Free variable

free occurrence of a variable

The occurrence of a variable in a linguistic expression as a parameter of this expression. A rigorous definition of this concept can be given only for a formalized language. Every language has its own definition of a free variable, depending on the rules for forming expressions in the particular language. The semantic criterion is the following condition: The substitution of any object from an implicit interpretation in place of the given occurrence(s) of a variable must not lead to an absurd expression. For example, in the expression $ \{ {( x, y) } : {x ^ {2} + y ^ {2} = z ^ {2} } \} $, which denotes the set of points of a circle of radius $ z $, the variable $ z $ is free while $ x $ and $ y $ are not (see Bound variable). If $ f $ denotes a mapping of the form $ X \times Y \rightarrow Z $, and the variables $ x $ and $ y $ range over $ X $ and $ Y $, respectively, then in the expression $ f ( x, y) $ the variables $ x $ and $ y $ are free (and so is $ f $, if it is considered as a variable with respect to functions). For a fixed $ x $ and by varying $ y $ one obtains a function of the form $ Y \rightarrow Z $, which is denoted by $ \lambda yf ( x, y) $. In this expression $ x $ is free and $ y $ is not. In the expression $ ( \lambda yf ( x, y)) ( y) $, which denotes the value of the function $ \lambda yf ( x, y) $ at an arbitrary point $ y $, the last occurrence of $ y $ is free while the two others are not. The first occurrence is called an operator occurrence (it is under the sign of an operator), and the second a bound occurrence.

For a non-formalized language, that is, in actual mathematical texts, for an individual expression it is not always possible to definitely identify the free variables and the bound ones. For example, in $ \sum _ {i < k } a _ {ik} $, depending on the context, the variable $ i $ can be free and $ k $ bound, or vice-versa, but they cannot both be free. An indication of which variable is assumed to be free is given by using additional means. For example, if this expression is met in a context of the form "let fk=i< kaik" , then $ k $ is free. If there is agreement that there is no summation over $ k $, then $ k $ is a parameter. The expression $ \{ a _ {i} \} $, often used in mathematics, sometimes denotes a one-element set, in which case the variable $ i $ occurs freely, and sometimes denotes the set of all $ a _ {i} $ where $ i $ runs over an assigned domain of objects, in which case $ i $ is a bound variable.

References

[a1]	S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)