X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week5_simply_typed_lambda.mdwn;h=047ee8b368a8a6f62c3111031a3b59d4ba06c559;hp=14e21729d3c573984875fcdd4dca223008622f1a;hb=a4d2693effe839524592f4427465ff8d97625302;hpb=bcf5a368b61427727409d77f7ee6da99bcf3b526 diff --git a/topics/_week5_simply_typed_lambda.mdwn b/topics/_week5_simply_typed_lambda.mdwn index 14e21729..047ee8b3 100644 --- a/topics/_week5_simply_typed_lambda.mdwn +++ b/topics/_week5_simply_typed_lambda.mdwn @@ -264,3 +264,47 @@ the structure of the Church numbers (and lists). Most importantly for 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*. 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, is a type + +So `>` and `,t>>` are types. As we have mentioned, +this paper is the source for the convention in linguistics that a type +of the form `` corresponds to a functional type that we will +write here as `a -> b`. So the type `` 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 *α* is an expression of type **, 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 . + +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.