Universal quantification
Universal quantification is one way of indicating a perhaps infinite number of statements in a small, finite amount of space; the statements must all be expressable with a "forall" template.
Without universal quantification, you could write 1 = 1, 2 = 2, 3 = 3, etc.. But with universal quantification, you can capture this whole family of statements with just one statement: For all x, x = x
Quite often, it is important to specify what "all" is meant in "for all". The standard way to do this is by saying "for each member x of set S..." This allows us to capture the difference between, for example, Everything is evil (For all x, evil(x)) and All humans are evil (For all x in the set of humans, evil(x)).
to do:
- add the relevant symbols
- fix first paragraph
- links to "quantifier" and/or "generalized quantifiers"