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Formal system

A formal system is an abstract structure and formalization of an axiomatic system used for inferring theorems from axioms by a set of inference rules.[1]

In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in mathematics.[2]


The term formalism is sometimes a rough synonym for formal system, but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation.

which is a set of well-formed formulas, which are strings of symbols from an alphabet, formed by a formal grammar (consisting of production rules or formation rules).

Formal language

Deductive system, deductive apparatus, or , which has rules of inference that take axioms and infers theorems, both of which are part of the formal language.

proof system

List of formal systems

 – Mathematical program specifications

Formal method

 – Branch of science

Formal science

 – Translation of a text into a logical system

Logic translation

 – Replacing subterm in a formula with another term

Rewriting system

 – Concept in logic

Substitution instance

 – Set of sentences in a formal language

Theory (mathematical logic)

1961. Theory of Formal Systems: Annals of Mathematics Studies, Princeton University Press (April 1, 1961) 156 pages ISBN 0-691-08047-X

Raymond M. Smullyan

1967. Mathematical Logic Reprinted by Dover, 2002. ISBN 0-486-42533-9

Stephen Cole Kleene

1979. Gödel, Escher, Bach: An Eternal Golden Braid ISBN 978-0-465-02656-2. 777 pages.

Douglas Hofstadter

Media related to Formal systems at Wikimedia Commons

Encyclopædia Britannica, definition, 2007.

Formal system

Daniel Richardson,

Formal systems, logic and semantics

William J. Rapaport,

Syntax & Semantics of Formal Systems

PlanetMath,

Formal System

Pr∞fWiki,

Definition:Formal System

Pr∞fWiki,

Definition:Deductive Apparatus

Encyclopedia of Mathematics,

Formal system

Peter Suber, Archived 2011-05-24 at the Wayback Machine, 1997.

Formal Systems and Machines: An Isomorphism

Ray Taol,

Formal Systems

: Some quotes from John Haugeland's `Artificial Intelligence: The Very Idea' (1985), pp. 48–64.

What is a Formal System?