logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

, syntax is anything having to do with

formal language In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also c ...

s or

formal system A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in ma ...

s without regard to any interpretation or 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

semantics Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...

of a language which is concerned with its meaning. The

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 different concep ...

, formulas,

system A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its open system (systems theory), environment, is described by its boundaries, str ...

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

s 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 formula In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. The abbreviation wf ...

s of a formal system. In

computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...

, the term ''

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 (constituenc ...

'' refers to the rules governing the composition of well-formed expressions in a

programming language A programming language is a system of notation for writing computer programs. Programming languages are described in terms of their Syntax (programming languages), syntax (form) and semantics (computer science), semantics (meaning), usually def ...

. As in mathematical logic, it is independent of semantics and interpretation.

Syntactic entities

Symbols

A symbol is an

idea In philosophy and in common usage, an idea (from the Greek word: ἰδέα (idea), meaning 'a form, or a pattern') is the results of thought. Also in philosophy, ideas can also be mental representational images of some object. Many philosophe ...

abstraction Abstraction is a process where general rules and concepts are derived from the use and classifying of specific examples, literal (reality, real or Abstract and concrete, concrete) signifiers, first principles, or other methods. "An abstraction" ...

concept A concept is an abstract idea that serves as a foundation for more concrete principles, thoughts, and beliefs. Concepts play an important role in all aspects of cognition. As such, concepts are studied within such disciplines as linguistics, ...

, 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

A ''formal language'' is a syntactic entity which consists of a set of finite strings of

symbol A symbol is a mark, Sign (semiotics), sign, or word that indicates, signifies, or is understood as representing an idea, physical object, object, or wikt:relationship, relationship. Symbols allow people to go beyond what is known or seen by cr ...

s which are its words (usually called its

s). Which strings of symbols are words is determined by the creator of the language, usually by specifying a set of formation rules. Such a language can be defined without

reference A reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to ''refer to'' the second object. It is called a ''nam ...

to any 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

''Formation rules'' are a precise description of which strings of

s are the

s of a formal language. It is synonymous with the set of strings over the

alphabet An alphabet is a standard set of letter (alphabet), letters written to represent particular sounds in a spoken language. Specifically, letters largely correspond to phonemes as the smallest sound segments that can distinguish one word from a ...

of the formal language which constitute well formed formulas. However, it does not describe their

(i.e. what they mean).

Propositions

A proposition is a sentence expressing something true or false. A proposition is identified ontologically as an

whose token instances are patterns of

, marks, sounds, or strings of words. Propositions are considered to be syntactic entities and also truthbearers.

Formal theories

A formal theory is a set of sentences in a

Formal systems

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 rules (also called ''inference rules'') or 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 ...

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 given to it (as being, for instance, a system of arithmetic).

Syntactic consequence within a formal system

A formula A is a syntactic consequence within some formal system

\mathcal

of a set Г of formulas if there is a derivation in

\mathcal

of A from the set Г. :

\Gamma \vdash_ A

Syntactic consequence does not depend on any interpretation of the formal system.Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971, p. 75.

Syntactic completeness of a formal system

A formal system

\mathcal

is ''syntactically complete'' (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

\mathcal

. In another sense, a formal system is syntactically complete iff no unprovable axiom can be added to it as an axiom without introducing an inconsistency. Truth-functional

propositional logic The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...

and first-order

predicate logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables ove ...

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 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

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

truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Truth values are used in ...

s to the sentences of a formal system. The study of interpretations is called formal semantics. ''Giving an interpretation'' is synonymous with ''constructing a

model A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided in ...

''. An interpretation is expressed in a metalanguage, which may itself be a formal language, and as such itself is a syntactic entity.

References

External links

{{DEFAULTSORT:Syntax (Logic) Formal languages Metalogic Concepts in logic Philosophy of logic