# Variable (logic)

In formal logic, a variable is a “linguistic sign for which any expressions of a certain type can be used”. In contrast to logical constants , variables have "no independent meaning" and are "meaningless characters that only serve to indicate the places where the meaningful constants ... are to be inserted."

Which expressions can be used for a variable is determined by a given set of elements. This is called the basic, object, definition or variability area or extension of a variable.

The expressions that can be used for certain variables are also called the values ​​of these variables (see mathematical logic ). Variables represent their values. It is also said that the variables pass through the set of objects denoted by the constants (their values).

The range of variability also specifies which type the expressions can belong to (individual names, predicate names, statements, etc.). Variables have the syntactic category of their values.

For the variables in the logic, the rule applies that only the same constant may be used for all occurrences of a variable in a context (“context condition”, “reference condition”).

If one obtains a true proposition by substituting constants for variables in a proposition function, “one says that the things which are described by these constants fulfill the given proposition function.” Example : The numbers 1 and 2 fulfill the proposition function “x < 3 ".

## meaning

Logic is the earliest science to introduce variables. Aristotle already introduced name variables. Variables were not used in algebra until the 16th century. It was only thanks to the introduction of the term quantifier that the role of variables in scientific language was fully recognized. This was mainly thanks to Charles S. Peirce . "The use of variables in logic serves [...] the same purpose as the corresponding use in mathematics."

## species

### Individual variable

The term individual variable (synonym: object variable ; individual variable) are variables for objects .

With the introduction of individual variables , quantified predications can be represented. They are therefore regarded as “guarantors of the general public”.

In logic, individual variables are mostly symbolized by small Latin letters from the end of the alphabet (x, y, z).

### Predicate variable

The predicate variable (predicate variable) is in the predicate logic a "schematic letter that represents any predicates of a certain arity ".

The narrower quantifier logic ( first-level predicate logic ) contains only predicate constants, but no predicate variables.

Predicates are conventionally symbolized mostly by Latin capital letters. In detail there is arbitrariness. You start with "A, B, C ..."; "F, G, H ..." or "P, Q, R ...". In some cases, other capital letters are reserved for single-digit predicates (properties) than for multi-digit predicates (relations). Arity can be identified by indices (e.g. P²) or by spaces, be it by points (P ..), underscores (P_ _) or individual variables (P (a, b)) (e.g. P = loves; P (a, b) = a loves b).

### Proposition variable

The proposition variable (synonymous: sentence variable, schema letter, truth value variable) is a variable that stands for statements (sentences, judgments).

A proposition variable must be distinguished from

• an abbreviation : "A statement variable is a sign that does not stand for any special statement, but that occupies a place that can be filled by any special statement."
• a proposition constant : Special statements are “special values ​​of proposition variables”.

In the two-valued logic, the sentence variables have the definition range {1,0}.

Small Latin letters from the middle of the alphabet beginning with the letters p, q, r… are mostly used as symbols for statement variables.

## Classifications

### Free and bound variables

A distinction must be made between free , fully free and bound variables. A free variable is a "variable that is not quantified in a sentence". A bound variable is a variable that is within the scope of a quantifier. "Bound variables denote ... no objects, but only help to indicate which places in the sentence the quantifier refers to."

### Syntactic and semantic variables

A distinction is also made between syntactic and semantic variables. Semantic variables stand for any real statements. Syntactic variables for any form of statement.

### Object language and metal language variables

A distinction is made between object-language and metal-language variables (also: meta-variables). Metalinguistic variables belong to a metalanguage . You can use names of expressions from the corresponding object language for them, e.g. B. a proposition variable. With the help of meta-variables one can formulate general laws that apply to all sentences of a certain shape.

With regard to the language level, the following applies: "Statement variables belong to the same language as the statements that make up their special values." B. Quotation marks give you metalinguistic variables. Reichenbach calls this "sentence name variable ... because its special values ​​are names of statements."

## Individual evidence

1. a b Detel: Basic Philosophy Course. Volume I: Logic. 2007
2. a b c d Tarski: Introduction to Mathematical Logic. 5th edition. 1977, p. 18f and 27.
3. Lorenzen: Formal Logic. 4th edition. 1970, p. 4f.
4. Reichenbach: Grundzüge der Symbolischen Logic (1999), pp. 10-12.
5. Hoyningen-Huene: Logic. 1998, p. 178; Tugendhat, Wolf: Logical-semantic propaedeutics. 1983, p. 46.
6. a b Essler, Martínez: Fundamentals of logic. Volume I. 4th edition. 1991, p. 174.
7. Copi: Introduction to Logic. 1998, p. 172; Wunderlich: Semantics workbook. 2nd Edition. 1991, p. 345.
8. a b c Hilbert, Ackermann: Fundamentals of theoretical logic. 6th edition. 1972, p. 69 and 11.
9. ^ Muhr: Logic. 1992, p. 56
10. predicate variable. In: Regenbogen, Meyer (Ed.): Dictionary of philosophical terms. 2005.
11. a b Czayka: Logic. 1991, p. 6.
12. Strobach: Introduction to Logic. 2005, p. 87.
13. Quine: Principles of Logic. 8th edition. 1993, p. 173.
14. ^ Wilhelm K. Essler: Introduction to Logic (= Kröner's pocket edition . Volume 381). 2nd, expanded edition. Kröner, Stuttgart 1969, DNB 456577998 , p. 102.