Syntax (logic)

Source: Wikipedia, the free encyclopedia.
This diagram shows the syntactic entities which may be constructed from formal languages.[1] The symbols and strings of symbols may be broadly divided into nonsense and well-formed formulas. A formal language is identical to the set of its well-formed formulas. The set of well-formed formulas may be broadly divided into theorems and non-theorems.
Part of
Formal languages
Key concepts
Applications

In

semantics
of a language which is concerned with its meaning.

The symbols, formulas, systems, theorems and proofs expressed in formal languages are syntactic entities whose properties may be studied without regard to any meaning they may be given, and, in fact, need not be given any.

Syntax is usually associated with the rules (or grammar) governing the composition of texts in a formal language that constitute the well-formed formulas of a formal system.

In computer science, the term syntax refers to the rules governing the composition of well-formed expressions in a programming language. As in mathematical logic, it is independent of semantics and interpretation.

Syntactic entities

Symbols

Main article: Symbol (formal)

A symbol is an

tokens of which may be marks or a metalanguage of marks which form a particular pattern. Symbols of a formal language need not be symbols of anything. For instance there are logical constants
which do not refer to any idea, but rather serve as a form of punctuation in the language (e.g. parentheses). A symbol or string of symbols may comprise a well-formed formula if the formulation is consistent with the formation rules of the language. Symbols of a formal language must be capable of being specified without any reference to any interpretation of them.

Formal language

Main article: Formal language

A formal language is a syntactic entity which consists of a

meanings of any of its expressions; it can exist before any interpretation
is assigned to it – that is, before it has any meaning.

Formation rules

Main article: Formation rule

Formation rules are a precise description of which strings of symbols are the well-formed formulas of a formal language. It is synonymous with the set of strings over the alphabet of the formal language which constitute well formed formulas. However, it does not describe their semantics (i.e. what they mean).

Propositions

Main article: Proposition

A proposition is a

truthbearers
.

Formal theories

Main article: Theory (mathematical logic)

A formal theory is a set of sentences in a formal language.

Formal systems

Main article: Formal system

A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a

transformation rules (also called inference rules) or a set of axioms, or have both. A formal system is used to derive one expression from one or more other expressions. Formal systems, like other syntactic entities may be defined without any interpretation
given to it (as being, for instance, a system of arithmetic).

Syntactic consequence within a formal system

A formula A is a syntactic consequence[3][4][5][6] within some formal system of a set Г of formulas if there is a derivation in formal system of A from the set Г.

Syntactic consequence does not depend on any interpretation of the formal system.[7]

Syntactic completeness of a formal system

Main article: Completeness (logic)

A formal system is syntactically complete[8][9][10][11] (also deductively complete, maximally complete, negation complete or simply complete) iff for each formula A of the language of the system either A or ¬A is a theorem of . In another sense, a formal system is syntactically complete iff no unprovable axiom can be added to it as an axiom without introducing an

Gödel's incompleteness theorem shows that no recursive system that is sufficiently powerful, such as the Peano axioms
, can be both consistent and complete.

Interpretations

Main articles:
Formal semantics (logic) and Interpretation (logic)

An interpretation of a formal system is the assignment of meanings to the symbols, and

formal semantics. Giving an interpretation is synonymous with constructing a model. An interpretation is expressed in a metalanguage
, which may itself be a formal language, and as such itself is a syntactic entity.

See also

References

  1. ^ Dictionary Definition
  2. ^ Metalogic, Geoffrey Hunter
  3. ISBN 9780674319318
    . Retrieved 2014-10-15.
  4. ISBN 9780521311786
    . Retrieved 2014-10-15.
  5. ISBN 9780521840156
    . Retrieved 2014-10-15.
  6. ^ "syntactic consequence from FOLDOC". swif.uniba.it. Archived from the original on 2013-04-03. Retrieved 2014-10-15.
  7. ^ Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971, p. 75.
  8. ^ "A Note on Interaction and Incompleteness" (PDF). Retrieved 2014-10-15.
  9. doi:10.1016/S0304-3975(00)00195-X
    .
  10. ISBN 9780080933641
    . Retrieved 2014-10-15.
  11. ^ "syntactic completeness from FOLDOC". swif.uniba.it. Archived from the original on 2001-05-02. Retrieved 2014-10-15.

External links

Media related to Syntax (logic) at Wikimedia Commons

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