MATH 705
Set Theory and Logic
Module 2
A
rguments
Objectives
After studying this module, you should be able to:
1. explain the concept of arguments;
2. construct diagram that will represent the argument using Venn diagram and truth
table; and
3. determine the validity and invalidity of an argument using Venn diagram, truth
table, and shortened truth table.
Introduction
Symbolic logic is mainly a study of arguments; and since argument is made up
of propositions, there is a need to determine the truthfulness of falsity or falsity of these
propositions in order to know the validity or invalidity of an argument.
Arguments
An argument is a collection of statements where it is claimed that one of the
statements called conclusion follows from the other statements called the premises of
the argument and is denoted by
P1, P2, …, Pn / Q
Thus, it has a truth value true if it is a valid and false if it is an invalid argument or fallacy.
Module 2: Arguments
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Example:
Set Theory and Logic
Consider the following argument
Some animals can reason.
Man is an animal.
-----------------------------------Therefore, man can reason.
The statement below the line denotes the conclusion and the statements above
the line denote the premises. Although each statement is true, this argument is a fallacy.
Example:
Consider this argument
Babies are irrational.
Nobody is despised who can mage a crocodile.
Irrational people are despised.
--------------------------------------------------------------Therefore, babies cannot manage crocodiles.
This argument represents a valid argument. Note that similar with the first
example,
an argument does not depend upon the particular truth value of each
statement in the argument.
Arguments and Venn Diagram
Many verbal statements can be translated into equivalent statements about sets,
which can then be described by Venn diagrams. Hence Venn diagram is very often used
to determine the validity of an argument.
Example: Consider the second example. By first premise, the set of babies is a subset of
irrational people, i.e.,
irrational people
babies
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Set Theory and Logic
by third premise, the set of irrational people is contained in the set of despised people,
i.e.,
despised people
irrational people
babies
by second premise, the set of despised people and the set of people who can manage
crocodile are disjoint, i.e.,
despised people
irrational people
people who can
manage crocodile
babies
Note that the set of babies and the set of people who can manage crocodiles are
disjoint. In other words, “Babies cannot manage crocodiles” is a consequence of the first,
second, and third premises, and represent a valid argument.
Module 2: Arguments
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Set Theory and Logic
Arguments and Propositions
The set of propositions P1, P2, P3,…, Pn yield another proposition Q, denoted by
P1, P2, P3, …,Pn / Q
is called argument on propositions.
The argument P1, P2, P3 …, Pn / Q is valid if and only if the proposition (P1 
P2  P3 . . . Pn)  Q is a tautology.
Example:
If a man is a bachelor , he is unhappy.
If a man is unhappy, he dies young.
So, bachelor dies young.
Assuming p be the statement “He is a bachelor.”, q be “He is unhappy.”, and r be “He
dies young.” Thus, the given argument in symbol can be written as [(pq)(qr)] 
(pr) and can be verified by the truth table below.
Let
P1 = (pq),
P2 = (qr),
P4 = (pq)(qr),
and P5 = [(pq)(qr)]  (pr)
Module 2: Arguments
P3 = (pr),
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Set Theory and Logic
p
q
r
P1
P2
P3
P4
P5
T
T
T
T
T
T
T
T
T
T
F
T
F
F
F
T
T
F
T
F
T
T
F
T
T
F
F
F
T
F
F
T
F
T
T
T
T
T
T
T
F
T
F
T
F
T
F
T
F
F
T
T
T
T
T
T
F
F
F
T
T
T
T
T
Thus, the given argument is valid.
Wait! … pause for a while, answer first the following question.
SAQ1
1. Show that the following arguments are not valid by constructing diagrams in which
the premises hold but the conclusion does not hold.
a) Some students are lazy.
b) All students are lazy.
All males are lazy.
Nobody who is wealthy is a student.
----------------------------------
-------------------------------------------------
Some students are males.
Lazy people are not wealthy.
2. Determine the validity of each of the following arguments using truth tables.
a) If it rains, Jen will be sick.
b) If it rains, Ashlene will be sick.
It did not rain.
Ashlene was not sick.
Therefore, Jen was not sick.
Therefore, it did not rain.
Module 2: Arguments
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MATH 705
Set Theory and Logic
ASAQ1
1. a.
lazy people
lazy people
males
b.
irrational people
Wealthy people
babies
2. a.
Let p be “It rains” and let q be “Jen is sick”. The given statement can be can be
written [(p  q)  ~p]  ~q which, by constructing truth table, is invalid.
b. Let p be “It rains”, and let q be “Ashlene is sick”. Then the given argument
can be written [(p  q)  ~p]  ~p which, by constructing truth tale, is valid.
The invalidity of an argument may be verified by showing that its propositional
form is not a tautology. Since the propositional form of an argument is an implication,
then we should be able to show an instance when the premise is true but the conclusion
is false. We do not have to construct the whole truth table for the propositional form to do
this. All we have to do is to determine the combination of values that makes the
propositional form of the argument false. This simplified process of constructing a truth
table is called shortened truth table method.
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Set Theory and Logic
Example:
Prove the invalidity of the argument
PQ
RS
QR
P  S
The propositional form of this argument is
[ (P  Q)(R  S)(Q  R)]  (P  S)
This is false when the propositional variables have the following truth values:
P
Q
R
S
F
T
F
F
Since the propositional form of the argument is not tautology, thus the argument
is not valid. The proof of an argument consists of finding the truth values of the
propositional variable which makes the premises true and the conclusion false.
Wait! … pause for a while, answer first the following question.
SAQ2
Use the shortened truth table method to prove the invalidity of the following arguments:
1) A  P
2) E  (F  G)
CD
G  (H  I)
BC
~H
A  D
Module 2: Arguments
E  I
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Set Theory and Logic
ASAQ2
The truth values of the variables that make the set of the premises true but the
conclusion false are:
1)
A
B
C
D
F
F
T
T
Module 2: Arguments
2)
E
F
G
H
I
T
T
F
F
F
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Set Theory and Logic
Activity No. 2
1. Identify the premises and conclusion of the following arguments and then
symbolize the following sentences using the indicated abbreviations:
Let
L: Dandie learned to read and write.
J: Dandie got a good job.
M: Dandie made lots of money.
E: Dandie had eight children.
a) If Dandie learned to read and write well, he got a good job and made lots of
money.
b) If Dandie learned to read and write well, he got a good job and made lots of
money – if he didn’t have eight children.
c) If Dandie didn’t make a lot of money, then either he didn’t learn to read and
write well, or he didn’t get a good job, or he had eight children.
d) If Dandie didn’t make lots of money or have eight children, but he did learn to
read an write well then he did a good job.
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Set Theory and Logic
2. For each set of premises, find a conclusion such that the argument is valid and
such that each premise is necessary for the conclusion.
a) P1: No student is lazy.
b) P1: All lawyers are wealthy.
P2: Keith is an artist.
P2: Poets are temperamental.
P3: All artists are lazy.
P3: Nelson is a lawyer.
Q:
P4: No temperamental person is wealthy.
Q:
3. Verify the validity of the following argument using truth table.
p~q, rq, r / ~p
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Set Theory and Logic
4. Prove the invalidity of each of the following arguments using the shortened truth
table.
a) M  (N  C)
b) J  (K  L)
c) T  U
N  (P  Q)
K  (~L  M)
U  (VW)
QR
(L  M)  N
V  (TX)
~(R  P)
---------------------
TX
JN
----------------
-------------------- ~M
Module 2: Arguments
TX
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