Any ACG generates two languages, an abstract language and an object language.
the abstract language generated by G (A(G )) is defined as follows: In words, the abstract language generated by G is the set of closed linear λ-terms, built upon the abstract vocabulary Σ 1 , whose type is the distinguished type s. the object language generated by G (O(G )) is defined to be the image of the abstract language by the term homomorphism induced by the lexicon L : It may be useful of thinking of the abstract language as a set of abstract grammatical structures, and of the object language as the set of concrete forms generated from these abstract structures.