Formal System (redirected from Formal systems)

formal system

In logic, a formal language together with a deductive apparatus by which some well-formed formulas can be derived from others. Each formal system has a formal language composed of primitive symbols that figure in certain rules of formation (statements concerning the expressions allowable in the system) and a set of theorems developed by inference from a set of axioms. In an axiomatic system, the primitive symbols are undefined and all other symbols are defined in terms of them. In Euclidean geometry, for example, such concepts as “point,” “line,” and “lies on” are usually posited as primitive terms. From the primitive symbols, certain formulas are defined as well formed, some of which are listed as axioms; and rules are stated for inferring one formula as a conclusion from one or more other formulas taken as premises. A theorem within such a system is a formula capable of proof through a finite sequence of well-formed formulas, each of which either is an axiom or is validly inferred from earlier formulas.

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Formal System 

an uninterpreted calculus such that (1) the class of its expressions, or formulas, is usually defined inductively through the specification of primitive (elementary, atomic) formulas and rules for the formation (construction) of formulas and (2) the subclass of provable formulas (theorems) is defined through the specification of a system of axioms and of rules for the derivation of theorems from axioms and already proved theorems (seeRULE OF INFERENCE).

The term “formal system” has many synonyms, which, however, are sometimes used to refer to concepts related to, but not identical with, formal systems. Examples of these synonyms are “formal theory,” “formal mathematics,” “formalism,” “formal calculus,” “abstract calculus,” “syntactic system,” “axiomatic system,” “logistic system,” “formal language,” “formal logic,” “codification,” and “deductive system.”