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In formal logic
Logic

Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical Trivium . The word derives from Greek language ?????? , fem....
, a formal system (also called a logical system, a logistic system, a logical calculus, or simply a logic) consists of a formal language
Formal language

A formal language is a set of words, i.e. finite string of letters, or symbols. The inventory from which these letters are taken is called the alphabet over which the language is defined....
 together with a deductive system
Deductive system

A deductive system consists of the axioms and rules of inference that can be used to formal proof the theorems of the system.Such a deductive system is intended to preserve deduction qualities in the formula s that are expressed in the system....
 (also called a deductive apparatus) which consists of a set
Set

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics....
 of inference rules and/or axiom
Axiom

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evidence, or subject to necessary decision....
s. A formal system is used to derive
Formal proof

A formal proof or derivation is a finite sequence of proposition each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference....
 one expression from one or more other expressions antecedently expressed in the system.






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In formal logic
Logic

Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical Trivium . The word derives from Greek language ?????? , fem....
, a formal system (also called a logical system, a logistic system, a logical calculus, or simply a logic) consists of a formal language
Formal language

A formal language is a set of words, i.e. finite string of letters, or symbols. The inventory from which these letters are taken is called the alphabet over which the language is defined....
 together with a deductive system
Deductive system

A deductive system consists of the axioms and rules of inference that can be used to formal proof the theorems of the system.Such a deductive system is intended to preserve deduction qualities in the formula s that are expressed in the system....
 (also called a deductive apparatus) which consists of a set
Set

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics....
 of inference rules and/or axiom
Axiom

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evidence, or subject to necessary decision....
s. A formal system is used to derive
Formal proof

A formal proof or derivation is a finite sequence of proposition each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference....
 one expression from one or more other expressions antecedently expressed in the system. These expressions are called axiom
Axiom

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evidence, or subject to necessary decision....
s, in the case of those previously supposed to be true, or theorem
Theorem

In mathematics, a theorem is a statement Mathematical proof on the basis of previously accepted or established statements such as axioms.In formal mathematical logic, the concept of a theorem may be taken to mean a formula that can be formal proof according to the deductive system of a fixed formal system....
s, in the case of those derived. A formal system may be formulated and studied for its intrinsic properties, or it may be intended as a description (i.e. a model
Model (abstract)

In mathematical logic, the formal languages, formal systems, and theory which are studied have no meaningful content until they are given an interpretation within some other system....
) of external phenomena.

Overview

Each formal system has a formal language
Formal language

A formal language is a set of words, i.e. finite string of letters, or symbols. The inventory from which these letters are taken is called the alphabet over which the language is defined....
, which is composed by primitive symbols. These symbols act on certain rules of formation and are developed by inference from a set of axiom
Axiom

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evidence, or subject to necessary decision....
s. 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.

Formal systems in mathematics consist of the following elements:
  1. A finite set of symbols (i.e. the alphabet
    Alphabet (computer science)

    In computer science, an alphabet is a, usually finite, set of characters or digits. The most common alphabet is , the binary alphabet. A finite String is a finite sequence of characters from an alphabet; for instance a binary string is a string drawn from the alphabet ....
    ), that can be used for constructing formulas (i.e. finite strings of symbols).
  2. A grammar
    Grammar

    Grammar is the field of linguistics that covers the conventions governing the use of any given natural language. It includes morphology and syntax, often complemented by phonetics, phonology, semantics, and pragmatics....
    , which tells how well-formed formula
    Well-formed formula

    In computer science and mathematical logic, a well-formed formula or simply formula is a symbol or string of symbols that is generated by the formal grammar of a formal language....
    s (abbreviated wff) are constructed out of the symbols in the alphabet. It is usually required that there be a decision procedure for deciding whether a formula is well formed or not.
  3. A set of axioms or axiom schema
    Axiom schema

    In mathematical logic, an axiom schema generalizes the notion of axiom.An axiom schema is a well-formed formula in the language of an axiomatic system, in which one or more schematic variables appear....
    ta: each axiom must be a wff.
  4. A set of inference rules
    Rule of inference

    In logic, a rule of inference is a function from sets of formulae to formulae. The argument is called the premise set and the value the conclusion....
    .


A formal system is said to be recursive
Recursive set

In computability theory, a Set of natural numbers is called recursive, computable or decidable if there is an algorithm which terminates after a finite amount of time and correctly decides whether or not a given number belongs to the set....
 (i.e. effective) if the set of axioms and the set of inference rules are decidable sets or semidecidable sets
Recursively enumerable set

In computability theory, traditionally called recursion theory, a set S of natural numbers is called recursively enumerable, computably enumerable, semidecidable, provable or Turing-recognizable if:...
, according to context.

Some theorists use the term formalism as a rough synonym for formal system, but the term is also used to refer to a particular style of notation, for example, Paul Dirac
Paul Dirac

Paul Adrien Maurice Dirac, Order of Merit , Royal Society was a United Kingdom theoretical physicist. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics....
's bra-ket notation
Bra-ket notation

Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of bracket and vertical bars....
.

Related subjects


Formal language


A formal language is a set A of strings (finite sequences) on a fixed alphabet α.

Formal grammar

In computer science
Computer science

Computer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems....
 and linguistics
Linguistics

Linguistics is the science study of natural language. Linguistics encompasses a number of sub-fields. An important topical division is between the study of language structure and the study of Meaning ....
 a formal grammar is a precise description of a formal language
Formal language

A formal language is a set of words, i.e. finite string of letters, or symbols. The inventory from which these letters are taken is called the alphabet over which the language is defined....
: a set
Set

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics....
 of strings
String (computer science)

In computer programming and some branches of mathematics, a string is an ordered sequence of symbols. These symbols are chosen from a predetermined set or alphabet....
. The two main categories of formal grammar are that of generative grammar
Generative grammar

In theoretical linguistics, generative grammar refers to a particular approach to the study of syntax. A generative grammar of a language attempts to give a set of rules that will correctly predict which combinations of words will form grammatical sentences....
s, which are sets of rules for how strings in a language can be generated, and that of analytic grammars, 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.

Formal proofs

Formal proofs are sequences of strings. For a string to qualify as part of a proof, it might either be an axiom
Axiom

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evidence, or subject to necessary decision....
 or be the product of applying an inference rule on previous strings in the proof sequence. The last string in the sequence is recognized as a theorem
Theorem

In mathematics, a theorem is a statement Mathematical proof on the basis of previously accepted or established statements such as axioms.In formal mathematical logic, the concept of a theorem may be taken to mean a formula that can be formal proof according to the deductive system of a fixed formal system....
.

The point of view that generating formal proofs is all there is to mathematics is often called formalism
Philosophy of mathematics

The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics....
. David Hilbert
David Hilbert

David Hilbert was a Germany mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries....
 founded metamathematics
Metamathematics

Metamathematics is `mathematics used to study mathematics', or it involves the application of a philosophy of mathematics. The first part of this general description appears tautological, or is perhaps open to Bertrand Russell's and Alfred Whitehead's types of antimonies , as described in their famous "Principia Mathematica"....
 as a discipline for discussing formal systems. Any language that one uses to talk about a formal system is called a metalanguage
Metalanguage

In logic and linguistics, a metalanguage is a language used to make statements about statements in another language which is called the object language....
. The metalanguage may be nothing more than ordinary 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
Object language

Object language has meaning in contexts of computer programming and operation, and in linguistics and logic....
, 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 strings for which there is a proof. Thus all axioms are considered theorems. Unlike the grammar for strings, there is no guarantee that there will be a decision procedure
Decidability (logic)

In logic, the term decidable refers to the existence of an effective method for determining membership in a set of formulas. Logical systems such as propositional logic are decidable if membership in their set of logical validity formulas can be effectively determined....
 for deciding whether a given string 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
Metatheorem

In mathematical logic, a metatheorem is a statement about theorems or about some axiomatic theory. It is crucial to realize that a metatheorm is not a statement in the theory....
s.

Formal interpretations


A formal interpretation of a formal system is the assignment of meanings, to the symbols, and truth-values to the sentences of the formal system. The study of formal interpretations is called Formal semantics. Giving an interpretation is synonymous with constructing a model
Structure (mathematical logic)

In universal algebra and in model theory, a structure consists of an underlying Set along with a collection of finitary functions and relations which are defined on it....
.

An interpreted formal system is a formal language for which both syntactical rules for deduction
Deductive system

A deductive system consists of the axioms and rules of inference that can be used to formal proof the theorems of the system.Such a deductive system is intended to preserve deduction qualities in the formula s that are expressed in the system....
, and semantical rules of interpretation are given. An interpreted formal system can be expressed as the ordered quadruple . Where, in the case of extension
Extension

Extension may refer to:* A List of cheerleading stunts* The building of community capacity by outsiders, for instance agricultural extension* Extension , relating to the pulling apart of the Earth's crust and lithosphere...
al metalanguages, is the relation of value assignment for the sentences of the language and in the case of intension
Intension

Intension refers to the possible things a word or phrase could describe. It stands in contradistinction to extension , which refers to the actual things the word or phrase does describe....
al metalanguages, it is relation of designation, i.e., the relation between an expression and its intension; and where d is the relation of direct derivability
Formal proof

A formal proof or derivation is a finite sequence of proposition each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference....
. This relation is understood in a comprehensive sense such that the primitive sentences of the formal system are taken as directly derivable from the empty set of sentences. Here axioms are stated, some similar to those stated for a formal system, and some like those for an interpreted formal language. Usually, we wish for d to be truth-preserving (that is, any sentence which is directly derivable from true sentences is itself true), however other modalities
Modal logic

A modal logic is any system of mathematical logic#Formal logic that attempts to deal with notions of possibility and necessity. Traditionally, there are three "modes" or "moods" or "modalities" of the Copula to be, namely, Logical possibility, probability, and Necessary_and_sufficient_conditions#Necessary_conditions....
 can also preserved in such a system. We can formulate an axiom for these purposes with use of the term "true". For any i1,...,in, j, p1,...,pn,q if d(j,), (i1,p1) and ... and (in,pn) and p1 and ... and pn, q.

For interpreted formal systems there are also alternative, more explicit definitions which include ds, or both ds and D, analogous to those given for interpreted formal languages.

Further reading

  • Raymond M. Smullyan, Theory of Formal Systems: Annals of Mathematics Studies, Princeton University Press (April 1, 1961) 156 pages ISBN 069108047X
  • S. C. Kleene, 1967. Mathematical Logic Reprinted by Dover, 2002. ISBN 0486425339


See also

Examples of formal systems
  • Axiomatic system
    Axiomatic system

    In mathematics, an axiomatic system is any Set of axioms from which some or all axioms can be used in conjunction to logically derive theorems....
  • Proof calculus
    Proof calculus

    In mathematical logic, a proof calculus corresponds to a family of formal systems that use a common style of formal inference for its inference rules....
  • Formal ethics
    Formal ethics

    Formal ethics is a formal logical system for describing and evaluating the form as opposed to the content of ethical principles. Formal ethics was introduced by Harry J....
  • Lambda calculus
    Lambda calculus

    In mathematical logic and computer science, lambda calculus, also written as ?-calculus, is a formal system designed to investigate function definition, function application and recursion....
  • Propositional calculus
    Propositional calculus

    In logic and mathematics, a propositional calculus or logic is a formal system in which formulae representing propositional formulas can be formed by combining atomic formula propositions using logical connectives, and a system of formal proof rules allows certain formul? to be established as "theorem"....
Other related topics
  • Axiom
    Axiom

    In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evidence, or subject to necessary decision....
  • Formal
    Formal

    The term formal has a number of uses, including:...
  • Formal language
    Formal language

    A formal language is a set of words, i.e. finite string of letters, or symbols. The inventory from which these letters are taken is called the alphabet over which the language is defined....
  • Formal method
  • Formal science
    Formal science

    A formal science is a branch of knowledge that is concerned with formal systems, for instance, logic, mathematics, systems theory and the theoretical aspects of theoretical computer science, information theory, statistics, and linguistics....
  • Gödel's incompleteness theorems
    Gödel's incompleteness theorems

    In mathematical logic, G?del's incompleteness theorems, proved by Kurt G?del in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest....
  • Institution
    Institution (computer science)

    The notion of institution has been created by Joseph Goguen and Rod Burstall in the late 1970'sin order to deal with the "population explosion among the logical systems used in...
  • substitution instance
    Substitution instance

    In propositional logic, a substitution instance of a propositional formula is a second formula obtained by replacing symbols of the original formula by other formulas....


External links

  • Encyclopædia Britannica, definition, 2007.
  • Christer Blomqvist, , webpage 1997.
  • : Some quotes from John Haugeland's `Artificial Intelligence: The Very Idea' (1985), pp. 48-64.
  • Heinrich Herre , 1997.
  • Peter Suber, , 1997.