metatheory

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metatheory

all or any second-order accounts of theories or second-order theories of theories.

Metatheory

a theory that analyzes the structure, methods, and properties of some other theory, called the object theory. The term “metatheory” is meaningful only with respect to a given concrete object theory. Thus, the metatheory of logic is called metalogic, and that of mathematics metamathematics. Such terms as “metachemistry” and “metabiology” have a similar meaning (but not “metaphysics”).

In principle, one may speak of the metatheory of any deductive or nondeductive scientific discipline (for example, philosophy in a certain sense plays a metatheoretical role). However, the concept of a metatheory is properly productive only when applied to the deductive sciences, such as mathematics, logic, and the mathematized fragments of the natural and other sciences, for example, linguistics. Furthermore, the actual object of consideration in a metatheory, as a rule, proves not to be some informal scientific theory but its formal analog and explicate— the precise concept of a calculus (formal system). If the theory being investigated in metatheory is informal, it is first subjected to formalization. Thus, the part of metatheory that studies the structure of its own object theory deals with the theory as a formal system; that is, it perceives the elements of that theory as purely formal constructive objects that are devoid of any “content” (meaning), that can be strictly identified with one another (or distinguished from one another), and from which one can construct, using clearly formulated rules of formation, combinations of symbols that are “expressions” (formulas) of the given formal system.

This part of metatheory—known as syntax—also studies the deductive methods of the given object theory. In particular, the concept of a (formal) proof of a given object theory and the more general concept of inference from given premises are defined. Metatheory itself, unlike the object theory, is an informal theory: the nature of the methods of description, inference, and proof used can be stipulated or limited in some special way, and in any case, these methods themselves are intuitively graspable elements of ordinary (natural) language and of the “logic of common sense.” Metatheorems, or “theorems about theorems,” constitute the fundamental content of a metatheory. An example of a syntactic metatheorem is the deduction theorem, which establishes a relation between the concept of deducibility (provability) in a given object theory (for example, in the propositional calculus or the predicate calculus) and the logical operation of implication occurring in the “alphabet” of a given object theory.

Another problem of metatheory is the examination of all possible interpretations of a given formal system. The corresponding part (or aspect) of a metatheory that perceives the object theory as a formalized language is called semantics. An example of a semantic metatheorem is the completeness theorem for the classical propositional calculus, according to which in this calculus the concept of a provable formula (formal theorem) and the concept of a formula that is true under some “natural” interpretation of it coincide.

Many concepts (and the related metatheorems) of metatheory are of “mixed” character: both syntactic and semantic. Such is, for example, the very important concept of consistency, defined both as the nondeducibility in the object theory of a formal contradiction—that is, the conjunction of a given formula and its negation (so-called internal consistency)—and as the “satisfaction” of a given object theory by some natural interpretation (external, or semantic, consistency). That both these concepts coincide in extension is a nontrivial fact of metatheory, a fact that evidently pertains to both the syntax and semantics of a given theory. Gödel’s incompleteness theorem for formal arithmetic (and richer logical-mathematical calculi containing it) and his theorem on the impossibility of proving the consistency of a broad class of calculi using methods formalizable in these calculi are classical examples of metatheorems that relate a number of important syntactic and semantic concepts. The concept of decidability of a formal theory, on the other hand, is purely syntactic, while the concept of completeness is predominantly semantic. Of course, a metatheory itself can be formalized and become the object of study of a given metametatheory.

The concept of “metatheory” was first advanced by D. Hilbert as part of his program of providing foundations of classical mathematics based on methods of proof theory (metamathematics) created by Hilbert’s school. Some very important meta-theoretical results, primarily semantic, were obtained by A. Tarski. In developing the ideas of Tarski and R. Carnap, H. B. Curry referred to metatheory as “epitheory,” reserving the term “metatheory” for somewhat more specialized usage.

REFERENCES

Kleene, S. C. Vvedenie v metamatematiku. Moscow, 1957. Chapters 3–8, 14, 15. (Translated from English.)
Kleene, S. C. Matematicheskaia logika. Moscow, 1973. (Translated from English.)
Church, A. Vvedenie v matematicheskuiu logiku, vol. 1. Moscow, 1960. See Introduction. (Translated from English.)
Curry, H. B. Osnovaniia matematicheskoi logiki. Moscow, 1969. Chapters 2 and 3. (Translated from English.)

IU. A. GASTEV

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