Generalization

Let P(x) and Q(x) be any predicate, D be the domain of x and consider the statement:

∀ x ∈ D, if P(x) then Q(x)

and its contrapositive

∀ x ∈ D, if ~Q(x) then ~P(x)

Any particular x in D that makes

"if P(x) then Q(x)"

true also makes

"if ~Q(x) then ~P(x)"

is true for all x in D if, and only if, the sentence

"if ~Q(x) then ~P(x)"

if true for all x in D.

In other words, the statement

∀ x ∈ D, if P(x) then Q(x)

is logically equivalent to the statement

∀ x ∈ D, if ~Q(x) then ~P(x)

and vice versa.

Therefore, we say that a universal conditional statement is logically equivalent to its contrapositive. Symbolically,

∀ x ∈ D, if P(x) then Q(x) ≡ ∀ x ∈ D, if ~Q(x) then ~P(x)

Also, a universal conditional statement is not logically equivalent to its converse. That is,

∀ x ∈ D, if P(x) then Q(x) (not ≡) ∀ x ∈ D, if Q(x) then P(x)

Similarly, a universal conditional statement is not logically equivalent to its inverse.

∀ x ∈ D, if P(x) then Q(x) (not≡) ∀ x ∈ D, if ~P(x) then ~Q(x)

Necessary and Sufficient Conditions

The definition of necessary and sufficient conditions can also be extendend to universal conditionals.

1.

∀ x, r(x) is a sufficient condition for s(x)

means

∀ x, if r(x) then s(x)

2.

∀ x, r(x) is a necessary condition for s(x)

means

if ~r(x) then ~s(x)

or equivalently,

if s(x) then r(x)

3.

∀ x, r(x) only if s(x)

means

∀ x, if ~s(x) then ~r(x)

or equivalently,

∀ x, if r(x) then s(x)