Syntax (logic)
Part of Formal languages |
In
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
A symbol is an
Formal language
A formal language is a syntactic entity which consists of a
Formation rules
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
A proposition is a
Formal theories
A formal theory is a set of sentences in a formal language.
Formal systems
A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a
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
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
Interpretations
An interpretation of a formal system is the assignment of meanings to the symbols, and
See also
- Symbol (formal)
- Formation rule
- Formal grammar
- Syntax (linguistics)
- Syntax (programming languages)
- Mathematical logic
- Well-formed formula
References
- ^ Dictionary Definition
- ^ Metalogic, Geoffrey Hunter
- ISBN 9780674319318. Retrieved 2014-10-15.
- ISBN 9780521311786. Retrieved 2014-10-15.
- ISBN 9780521840156. Retrieved 2014-10-15.
- ^ "syntactic consequence from FOLDOC". swif.uniba.it. Archived from the original on 2013-04-03. Retrieved 2014-10-15.
- ^ Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971, p. 75.
- ^ "A Note on Interaction and Incompleteness" (PDF). Retrieved 2014-10-15.
- .
- ISBN 9780080933641. Retrieved 2014-10-15.
- ^ "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