propositions

2. Proposition
Proposition is a declarative
sentence or statement
which is either true or false,
but cannot be both.

3. Arguments
• Arguments in everyday
situations take place
between people.
• Arguments give reasons for
believing the truth of a
proposition.
• Logic studies the
information content of

4. An argument is a list of propositions,
called the premises, followed by a word such
as ‘therefore’, or ‘so’, and then another
proposition, called the conclusion.
The reason for accepting the information in
the conclusion is based on the premises.
If everything is determined, people are not free.
Premise
People are free. Premise
So not everything is determined. Conclusion
Arguments

5. • An argument is valid when its
conclusion necessarily follows
from a given set of premises.
• An argument is invalid (or a
fallacy) when the conclusion
does not necessarily follow
from the given set of premises.
Arguments

6. Truth Table
• A truth table for a compound
statement is a list of the truth
or falsity of the statement for
every possible combination of
truth and falsity of its
components.
• In other words, a truth table

7. • To find the number of rows used in a
truth table, take the number 2
raised to the power of the number of
variables.
• For example, if there was a p
statement and a q statement, there
would be 2 variables, 22 is 4.
• If there were three statements, it
Truth Table

8. Negation Truth Table
Negation
P ~ P
T F
F T
Negation(not): Opposite truth value from the statement

9. Conjunction (and): Only true when both statements are true.
∧ Conjunction
P Q P ∧ Q
T T T
T F F
F T F
F F F
Conjunction Truth
Table

10. Disjunction Truth
Table
Disjunction (or): Only false when both statements are false.
∨ Disjunction
P Q P ∨ Q
T T T
T F T
F T T
F F F

11. Example:
Let p and q represent the following simple statements.
p : All triangles have three sides.
q : All right angles measure 90 degrees.
1. P ∧ Q
2. ~P ∧ Q
T T
F T
TRUE
FALSE
Determine the truth value for each statement.

12. Example:
Let p and q represent the following simple statements.
P : All triangles have three sides.
Q : All right angles measure 90 degrees.
1. P ∨ ~Q
2. ~P ∨ ~Q
T F
F F
TRUE
FALSE
Determine the truth value for each statement.

13. CONSTRUCTING TRUTH TABLES
Construct a truth table for ~ ( p ∧ q )
∧ Conjunction
p q p ∧ q ~ ( p ∧ q )
T T T
T F F
F T F
F F F
F
T
T
T

14. (~ P v Q ) ∧ ~ Q
CONSTRUCTING TRUTH TABLES
Construct a truth table for (~ p V q ) ∧ ~ q
P Q
T T
T F
F T
F F
~ P
F
F
T
T
~ P V Q
T
~ Q
F
T
F
T
T
T
F
F
F
F
T

15. → Implication
P Q 𝑷 → 𝑸
T T T
T F F
F T T
F F T
Implication Truth
Table

16. • In case all the substitution instances of an
argument are all true, the argument is said to be
tautologous or a tautology.
• A statement formed that has only false
substitution instances is said to be contradictory
or a contradiction.
• Contingency is neither a tautology nor a
contradiction.
Tautology

17. Using the truth table of the given symbolic
statement,the following hold.
1. If the truth values in the column of the given
symbolic statement are all true (T), then the
given statement is a tautology.
2. If the truth values in the column of the given
symbolic statement are all false (F), then the
given statement is a contradiction.
Tautology

18. Show that the statement p ∨ ∼ p is a
tautology and show that the
statement p ∧ ∼ p is a contradiction.
EXAMPLE

19. Exercise
Construct a truth table for the given expression:
1. ~ (p v q )
2. ( p v ~q ) ∧ ~p
3. (~p ∧ q ) v ( p ∧ ~q )