Valid and Invalid arguments
1
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.
2
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.
3
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.

4
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.
5
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.
6
Example of valid argument form

p and q;
if p then q;
q.
Critical row
premises
conclusion
p
q
p and q
if p then q
q
T
T
T
T
T
T
F
F
F
T
F
F
F
F
7
Example of invalid argument form

p or q;
if p then q;
p.
Critical row
Critical row
premises
conclusion
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
8
Valid Argument Forms
• Modus ponens:
If p then q;
p;
 q.
• Modus tollens:
If p then q;
~q;
 ~p.
9
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.




10
Valid Argument Forms
• Proof by division into cases:
• Rule of contradiction:

p or q
pr
q r
r
~p  c
p

11
A more complex deduction
• Knights always tell the truth,
and knaves always lie.
• U says: None of us is a knight.
V says: At least three of us are knights.
W says: At most three of us are knights.
X says: Exactly five of us are knights.
Y says: Exactly two of us are knights.
Z says: Exactly one of us is a knight.
 Which are knights and which are knaves?
12