Variable: Term that does not also belong to the other terms of a language, and that may be replaced in statements of the language for a certain kind (or kinds) of term in the language, to produce what is called a formula of the language.
Thus, if one declares "X" to be a variable and not an English word, one may use it in the English sentences "Romeo is human" and "Romeo loves Juliette" to obtain the formulas "X is human" and "X is loves Juliette".
The point of doing so is that one now has a means to handle generality in various ways, for formulas now have the property that all, some, none, or any (arbitrary) constants of the kind the variable replaces may produce true statements.
It should be noted that adding variable to a language to enable one to have formulas extends the language. The idea was first used for mathematics, and was introduced by Aristotle in logic.
Anyone who knows a little algebra knows of algebraic formulas, like "2x=x+x", "(x+y)^{2}=(x^{2}+2xy+y^{2})".
Also it should be noted that "the meaning of a variable" consists in its standing for any arbitrary member of a collection of constants, that when substituted for the variable produce a meaningful statement. This statement then may be true or not.
Furthermore, having produced formulas from statements by putting one or more variables for one or more constants in it one may do the converse, put constants for variables in formulas to obtain statements.
Here it should be carefully noted that there arise three cases of formulas here, as to the sort of statements that can be produced from formulas, and that deserve their own names and examples. Here they are, after specifying the sort of the sort of replacements to be used:
The replacements to be used to obtain formulas and statements from formulas must have two characteristics in order to qualify as proper replacements:
A. Kind preserving: If there are variables of several kinds, the variables of a given kind may only be replaced by constants of the same kind.
B. Pattern preserving: If the same variable occurs at several places in the formula, and one of these is replaced by a constant, all of these are to be replaced by the same constant.
The first of these is intended to preserve a kind of general meaning: A formula was obtained from a statement and is a kind of summary of all statements that have the same terms on the same places, but to return to the kind of statement it came from variables for nouns should be replaced by nouns and not by adjectives or verbs etc.
The second of these is intended to preserve a kind of structural meaning: A formula was obtained from a statement and is a kind of structure that all statements that have the same terms on the same places and the same kinds of variables on the same places share.
Now to the kinds of formulas that result from replacing variables with constants of the same kinds:
Valid formulas: Those formulas that turn into truths for any proper replacements of their variables by constants. Algebraical examples are "2x=x+x", "(x+y)^{2}=(x^{2}+2xy+y^{2})". Linguistic examples are "v=v" and "t is human iff t is human".
Contingent formulas: Those formulas that turn into truths for some proper replacements of their variables by constants but not for others. Algebraic examples are "x+y=x" (holds if y=0 but not else); "x+4=7" (holds if x=3 but not else); and "x*x = 25" (holds if x=5 or x=5 but not else). Linguistic example is: "x is queen of Holland in 2000 A.D." (holds for Beatrix of Orange but no one else); "x is a member of the Dutch parliament in 2000 A.D." (holds for a number of Dutchmen but no other Dutchmen or nonDutchmen) etc.
Contradictory formulas: Those formulas that turn into falsities for any proper replacements of their variables by constants. Algebraical examples are "x+5=x5", "(x+y)^{2}=1+(x+y)". Linguistic examples are "if v is a boar, then v is not a boar" and "v is great and v is not great".
Mext, it should be noted that variables, so to speak, are not natural elements of a natural language. There are a few terms in a natural language that are rather like variables in algebra, such as "John Doe" and "Joe Sixpack", when used in the sense of "an arbitrary member of society (or some part of it)", but they are a bit contrived. Also, in natural language one handles generality rather as in "all Greeks are men" rather than "for all x, if x is a Greek, then x is a man", though the latter is more subtle and explicit than the former.
Hence, adding variables to a language indeed extends it as regards what are terms of the language and as regards to the complexity, subtlety and and expressive ability of the language.
The price for this is that one has to think of and introduce rules to reason with variables and formulas, which is done in logic.
