**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. |