From: Chris Barker <barker@kappa.(none)>
Date: Mon, 27 Sep 2010 00:17:10 +0000 (-0400)
Subject: edits
X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=3759f037b7715c05c042c7ec1cad77a590a610bc

edits
---

diff --git a/week3.mdwn b/week3.mdwn
index 27bdc53e..653b3076 100644
--- a/week3.mdwn
+++ b/week3.mdwn
@@ -493,16 +493,16 @@ As any functional programmer quickly learns, writing a recursive
 function divides into two tasks: figuring out how to handle the
 recursive case, and remembering to insert a base case.  The
 interesting and enjoyable part is figuring out the recursive pattern,
-but leaving out the base case creates a program that runs forever.
-For instance, consider computing a factorial: `n!` is `n * (n-1) *
-(n-2) * ... * 1`.  The recursive case says that the factorial of a
-number `n` is `n` times the factorial of `n-1`.  But if we leave out
-the base case, we get
+but the base case cannot be ignored, since leaving out the base case
+creates a program that runs forever.  For instance, consider computing
+a factorial: `n!` is `n * (n-1) * (n-2) * ... * 1`.  The recursive
+case says that the factorial of a number `n` is `n` times the
+factorial of `n-1`.  But if we leave out the base case, we get
 
     3! = 3 * 2! = 3 * 2 * 1! = 3 * 2 * 1 * 0! = 3 * 2 * 1 * 0 * -1! ...
 
-That's why it's crucial to declare that 0! = 1, and the recursive rule
-does not apply.  In our terms,
+That's why it's crucial to declare that 0! = 1, in which case the
+recursive rule does not apply.  In our terms,
 
     fac = Y (\fac n. iszero n 1 (fac (predecessor n)))
 
@@ -514,12 +514,12 @@ case: the truth-teller paradox (and related paradoxes).
 
 (1)    This sentence is true.
 
-If we assume that "this sentence" can refer to (1), then the
-proposition expressed by (1) will be true just in case the thing
-referred to by "this sentence is true".  Thus (1) will be true just in
-case (1) is true, and (1) is true just in case (1) is true, and so on.
-If (1) is true, then (1) is true; but if (1) is not true, then (1) is
-not true.
+If we assume that the complex demonstrative "this sentence" can refer
+to (1), then the proposition expressed by (1) will be true just in
+case the thing referred to by *this sentence* is true.  Thus (1) will
+be true just in case (1) is true, and (1) is true just in case (1) is
+true, and so on.  If (1) is true, then (1) is true; but if (1) is not
+true, then (1) is not true.
 
 Without pretending to give a serious analysis of the paradox, let's
 assume that sentences can have for their meaning boolean functions
@@ -529,27 +529,31 @@ is John* might denote the function `\x y. x`, our `true`.
 Then (1) denotes a function from whatever the referent of *this
 sentence* is to a boolean.  So (1) denotes `\f. f true false`, where
 the argument `f` is the referent of *this sentence*.  Of course, if
-`f` is a boolean, `f true false <~~> f`, (1) denotes the identity
-function `I`.
+`f` is a boolean, `f true false <~~> f`, so for our purposes, we can
+assume that (1) denotes the identity function `I`.
 
 If we use (1) in a context in which *this sentence* refers to the
 sentence in which the demonstrative occurs, then we must find a
 meaning `m` such that `I m = I`.  But since in this context `m` is the
-same as the meaning `I`, we have `m = I m`, so `m` is a fixed point
-for the denotation of the sentence (when used in the appropriate context).
+same as the meaning `I`, so we have `m = I m`.  In other words, `m` is
+a fixed point for the denotation of the sentence (when used in the
+appropriate context).
 
 That means that in a context in which *this sentence* refers to the
 sentence in which it occurs, the sentence denotes a fixed point for
-the identity function, `Y I`.
+the identity function.  Here's a fixed point for the identity
+function:
 
+<pre>
     Y I
     (\f. (\h. f (h h)) (\h. f (h h))) I
     (\h. I (h h)) (\h. I (h h)))
     (\h. (h h)) (\h. (h h)))
     &omega; &omega;
     &Omega
+</pre>
 
-Oh.  Well.  That feels right!  The meaning of *This sentence is true*
+Oh.  Well!  That feels right.  The meaning of *This sentence is true*
 in a context in which *this sentence* refers to the sentence in which
 it occurs is &Omega;, our prototypical infinite loop...