From: Jim Date: Sun, 1 Feb 2015 02:38:00 +0000 (-0500) Subject: update week1 notes X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=a57738ceb3c389a252958613dcc6d125a64384e0 update week1 notes --- diff --git a/week1.mdwn b/week1.mdwn index dffe0605..7a6050ef 100644 --- 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 ∀`x < M.` φ, where `M` and φ 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 ∀ x : ψ . φ in my informal metalanguage, where the ψ clause represents the quantifier's *restrictor*. Other people write this like `[`∀ x : ψ `]` φ, or in various other ways. My notation is meant to parallel the notation some linguists (for example, Heim & Kratzer) use in writing λ x : ψ . φ, where ψ clause restricts the range of arguments over which the function designated by the λ 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 ∀ x : ψ . φ in my informal metalanguage, where the ψ clause represents the quantifier's *restrictor*. Other people write this like `[`∀ x : ψ `]` φ, or in various other ways. My notation is meant to parallel the notation some linguists (for example, Heim & Kratzer) use in writing λ x : ψ . φ, where ψ clause restricts the range of arguments over which the function designated by the λ-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 ∀ `x < 10.` φ. Obviously we could also make sense of ∀ `x == 5.` φ in just the same way. This would evaluate φ 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` φ. +So we have bounded quantification as in ∀ `x < 10.` φ. Obviously we could also make sense of ∀ `x == 5.` φ in just the same way. This would evaluate φ 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` φ. (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 φ evaluates to a boolean. I also permitted myself expressions like this: