LogicLogicLogicLogicLogicLogicLogicLogicLogic

An Introduction to Logic
Chuck Cusack

Logic: Basic Definitions
• Definition: A proposition is a statement that is either true
or false, but not both.
• Defintion: The value of a proposition is called its truth
value. Denoted by T if it is true, F if it is false
Example 1: The statement “John Cusack is the president of
the U.S.A.” is a proposition with truth value false.
Example 2: The statement “Do your homework” is not a
proposition because it is not a statement that can be true or
false.

Logical Connectives
• Connectives are used to create a proposition from
several other propositions.
• Such propositions are called compound
propositions
• The most common connectives are:
– NEGATION ( or )
– AND ()
– OR ()
– XOR ()
– IMPLICATION ()
– BICONDITIONAL or IF AND ONLY IF ()

Connective Examples
• Let p be the proposition “The sky is clear.”
• Let q be the proposition “It is raining.”
• Some examples that combine these are:
– The sky is clear and it is raining. (pq)
– The sky is clear and it is not raining. (pq)
– It is raining if and only if the sky is not clear.
(q   p)

Truth Tables
• Truth Tables are used to show the
relationship between the truth values of
individual propositions and the compound
propositions based on them.
• Example: p q p q
T T T
T F F
F T F
F F F

NEGATION
• If p is a proposition, the negation of p, denoted p, is “it
is not the case that p.”
• Example: Let p be the statement “this class has 30
students.” Then p is the statement “this class does not
have 30 students.”
• It should be obvious that the negation of a proposition has
the opposite truth value. In other words, if p is true, then
p is false.
• The truth table for p is p p
T F
F T

AND
• Let p and q be propositions. The proposition “p
and q,” denoted by pq, is true if and only if both p
and q are true.
• pq is called the conjunction of p and q.
• The truth table for pq is p q p q
T T T
T F F
F T F
F F F

OR
• Let p and q be propositions. The proposition “p or
q,” denoted by pq, is false if and only if both p
and q are false. In other words, it is true if either p
or q is true, and false otherwise.
• pq is called the disjunction of p and q.
• The truth table for pq is p q pq
T T T
T F T
F T T
F F F

XOR
exclusive or q,” denoted by pq, is true if and
only if either p or q is true, but not both.
• When the term OR is used in conversation, often
the correct interpretation is XOR.
• The truth table for pq is p q pq
T T F
T F T
F T T
F F F

IMPLICATION
implies q,” denoted by pq, is false if and only if
p is true and q is false.
• pq is called an implication.
• The truth table for pq is p q pq
T T T
T F F
F T T
F F T

BICONDITIONAL
• Let p and q be propositions. The proposition “p if
and only if q,” denoted by pq, is true if and only
if p and q have the same truth value.
• pq is called a biconditional.
• The truth table for pq is p q pq
T T T
T F F
F T F
F F T

Constructing Truth Tables
• Construct the truth table for the proposition
((pq)q)
• We do this step by step as follows:
p q
T T
T F
F T
F F
p q
T
F
F
F
q
F
T
F
T
((pq)q)
T
T
F
T

Everyday Logic
• Logic is used in many places:
– Writing
– Speaking
– Search engines
– Mathematics
– Computer Programs
• A proper understanding of logic is useful, as the following
examples will demonstrate.

Logic in Searching I
• Situation: You want to find out all you can about
disc golf.
• Problem: When you search for “disc golf,” you
get many hits about golf and some about discs, but
can’t find those about “disc golf.”
• Solution: You need to find sites which mention
both disc and golf, not either word. Search for
disc AND golf

Logic in Searching II
• Situation: You just bought some fresh corn, and
you need a cornhusker to husk it, so you search for
“cornhusker” on the Internet
• Problem: Most of the results you get are about
UNL’s football team.
• Solution: You need to find sites which mention
cornhusker, but not UNL or football. Search for
cornhusker AND NOT (UNL OR football)

Logic at Home
• Situation: Your mom said “If you are good, you
can have some ice cream or some cake.”
• Problem: You were good, so you ate some ice
cream and some cake. Your mom got mad
because you had both.
• Solution: A simple miscommunication. By
having ice cream and cake, you had ice cream or
cake. But as is often the case in conversation, she
really meant XOR, not OR.

Logic in School
• Situation: You have 3 tests for a class. If you get
an A on any two of them, or get an A on at least
one but do not fail any of them, you will get an A
for the course.
• Problem: You are lazy, but want an A.
• Solution: Because of the OR condition, the
minimal you can do is get an A on two exams and
fail the third, or get an A on one exam and Ds on
the other two. I’ll pick one A and 2 Ds.

Logic in Programming I
• Situation: If x is greater than 0 and is less than or
equal to 10, you need to increment it.
• Problem: You tried the following, but it seems
too complicated, and doesn’t compile.
if(0<x<10 OR x=10) x++;
• Solution: Try:
if(x>0 AND x<=10) x++;

Logic and e-Mail Filtering I
• Situation: You are tired of getting spam about
losing weight and making money on eBay.
• Problem: You tried the following filter:
If(subject contains weight and subject
contains eBay) Delete message
• Solution: You meant:
If(subject contains weight or subject
contains eBay) Delete message

Logic in Programming II
• Situation: Consider the following loop:
while(NOT(A[i]!= 0 AND NOT(A[i]>= 10)))
• Problem: You are convinced this is way too
complicated.
• Solution: Well, it’s hard to say at this point…

Logic in Programming III
while( (i<size AND A[i]>10) OR (i<size
AND A[i]<0) OR NOT (A[i]!= 0 AND
NOT (A[i]>= 10) ) )
• Problem: You are convinced this is way too
complicated.
• Solution: Yet another example we can’t solve—
yet. But we will get there shortly.

Logic and e-mail Filtering II
• Situation: You get so much spam, you decide to delete
any message not sent (or copied) to you (blah), or a group
to which you belong (foo). You try:
if( TO or CC does not contain blah OR
TO or CC does not contain foo)
Delete message
• Problem: It seems like all of your e-mail is being deleted.
• Solution: This one is confusing, and we will solve it later.

Logic and Medication
• Situation: Your new medication has the following label:
Take 1 or 2 pills every 4-6 hours until condition improves. Do not exceed 6 pills per day, or take for
longer than 7 days unless directed by a doctor. Do not take this drug during the last 3 months of
pregnancy, unless directed by a doctor. If you are taking XYZs and PDQs or have taken either an
XYZ or a PDQ within the last 90 days, and the other within the last 30 days, or are taking an ABC or
have taken an ABC within the last 60 days, you should not take this drug. Do not drink alcohol or
smoke while on this drug. If you have a heart condition, asthma, diabetes, or have an IQ below 25,
do not take this drug. Do not take this drug if you have a high fever, cold sweats, runny nose,
headache, or sore throat, and discontinue use if you develop any of these symptoms, and dizziness,
nervousness, or sleeplessness occur.
• Problem: Under what conditions can you take it?
• Solution: Well, this one may take more than
simple logic.

Why Logic?
• Hopefully the last several examples have
convinced you that knowing more logic is
important for computer science, and life in
general.
• If you are still not convinced, tough. You are
going to learn it anyway.

Propositional Equivalences
• Many problems, included the last several
examples, can be solved by understanding
the concept of propositional equivalences.
• Example: The statement “I am not a student and I
am not living in the dorm” is equivalent to “I am
not a student or living in the dorm.”
• Example: “You will pass this class or you will not
take CSCI260 next spring” is equivalent to “If you
want to take CSCI260 next spring, then you must
pass this class.”

Some Terminology
• Definition: A tautology is a proposition that
is always true.
• Definition: A contradiction is a proposition
that is always false.
• Definition: A proposition that is not a
tautology or a contradiction is a
contingency.

Propositional Equivalence
• Definition #1: Propositions p and q are called logically
equivalent if pq is a tautology.
• Definition #2: Propositions p and q are logically
equivalent if and only if they have the same truth table.
• Notation: If p and q are equivalent, we write pq
• There are three common technique to prove that
propositions are logically equivalent.
– Construct the truth table for both and state “They have the same
truth table so they are logically equivalent” (or similar statement).
– Argue that they have the same truth values for all inputs without
explicitly constructing a truth table.
(This can be tricky since you have to make sure you have taken
into account all possibilities.)
– Use other logical equivalences to transform from one to the other.

Proving Equivalence
• Example: Prove that pq and pq are logically
equivalent.
• Proof #1: By definition pq is only false if p is true and q
is false. Similarly, pq is false only when both p and q
are false. This is the same thing as saying p is true and q is
false. Since both statements are true in all other cases, they
are logically equivalent.
• Proof #2: They are equivalent because the truth tables are
the same for both (see below).
p q
T T
T F
F T
F F
pq
T
F
T
T
p
F
F
T
T
pq
T
F
T
T

Another Example
• Prove that p(qr) and (pq)(pr) are logically equivalent.
• Proof: The truth table shows that they have the same truth
value for all inputs. Thus they are logically equivalent.
p q r
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
(pq)(pr)
T
T
T
T
T
F
F
F
pr
T
T
T
T
T
F
T
F
pq
T
T
T
T
T
T
F
F
p(qr)
T
T
T
T
T
F
F
F
qr
T
F
F
F
T
F
F
F
• Note: A truth table alone is not a proof. The truth table is part of the
proof, but on its own it is just a table. You need an explanation as to why
the truth table proves they are equivalent.

Simple Logical Equivalences
Equivalences involving one proposition
Equivalence Name
pT  T Domination laws
pF  F
pT  p Identity laws
pF  p
pp  p Idempotent laws
pp p
(p) p Double negation law
pp  T Cancellation laws
pp  F (Not an offical name)

Logical Equivalences
Equivalences involving multiple propositions
Equivalence Name
pq  qp Commutative laws
pq  qp
(pq)r  p(qr) Associative laws
(pq)r  p(qr)
p(qr)  (pq)(pr) Distributive laws
p(qr)  (pq)(pr)
(pq)  pq De Morgan’s laws
(pq)  pq
(pq)  (pq) Implication law

Using Logical Equivalences I
Example 1:
• Show that (pq)q is a tautology using logical
equivalences.
q
q
p 



 )
(
Implication law
De Morgan’s Law
Associative law
Cancellation Law
Domination Law
)
(
)
)
(( q
q
p
q
q
p 





)
( q
q
p 




T


 p
T


Using Logical Equivalences II
Example 2:
Show that ((qp))(pq) is logically equivalent to q
 Implication law
De Morgan’s and double negation
Commutative law
Distributive law
)
(
))
(
(
)
(
))
(
( q
p
p
q
q
p
p
q 









)
(
)
( q
p
p
q 




)
(
)
( p
q
p
q 




)
( p
p
q 



T

 q
q

Cancellation law
Identity law

Logic in Programming II
while(NOT(A[i]!=0 AND NOT(A[i]>= 10)))
• Problem: You are still convinced this is way too
complicated, and now you think you can simply it.
• Solution: We can use De Morgan’s law and the
double negation law to obtain
while( A[i]==0 OR A[i]>= 10)

Logic in Programming III
while( (i<size AND A[i]>10) OR (i<size AND
A[i]<0) OR NOT (A[i]!= 0 AND NOT (A[i]>= 10)))
• Problem: You are convinced this is way too complicated,
and with some work, you can simplify it.
• Solution: Start by simplifying the last part is in the last
example:
while( (i<size AND A[i]>10) OR (i<size AND
A[i]<0) OR (A[i]==0 OR A[i]>= 10) )
• Then, use the distributive law:
while( (i<size AND ( A[i]>10 OR A[i]<0) ) OR
(A[i]==0 OR A[i]>= 10) )

An Important Note
• In many programming languages, including Java, C++,
and C, applying the commutative law to a proposition may
or may not be a good idea.
• The reason for this is that these languages use a technique
sometimes call “short circuiting.”
• For instance, if A is an array of n elements, the statements
if(i<n AND A[i]==0)
and
if(A[i]==0 AND i<n)
are NOT equivalent. Why?

Logic and e-Mail Filtering II
• Situation: You get so much spam, you decide to delete any
message not sent (or copied) to you (blah), or a group to
which you belong (foo). You try:
if( TO or CC does not contain blah OR TO or CC
does not contain foo) Delete message
• Problem: It seems like all of your e-mail is being deleted.
• Solution: This one is a little more complicated. We start
by applying De Morgan’s law:
if(NOT (TO or CC contains blah AND TO or CC
contains foo) ) Delete message
• Let p=“TO or CC contains blah” and q=“TO or CC
contains foo”.

• The statement becomes
if(NOT (p AND q)) Delete message
• The truth table for NOT(p AND q)is:
p q pq
T T T
T F F
F T F
F F F
(pq)
F
T
T
T
• So e-mail is deleted unless TO or CC contain both
blah and foo, which is clearly not what we wanted.
• If TO or CC contains either blah or foo, we do not
want to delete. The truth table we want is…
?
F
F
F
T
pq
T
T
T
F
• Applying negation, we now recognize this as:

• Then what we really want is:
if(NOT(p OR q)) Delete message
p q (pq)
T T F
T F F
F T F
F F T
•Using DeMorgan’s Law, it becomes:
if(NOT p AND NOT q)) Delete message
•Retranslating, we seem to have wanted
if( TO or CC does not contain blah AND TO
or CC does not contain foo) Delete message
•Great. We did it.
•Or did we?

• We now think that the following filter should work:
if( TO or CC does not contain blah AND TO or CC
does not contain foo) Delete message
• Unfortunately, all of your e-mail is still being deleted.
• Let’s keep trying. Let
p = “TO contains blah” q = “CC contains blah”
r = “TO contains foo” s = “CC contains foo”
• The filter is
if( (NOT p OR NOT q ) AND (NOT r OR NOT s) )
Delete message
• This is where we went wrong earlier…

• Continuing, we had
if( (NOT p OR NOT q ) AND (NOT r OR NOT s) )
Delete message
• Applying De Morgan’s Law, we have
if( NOT (p AND q ) AND NOT( r AND s) )
Delete message
• Applying De Morgan’s Law again, we have
if( NOT( (p AND q ) OR ( r AND s) ) )
Delete message
• In English, the condition is loosely translated
“If it is not the case that either
both TO AND CC contains blah OR
both TO AND CC contains foo”

• In other words, unless the person sending the message put
your address in both the TO and CC fields, it will be deleted.
• The problem is that the filter parses “if TO OR CC (X)” as
“if TO (X) OR CC (X).”
• This means that “if TO OR CC (NOT X)” is parsed as
“if TO (NOT X) OR CC (NOT X),”
not as
“if NOT ( TO (X) OR CC (X)),”
which is what we did earlier in the example.
• Because of this, you should not use the “if TO OR CC…”
filter with the “does not contain” condition, since it is most
likely not what you intended.

• Now that we know the problem, we can fix it.
• What we want is really:
If(TO does not contain blah AND
CC does not contain blah AND
TO does not contain foo AND
CC does not contain foo) Delete message
• Recall the original filter was:
if(TO or CC does not contain blah OR
TO or CC does not contain foo) Delete
message

• This example should help illustrate the following
1. Sometimes we can think a statement means one thing
when it actually means another.
2. Sometimes, we simply can’t figure out what a statement
means at all.
• In these cases, we can use logic to assist us in determining the
true meaning of statements.

Logic and Medication
• Situation: Your new medication has the following label:
Take 1 or 2 pills every 4-6 hours until condition improves. Do not exceed 6 pills per
day, or take for longer than 7 days unless directed by a doctor. Do not take this
drug during the last 3 months of pregnancy, unless directed by a doctor. If you are taking
XYZs and PDQs or have taken either an XYZ or a PDQ within the last 90 days, and the
other within the last 30 days, or are taking an ABC or have taken an ABC within the last
60 days, you should not take this drug. Do not drink alcohol or smoke while on this
drug. If you have a heart condition, asthma, diabetes, or have an IQ below 25, do not
take this drug. Do not take this drug if you have a high fever, cold sweats, runny nose,
headache, or sore throat, and discontinue use if you develop any of these symptoms, and
dizziness, nervousness, or sleeplessness occur.
• Problem: Under what conditions can you take it?
• Solution: O.K., I give up. But seriously, there may be a
time when you really need to solve a similar problem.

Some Exercises
1. Construct the truth table for the following propositions
a) (pq)p
b) (pq) (qr)
c) p(q (r  q))
2. I do not want any e-mail that contains the words puke,
ralph, or hurl, unless it was specifically sent to me (blah).
How do I do it?
3. Is (pq)q a tautology? Give two different proofs.
4. Show that pq and (pq)(pq) are logically
equivalent.
5. Show that [(pq)(pr)(qr)]r is a tautology.
Give a proof using equivalences and a truth table.

6. Can you enter this contest? Can I? Can your friends?
Rules: Must be at least 18 years old to enter. Must be enrolled in a
computer science course at Hope, but cannot be enrolled in a Japanese
course in the. Full-time employees of Hope and Calvin College are not
allowed to enter, unless they are faculty of the computer science
department, or work in information systems support.
7. Your mom says “If you don’t eat your meat, you can’t have any
pudding.” What should you do? Why?
8. Your friend offers “Heads I win, tails you lose.” Do you take the
bet?
9. If you can’t understand this problem, then you didn’t learn the
material from these notes. Did you learn the material from these
notes?
10. If you don’t want to not learn the material from this class, you
should not fail to not skip doing problems. Or should you?
11. In an episode of The Simpsons Bart said something similar to “All I
know is that I’m getting straight A’s, and that ain’t not bad.” Does
he deserve all A’s?

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