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Re2: MOPT04  identification, part 2by mstone (Deacon) 
on Jan 08, 2003 at 01:10 UTC ( #225132=note: print w/ replies, xml )  Need Help?? 
I just realized this morning that I'd used the word 'closure' without explaining the two different meanings of that word.. my bad. Set closures and 'value capturing' closures are entirely different creatures, which trace their origins back to different mathematical definitions for the word 'closed'. Regarding set closures, we say that a set S is closed for a certain operation '.' (where '.' is the traditional symbol for some unspecified operation) if 'a.b' is a member of S for every a and b in S. A set is open if a and b are in S, but 'a.b' isn't. The set closure of S for 'a' under '.' is the smallest closed subset of S that contains a for operation '.': {'a', 'a.a', 'a.a.a', ...}
Different critter entirely.. while the set theory people were coming up with their definition of the word 'closed', the predicate calculus people were using it to mean something else. Predicate calculus (aka: formal logic) revolves around statements called sentences. Each sentence indicates some relationship between basic entities, known as predicates. Without getting too technical, a predicate is a statement that has a truth value, and the sentence says something about the relationship between the truth values of its predicates. Sentences can also contain variables, which work pretty much like function parameters. A sentence where every variable is bound to some predicate is called a closed sentence. A sentence with an undefined variable is called an open sentence. A 'value capturing' closure binds an entity to one of a function's free variables, so in predicate calculus terms, it closes one variable in the function.. hence the term closure.
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