In symbolic logic, quantifiers are words or symbols which indicate quantity in the sense of "all", "some", "none", or "one". The two main logical quantifiers are "For all", usually symbolized by ∀, and "There exists", symbolized by ∃. Thus, ∀x is read "For all x" and ∃x is read "There exists an x". The quantifier ∀ is called the universal quantifier because, when it is used, it indicates that every member of the "universe of discourse" has some property which will follow the symbol. The universe of discourse is just the set of all the entities of which we are making statements. So let us suppose that our universe of discourse is the set of integers; then "∀x[(x is even)(x+1 is odd)]" states that every even integer has the property that if 1 is added to it, the result is an odd integer. The...