# 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

*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

#### 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

*Giving an interpretation*is synonymous with

*constructing a model*. An interpretation is expressed in a metalanguage

## 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.
**. Retrieved 2014-10-15.****. 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.**.****. 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