Section 1.3
The basics of propositional logic
Liars and Truthtellers
Raymond Smullyan has written a number of
wonderful books filled with challenging logic
puzzles. A recurring theme in these books is an
Island where each inhabitant is either a Liar or a
Truthteller. Here is an example:
 You come across two inhabitants of the
island, let’s call them A and B. A says, “We
are both telling the truth,” and B says, “A is
lying.” Can you tell who is a liar and who is a
truthteller?
Liars and Truthtellers
To solve the puzzle, we consider in table form
the four possibilities for the types of people A
and B are:
Scenario #1 A is a TruthTeller
B is a TruthTeller
Scenario #2 A is a TruthTeller
B is a Liar
Scenario #3 A is a Liar
B is a TruthTeller
Scenario #4 A is a Liar
B is a Liar
Liars and Truthtellers
In each of these situations, we consider the two statements that
were made. To keep from getting dizzy, we will initially make our
assessment purely with regard to the actual statements and with
no regard to who makes the statements.
Both A and B
are TruthTellers
Scenario #1
A is a
TruthTeller
B is a
TruthTeller
Scenario #2
A is a
TruthTeller
B is a Liar
Scenario #3 A is a Liar
B is a
TruthTeller
Scenario #4 A is a Liar
B is a Liar
A is a Liar
Propositional logic
Propositional logic is essentially a shorthand for
the expression of sentences along with some
rules that govern when compound expressions
are true or false.
 In the previous example, we might use the
propositional variable p for the statement, “A
is a truthteller,” and the propositional variable q
for the statement, “B is a truthteller.”
Propositional logic
Propositional logic is essentially a shorthand for
the expression of sentences along with some
rules that govern when compound expressions
are true or false.
 The logical connectives , , and  stand for
the English words “and,” “or,” and “not,”
respectively. So the statement “A and B are
both Truthtellers” can be expressed p  q, and
the statement, “A is lying” can be expressed as
 p.
Truth tables for formal propositions
Using these shorthand conventions, we can write the
table that allowed us to solve the previous problem in
the following concise form:
p
q
pq
p
T
T
T
F
T
F
F
F
F
T
F
T
F
F
F
T
Who is the truthteller? Who is the liar?
Now consider which person, A or B, made
each statement.
Consider each row in the table and
determine if each outcome could represent
the real situation.
Problem 2:
Suppose that you meet three inhabitants,
A, B, and C and that A says, “B or C is
lying,” B says, “C is lying,” and C says, “A
and I are both telling the truth.” Who if
anyone is telling the truth. Set up a truth
table to solve this problem.
Propositional logic
Practice Problem. Write each of the following statements
in propositional logic using the propositional variable
c for “Sasha is a chess player” and the variable s for
“Sasha plays soccer.”
1. Sasha is a soccer player who also plays
chess.
2. Sasha is a soccer player but has not learned
chess yet.
3. Sasha has time to participate in soccer or
chess but not both.
Truth tables for compound statements
The key to making a truth table is to analyze the truth of
statements from simple to complex in each of the possible
scenarios for the values of the propositional variables.
Example. Give the truth table for the statement,
“ (p  q)”
p
q
T
T
T
F
F
T
F
F
pq
 (p  q)
Truth tables for compound statements
Practice. Give the truth table for the statement,
“( p)  ( q)”
p
q
T
T
T
F
F
T
F
F
p
q
( p)  ( q)
Negations of formal propositions
When a propositional statement is false, its
negation is a true propositional statement. For
example, if the statement “Sasha plays soccer
and chess” is false, then the statement “Sasha
doesn’t play chess or Sasha doesn’t play
soccer” is true.
An advantage of formal logic is that we can
present formal rules for negating compound
statements like this.
Negations of formal propositions
The truth table examples we saw earlier
give us one such rule for negation. The
truth tables for the statements (p  q)
and ( p)  ( q) were identical.
Logical equivalence
When two statements have exactly the
same truth table, we say that the two
statements are logically equivalent.
The pair (p  q) and ( p)  ( q) of
logically equivalent statements is one of
DeMorgan’s Laws, and it gives us a
handy way to rewrite potentially clumsy
negated statements.
Which of the following statements are
true?

p  ( q  r )  ( p  q)  r
p  (q  r )  ( p  q)  r
Before next time you should…
Read Section 1.3 of the text completing all
Practice Problems as you go.
Practice truth tables by going to the Flash
section of the Discrete Math website and
completing all problems from Section 1.3.
See if you have questions on any of the
assigned questions from Section 1.3.