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 " axiomatic system". In 1921,

David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...

proposed to use such a system as the foundation for the knowledge in

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

. A formal system may represent a well-defined system of abstract thought. 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 In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathema ...

Background

Each formal system is described by primitive

symbols A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...

(which collectively form an alphabet) to finitely construct a formal language from a set of

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

s through inferential

rules of formation In formal language theory, a grammar (when the context is not given, often called a formal grammar for clarity) describes how to form strings from a language's alphabet that are valid according to the language's syntax. A grammar does not describe ...

. 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 consisting of rules to form formulas from simpler formulas. A formula is said to be 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 schemata, consisting of well-formed formulas. # A set of inference rules. 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 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.

Inference and entailment

The 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 In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...

Formal language

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 In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure ( constituency) ...

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 * the semantics 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 and linguistics usually only the syntax of a formal language is considered via the notion of a formal grammar. A formal grammar is a precise description of the syntax of a formal language: a set of strings. 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 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

, 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 the

s (or axiom schemata) and rules of inference that can be used to derive theorems of the system. Such deductive systems preserve deductive qualities in the

formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...

s 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 Justification may refer to: * Justification (epistemology), a property of beliefs that a person has good reasons for holding * Justification (jurisprudence), defence in a prosecution for a criminal offenses * Justification (theology), God's act of ...

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

syntactic consequence Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statement (logic), statements that hold true when one statement logically ''follows from'' one or more statements. A Validity (lo ...

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 ...

of the language that gets involved with the deductive nature of the system. An example of deductive system is first order predicate logic.

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 commonly first order predicate logic) together with additional (non-logical) axioms. According to

, a logical system may be given one or more semantics or

s which 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 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 (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 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 mathematician Giuseppe Peano. These axioms have been used nearly u ...

. The standard model of arithmetic sets the

domain of discourse In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range. Overview The domain ...

to be the

nonnegative integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...

s and gives the symbols their usual meaning. There are also non-standard models of arithmetic.

History

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.

Formalism

Hilbert's program

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 is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research i ...

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 in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation ...

* Predicate calculus * Propositional calculus

Variants

The following systems are variations of formal systems.

Proof system

Formal proofs are sequences of well-formed formulas (or wff for short). For a wff to qualify as part of a proof, it might either be an

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

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 ter ...

as a discipline for discussing formal systems. Any language that one uses to talk about a formal system is called a ''

metalanguage In logic and linguistics, a metalanguage is a language used to describe another language, often called the ''object language''. Expressions in a metalanguage are often distinguished from those in the object language by the use of italics, quot ...

''. 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. 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 metatheorems.

References

External links

* * Encyclopædia Britannica
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)

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 (systems), environment, is described by its boundaries, ...

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