formal language a language in which an expression’s grammaticality and interpretation (if any) are determined by precisely defined rules that appeal only to the form or shape of the symbols that constitute it (rather than, for example, to the intention of the speaker). It is usually understood that the rules are finite and effective (so that there is an algorithm for determining whether an expression is a formula) and that the grammatical expressions are uniquely readable, i.e., they are generated by the rules in only one way. A paradigm example is the language of firstorder predicate logic, deriving principally from the Begriffsschrift of Frege. The grammatical formulas of this language can be delineated by an inductive definition: (1) a capital letter ‘F’, ‘G’, or ‘H’, with or without a numerical subscript, followed by a string of lowercase letters ‘a’, ‘b’, or ‘c’, with or without numerical subscripts, is a formula; (2) if A is a formula, so is -A; (3) if A and B are formulas, so are (A & B), (A P B), and (A 7 B); (4) if A is a formula and v is a lowercase letter ‘x’, ‘y’, or ‘z’, with or without numerical subscripts, then DvA’ and EvA’ are formulas where A’ is obtained by replacing one or more occurrences of some lowercase letter in A (together with its subscripts if any) by v; (5) nothing is a formula unless it can be shown to be one by finitely many applications of the clauses 1–4. The definition uses the device of metalinguistic variables: clauses with ‘A’ and ‘B’ are to be regarded as abbreviations of all the clauses that would result by replacing these letters uniformly by names of expressions. It also uses several naming conventions: a string of symbols is named by enclosing it within single quotes and also by replacing each symbol in the string by its name; the symbols ‘7’, ‘(‘,’)’, ‘&’, ‘P’, ‘-‘ are considered names of themselves. The interpretation of predicate logic is spelled out by a similar inductive definition of truth in a model. With appropriate conventions and stipulations, alternative definitions of formulas can be given that make expressions like ‘(P 7 Q)’ the names of formulas rather than formulas themselves. On this approach, formulas need not be written symbols at all and form cannot be identified with shape in any narrow sense. For Tarski, Carnap, and others a formal language also included rules of ‘transformation’ specifying when one expression can be regarded as a consequence of others. Today it is more common to view the language and its consequence relation as distinct. Formal languages are often contrasted with natural languages, like English or Swahili. Richard Montague, however, has tried to show that English is itself a formal language, whose rules of grammar and interpretation are similar to – though much more complex than – predicate logic. See also FORMAL LOGI. S.T.K.