the discussion of the simply-typed lambda calculus, it demonstrates
that even fairly basic recursive computations are beyond the reach of
a simply-typed system.
+
+
+## Montague grammar is based on a simply-typed lambda calculus
+
+Systems based on the simply-typed lambda calculus are the bread and
+butter of current linguistic semantic analysis. One of the most
+influential modern semantic formalisms---Montague's PTQ
+fragment---included a simply-typed version of the Predicate Calculus
+with lambda abstraction.
+
+Montague called the semantic part of his PTQ fragment *Intensional
+Logic*. Montague's IL had three base types: `e`, for individuals,
+`t`, for truth values, and `s` for evaluation indicies (world-time
+pairs). The set of types was defined recursively:
+
+ e, t, s are base types
+ if a and b are types, <a,b> is a type
+ if a is a type, <s,a> is a type
+
+So `<e,<e,t>>` and `<s,<<s,e>,t>>` are types, but `<e,s>` is not a
+type. As we have mentioned, this paper is the source for the
+convention in linguistics that a type of the form `<a, b>` corresponds
+to a functional type that we will write here as `a -> b`.
+
+Montague gave rules for the types of various logical formulas. Of
+particular interest here, he gave the following typing rules for
+functional application and for lambda abstracts:
+
+* If *α* is an expression of type *<a, b>*, and *β* is an
+expression of type b, then *α(β)* has type *b*. * If
+*α* is an expression of type *a*, and *u* is a variable of type
+*b*, then *λuα* has type <code><b, a></code>.
+
+When we talk about monads, we will investigate Montague's treatment of
+intensionality in some detail. In the meantime, Montague's PTQ
+fragment is responsible for making the simply-typed lambda calculus
+the baseline semantic analysis for linguistics.
+