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Formal system

October 20, 2023

A formal system is an abstract structure or 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 the formal system as the foundation for the 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.

Concepts Edit

 
This diagram shows the syntactic entities that may be constructed from formal languages. The symbols and strings of symbols may be broadly divided into nonsense and well-formed formulas. A formal language can be thought of as identical to the set of its well-formed formulas, which may be broadly divided into theorems and non-theorems.

A formal system has the following:[3][4][5]

A formal system is said to be recursive (i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules are decidable sets or semidecidable sets, respectively.

Formal language Edit

Main articles: Formal language, Formal grammar, Syntax (logic), and Logical form

A formal language is a language that is defined by a formal system. Like languages in linguistics, formal languages generally have two aspects:

  • the syntax is what the language looks like (more formally: the set of possible expressions that are valid utterances in the language)
  • the semantics are what the utterances of the language mean (which is formalized in various ways, depending on the type of language in question)

Usually only the syntax of a formal language is considered via the notion of a formal grammar. The two main categories of formal grammar are that of generative grammars, which are sets of rules for how strings in a language can be written, and that of analytic grammars (or reductive grammar[6][7]), which are sets of rules for how a string can be analyzed to determine whether it is a member of the language.

Deductive system Edit

Main articles: Inference, Logical consequence, and Deductive reasoning

A deductive system, also called a deductive apparatus,[8] consists of the axioms (or axiom schemata) and rules of inference that can be used to derive theorems of the system.[9]

Such deductive systems preserve deductive qualities in the formulas that are expressed in the system. Usually the quality we are concerned with is truth as opposed to falsehood. However, other modalities, such as justification or belief may be preserved instead.

In order to sustain its deductive integrity, a deductive apparatus must be definable without reference to any intended interpretation of the language. The aim is to ensure that each line of a derivation is merely a logical consequence of the lines that precede it. There should be no element of any interpretation of the language that gets involved with the deductive nature of the system.

The logical consequence (or entailment) of the system by its logical foundation is what distinguishes a formal system from others which may have some basis in an abstract model. Often the formal system will be the basis for or even identified with a larger theory or field (e.g. Euclidean geometry) consistent with the usage in modern mathematics such as model theory.[clarification needed]

An example of deductive system is first order logic.

The two main types of deductive systems are proof systems and formal semantics.[8]

Proof system Edit

Main articles: Proof system and Formal proof

Formal proofs are sequences of well-formed formulas (or WFF for short) that might either be an axiom or be the product of applying an inference rule on previous WFFs in the proof sequence. The last WFF in the sequence is recognized as a theorem.

Once a formal system is given, one can define the set of theorems which can be proved inside the formal system. This set consists of all WFFs for which there is a proof. Thus all axioms are considered theorems. Unlike the grammar for WFFs, there is no guarantee that there will be a decision procedure for deciding whether a given WFF is a theorem or not.

The point of view that generating formal proofs is all there is to mathematics is often called formalism. David Hilbert founded metamathematics as a discipline for discussing formal systems. Any language that one uses to talk about a formal system is called a metalanguage. The metalanguage may be a natural language, or it may be partially formalized itself, but it is generally less completely formalized than the formal language component of the formal system under examination, which is then called the object language, that is, the object of the discussion in question. The notion of theorem just defined should not be confused with theorems about the formal system, which, in order to avoid confusion, are usually called metatheorems.

Formal semantics of Logical system Edit

Main articles: Semantics of logic, Interpretation (logic), and Model theory

A logical system is a deductive system (most commonly first order logic) together with additional non-logical axioms. According to model theory, a logical system may be given interpretations which describe whether a given structure - the mapping of formulas to a particular meaning - satisfies a well-formed formula. A structure that satisfies all the axioms of the formal system is known as a model of the logical system.

A logical system is:

  • Sound, if each well-formed formula that can be inferred from the axioms is satisfied by every model of the logical system.
  • Semantically complete, if each well-formed formula that is satisfied by every model of the logical system can be inferred from the axioms.

An example of a logical system is Peano arithmetic. The standard model of arithmetic sets the domain of discourse to be the nonnegative integers and gives the symbols their usual meaning.[10] There are also non-standard models of arithmetic.

History Edit

Main articles: Formalism (philosophy of mathematics) and Formal logical systems

Early logic systems includes Indian logic of Pāṇini, syllogistic logic of Aristotle, propositional logic of Stoicism, and Chinese logic of Gongsun Long (c. 325–250 BCE) . In more recent times, contributors include George Boole, Augustus De Morgan, and Gottlob Frege. Mathematical logic was developed in 19th century Europe.

David Hilbert instigated a formalist movement called Hilbert’s program as a proposed solution to the foundational crisis of mathematics, that was eventually tempered by Gödel's incompleteness theorems.[2] The QED manifesto represented a subsequent, as yet unsuccessful, effort at formalization of known mathematics.

See also Edit

References Edit

  1. ^ "Formal system | Logic, Symbols & Axioms | Britannica". www.britannica.com. Retrieved 2023-10-10.
  2. ^ a b Zach, Richard (31 July 2003). "Hilbert's Program". Hilbert's Program, Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
  3. ^ "formal system". planetmath.org. Retrieved 2023-10-10.
  4. ^ Rapaport, William J. (25 March 2010). "Syntax & Semantics of Formal Systems". University of Buffalo.
  5. ^ "Definition:Formal System - ProofWiki". proofwiki.org. Retrieved 2023-10-16.
  6. ^ Reductive grammar: (computer science) A set of syntactic rules for the analysis of strings to determine whether the strings exist in a language. "Sci-Tech Dictionary McGraw-Hill Dictionary of Scientific and Technical Terms" (6th ed.). McGraw-Hill.[unreliable source?] About the Author Compiled by The Editors of the McGraw-Hill Encyclopedia of Science & Technology (New York, NY) an in-house staff who represents the cutting-edge of skill, knowledge, and innovation in science publishing. [1]
  7. ^ "There are two classes of formal-language definition compiler-writing schemes. The productive grammar approach is the most common. A productive grammar consists primarrly of a set of rules that describe a method of generating all possible strings of the language. The reductive or analytical grammar technique states a set of rules that describe a method of analyzing any string of characters and deciding whether that string is in the language." "The TREE-META Compiler-Compiler System: A Meta Compiler System for the Univac 1108 and General Electric 645, University of Utah Technical Report RADC-TR-69-83. C. Stephen Carr, David A. Luther, Sherian Erdmann" (PDF). Retrieved 5 January 2015.
  8. ^ a b "Definition:Deductive Apparatus - ProofWiki". proofwiki.org. Retrieved 2023-10-10.
  9. ^ Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971
  10. ^ Kaye, Richard (1991). "1. The Standard Model". Models of Peano arithmetic. Oxford: Clarendon Press. p. 10. ISBN 9780198532132.

Further reading Edit

External links Edit

 
Look up formalisation in Wiktionary, the free dictionary.
  •   Media related to Formal systems at Wikimedia Commons
  • Encyclopædia Britannica, Formal system definition, 2007.
  • 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, Formal Systems and Machines: An Isomorphism 2011-05-24 at the Wayback Machine, 1997.
  • Ray Taol, Formal Systems
  • What is a Formal System?: Some quotes from John Haugeland's `Artificial Intelligence: The Very Idea' (1985), pp. 48–64.

October 20, 2023
formal, system, formal, system, abstract, structure, formalization, axiomatic, system, used, inferring, theorems, from, axioms, inference, rules, 1921, david, hilbert, proposed, formal, system, foundation, knowledge, mathematics, term, formalism, sometimes, ro. A formal system is an abstract structure or 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 the formal system as the foundation for the 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 Contents 1 Concepts 1 1 Formal language 1 2 Deductive system 1 2 1 Proof system 1 2 2 Formal semantics of Logical system 2 History 3 See also 4 References 5 Further reading 6 External linksConcepts Edit nbsp This diagram shows the syntactic entities that may be constructed from formal languages The symbols and strings of symbols may be broadly divided into nonsense and well formed formulas A formal language can be thought of as identical to the set of its well formed formulas which may be broadly divided into theorems and non theorems A formal system has the following 3 4 5 Formal language 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 Deductive system deductive apparatus or proof system which have rules of inference which take axioms and infer a theorem both of which are part of the formal language A formal system is said to be recursive i e effective or recursively enumerable if the set of axioms and the set of inference rules are decidable sets or semidecidable sets respectively Formal language Edit Main articles Formal language Formal grammar Syntax logic and Logical form A formal language is a language that is defined by a formal system Like languages in linguistics formal languages generally have two aspects the syntax is what the language looks like more formally the set of possible expressions that are valid utterances in the language the semantics are what the utterances of the language mean which is formalized in various ways depending on the type of language in question Usually only the syntax of a formal language is considered via the notion of a formal grammar The two main categories of formal grammar are that of generative grammars which are sets of rules for how strings in a language can be written and that of analytic grammars or reductive grammar 6 7 which are sets of rules for how a string can be analyzed to determine whether it is a member of the language Deductive system Edit This section may require cleanup to meet Wikipedia s quality standards The specific problem is This section needs better organization and more citations Please help improve this section if you can October 2023 Learn how and when to remove this template message Main articles Inference Logical consequence and Deductive reasoning A deductive system also called a deductive apparatus 8 consists of the axioms or axiom schemata and rules of inference that can be used to derive theorems of the system 9 Such deductive systems preserve deductive qualities in the formulas that are expressed in the system Usually the quality we are concerned with is truth as opposed to falsehood However other modalities such as justification or belief may be preserved instead In order to sustain its deductive integrity a deductive apparatus must be definable without reference to any intended interpretation of the language The aim is to ensure that each line of a derivation is merely a logical consequence of the lines that precede it There should be no element of any interpretation of the language that gets involved with the deductive nature of the system The logical consequence or entailment of the system by its logical foundation is what distinguishes a formal system from others which may have some basis in an abstract model Often the formal system will be the basis for or even identified with a larger theory or field e g Euclidean geometry consistent with the usage in modern mathematics such as model theory clarification needed An example of deductive system is first order logic The two main types of deductive systems are proof systems and formal semantics 8 Proof system Edit Main articles Proof system and Formal proof Formal proofs are sequences of well formed formulas or WFF for short that might either be an axiom or be the product of applying an inference rule on previous WFFs in the proof sequence The last WFF in the sequence is recognized as a theorem Once a formal system is given one can define the set of theorems which can be proved inside the formal system This set consists of all WFFs for which there is a proof Thus all axioms are considered theorems Unlike the grammar for WFFs there is no guarantee that there will be a decision procedure for deciding whether a given WFF is a theorem or not The point of view that generating formal proofs is all there is to mathematics is often called formalism David Hilbert founded metamathematics as a discipline for discussing formal systems Any language that one uses to talk about a formal system is called a metalanguage The metalanguage may be a natural language or it may be partially formalized itself but it is generally less completely formalized than the formal language component of the formal system under examination which is then called the object language that is the object of the discussion in question The notion of theorem just defined should not be confused with theorems about the formal system which in order to avoid confusion are usually called metatheorems Formal semantics of Logical system Edit Main articles Semantics of logic Interpretation logic and Model theory A logical system is a deductive system most commonly first order logic together with additional non logical axioms According to model theory a logical system may be given interpretations which describe whether a given structure the mapping of formulas to a particular meaning satisfies a well formed formula A structure that satisfies all the axioms of the formal system is known as a model of the logical system A logical system is Sound if each well formed formula that can be inferred from the axioms is satisfied by every model of the logical system Semantically complete if each well formed formula that is satisfied by every model of the logical system can be inferred from the axioms An example of a logical system is Peano arithmetic The standard model of arithmetic sets the domain of discourse to be the nonnegative integers and gives the symbols their usual meaning 10 There are also non standard models of arithmetic History EditMain articles Formalism philosophy of mathematics and Formal logical systems Early logic systems includes Indian logic of Paṇini syllogistic logic of Aristotle propositional logic of Stoicism and Chinese logic of Gongsun Long c 325 250 BCE In more recent times contributors include George Boole Augustus De Morgan and Gottlob Frege Mathematical logic was developed in 19th century Europe David Hilbert instigated a formalist movement called Hilbert s program as a proposed solution to the foundational crisis of mathematics that was eventually tempered by Godel s incompleteness theorems 2 The QED manifesto represented a subsequent as yet unsuccessful effort at formalization of known mathematics See also Edit nbsp Systems science portal nbsp Philosophy portalList of formal systems Formal method Mathematical program specificationsPages displaying short descriptions of redirect targets Formal science Branch of science Logic translation Translation of a text into a logical system Rewriting system Replacing subterm in a formula with another termPages displaying short descriptions of redirect targets Substitution instance concept in logic syntactic transformation on formal expressionsPages displaying wikidata descriptions as a fallback Theory mathematical logic Set of sentences in a formal languageReferences Edit Formal system Logic Symbols amp Axioms Britannica www britannica com Retrieved 2023 10 10 a b Zach Richard 31 July 2003 Hilbert s Program Hilbert s Program Stanford Encyclopedia of Philosophy Metaphysics Research Lab Stanford University formal system planetmath org Retrieved 2023 10 10 Rapaport William J 25 March 2010 Syntax amp Semantics of Formal Systems University of Buffalo Definition Formal System ProofWiki proofwiki org Retrieved 2023 10 16 Reductive grammar computer science A set of syntactic rules for the analysis of strings to determine whether the strings exist in a language Sci Tech Dictionary McGraw Hill Dictionary of Scientific and Technical Terms 6th ed McGraw Hill unreliable source About the Author Compiled by The Editors of the McGraw Hill Encyclopedia of Science amp Technology New York NY an in house staff who represents the cutting edge of skill knowledge and innovation in science publishing 1 There are two classes of formal language definition compiler writing schemes The productive grammar approach is the most common A productive grammar consists primarrly of a set of rules that describe a method of generating all possible strings of the language The reductive or analytical grammar technique states a set of rules that describe a method of analyzing any string of characters and deciding whether that string is in the language The TREE META Compiler Compiler System A Meta Compiler System for the Univac 1108 and General Electric 645 University of Utah Technical Report RADC TR 69 83 C Stephen Carr David A Luther Sherian Erdmann PDF Retrieved 5 January 2015 a b Definition Deductive Apparatus ProofWiki proofwiki org Retrieved 2023 10 10 Hunter Geoffrey Metalogic An Introduction to the Metatheory of Standard First Order Logic University of California Press 1971 Kaye Richard 1991 1 The Standard Model Models of Peano arithmetic Oxford Clarendon Press p 10 ISBN 9780198532132 Further reading EditRaymond M Smullyan 1961 Theory of Formal Systems Annals of Mathematics Studies Princeton University Press April 1 1961 156 pages ISBN 0 691 08047 X Stephen Cole Kleene 1967 Mathematical Logic Reprinted by Dover 2002 ISBN 0 486 42533 9 Douglas Hofstadter 1979 Godel Escher Bach An Eternal Golden Braid ISBN 978 0 465 02656 2 777 pages External links Edit nbsp Look up formalisation in Wiktionary the free dictionary nbsp Media related to Formal systems at Wikimedia Commons Encyclopaedia Britannica Formal system definition 2007 Daniel Richardson Formal systems logic and semantics William J Rapaport Syntax amp Semantics of Formal Systems PlanetMath Formal System Pr fWiki Definition Formal System Pr fWiki Definition Deductive Apparatus Encyclopedia of Mathematics Formal system Peter Suber Formal Systems and Machines An Isomorphism Archived 2011 05 24 at the Wayback Machine 1997 Ray Taol Formal Systems What is a Formal System Some quotes from John Haugeland s Artificial Intelligence The Very Idea 1985 pp 48 64 Retrieved from https en wikipedia org w index php title Formal system amp oldid 1180822652 Deductive system, wikipedia, wiki, book, books, library,

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