-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`.
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.
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]
+<pre><code>
+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]
+</code></pre>
-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.
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.