**Description: **Extend wff definition to
include the universal quantifier ('for all').
∀*x**φ* 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. |