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[lambda.git] / topics / _week5_simply_typed_lambda.mdwn

diff --git a/topics/_week5_simply_typed_lambda.mdwn b/topics/_week5_simply_typed_lambda.mdwn
index 4b1bde5..047ee8b 100644 (file)
--- a/topics/_week5_simply_typed_lambda.mdwn
+++ b/topics/_week5_simply_typed_lambda.mdwn
@@ -266,38 +266,45 @@ that even fairly basic recursive computations are beyond the reach of
 a simply-typed system.
 
 
-## Montague grammar is a simply-typed
+## 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---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
+fragment---included a simply-typed version of the Predicate Calculus
+with lambda abstraction.
+
+Montague called the semantic part of his PTQ fragment *Intensional
+Logic*.  Without getting too fussy about details, we'll present the
+popular Ty2 version of the PTQ types, roughly as proposed by Gallin
+(1975).  [See Zimmermann, Ede. 1989. Intensional logic and two-sorted
+type theory.  *Journal of Symbolic Logic* ***54.1***: 65--77 for a
+precise characterization of the correspondence between IL and
+two-sorted Ty2.]
+
+We'll need three base types: `e`, for individuals, `t`, for truth
+values, and `s` for evaluation indicies (world-time pairs).  The set
+of types is defined recursively:
+
+    the base types e, t, and 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`.  
+So `<e,<e,t>>` and `<s,<<s,e>,t>>` are types.  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`.  So the type `<a,b>` is the type of a function
+that maps objects of type `a` onto objects of type `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 *&alpha;* is an expression of type *a*, and *&beta;* is an
-expression of type b, then *&alpha;(&beta;)* has type *b*.
-*    If *&alpha;* is an expression of type *a*, and *u* is a variable of
-type *b*, then *&lambda;u&alpha;* has type <code><b, a></code>.
+* If *&alpha;* is an expression of type *<a, b>*, and *&beta;* is an
+expression of type b, then *&alpha;(&beta;)* has type *b*.  
 
-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.
+* If *&alpha;* is an expression of type *a*, and *u* is a variable of type *b*, then *&lambda;u&alpha;* has type <code><b, a></code>.
 
+When we talk about monads, we will consider Montague's treatment of
+intensionality in some detail.  In the meantime, Montague's PTQ is
+responsible for making the simply-typed lambda calculus the baseline
+semantic analysis for linguistics.