A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A formal system is essentially an "

Formal system

definition, 2007.

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

1997. {{DEFAULTSORT:Formal System Metalogic Syntax (logic)

axiomatic system
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

".
In 1921, David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Gr ...

proposed to use such a system as the foundation for the knowledge in mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

. A formal system may represent a well-defined .
The term ''formalism'' is sometimes a rough synonym for ''formal system'', but it also refers to a given style of notation
In linguistics
Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them.
The traditional areas of linguistic analysis include p ...

, for example, Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century.
Dirac made fundamental contributions to the early developme ...

's bra–ket notation
In quantum mechanics
Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum ...

.
Background

Each formal system uses primitivesymbols
A symbol is a mark, sign, or word
In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meaning (linguistics), meanin ...

(which collectively form an alphabet
An alphabet is a standardized set of basic written symbols
A symbol is a mark, sign, or word
In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semanti ...

) to finitely construct a formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formedness, well-formed a ...

from a set of axiom
An axiom, postulate or assumption is a statement that is taken to be , to serve as a or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which comm ...

s through inferential rules of formation
In formal language theory
In mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formed ...

.
The system thus consists of valid formulas built up through finite combinations of the primitive symbols—combinations that are formed from the axioms in accordance with the stated rules.
More formally, this can be expressed as the following:
# A finite set of symbols, known as the alphabet, which concatenate formulas, so that a formula is just a finite string of symbols taken from the alphabet.
# A grammar
In linguistics
Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Linguistics encompasses the analysis of every aspect of language, as well as the ...

consisting of rules to form formulas from simpler formulas. A formula is said to be well-formed
Well-formed or wellformed indicate syntactic correctness and may refer to:
* Well-formedness, quality of linguistic elements that conform to grammar rules
* Well-formed formula, a string that is generated by a formal grammar in logic
* Well-formed ...

if it can be formed using the rules of the formal grammar. It is often required that there be a decision procedure for deciding whether a formula is well-formed.
# A set of axioms, or axiom schemaIn mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical prop ...

ta, consisting of well-formed formulas.
# A set of inference rules
A rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax
In linguistics, syntax () is the set of rules, principles, and processes that govern the structu ...

. A well-formed formula that can be inferred from the axioms is known as a theorem of the formal system.
Recursive

A formal system is said to berecursive
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics
Linguistics is the science, scientific study of language. It e ...

(i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules are decidable set
The word ''decidable'' may refer to:
* Decidable language
*Decidability (logic) for the equivalent in mathematical logic
*Decidable problem and Undecidable problem
*Gödel's incompleteness theorem, a theorem on the indecidability of languages cons ...

s or semidecidable sets, respectively.
Inference and entailment

Theentailment
Logical consequence (also entailment) is a fundamental concept
Concepts are defined as abstract ideas
A mental representation (or cognitive representation), in philosophy of mind
Philosophy of mind is a branch of philosophy that studies th ...

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
A theory is a rational
Rationality is the quality or state of being rational – that is, being based on or agreeable to reason
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, G ...

or field (e.g. Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandria
Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic
Coptic may refer to:
Afro-Asia
* Copts, an ethnoreligious group mainly in the area of modern ...

) consistent with the usage in modern mathematics such as model theory
In mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...

.
Formal language

Aformal language
In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formedness, well-formed a ...

is a language that is defined by a formal system. Like languages in linguistics
Linguistics is the scientific study of language
A language is a structured system of communication
Communication (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo ...

, formal languages generally have two aspects:
* the syntax
In linguistics
Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Linguistics encompasses the analysis of every aspect of language, as well as the ...

of a language is what the language looks like (more formally: the set of possible expressions that are valid utterances in the language) studied in formal language theory
In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science), alphabet and are well-formedness, well-formed ...

* the semantics
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference
Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another ...

of a language are what the utterances of the language mean (which is formalized in various ways, depending on the type of language in question)
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of computation, automation, a ...

and linguistics
Linguistics is the scientific study of language
A language is a structured system of communication
Communication (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo ...

usually only the syntax of a formal language is considered via the notion of a formal grammar
In formal language theory
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λ ...

. A formal grammar is a precise description of the syntax of a formal language: a set of strings
String or strings may refer to:
*String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects
Arts, entertainment, and media Films
* Strings (1991 film), ''Strings'' (1991 fil ...

. The two main categories of formal grammar are that of generative grammar
Generative grammar, or generativism , is a linguistic theory that regards linguistics
Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying ...

s, which are sets of rules for how strings in a language can be generated, and that of analytic grammars (or reductive grammar,) which are sets of rules for how a string can be analyzed to determine whether it is a member of the language. In short, an analytic grammar describes how to ''recognize'' when strings are members in the set, whereas a generative grammar describes how to ''write'' only those strings in the set.
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a formal language is usually not described by a formal grammar but by (a) natural language, such as English. Logical systems are defined by both a deductive system and natural language. Deductive systems in turn are only defined by natural language (see below).
Deductive system

A ''deductive system'', also called a ''deductive apparatus'' or a ''logic'', consists of theaxiom
An axiom, postulate or assumption is a statement that is taken to be , to serve as a or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which comm ...

s (or axiom schemaIn mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical prop ...

ta) and rules of inference
In the philosophy of logic
Following the developments in formal logic with symbolic logic in the late nineteenth century and mathematical logic in the twentieth, topics traditionally treated by logic not being part of formal logic have tended to ...

that can be used to derive
Derive may refer to:
*Derive (computer algebra system), a commercial system made by Texas Instruments
*Dérive (magazine), ''Dérive'' (magazine), an Austrian science magazine on urbanism
*Dérive, a psychogeographical concept
See also
*
*Deri ...

theorem
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

s of the system.Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971
Such deductive systems preserve deductive
Deductive reasoning, also deductive logic, is the process of reasoning
Reason is the capacity of consciously applying logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making ...

qualities in the formula
In , a formula is a concise way of expressing information symbolically, as in a mathematical formula or a . The informal use of the term ''formula'' in science refers to the .
The plural of ''formula'' can be either ''formulas'' (from the mos ...

s that are expressed in the system. Usually the quality we are concerned with is truth
Truth is the property of being in accord with fact
A fact is something that is true
True most commonly refers to truth
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In ...

as opposed to falsehood. However, other modalities, such as justification or belief
A belief is an attitude
Attitude may refer to:
Philosophy and psychology
* Attitude (psychology)
In psychology
Psychology is the science of mind and behavior. Psychology includes the study of consciousness, conscious and Unconsci ...

may be preserved instead.
In order to sustain its deductive integrity, a ''deductive apparatus'' must be definable without reference to any intended interpretation
An interpretation is an assignment of meaning to the symbols
A symbol is a mark, sign, or word
In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, o ...

of the language. The aim is to ensure that each line of a derivation
Derivation may refer to:
* Derivation (differential algebra), a unary function satisfying the Leibniz product law
* Derivation (linguistics)
* Formal proof or derivation, a sequence of sentences each of which is an axiom or follows from the precedi ...

is merely a syntactic consequence of the lines that precede it. There should be no element of any interpretation
Interpretation may refer to:
Culture
* Aesthetic interpretation, an explanation of the meaning of a work of art
* Allegorical interpretation, an approach that assumes a text should not be interpreted literally
* Dramatic Interpretation, an event i ...

of the language that gets involved with the deductive nature of the system.
An example of deductive system is first order predicate logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system
A formal system is an used for inferring theorems from axioms according to a set of rules. These rul ...

.
Logical system

A ''logical system'' or ''language'' (not be confused with the kind of "formal language" discussed above which is described by a formal grammar), is a deductive system (see section above; most commonlyfirst order predicate logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system
A formal system is an used for inferring theorems from axioms according to a set of rules. These rul ...

) together with additional (non-logical) axioms and a semantics
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference
Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another ...

. According to model-theoretic interpretation
Interpretation may refer to:
Culture
* Aesthetic interpretation, an explanation of the meaning of a work of art
* Allegorical interpretation, an approach that assumes a text should not be interpreted literally
* Dramatic Interpretation, an event i ...

, the semantics of a logical system describe whether a well-formed formula is satisfied by a given structure. 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
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

if each well-formed formula that can be inferred from the axioms is satisfied by every model of the logical system. Conversely, a logic system is 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
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian people, Italian mathematician Giuseppe Peano. These axioms have been ...

.
History

Early logic systems includes Indian logic ofPāṇini
(Devanagari: पाणिनि, ) was a Sanskrit
Sanskrit (; attributively , ; nominalization, nominally , , ) is a classical language of South Asia that belongs to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European language ...

, syllogistic logic of Aristotle, propositional logic of Stoicism, and Chinese logic of Gongsun Long
Gongsun Long (, BCLiu 2004, p. 336), courtesy name Zibing (子秉), was a Chinese philosopher and writer who was a member of the School of Names
The School of Names (), sometimes called the School of Forms and Names (), was a school of Chinese ...

(c. 325–250 BCE) . In more recent times, contributors include George Boole
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught
Autodidacticism (also autodidactism) or self-education (also self-learning and self-teaching) is education
Education is the process of facil ...

, Augustus De Morgan
Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmeti ...

, and Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analy ...

. Mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal sys ...

was developed in 19th century Europe
Europe is a continent
A continent is any of several large landmass
A landmass, or land mass, is a large region
In geography
Geography (from Greek: , ''geographia'', literally "earth description") is a field of scienc ...

.
Formalism

Hilbert's program

David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Gr ...

instigated a formalist movement that was eventually tempered by Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic
Mathematical logic, also called formal logic, is a subfield of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (nu ...

.
QED manifesto

The QED manifesto represented a subsequent, as yet unsuccessful, effort at formalization of known mathematics.Examples

Examples of formal systems include: *Lambda calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system
A formal system is an used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, ar ...

* Predicate calculus
Predicate or predication may refer to:
Computer science
*Syntactic predicateA syntactic predicate specifies the syntactic validity of applying a Formal grammar#The syntax of grammars, production in a formal grammar and is analogous to a semantic P ...

* Propositional calculus
Propositional calculus is a branch of logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fal ...

Variants

The following systems are variations of formal systems.Proof system

Formal proofs are sequences ofwell-formed formula
In mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...

s (or wff for short). For a wff to qualify as part of a proof, it might either be an axiom
An axiom, postulate or assumption is a statement that is taken to be , to serve as a or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which comm ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

.
The point of view that generating formal proofs is all there is to mathematics is often called ''formalism
Formalism may refer to:
* Form (disambiguation)
* Formal (disambiguation)
* Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary
* Formalism (linguistics)
* Scienti ...

''. David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Gr ...

founded metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the term it ...

as a discipline for discussing formal systems. Any language that one uses to talk about a formal system is called a ''metalanguage
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument ...

''. 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
An object language is a language
A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) and writing. Most languages have a writing system composed of gly ...

'', that is, the object of the discussion in question.
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 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
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, a ...

s.
See also

* Formal method *Formal science
Formal science is a branch of science
Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general), organizes knowledge in the form of Testability, testable ...

* Rewriting system
In mathematics, computer science, and logic, rewriting covers a wide range of (potentially deterministic computation, non-deterministic) methods of replacing subterms of a well-formed formula, formula with other terms. The objects of focus for this ...

* Substitution instance
* Theory (mathematical logic)In mathematical logic, a theory (also called a formal theory) is a set of sentence (mathematical logic), sentences in a formal language. In most scenarios, a deductive system is first understood from context, after which an element \phi\in T of a the ...

References

Further reading

* Raymond M. Smullyan, 1961. ''Theory of Formal Systems: Annals of Mathematics Studies'', Princeton University Press (April 1, 1961) 156 pages *Stephen Cole Kleene
Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...

, 1967. ''Mathematical Logic'' Reprinted by Dover, 2002.
* Douglas Hofstadter
Douglas Richard Hofstadter (born February 15, 1945) is an American scholar of cognitive science
Cognitive science is the interdisciplinary
Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic ...

, 1979. '' Gödel, Escher, Bach: An Eternal Golden Braid'' . 777 pages.
External links

* * Encyclopædia BritannicaFormal system

definition, 2007.

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

1997. {{DEFAULTSORT:Formal System Metalogic Syntax (logic)

System
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A system, surrounded and influenced by its environment, is described by its boundaries, structure and purp ...

1st-millennium BC introductions
4th century BC in India