# formal system

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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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logic
also called  logistic system

in logic and mathematics, abstract, theoretical organization of terms and implicit relationships that is used as a tool for the analysis of the concept of deduction. Models—structures that interpret the symbols of a formal system—are often used in conjunction with formal systems.

Each formal system has a formal language composed of primitive symbols acted on by certain rules of formation (statements concerning the symbols, functions, and sentences allowable in the system) and developed by inference from a set of axioms. The system thus consists of any number of formulas built up through finite combinations of the primitive symbols—combinations that are formed from the axioms in accordance with the stated rules.

In an axiomatic system, the primitive symbols are undefined; and all other symbols are defined in terms of them. In the Peano postulates for the integers, for example, 0 and ′ are taken as primitive, and 1 and 2 are defined by 1 = 0′ and 2 = 1′. Similarly, in geometry 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 inferred from earlier formulas.

A formal system that is treated apart from intended interpretation is a mathematical construct and is more properly called logical calculus; this kind of formulation deals rather with validity and satisfiability than with truth or falsity, which are at the root of formal systems.

In general, then, a formal system provides an ideal language by means of which to abstract and analyze the deductive structure of thought apart from specific meanings. Together with the concept of a model, such systems have formed the basis for a rapidly expanding inquiry into the foundations of mathematics and of other deductive sciences and have even been used to a limited extent in analyzing the empirical sciences. See also deontological ethics; metalogic; metatheory.

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Universalium. 2010.

### Look at other dictionaries:

• Formal system — In formal logic, a formal system (also called a logical system,Audi, Robert (Editor). The Cambridge Dictionary of Philosophy . Second edition, Cambridge University Press, 1999. ISBN 978 0521631365 (hardcover) and ISBN 978 0521637220 (paperback).] …   Wikipedia

• formal system — noun The grouping of a formal language and a set of inference rules and/or axioms. Syn: logical system, logical calculus, logic …   Wiktionary

• formal system — (or theory) A theory whose sentences are well formed formulae of a logical calculus, and in which axioms or rules governing particular terms correspond to the principles of the theory being formalized. The theory is said to be couched or framed… …   Philosophy dictionary

• formal systems — formal system …   Philosophy dictionary

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• formal logic — the branch of logic concerned exclusively with the principles of deductive reasoning and with the form rather than the content of propositions. [1855 60] * * * Introduction       the abstract study of propositions, statements, or assertively used …   Universalium

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• Formal language — A formal language is a set of words , i.e. finite strings of letters , or symbols . The inventory from which these letters are taken is called the alphabet over which the language is defined. A formal language is often defined by means of a… …   Wikipedia