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This chapter describes the underlying formalism of the theorem provers, as it is visible during an interactive proof trace, and present the general strategy followed by the theorem provers.  The rest of this chapter is structured as follows. Section "Type-logical grammars" presents a general introduction to type-logical grammars and illustrates its basic concepts using the Lambek calculus, ending the section with some problems at the syntax-semantics interface for the Lambek calculus. Type-logical grammars are a family of grammar formalisms built on a foundation of logic and type theory. Type-logical grammars originated when BIBREF4 introduced his Syntactic Calculus (called the Lambek calculus, L, by later authors). Though Lambek built on the work of BIBREF5 , BIBREF6 and others, Lambek's main innovation was to cast the calculus as a logic, giving a sequent calculus and showing decidability by means of cut elimination. 
 Question: What formalism does Grail use?

a family of grammar formalisms built on a foundation of logic and type theory. Type-logical grammars originated when BIBREF4 introduced his Syntactic Calculus (called the Lambek calculus, L, by later authors).