This is a topic under Grade 11 General Mathematics. It will discuss logical propositions, truth tables and tautologies and fallacies

1. logic

2. OBJECTIVES
At the end of the lesson, the learner
o Illustrates and symbolizes propositions
o Distinguishes between simple and compound
propositions
o Performs the different types of propositions
o Determines the truth value of propositions

3. logic
 is the study of the techniques and principles
used to differentiate accurate reasoning from
inaccurate reasoning.
 a tool for evaluating the reasoning of an
argument
 the study of logic is considered fundamental to
every field of education.
 the proofs of algebra, geometry and calculus
depend upon the rules of logic

4. Proposition
- Is a declarative sentence that is either true or false, but not
both.
Examples:
1. “All dolphins are mammals.”
2. “All mammals are dolphins.”
3. 5 + 5 = 10
4. 7 + 7 = 77
5. “Jon Snow knows nothing.”

5. Not examples of a
proposition
1) Am I pretty?
2) Excuse me.
3) Senator Trillanes, can you just resign?
4) WOOHOO!!
5) Sit!
6) x + 1 = 20
These are just sentences/statements and they do not assert if
something is TRUE or FALSE.

6. A Compound Proposition
- Is a proposition composed of simpler propositions using
propositional connectives.
- A propositional connective is an operation that combines
two propositions p and q to yield a new proposition whose
truth value
- the truth value depends only on the truth values of the two
original propositions.
The 5 important logical operations are :
 Conjunction ( p ˄ q )
 Disjunction ( p ˅ q )
 Negation ( ~p )
 Conditional/Implication ( p -> q )
 Biconditional ( p <-> q )

7. Conjunction
- Let P be a proposition and Q be another proposition. The operation “and” is
denoted by the symbol “ ˄ ” for Conjunction.
- In order for the proposition P ˄ Q to be true, both P ˄ Q must be
true.
Truth Table for Conjunction
P Q P ˄ Q
T T T
T F F
F T F
F F F

8. Disjunction
- Let P be a proposition and Q be another proposition . The operation “or” is
denoted by the symbol “ ˅ ” for Disjunction.
- In order for the proposition P ˅ Q to be true, EITHER P must be true or Q must be true, not
necessarily both.
Truth Table for Disjunction
P Q P ˄ Q
T T T
T F T
F T T
F F F

9. NEGATION
Let P be a proposition. The “negation” of P is written as ~P. Observe that P and ~P cannot
both have the same truth value.
TRUTH TABLE FOR NEGATION

10. CONDITIONAL/ IMPLICATION
LOGICAL IMPLICATION, symbolized by “  “ is another relation between two
propositions. It makes use of the “If-then” statement. Then the implication “ P
implies Q “ is denoted by
“ P  Q ”
TRUTH TABLE FOR CONDITIONAL/IMPLICATION
P Q P  Q
T T T
T F F
F T T
F F T

11. BICONDITIONAL
It is possible for P  Q and Q  P to be both true. If this is so, we use the symbol “ < -- > “
(double-sided arrow) to mean “P if and only if Q”. When this condition is true, we say that P and Q
are logically equivalent.
TRUTH TABLE FOR BICONDITIONAL
P Q P < -- > Q
T T T
T F F
F T F
F F T

12. p q p ^ q p ˅ q ~p p -> q p <-> q
T T
T F
F T
F F
REVIEW
LOGICAL OPERATIONS
p q p ^ q p ˅ q ~p p -> q p <-> q
T T T T F T T
T F F T F F
F T F T T T F
F F F F T T

13. TAUTOLOGIES AND
FALLACIES
With Converse,
Contrapositive, and
Inverse of a Conditional

14. p q ~p ~q p ->
q
q ->
p
~p ->
~q
~q -> ~p
T T
T F
F T
F F
CONDITIONAL OR p ->q
- It makes use of the “If – then” statement.
- There are 4 Types of Conditional Statements.
CONDITIONAL
p -> q
CONVERSE
q -> p
INVERSE
~p -> ~q
CONTRAPOSITIV
E
~q -> ~p
p q ~p ~q p ->
q
q ->
p
~p ->
~q
~q -> ~p
T T F F T T T T
T F F T F T T F
F T T F T F F T
F F T T T T T T
Truth Table
Applications:

15. TAUTOLOGIES AND FALLACIES
TAUTOLOGIES
- In logic, a Tautology is a formula (or compound proposition) that is
TRUE in every possible interpretation (or T/F Combination)
FALLACIES
- The NEGATION of a Tautology, wherein the result is FALSE in every
possible combination
Examples: Construct the Truth Table of each Compound Proposition
1. ) p v ~(p ^ q)
2.) (p ^ q) ^ ~(p v q)

16. TRUTH TABLE CONSTRUCTION
1. Write out the number of variables. If p and q, then 2. If p, q,
and r then 3. If p, q, r, and s then 4.
2. The number of rows needed is 2n
, where n is the number of
variables.
3. Start in the right-most column and alternate T and F values
until the last row.
4. Move to the left column then place alternate pairs of T’s and
F’s until the last row.
5. Then continue to the next left column then double the
number T’s and F’s until table is complete.

17. CONSTRUCT THE TRUTH TABLE OF
EACH COMPOUND PROPOSITION
1. ) p v ~(p ^ q)
STEP1: n = 2 variables (p and q)
STEP2: 2 𝑛 = 22 = 4 (rows)
STEP3-5: p q
1. Write out the
number of variables. If
p and q, then 2. If p,
q, and r then 3. If p,
q, r, and s then 4.
2. The number of
rows needed is 2n
,
where n is the number
of variables.
3. Start in the right-
most column and
alternate T and F
values until the last
row.
4. Move to the left
column then place
alternate pairs of T’s
and F’s until the last
row.
5. Then continue to
the next left column
then double the
number T’s and F’s
until table is
T
F
T
F
T
T
F
F
T
F
F
F
p ^ q ~(p ^ q)
F
T
T
T
p v ~(p ^
q)
T
T
T
T
Since all the logical combinations has a result of TRUE, then
the logical proposition is a TAUTOLOGY

18. CONSTRUCT THE TRUTH TABLE OF
EACH COMPOUND PROPOSITION
2. ) (p ^ q) ^ ~(p v q)
1. Write out the
number of variables. If
p and q, then 2. If p,
q, and r then 3. If p,
q, r, and s then 4.
2. The number of
rows needed is 2n
,
where n is the number
of variables.
3. Start in the right-
most column and
alternate T and F
values until the last
row.
4. Move to the left
column then place
alternate pairs of T’s
and F’s until the last
row.
5. Then continue to
the next left column
then double the
number T’s and F’s
until table is

19. EVALUATION OF PROPOSITIONAL LOGIC
STATEMENTS
Suppose the proposition P is true and Q is false.
Evaluate ~ (P ˄ Q) < -- > ~ P ˅ ~Q
Answer:
~ (T ˄F) < --> ~ T ˅ ~ F
~ (F) < -- > F ˅ T
T < -- > T
T
The truth value of the proposition is true.

20. SEATWORK
If p is true and q is false, solve the following:
1.) (p ^ q) v (~q v q)
2.) ~(q v q)  ~(~p)
3.) (p v p)  ~(q ^ q)