Logical Calculus

a calculus (formal system) that can be interpreted in terms of a particular fragment of deductive logic. Various logical calculi are used as a basis for constructing richer “nonlogical,” for example, mathematical, theories. Examples of logical calculi that can be used for this purpose are the propositional calculus and the predicate calculus and various weakened versions of them, as well as extensions obtained by adding modal operators (such as possibility or necessity) or the predicate of equality to them.

When constructing a particular theory on the basis of the logical calculus, different individual, predicate, and (or) functional constants and postulates (axioms and, perhaps, rules of inference) that characterize these constants are adjoined to the “pure” logical calculus. The simplest and most important example of “applied” logical calculus obtained as a result of it is the already mentioned predicate calculus with equality (classified as a logical calculus depending on whether equality is attributed to “purely logical” or “mathematical” predicates), which is a constituent of all more developed and richer axiomatic mathematical theories. Of the latter, of particular importance are arithmetic calculi, whose interpretation is that of a natural number series with various relations defined in it (equality, “greater than,” and “less than”) and operations (addition, multiplication) defined in the series, as well as different systems of axiomatic set theory. The study of such logical calculi with natural mathematical interpretations is an extremely important problem in the foundations of logic and mathematics. At the same time, their theory, from some viewpoints shared by representatives of the constructive trend in mathematics and logic, is more “elementary” than that of the “pure” logical calculi, since the concepts of such logical calculi are products of higher-level abstractions.

The term “logical calculus” allows several broader interpretations in addition to that given above. Thus, besides the logical calculi based on “two-valued” logic (in which only two “truth values” of propositions, namely, “true” and “false,” are allowed), different systems of multivalued logic have found widespread use. Also included in the logical calculi are all the possible modifications of the theory of types introduced by B. Russell, that is, calculi with several “sorts” (types, levels, stages) of variables: individuals, predicates, predicates of predicates, and so on. It is customary to call all the logical calculi mentioned up to now “Hilbert-type systems” after D. Hilbert. However, the concept of a logical calculus is broader; under it are subsumed different modifications of the “sequent” calculus and the “natural deduction” calculus introduced by the German logician G. Gentzen. Fragments of logic that are not constructed axiomatically but on the basis of an informal intuitive (“tabular,” or “matrix”) definition of the logical operations are also called logical calculi.