A type of formal language set up by the logician Richard Montague in his seminal ‘The Proper Treatment of Quantification in English’ (1974). A Montague grammar is based on a first-order language to which are added powerful logical tools for the construction and evaluation of sentences. The formal language includes:

1. (i) first-order predicate logic,

2. (ii) modal operators,

3. (iii) tense operators,

4. (iv) lambda abstraction,

5. (v) operators forming the intensions and extensions of predicates.

Montague provides a type-theoretic structure, but permits quantification over every type of expression. This language is called IL (intensional language). He gives IL a model theory, in terms of individuals, truth-values, coordinates of possible worlds and times (indexes), and functions of all these. IL enables us to give an indirect interpretation of any sentence of a natural language: first the sentence is mapped onto a translation in IL, and then the interpretation of this sentence is given. Montague’s ideal was a full translation of natural language into the interpreted formal language thus generated.