update week1 notes

diff --git a/week1.mdwn b/week1.mdwn
index dffe060..7a6050e 100644 (file)
--- a/week1.mdwn
+++ b/week1.mdwn
@@ -99,9 +99,9 @@ Functions are another class of values we'll have in our language. They aren't "l
 
 I said we wanted to be starting with a fragment of arithmetic, so we'll keep the function values off-stage for the moment, and also all the symbolic atoms except for `'true` and `'false`. So we've got numbers, truth-values, and some functions and relations (that is, boolean functions) defined on them. We also help ourselves to a notion of bounded quantification, as in &forall;`x < M.` &phi;, where `M` and &phi; are (simple or complex) expressions that evaluate to a number and a boolean, respectively. We limit ourselves to *bounded* quantification so that the fragment we're dealing with can be "effectively" or mechanically decided. (As we extend the language, we will lose that property, but it will be a topic for later discussion exactly when that happens.)
 
 
 I said we wanted to be starting with a fragment of arithmetic, so we'll keep the function values off-stage for the moment, and also all the symbolic atoms except for `'true` and `'false`. So we've got numbers, truth-values, and some functions and relations (that is, boolean functions) defined on them. We also help ourselves to a notion of bounded quantification, as in &forall;`x < M.` &phi;, where `M` and &phi; are (simple or complex) expressions that evaluate to a number and a boolean, respectively. We limit ourselves to *bounded* quantification so that the fragment we're dealing with can be "effectively" or mechanically decided. (As we extend the language, we will lose that property, but it will be a topic for later discussion exactly when that happens.)
 

-As I mentioned in class, I will sometimes write &forall; x : &psi; . &phi; in my informal metalanguage, where the &psi; clause represents the quantifier's *restrictor*. Other people write this like `[`&forall; x : &psi; `]` &phi;, or in various other ways. My notation is meant to parallel the notation some linguists (for example, Heim &amp; Kratzer) use in writing &lambda; x : &psi; . &phi;, where &psi;  clause restricts the range of arguments over which the function designated by the &lambda; expression is defined. Later we will see the colon used in a somewhat similar (but also somewhat different) way in our programming languages. But that's just foreshadowing.
+As I mentioned in class, I will sometimes write &forall; x : &psi; . &phi; in my informal metalanguage, where the &psi; clause represents the quantifier's *restrictor*. Other people write this like `[`&forall; x : &psi; `]` &phi;, or in various other ways. My notation is meant to parallel the notation some linguists (for example, Heim &amp; Kratzer) use in writing &lambda; x : &psi; . &phi;, where &psi;  clause restricts the range of arguments over which the function designated by the &lambda;-expression is defined. Later we will see the colon used in a somewhat similar (but also somewhat different) way in our programming languages. But that's just foreshadowing.

 
 

-So we have bounded quantification as in &forall; `x < 10.` &phi;. Obviously we could also make sense of &forall; `x == 5.` &phi; in just the same way. This would evaluate &phi; but with the variable `x` now bound to the value `5`, ignoring whatever it may be bound to in broader contexts. I will express this idea in a more perspicuous vocabulary, like this: `let x be 5 in` &phi;.
+So we have bounded quantification as in &forall; `x < 10.` &phi;. Obviously we could also make sense of &forall; `x == 5.` &phi; in just the same way. This would evaluate &phi; but with the variable `x` now bound to the value `5`, ignoring whatever it may be bound to in broader contexts. I will express this idea in a more perspicuous vocabulary, like this: `let x be 5 in` &phi;. (I say `be` rather than `=` because, as I mentioned before, it's too easy for the `=` sign to get used for too many subtly different jobs.)

 
 As one of you was quick to notice in class, though, when I shift to the `let`-vocabulary, I no longer restricted myself to just the case where &phi; evaluates to a boolean. I also permitted myself expressions like this:
 
 
 As one of you was quick to notice in class, though, when I shift to the `let`-vocabulary, I no longer restricted myself to just the case where &phi; evaluates to a boolean. I also permitted myself expressions like this: