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 a simply-typed
+
+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---involved a simply-typed version of the Predicate Calculus
+with lambda abstraction. More specifically, Montague called the
+semantic part of the 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 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 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 `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*, 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>.
+
+In future discussions 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.
+