From: Jim Date: Thu, 12 Feb 2015 17:55:56 +0000 (-0500) Subject: combinatory tweaks and formatting X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=662133800820b7973cc895c474ec52e98b74a9b1 combinatory tweaks and formatting --- diff --git a/topics/week3_combinatory_logic.mdwn b/topics/week3_combinatory_logic.mdwn index 0b3390ee..1c6537e4 100644 --- a/topics/week3_combinatory_logic.mdwn +++ b/topics/week3_combinatory_logic.mdwn @@ -51,7 +51,10 @@ S, K, I, B also known --> -It's possible to build a logical system equally powerful as the lambda calculus (and readily intertranslatable with it) using just combinators, considered as atomic operations. Such a language doesn't have any variables in it: not just no free variables, but no variables at all. +It's possible to build a logical system equally powerful as the lambda calculus +(and readily intertranslatable with it) using just combinators, considered as +atomic operations. Such a language doesn't have any variables in it: not just +no free variables, but no variables (or "bound positions") at all. One can do that with a very spare set of basic combinators. These days the standard base is just three combinators: `S`, `K`, and `I`. @@ -70,15 +73,16 @@ natural language denotation is a combinator. For instance, Szabolcsi 1987 argues that reflexive pronouns are argument duplicators. - everyone hit himself - S/(S!NP) (S!NP)/NP (S!NP)!((S!NP)/NP) - \fAx[fx] \y\z[HIT y z] \h\u[huu] - --------------------------------- - S!NP \u[HIT u u] - -------------------------------------------- - S Ax[HIT x x] +
``````
+everyone   hit           himself
+S/(S!NP)   (S!NP)/NP     (S!NP)!((S!NP)/NP)
+\fAx[fx]   \y\z[HIT y z] \h\u[huu]
+           ---------------------------------
+                  S!NP     \u[HIT u u]
+--------------------------------------------
+                  S        ∀x[HIT x x]
+``````
-Here, "A" is our crude markdown approximation of the universal quantifier. Notice that the semantic value of *himself* is exactly `W`. The reflexive pronoun in direct object position combines with the transitive verb. The result is an intransitive verb phrase that takes a subject argument, duplicates that argument, and feeds the two copies to the transitive verb meaning.