Part 2 Module 2
Extension: The Biconditional and
the Exclusive Or
Two common sources of error in logic involve misusing conditional statements and misuing disjunctions.
A typical misuse of conditional statements is confusing a conditional with its converse or its inverse.
A typical misuse of disjunctions is failure to realize that or in logic is inclusive.
Anothrer way of stating it is to say that a typical error in logic is confusing a conditional statement with a biconditional statement, and a second typical error is confusing a disjunction with an exclusive disjunction.
Biconditional statements
A biconditional statement is a statement of the form p, if and only if q.
This is denoted p q, and is sometimes abbreviated p iff q.
Definition: A
biconditional statement is true, only when the two terms have the same value.
Exclusive disjunctions
An exclusive disjunction, more simply called an exlcusive or, is a statement of the form p or q (but not both).
This is denoted p q, and is sometimes abbreviated p xor q.
Definition: An
exclusive or statement is true, only when exactly one of the two terms is true.
This truth table illustrates the definitions of the biconditional and exclusive or propositions.
|
p |
q |
p q |
p q |
|
T |
T |
T |
F |
|
T |
F |
F |
T |
|
F |
T |
F |
T |
|
F |
F |
T |
F |
Exercises
1-3: Use a truth table to prove each of the following:
1. 
2. 
3. 
4. Let p be the statement Thomasville is the capitol of Georgia.
Let q be the statement Sopchoppy is the capitol of Florida.
Determine the truth values of:
a.
b. 
c. 
d. 
e. 
f. 
g. 
h. 
Answers
1.
|
p |
q |
~p |
~q |
pq |
pq |
qp |
(pq)(qp) |
|
T |
T |
F |
F |
T |
T |
T |
T |
|
T |
F |
F |
T |
F |
F |
T |
F |
|
F |
T |
T |
F |
F |
T |
F |
F |
|
F |
F |
T |
T |
T |
T |
T |
T |
2.
|
p |
q |
~p |
~q |
pq |
pq |
~p~q |
(pq)( ~p~q) |
|
T |
T |
F |
F |
T |
T |
T |
T |
|
T |
F |
F |
T |
F |
F |
T |
F |
|
F |
T |
T |
F |
F |
T |
F |
F |
|
F |
F |
T |
T |
T |
T |
T |
T |
3.
|
p |
q |
~p |
~q |
pq |
~ pq |
pq |
|
T |
T |
F |
F |
T |
F |
F |
|
T |
F |
F |
T |
F |
T |
T |
|
F |
T |
T |
F |
F |
T |
T |
|
F |
F |
T |
T |
T |
F |
F |
4.
a.
T
b. 
F
c. 
T
d. 
F
e. 
F
f. 
T
g. 
F
h. 
F