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{{Short description|Mathematical model for deduction or proof systems}}
{{Tertiary sources|date=December 2024}}
A '''formal system''' (or '''deductive system''') is an [[abstract structure]] and [[Formalism (philosophy of mathematics)|formalization]] of an [[axiomatic system]] used for [[Deductive reasoning|deducing]], using [[rule of inference|rules of inference]], [[theorem]]s from [[axioms]].{{sfn|Hunter|1996|p=7}}

In 1921, [[David Hilbert]] proposed to use formal systems as the foundation of knowledge in [[mathematics]].<ref name=":0">{{cite book
| title = Hilbert's Program, Stanford Encyclopedia of Philosophy 
| date = 31 July 2003
| chapter-url = https://plato.stanford.edu/archives/spr2016/entries/hilbert-program
| last1 = Zach
| first1 = Richard
| chapter = Hilbert's Program
| publisher = Metaphysics Research Lab, Stanford University
}}</ref>
However, in 1931 [[Kurt Gödel]] proved that any [[Consistency|consistent]] formal system sufficiently powerful to express basic arithmetic cannot prove its own [[Completeness (logic)|completeness]]. This effectively showed that [[Hilbert's program]] was impossible as stated.

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 ==
[[File:Formal languages.svg|thumb|300px|This diagram shows the [[Syntax (logic)|syntactic entities]] that may be constructed from [[formal language]]s. The symbols and [[string (computer science)|strings of symbols]] may be broadly divided into [[nonsense]] and [[Well-formed formula|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 [[theorem]]s and non-theorems.]]

A formal system has the following components, as a minimum:<ref>{{planetmath reference|urlname=formalsystem|title=Formal System}}</ref><ref>{{Cite web |last=Rapaport |first=William J. |date=25 March 2010 |title=Syntax & Semantics of Formal Systems |url=https://cse.buffalo.edu/~rapaport/formalsystems |website=University of Buffalo}}</ref><ref>{{proofwiki reference|id=Definition:Formal_System|name=Formal System}}</ref>

* [[Formal language]], which is a set of [[Well-formed formula|well-formed formulas]], which are strings of [[Symbol (formal)|symbols]] from an [[Alphabet (formal languages)|alphabet]], formed by a [[formal grammar]] (consisting of [[Production (computer science)|production rules]] or [[Formation rule|formation rules]]).
* Deductive system, deductive apparatus, or [[Proof calculus|proof system]], which has [[rule of inference|rules of inference]] that take [[Axiom|axioms]] and infers [[theorem]]s, both of which are part of the formal language.
* In some cases an [[Mathematical induction|inductive system]], used to derive a proof by first establishing a simple case, then generalizing it.

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

=== Formal language ===
{{Formal languages}}
{{Main|Formal language|Formal grammar|Syntax (logic)|Logical form}}

A [[formal language]] is a language that uses a set of strings whose symbols are taken from a specific alphabet, and operations used to form sentences from them. Like languages in [[linguistics]], formal languages generally have two aspects:
* the [[Syntax (logic)|syntax]] is what the language looks like (more formally: the set of possible expressions that are valid utterances in the language)
* the [[Semantics of logic|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 (logic)|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 grammar]]s, which are sets of rules for how strings in a language can be written, and that of [[analytic grammar]]s (or reductive grammar<ref>{{cite dictionary |dictionary=Dictionary of Scientific and Technical Terms |url=http://encyclopedia2.thefreedictionary.com/reductive+grammar |title=Reductive grammar |publisher=McGraw-Hill |edition=6th |quote=Reductive grammar:  (''computer science'') A set of syntactic rules for the analysis of strings to determine whether the strings exist in a language.}}</ref>{{unreliable source?|date=February 2015}}<ref>{{cite web|url=http://bitsavers.informatik.uni-stuttgart.de/pdf/sri/arc/rulifson/A_Tree_Meta_For_The_XDS_940_Appendix_D_Apr68.pdf|title=A Tree Meta for the XDS 940 | publisher=[[Augmentation Research Center]] |date=April 1968 |last=Rulifson |first=Johns F. |author-link=Jeff Rulifson |access-date=30 November 2024 |quote="There are two classes of formal-language definition compiler-writing schemes. The productive [[formal grammar|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 [[formal grammar#Analytic grammars|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."}}</ref>), which are sets of rules for how a string can be analyzed to determine whether it is a member of the language.

===Deductive system===
{{Main|Inference|Logical consequence|Deductive reasoning}}
{{Cleanup|date=October 2023|reason=This section needs better organization and more citations.|section}}
A ''deductive system'', also called a ''deductive apparatus'',<ref name=":1">{{proofwiki reference|id=Definition:Deductive_Apparatus |title=Deductive Apparatus |access-date=30 November 2024}}</ref> consists of the [[Axiom#Role_in_mathematical_logic|axiom]]s (or [[axiom schema]]ta) and [[rules of inference]] that can be used to [[formal proof|derive]] [[theorem]]s of the system.{{sfn|Hunter|1996|p=7}}

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 [[Mathematical proof|derivation]] is merely a [[logical consequence]] of the lines that precede it. There should be no element of any [[Interpretation (logic)|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]].{{clarify|reason=This section doesn't really do a group job stating what an entailment actually is.|date=September 2017}}

An example of a deductive system would be the rules of inference and [[First-order_logic#Equality_and_its_axioms|axioms regarding equality]] used in [[First-order logic|first order logic]].

The two main types of deductive systems are proof systems and formal semantics.<ref name=":1" /><ref>{{cite book|title=Formal Semantics and Logic|url=https://www.princeton.edu/~fraassen/books/pdfs/Formal%20Semantics%20and%20Logic.pdf |last=van Fraassen |first=Bas C. |author-link=Bas van Fraassen |year=2016 |orig-date=1971 |publisher=Nousoul Digital Publishers|page=12|quote=Metalogic can in turn be roughly divided into two parts: proof theory and formal semantics... The division is not exact; many questions have been dealt with from both points of view, and some proof-theoretic methods and results are indispensable in semantics.}}</ref>

==== Proof system ====
{{Main|Proof system|Formal proof}}
Formal proofs are sequences of [[well-formed formula]]s (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.

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 [[decidability (logic)|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 (philosophy of mathematics)|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 [[metatheorem]]s.

==== Formal semantics of logical system ====
{{main|Semantics of logic|Interpretation (logic)|Model theory}}

A ''logical system'' is a deductive system (most commonly [[First-order logic|first order logic]]) together with additional [[non-logical axioms]]. According to [[model theory]], a logical system may be given [[interpretation (logic)|interpretation]]s which describe whether a given [[Structure (mathematical logic)|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 (model theory)|model]] of the logical system. 

A logical system is:
*[[Soundness|Sound]], if each well-formed formula that can be inferred from the axioms is satisfied by every model of the logical system.
*[[Completeness (logic)#Semantic completeness|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 integer]]s and gives the symbols their usual meaning.<ref>{{cite book |last1=Kaye |first1=Richard |title=Models of Peano arithmetic |date=1991 |publisher=Clarendon Press |location=Oxford |isbn=9780198532132 |page=10 |chapter=1. The Standard Model}}</ref> There are also [[non-standard model of arithmetic|non-standard models of arithmetic]].

==History==
{{main|Formalism (philosophy of mathematics)|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).{{cn|date=August 2025}} 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 [[Formalism (philosophy of mathematics)|formalist]] movement called [[Hilbert's program|Hilbert’s program]] as a proposed solution to the [[foundational crisis of mathematics]], that was eventually tempered by [[Gödel's incompleteness theorems]].<ref name=":0" /> The [[QED manifesto]] represented a subsequent, as yet unsuccessful, effort at formalization of known mathematics.

== See also ==
{{Portal|Systems science|Philosophy}}
* {{ Annotated link |  List of formal systems}}
* {{ Annotated link |  Formal method}}
* {{ Annotated link |  Formal science}}
* {{ Annotated link |  Logic translation}}
* {{ Annotated link |  Rewriting system}}
* {{ Annotated link |  Substitution instance}}
* {{ Annotated link |  Theory (mathematical logic)}}

==References==
{{reflist}}

==Sources==
* {{cite book
  |last1=Hunter
  |first1=Geoffrey
  |author1-link=Geoffrey Hunter (logician)
  |title=Metalogic: An Introduction to the Metatheory of Standard First-Order Logic
  |orig-date=1971
  |date=1996
  |publication-date=1973
  |publisher=University of California Press 
  |isbn=9780520023567
  |oclc=36312727
  |page=
  |pages=
  |section=
}} ([https://archive.org/details/metalogicintrodu0000hunt accessible to patrons with print disabilities]){{sfn whitelist|CITEREFHunter1996}}

== Further reading ==
* [[Douglas Hofstadter|Hofstadter, Douglas]], 1979. ''[[Gödel, Escher, Bach: An Eternal Golden Braid]]'' {{ISBN|978-0-465-02656-2}}. 777 pages.
* [[Stephen Cole Kleene|Kleene, Stephen C.]], 1967. ''Mathematical Logic'' Reprinted by Dover, 2002. {{ISBN|0-486-42533-9}}
* [[Raymond M. Smullyan|Smullyan, Raymond M.]], 1961. ''Theory of Formal Systems: Annals of Mathematics Studies'', Princeton University Press (April 1, 1961) 156 pages {{ISBN|0-691-08047-X}}

==External links==
{{Wiktionary|formalisation}}
*{{Commons category-inline|Formal systems}}
* Encyclopædia Britannica, [https://www.britannica.com/eb/article-9034889/formal-system Formal system] definition, 2007.
* Daniel Richardson, [http://cs.bath.ac.uk/pb/EMCL/DS/DS-Ref-2011/c19.pdf Formal systems, logic and semantics]
* {{planetmath|formalsystem|Formal System}}
* Encyclopedia of Mathematics, [https://encyclopediaofmath.org/wiki/Formal_system Formal system]
* Peter Suber, [http://www.earlham.edu/~peters/courses/logsys/machines.htm Formal Systems and Machines: An Isomorphism] [[Category:4th century BC in India]]  {{Webarchive|url=https://web.archive.org/web/20110524103726/http://www.earlham.edu/~peters/courses/logsys/machines.htm |date=2011-05-24 }}, 1997.
* Ray Taol, [https://cs.lmu.edu/~ray/notes/formalsystems/ Formal Systems]
*[http://www.cs.indiana.edu/~port/teach/641/formal.sys.haug.html What is a Formal System?]: Some quotes from John Haugeland's `Artificial Intelligence: The Very Idea' (1985), pp.&nbsp;48–64.

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[[Category:1st-millennium BC introductions]]<!-- Pāṇini fl. 4th century BCE in ancient India -->