Conditionals and Biconditionals
Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.
Hermann Weyl (1885–1955)

Definitions. Given propositions p and q,  represents the conditional proposition "If p, then q." or "p implies q." The proposition p is called the antecedent and the proposition q is called the consequent.  The conditional has the truth table

 p q T T T T F F F T T F F T

Many people have problems understanding the truth values for the conditional. The following example will help illustrate the truth values for the conditional.

A person makes the promise: "If I find a \$20 bill, then I will take you to a movie." Now we consider when the person has broken the promise.

(1)  The person finds a \$20 bill and takes you to a movie.
is true, since the person did not break the promise. Note p and q are both true.

(2)  The person finds a \$20 bill and does not take you to a movie.
is false, since the person broke the promise.  Note p is true, but q is false.

(3)  The person did not find a \$20 bill, but took you to a movie anyway.
is true, since the person did not break the promise.  Note p is false,
but q is true.
(This is the case with which many people have trouble understanding.)

(4)  The person did not find a \$20 bill and did not take you to a movie.

is true, since the person did not break the promise. Note p and q

are both false.

Examples of true statements.
(a)  If x is an odd integer, then x + 1 is an even integer.
(b)  If f  is differentiable at a and has a relative extremum at a, then f '(a) = 0.
(c)  If two lines are perpendicular to the same line, then the two lines are parallel. (True in neutral geometry, but false in elliptical geometry.)
(d)  If 3 < 2, then 9 < 4.
(e)  If 2 < 3, then 4 < 9.
(f)  If 3 < 2, then 4 < 9.

Examples of false statements.
(a)  If x is an odd integer, then x + 2 is an even integer.
(b)  If 2 < 3, then 9 < 4.

Definitions.
The converse of  is .
The inverse of  is .
The contrapositive of  is

Examples.

conditional:  If a polygon is a square, then it has four sides of equal length.

converse:  If a polygon has four sides of equal length, then it is a square.
inverse:  If a polygon is not a square, then it does not have four sides of equal length.
contrapositive:  If a polygon does not have four sides of equal length, then it is not a square.

conditional: If f  is differentiable at a and has a relative extremum at a, then f '(a) = 0.
converse:  If f '(a) = 0, then f  is differentiable at a and has a relative extremum at a.
inverse:  If f  is not differentiable at a or does not have a relative extremum at a, then
contrapositive:  If  then f  is not differentiable at a or does not have a relative extremum at a.

A conditional and its contrapositive are equivalent.

 p q p q ~q ~p T T T T T F F T T F F T F T F F F T T F T F T T F F T F F T T T

A conditional is not equivalent to either its inverse or its converse. Note that the inverse of a conditional is the contrapositive of the converse.

Definition. p if and only if q is a biconditional statement and is denoted by   and often written as p iff  q. A biconditional is true only when p and q have the same truth value. The truth table for the biconditional is

 p q T T T T F F F T F F F T

Note that  is equivalent to

Biconditional statements occur frequently in mathematics. Definitions are usually biconditionals.

Examples.
(a)  A quadrilateral is a rectangle if and only if it has four right angles.

Theorem 1.  For propositions p, q and r:

(a)     is logically equivalent to

(b)    is logically equivalent to .(DeMorgan's)

(c)     is logically equivalent to . (DeMorgan's)

(d)    is logically equivalent to .

(e)     is logically equivalent to .

(f)     is logically equivalent to .

(g)    is logically equivalent to .

(h)     is logically equivalent to .

Examples.
(a)  "It is false that the transformation is an isometry and the transformation is a dilation." is equivalent to "The transformation is not an isometry or the transformation is not a dilation."

(b)  "It is not the case that if John is 16 years old, then John has a driver's license." is equivalent to "John is 16 years old and does not have a driver's license."

Definitions. Conditionals are often written in forms other than if-then. The terms necessary (only if) and sufficient (if) are often used when stating conditionals. The antecedent is the sufficient condition; the consequent is the necessary condition.

Examples that are equivalent forms of the same conditional.

"For f to be an isometry it is sufficient that f  is a rotation."

"If f  is a rotation, then f  is an isometry."

"For f  to be a rotation, it is necessary that f  is an isometry."

"f  is a rotation only if f is an isometry."

" For two triangles to be perspective from a point, it is necessary for the triangles to be perspective from a line."

"Two triangles are perspective from a point only if the two triangles are perspective from a line."
"If two triangles are perspective from a point, then they are perspective from a line."
"For two triangles to be perspective from a line, it is sufficient that the two triangles are perspective from a point."