2. 2
Definition of Argument
• Sequence of statements:
Statement 1;
Statement 2;
Therefore, Statement 3.
• Statements 1 and 2 are called
premises.
• Statement 3 is called conclusion.
3. 3
Examples of Arguments
• It is raining or it is snowing;
It is not snowing;
Therefore, it is raining.
• If x=2 then x<5;
x<5;
x is an even integer;
Therefore, x=2.
4. 4
Argument Form
• If the premises and the conclusion
are statement forms
instead of statements,
then the resulting form is called
argument form.
• Ex: If p then q;
p;
q.
5. 5
Validity of Argument Form
• Argument form is valid means that
for any substitution of statement
variables,
if the premises are true,
then the conclusion is also true.
• The example of previous slide is a
valid argument form.
6. 6
Checking the validity
of an argument form
1) Construct truth table for the premises and
the conclusion;
2) Find the rows in which all the premises are
true (critical rows);
3) a. If in each critical row the conclusion
is true
then the argument form is valid;
b. If there is a row in which conclusion
is false
then the argument form is invalid.
7. 7
Example of valid argument form
p and q;
if p then q;
q. premises conclusion
Critical row
p q p and q if p then q q
T T T T T
T F F
F T F
F F F
8. 8
Example of invalid argument
form
p or q;
if p then q;
p. premises conclusion
Critical row
Critical row
p q p or q if p then q p
T T T T T
T F T F
F T T T F
F F F
9. 9
Valid Argument Forms
• Modus ponens: If p then q;
p;
q.
• Modus tollens: If p then q;
~q;
~p.
10. 10
Valid Argument Forms
• Disjunctive addition: p;
p or q.
• Conjunctive simplification: p and q;
p.
• Disjunctive Syllogism: p or q;
~q;
p.
• Hypothetical Syllogism: p q;
q r;
p r.
11. 11
Valid Argument Forms
• Proof by division into cases:
p or q
p r
q r
r
• Rule of contradiction:
~p c
p
12. 12/8/2018 IT 201 12
Some more
• modus tollens
Q
P Q
---------
P
• Conjunctive Simplification
P Q
--------
P
• Conjunctive addition
P
Q
-------------
P Q
• Rule of contradiction
P c, where c is a contradiction
---------
P
13. 12/8/2018 CS 201 13
A complex example
1. If my glasses are on the kitchen table, then I saw
them at breakfast.
2. I was reading the newspaper in the living room or I
was reading the newspaper in the kitchen.
3. If I was reading the newspaper in the living room,
then my glasses are on the coffee table.
4. I did not see my classes at breakfast.
5. If I was reading my book in bed, then my glasses are
on the bed table.
6. If I was reading the newspaper in the kitchen, then
my glasses are on the kitchen table.
Where are the glasses?
14. 12/8/2018 CS 201 14
Translate them into symbols
• P = my glasses are on the kitchen table,
• Q = I saw my glasses at breakfast.
• R = I was reading the newspaper in the living room
• S = I was reading the newspaper in the kitchen.
• T = my glasses are on the coffee table.
• U = I was reading my book in bed.
• V= my glasses are on the bed table.
Statements in the previous slide are translated as follows:
1. P Q 2. R S
3. R T 4. Q
5. U V 6. S P
15. 12/8/2018 IT 201 15
Deductions
a. P Q by (1)
Q by (4)
P by modus tollens
b. S P by (6)
P by the conclusion of (a)
S by modus tollens
c. R S by (2)
S by the conclusion of (b)
R by disjunctive syllogism
d. R T by (3)
R by the conclusion of (c)
T by modus ponens
16. 12/8/2018 IT 201 16
Knights always tell truth Knaves
always lie.
3 people A B C.
A: We are all Knaves
B: Exactly one is a knight
What are ABC?
17. 12/8/2018 IT 201 17
This is called an Island of Knights and Knaves problem.
A neat feature of the problem is that no inhabitant is capable of
saying "I am a knave". If a knight said it he would be lying and
knights don't lie. If a knave said it he would be telling the truth and
knaves always lie.
So A is definitely a knave. The statement "We are all knaves." Can't
be true for reason similar to above.
So either both B and C are knights, or one is a knight and one is a
knave.
If both were knights then B would be lying when he said "Exactly
one of us is a knight", and he would be a knave.
If B were a knave and C were a knight then B would be telling the
truth when he said "Exactly one of us is a knight" and that would be
against his knavish character to say.
The only option then is for B to be the knight and A and C to be
knaves!
