**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. |