This chapter describes the framework of categorial type logic, a grammar architecture that can be seen as the logical development of the categorial approach to natural language analysis initiated in the Thirties in the work of Ajdukiewicz (||Ajdukiewicz 35|). The choice of grammatical number in the title stresses the pluralistic nature of the enterprise: this chapter systematically charts a landscape of systems of grammatical inference — the categorial counterpart of the Chomsky Hierarchy in the framework of phrase structure grammars.