Introduction to Propositional Logic

1. Introduction to
Propositional Logic
Jay Vaughn G. Pelonio, M.A.
Teacher 1

2. Short Quiz:
Let: A = {a,b,c,d,e,f}; B = {a,g,h}; C = {a,e,f}
1. What is A U B?
2. What is A U (B ∩ C)
3. Is C a subset of A?
A U B = {a,b,c,d,e,f,g,h}
B ∩ C = {a};
A U (B ∩ C) = {a,b,c,d,e,f}
False

3. What is Logic?
Logic is:
1. the study of the valid rules of inference
2. a proper way of thinking about something
3. the study of the principles governing correct or reliable
inference

4. Statements
Statement – declarative sentence that is either true or false.
Examples: True or False
1. The shops are closed. True
2. Wind is produced by difference True
in temperature and pressure.
3. The Earth is flat. False

5. Statements (cont.)
The following are not considered statements:
Questions; Imperatives; Requests, etc.
Examples: True or False
1. When will the quarantine end? -
2. Choose the correct answer. -
3. Why do we need to study? -

6. Activity 1:
Give 5 examples of statements:
1. _______________________________________________________
2. _______________________________________________________
3. _______________________________________________________
4. _______________________________________________________
5. _______________________________________________________

7. Activity 1(cont.)
Give 5 examples of “not” statements:
1. _______________________________________________________
2. _______________________________________________________
3. _______________________________________________________
4. _______________________________________________________
5. _______________________________________________________

8. Syntax and Proposition
Syntax – refers to “proper arrangement” of statements.
Proposition – refers to statements represented by letters like:
P, Q, R, S, etc.
Example: Let - P = I am hungry., Q = I will eat food.
P and Q are propositions; “I am hungry” and I will eat food” are
statements.

9. Connectives
Logical Connective – connect two statements together.
Types of connectives:
1. Negation (¬) - transform a statement into negative
Example: let: P = I am hungry.
¬P would mean: "I am not hugry"

10. Connectives (cont.)
2. Disjunction (v) - Also read as “or”
Example: Let - P = I am hungry., Q = I will eat food.
P v Q means: I am hungry or I will eat food.
3. Conjunction (^) – Also read as “and”
P ^ Q means: I am hungry and I will eat food.

11. Connectives (cont.)
4. Conditional (→) – also known as if, then statement.
P → Q means: If I am hungry, then I will eat food.
In this logical statement, P is called the antecedent or the hypothesis
while Q is called the consequent or the conclusion.
5. Biconditional (↔) – also known as iff (if and only if)
P ↔ Q means: I am hungry if and only if I will eat.

12. Exercises:
Translate the following into a formal logical statement:
1. If we follow the health protocols, then the cases of covid will decline.
Let P: We follow health protocols, Q: The cases of covid will decline.
Answer: P → Q

13. Exercises (cont.)
2. If we do not follow the rules, then the cases of covid will not decline.
Let R - We follow the Rules
Answer: ¬R → ¬Q
3. If we follow the rules and follow the health protocols, then we will
return to normal soon and the cases of covid will decline.
let S - We will return to normal soon
Answer: (R^P)→(S^Q)

14. Activity 2:
Transform the following propositions into statements:
Let: A- I Practice C- I will win E- I will get fired
B- I will get better D- I cheat
1. A→B __________________________________________________
2. (A ^ B) → C _____________________________________________
3. (D ^ ¬A) → E ____________________________________________
4. [(D ^ ¬A) v (¬B ^ ¬C)] → E __________________________________
5. E → (A ^ B) _____________________________________________
If I practice, then I will get better.
If I practice and get better, then I will win.
If I cheat and not practice, then I will get fired.
If I cheat and not practice or If I don’t get better and not win,
then I will get fired
If I will get fired, then I will practice and I will get better

15. Activity 3
Answer the following questions:
1. Write the symbols for the connectives negative, conjunction,
disjunction, conditional and biconditionals respectively ____, ____,
____, ____, _____
2. Translate this sentence into a logical statement:
If a vaccine is discovered, then face to face classes will resume and
everything will return to normal.

16. Summary
1. The study of the valid rules of inference
2. A statement is a declarative sentence that is either true or false
3. Syntax refers to the proper arrangement of statements.
4. Propositions are statement denoted by letters P,Q,R,S,…
5. Connectives connect two or more statements together.

17. Online Quiz
Check the google classroom link. Make sure to finish the assessment
before the deadline.