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

      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.

 

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

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."