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The answers given to these questions determine several consequences.  If variables are good things, things that ought to be retained in a purified formal system, then it must be possible to account for their valid uses in a sensible fashion.  If variables are bad things, things that ought to be eliminated from a purified formal system, then it must be possible to “explain away” their properties and utilities in terms of more basic concepts and operations.

The answers given to these questions determine several consequences.  If variables are good things, things that ought to be retained in a purified formal system, then it must be possible to account for their valid uses in a sensible fashion.  If variables are bad things, things that ought to be eliminated from a purified formal system, then it must be possible to “explain away” their properties and utilities in terms of more basic concepts and operations.

One approach is to eliminate variables altogether from the primitive conceptual basis of one's formalism, replacing every form of substitution with a form of application.  In the abstract, this makes applications of constant operators to one another the only type of combination that needs to be considered.  This is the strategy of the so-called ''combinator calculus''.

One approach is to eliminate variables altogether from the primitive conceptual basis of one's formalism, replacing every form of substitution with a form of application.  In the abstract, this makes applications of constant operators to one another the only type of combination that needs to be considered.  This is the strategy of the so called "combinator calculus".

If it is desired to retain a notion of variables in the formalism, and to maintain variables as objects of reference, then there are a couple of partial explanations of variables that still afford them with various measures of objective existence.

If it is desired to retain a notion of variables in the formalism, and to maintain variables as objects of reference, then there are a couple of partial explanations of variables that still afford them with various measures of objective existence.

In the "elemental construal" of variables, a variable x is just an existing object x that is an element of a set X, the catch being "which element?".  In spite of this lack of information, one is still permitted to write "x C X" as a syntactically well formed expression and otherwise to treat the variable name "x" as a pronoun on a grammatical par with a noun.  Given enough information about the contexts of usage and interpretation, this explanation of the variable x as an unknown object would complete itself in a determinate indication of the element intended, just as if a constant object had always been named by "x".

In the "elemental construal" of variables, a variable x is just an existing object x that is an element of a set X, the catch being "which element?".  In spite of this lack of information, one is still permitted to write "x C X" as a syntactically well formed expression and otherwise to treat the variable name "x" as a pronoun on a grammatical par with a noun.  Given enough information about the contexts of usage and interpretation, this explanation of the variable x as an unknown object would complete itself in a determinate indication of the element intended, just as if a constant object had always been named by "x".