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A formal system is 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".[1]

In 1921, David Hilbert proposed to use such system as the foundation for the knowledge in mathematics.[2] 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.

Background

Each formal system uses primitive symbols (which collectively form an alphabet) to finitely construct a formal language from a set of axioms through inferential rules of formation.

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.[3]

More formally, this can be expressed as the following:

  1. 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.
  2. 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.
  3. A set of axioms, or axiom schemata, consisting of well-formed formulas.
  4. 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

In 1921, David Hilbert proposed to use such system as the foundation for the knowledge in mathematics.[2] 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.

Each formal system uses primitive symbols (which collectively form an alphabet) to finitely construct a formal language from a set of axioms through inferential rules of formation.

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.[3]

More formally, this can be expressed as the following:

  1. 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.
  2. 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.
  3. A set of axioms, or axiom schemata, consisting of well-formed formulas.
  4. 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.[clarification needed]

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.[3]

More formally, this can be expressed as the following:

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.[clari

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.[clarification needed]

Formal language

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

  • the syntax 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

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,[4][5]) 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, 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

David Hilbert instigated a formalist movement that was eventually tempered by Gödel's incompleteness theorems.

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