| 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. |
