together an extremely powerful tool for metamathematical demonstrations, for example, of non-contradiction, completeness, and incompleteness.

Following the tradition starting from Leibniz, propositions of a language that are true in all models of the language (“in all possible worlds”) are called analytically true; accordingly, propositions not true in any model are called analytically false. Analytically true propositions are in contrast to synthetically (or factually) true propositions whose truth is said to depend on the properties of the “given universe.” In other words, synthetically true propositions are neither analytically true nor analytically false. They are satisfied in some but not in all models of a given language. For complete languages the concept of analytic truth, which is a semantic concept, can be described in purely syntactic terms, as through demonstrability. For incomplete languages, on the other hand, which are precisely the languages that are of greatest interest to science, no such reduction of logical semantics to syntax can be directly carried out.

Leibniz’ idea of the distinction between “possible worlds” and the “actual universe” as a foundation for logical semantics has been developed by the Dutch logician E. Beth, the English logician A. N. Prior, the Finnish logician J. Hintikka, and particularly by the American logician S. A. Kripke, who introduced the concept of a model structure. Such a model structure is the universe of the set of all models in classical propositional logic (“all possible universes”), a concrete model from this set (“actual universe”), and a reflexive binary relation on the set of models that relates the general validity (identical truth) of an arbitrary proposition in one model with the possibility of this proposition in other models. Depending on additional properties of such a relation (whether it is both symmetrical and transitive, or simply one or the other), different systems of modal logic correspond to a model of the “actual universe.” Contemporary work in logical semantics also draws on ideas and concepts of many-valued logic, axiomatic set theory, and abstract algebra.

The ideas, methods, and results of logical semantics have been utilized in various spheres of applied linguistics and in semiotics (automatic decoding, machine translation, automatic abstracting), in the construction of a semantic information theory, in problems arising in heuristic programming, in investigating problems of pattern recognition, and other questions dealt with in cybernetics.