View source for Syntax (logic)

{{short description|Rules used for constructing, or transforming the symbols and words of a language}}
[[File:Formal languages.svg|thumb|300px|right|This diagram shows the syntactic entities which may be constructed from [[formal language]]s.<ref>[http://dictionary.reference.com/browse/syntax Dictionary Definition]</ref> The [[symbol (formal)|symbols]] and [[string (computer science)|strings of symbols]] may be broadly divided into [[nonsense]] and [[well-formed formula]]s. A formal language is identical to the set of its well-formed formulas. The set of well-formed formulas may be broadly divided into [[theorem]]s and non-theorems.]]
{{Formal languages}}
In [[logic]], '''syntax''' is anything having to do with [[formal language]]s or [[formal system]]s without regard to any [[interpretation (logic)|interpretation]] or [[meaning (linguistics)|meaning]] given to them. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the [[Formal semantics (logic)|semantics]] of a language which is concerned with its meaning.

The [[symbol (formal)|symbols]], [[well-formed formula|formulas]], [[formal system|system]]s, [[theorem]]s and [[formal proof|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 formula]]s of a formal system.

In [[computer science]], the term ''[[Syntax (programming languages)|syntax]]'' refers to the rules governing the composition of well-formed [[Expression (mathematics)|expressions]] in a [[programming language]]. As in mathematical logic, it is independent of semantics and interpretation.

== Syntactic entities ==
=== Symbols ===
{{Main|Symbol (formal)}}

A symbol is an [[idea]], [[abstraction]] or [[concept]], [[Type-token distinction|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 constant]]s 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|Formal language}}

A ''formal language'' is a syntactic entity which consists of a [[Set (mathematics)|set]] of finite [[string (computer science)|strings]] of [[symbol (formal)|symbol]]s which are its words (usually called its [[well-formed formula]]s). Which strings of symbols are words is determined by the creator of the language, usually by specifying a set of [[formation rule]]s. Such a language can be defined without [[reference]] to any [[meaning (linguistics)|meaning]]s of any of its expressions; it can exist before any [[Interpretation (logic)|interpretation]] is assigned to it – that is, before it has any meaning.

=== Formation rules ===
{{main|Formation rule}}

''Formation rules'' are a precise description of which [[string (computer science)|strings]] of [[symbol (formal)|symbol]]s are the [[well-formed formula]]s of a formal language. It is synonymous with the set of [[String (computer science)|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|Proposition}}

A '''proposition''' is a [[Sentence (linguistics)|sentence]] expressing something [[truth|true]] or [[Deception|false]].<ref>{{Hunter 1996|p=7}}</ref> A proposition is identified [[ontology|ontologically]] as an [[idea]], [[concept]] or [[abstraction]] whose [[type-token distinction|token instances]] are patterns of [[symbol (formal)|symbols]], marks, sounds, or [[string (computer science)|strings]] of words. Propositions are considered to be syntactic entities and also [[truthbearer]]s.

=== Formal theories ===
{{main|Theory (mathematical logic)}}

A '''formal theory''' is a [[set (mathematics)|set]] of [[sentence (mathematical logic)|sentence]]s in a [[formal language]].

=== Formal systems ===
{{main|Formal system}}

A ''formal system'' (also called a ''logical calculus'', or a ''logical system'') consists of a formal language together with a [[deductive apparatus]] (also called a ''deductive system''). The deductive apparatus may consist of a set of [[transformation rule]]s (also called ''inference rules'') or a set of [[axiom]]s, 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 (logic)|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'''<ref name="google">{{cite book|title=Frege: Philosophy of Language|author=Dummett, M.|date=1981|publisher=Harvard University Press|isbn=9780674319318|url=https://books.google.com/books?id=EYP7uCZIRQYC|page=82|access-date=2014-10-15}}</ref><ref name="google2">{{cite book|title=Aristotle and Logical Theory|author=Lear, J.|date=1986|publisher=Cambridge University Press|isbn=9780521311786|url=https://books.google.com/books?id=lXI7AAAAIAAJ|page=1|access-date=2014-10-15}}</ref><ref name="google3">{{cite book|title=The Cambridge Companion to Carnap|author1=Creath, R.|author2=Friedman, M.|date=2007|publisher=Cambridge University Press|isbn=9780521840156|url=https://books.google.com/books?id=87BcFLgJmxMC|page=189|access-date=2014-10-15}}</ref><ref name="uniba">{{cite web|url=http://www.swif.uniba.it/lei/foldop/foldoc.cgi?syntactic+consequence|title=syntactic consequence from FOLDOC|publisher=swif.uniba.it|access-date=2014-10-15|url-status=dead|archive-url=https://web.archive.org/web/20130403201417/http://www.swif.uniba.it/lei/foldop/foldoc.cgi?syntactic+consequence|archive-date=2013-04-03}}</ref> within some formal system <math> \mathcal{FS}</math> of a set Г of formulas if there is a [[Formal proof|derivation]] in [[formal system]] <math> \mathcal{FS}</math> of A from the set Г.

:<math>\Gamma \vdash_{\mathrm FS} A</math>

Syntactic consequence does not depend on any [[interpretation (logic)|interpretation]] of the formal system.<ref>Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971, p. 75.</ref>

==== Syntactic completeness of a formal system ====
{{Main|Completeness (logic)}}

A formal system <math> \mathcal{S}</math> is ''syntactically complete''<ref name="oxfordjournals">{{cite web|url=http://jigpal.oxfordjournals.org/cgi/reprint/11/5/513.pdf|title=A Note on Interaction and Incompleteness|access-date=2014-10-15}}</ref><ref name="acm">{{cite journal|title=Normal forms and syntactic completeness proofs for functional independencies|year=2001|publisher=portal.acm.org|doi=10.1016/S0304-3975(00)00195-X|last1=Wijesekera|first1=Duminda|last2=Ganesh|first2=M.|last3=Srivastava|first3=Jaideep|last4=Nerode|first4=Anil|journal=Theoretical Computer Science|volume=266|issue=1–2|pages=365–405|doi-access=}}</ref><ref name="google4">{{cite book|title=Handbook of Mathematical Logic|author=Barwise, J.|author-link=Jon Barwise|date=1982|publisher=Elsevier Science|isbn=9780080933641|url=https://books.google.com/books?id=b0Fvrw9tBcMC|page=236|access-date=2014-10-15}}</ref><ref name="uniba2">{{cite web|url=http://www.swif.uniba.it/lei/foldop/foldoc.cgi?syntactic+completeness|archive-url=https://web.archive.org/web/20010502223539/http://www.swif.uniba.it/lei/foldop/foldoc.cgi?syntactic+completeness|url-status=dead|archive-date=2001-05-02|title=syntactic completeness from FOLDOC|publisher=swif.uniba.it|access-date=2014-10-15}}</ref> (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 <math> \mathcal{S}</math>. In another sense, a formal system is syntactically complete iff no unprovable axiom can be added to it as an axiom without introducing an [[consistency|inconsistency]]. Truth-functional [[propositional logic]] and first-order [[predicate logic]] are semantically complete, but not syntactically complete (for example the propositional logic statement consisting of a single variable "a" is not a theorem, and neither is its negation, but these are not [[tautology (logic)|tautologies]]). [[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|Formal semantics (logic)|Interpretation (logic)}}

An ''interpretation'' of a formal system is the assignment of meanings to the symbols, and [[truth value]]s to the sentences of a formal system. The study of interpretations is called [[Formal semantics (logic)|formal semantics]]. ''Giving an interpretation'' is synonymous with ''constructing a [[Structure (mathematical logic)|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==
*[[Symbol (formal)]]
*[[Formation rule]]
*[[Formal grammar]]
*[[Syntax|Syntax (linguistics)]]
*[[Syntax (programming languages)]]
*[[Mathematical logic]]
*[[Well-formed formula]]

==References==
{{Reflist}}

==External links==
{{Commons category-inline}}

{{Logic}}
{{Mathematical logic}}

{{DEFAULTSORT:Syntax (Logic)}}
[[Category:Syntax (logic)| ]]
[[Category:Formal languages]]
[[Category:Metalogic]]
[[Category:Concepts in logic]]
[[Category:Philosophy of logic]]