| Description: Extend wff definition to
include the universal quantifier ('for all').
∀xφ is read "φ (phi) is true for all x." Typically,
in its final application φ would
be replaced with a wff containing
a (free) occurrence of the variable x,
for example x = y.
In a universe with a finite number of objects, "for all" is
equivalent
to a big conjunction (AND) with one wff for each possible case of
x. When the universe is infinite (as
with set theory), such a
propositional-calculus equivalent is not possible because an infinitely
long formula has no meaning, but conceptually the idea is the
same. |
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