PROPOSITIONS AND TRUTH VALUES
• A PROPOSITION IS ANY SIMPLE DECLARATIVE STATEMENT THAT IS EITHER
TRUE OR FALSE, YES OR NO, 1 OR 0 BUT NOT BOTH. WE SHALL ASSIGN TO EACH
STATEMENT EXACTLY ONE OF TWO VALUES – TRUE (SYMBOLIZED BY “T”), OR
FALSE (SYMBOLIZED BY “F”).
LOGICAL CONNECTIVES
• THERE ARE FIVE BASIC OR FUNDAMENTAL CONNECTIVES IN SYMBOLIC LOGIC.
IT COMBINES TWO OR MORE PROPOSITIONS INTO A SINGLE PROPOSITION.
THE FOLLOWING TABLE SUMMARIZES THEIR NAMES AND SYMBOLS.
SIMPLE
COMPOUND
SYMBOL
NAME
P
Not P
~P
Negation
P, Q
P or Q
P⋁Q
Disjunction
P, Q
P and Q
P⋀Q
Conjunction
P, Q
If P, then Q
P→ Q
Conditional
P, Q
P iff Q
P↔Q
Biconditional
EXAMPLE 1
CONSIDER THE PROPOSITIONS:
H: JOHN IS HEALTHY
A.) H ⋀ W ⋀ ~ S
B.) ~ W ⋀ H ⋀ S
C.) ~ (H ⋁ W ⋁ S)
W: JOHN IS WEALTHY
S: JOHN IS WISE
EXAMPLE 2
CONSIDER THE PROPOSITIONS:
P: MANILA IS THE CAPITAL OF THE PHILIPPINES.
Q: 17 IS DIVISIBLE BY 3.
R: CEBU IS THE CAPITAL OF THE PHILIPPINES.
A.) MANILA IS THE CAPITAL OF THE PHILIPPINES AND 17 IS DIVISIBLE BY 3.
B.) MANILA IS NOT THE CAPITAL OF THE PHILIPPINES OR 17 IS DIVISIBLE BY 3.
C.) IF 17 IS DIVISIBLE BY 3 THEN MANILA IS THE CAPITAL OF THE PHLIPPINES BUT NOT CEBU.
CONDITIONAL STATEMENTS
CAN BE ALSO EXPRESSED AS:
• IF P, THEN Q
• P IMPLIES Q
• IF P, Q
• P ONLY IF Q
• P IS SUFFICIENT FOR Q
• NOT P UNLESS Q
• Q FOLLOWS FROM P
• Q IF P
• Q WHENEVER P
• Q IS NECESSARY FOR P
• Q PROVIDED THAT P
BICONDITIONAL STATEMENTS
CAN BE ALSO EXPRESSED AS:
• P IF AND ONLY IF Q
• P IS NECESSARY AND SUFFICIENT FOR Q
• P IS EQUIVALENT TO Q
• P PRECISELY WHEN Q
EXAMPLE 3
WRITE THE FOLLOWING STATEMENTS IN THE FORM “IF P, THEN Q” OR “P IFF Q”.
1.) THAT YOU GET A JOB IMPLIES THAT YOU HAVE THE BEST CREDENTIALS.
2.) YOUR GUARANTEE IS GOOD ONLY IF YOU BOUGHT YOUR CD LESS THAN 90 DAYS
AGO.
3.) THIS NUMBER IS DIVISIBLE BY 6 PRECISELY WHEN IT IS DIVISIBLE BY 2 AND 3.
4.) FOR YOU TO WIN THE CONTEST IT IS NECESSARY AND SUFFICIENT THAT YOU HAVE
THE ONLY WINNING TICKET.
TRUTH TABLE
• A TRUTH TABLE IS A CASE TABLE IN WHICH “T” REPRESENTS TRUE AND “F”
REPRESENTS FALSE THAT PROVIDES DEFINITION OF ANY PROPOSITIONAL
LOGIC.
FORMULA:
𝑹 = 𝟐𝑵
WHERE:
R – NUMBER OF ROWS
N – NUMBER OF PROPOSITIONS
TRUTH TABLE
• FOLLOWING ARE TRUTH TABLES FOR THE FIVE FUNDAMENTAL COMPOUND
STATEMENTS.
P
Q
P⋁Q
P⋀Q
P→ Q
P↔Q
F
F
F
F
T
T
F
T
T
F
T
F
T
F
T
F
F
F
T
T
T
T
T
T
EXAMPLE 4
DETERMINE THE TRUTH VALUE OF THE FOLLOWING COMPOUND PROPOSITIONS.
1. 2 ≤ 5 AND 6 IS AN ODD INTEGER.
2. IT IS NOT TRUE THAT 3+3=8 AND 5>7.
3. IF 1=1, THEN 3=3 AND 3=2.
4. IF 1=3 OR 1=2, THEN 3=3.
5. 33 IS DIVISIBLE BY 4 AND 1+1=3 IFF EARTH IS FLAT OR HORSE HAS FOUR LEGS.
TAUTOLOGIES, CONTRADICTIONS AND
CONTINGENCIES
• A COMPOUND PROPOSITION THAT IS TRUE FOR ALL POSSIBLE TRUTH VALUES
OF ITS PROPOSITIONAL VARIABLES IS CALLED TAUTOLOGY. A COMPOUND
PROPOSITION THAT IS FALSE FOR ALL POSSIBLE TRUTH VALUES OF ITS
PROPOSITIONAL VARIABLES IS CALLED CONTRADICTION. OTHERWISE, IT IS
CONTINGENCY.
EXAMPLE 5
JUDGE, BY CONSTRUCTING TRUTH TABLES, WHETHER EACH OF THE FOLLOWING
IS A TAUTOLOGY, CONTRADICTION OR CONTINGENCY.
1. 𝑌 ≡ 𝐴 ⋀[~(𝐴 ⋁ 𝐵)]
2. Z ≡ 𝑃 → 𝑄 ↔ [ ~𝑃 → 𝑅 ⋁ ~𝑄]
3. X ≡ 𝑃⋀𝑄 ⋁[~𝑃 ⋁(𝑃 ⋀ ~𝑄)]
OTHER RELATED CONDITIONALS
CONDITIONAL: 𝑃 → 𝑄
CONVERSE OF 𝑃 → 𝑄: 𝑄 → 𝑃
INVERSE OF 𝑃 → 𝑄: ~𝑃 → ~𝑄
CONTRAPOSITIVE OF 𝑃 → 𝑄: ~𝑄 → ~𝑃
EXAMPLE 6
FIND:
A.) CONVERSE OF ~𝑃 → ~𝑄
B.) INVERSE OF Q → ~𝑅
C.) INVERSE OF CONVERSE OF P → ~(𝑅 ⋀ ~𝑄)
SEATWORK #1
CONSTRUCT THE TRUTH TABLE AND DETERMINE WHETHER EACH OF THE
FOLLOWING COMPOUND PROPOSITION IS A TAUTOLOGY, CONTRADICTION OR
CONTINGENCY.
1. 𝑋 ≡ (𝑃 ⋀ ~𝑄) ⋀(~𝑃 ⋁ 𝑄)
2. Z ≡ [
𝐴 → 𝐵 ⋀ 𝐶 → 𝐷 ⋀ 𝐴⋁𝐶
⋀ ~𝐵] → 𝐷