CSCI2110 Tutorial 9: Propositional Logic

1. CSCI2110 Tutorial 9:Propositional Logic Chow Chi Wang (cwchow ‘at’ cse.cuhk.edu.hk)

2. Propositional Logic I am lying right now… Is he telling the truth or lying?

3. Propositional Statement • A Statement is a sentence that is either True or False. • Which of these are propositional statements? • The sun is shining. • 3 + 4 = 7. • It rained this morning. • . • for . • Is it raining? • Come to tutorial! • 2012 is a prime number. • Hello world! • Discrete Maths is very interesting.

4. Basic Logic Operators • We can construct more complicated statements from these basic operators. (e.g. ) , ~(NOT) – Truth Table ∧ (AND) – Truth Table ∨ (OR) – Truth Table

5. XOR (Exclusive-OR) • When OR is used in its exclusive sense, “p xor q” means “p or q but not both”. • How to express XOR in terms of AND, OR and NOT?  (XOR) – Truth Table

6. Writing Logical Formula for a Truth Table • Idea 1: Look at the true rows and take the OR. • For each true row: • Construct a clause that is an AND of the true variables. • Take the OR of all the clauses obtained from the for loop.

7. Writing Logical Formula for a Truth Table • Idea 2: Look at the false rows and take the AND. • For each false row: • Construct a clause that is an AND of the true variables. • Take the AND of all the negated clauses obtained from the for loop. ~

8. Logical Equivalence • Two statements are logically equivalent if they have the same truth table. • e.g.

9. Tautology, Contradiction • A tautology is a statement that is always true. • , • A contradiction is a statement that is always false. • ,

10. Summary of Basic Logical Rules = tautology, = contradiction

11. Simplifying Statements (Distribution law) (Idempotent law) (Identity law) ~ (De Morgan’s law) (Distributive law) (Idempotent law) (Identity law)

12. Conditional Statement If P then Q P implies Q p is called the hypothesis; q is called the conclusion – Truth Table

13. Conditional Statement • is equivalent to . • The contrapositive of “”is “”. • (Statement) Every CUHK student has CULINK. • (Contrapositive) If someone doesn’t have CULINK, then (s)he is not a CUHK student. Important fact A conditional statement is logically equivalent to its contrapositive.

14. Argument • An argument is a sequence of statements. • All statements but the final one are called assumptions or hypothesis. • The final statement is called the conclusion. • An argument is valid if whenever all the assumptions are true, then the conclusion is true. If today is Sunday, then it is holiday. Today is Sunday.  Today is holiday.

15. Valid Argument?  Invalid argument!

16. Valid Argument?  Valid argument!

17. Knights and Knaves

18. Knights and Knaves Suppose you are visiting an island containing two types of people – knights and knaves. Two natives and address you as follow: : Both of us are knights. : is a knave. Knights always tell the truth. Knaves always lie. What are and ?

19. Knights and Knaves : Both of us are knights. : is a knave. If is a knave • is a knight • Both of them are knights • In particular, is a knight Contradiction! Therefore must be a knight • is a knave Knights always tell the truth. Knaves always lie.

20. Knights and Knaves Another two natives and approach you but only speaks. : Both of us are knaves. Knights always tell the truth. Knaves always lie. What are and ?

21. Knights and Knaves : Both of us are knaves. If is a knight • Both of them are knaves • In particular, is a knave Contradiction! Therefore must be a knave • One of them is a knight • is a knight Knights always tell the truth. Knaves always lie.

22. Knights and Knaves You then encounter natives and . : is a knave. : is a knave. Knights always tell the truth. Knaves always lie. How many knaves are there?

23. Knights and Knaves : is a knave. : is a knave. If is a knight • is a knave • is a knight • There is a knight and a knave Similarly, if is a knight • is a knave • is a knight • There is a knight and a knave Knights always tell the truth. Knaves always lie.

24. Knights and Knaves Finally, you meet a group of six natives, and , who speak to you as follow: : None of us is a knight. : At least three of us are knights. : At most three of us are knights. : Exactly five of us are knights. : Exactly two of us are knights. : Exactly one of us is a knight. Knights always tell the truth. Knaves always lie. Which are knights and which are knaves?

25. Knights and Knaves Suppose is a knight • None of them is a knight • is a knave Contradiction! So can only be a knave. : None of us is a knight. : At least three of us are knights. : At most three of us are knights. : Exactly five of us are knights. : Exactly two of us are knights. : Exactly one of us is a knight. Knights always tell the truth. Knaves always lie.

26. Knights and Knaves Suppose is a knight • Exactly five of them are knights • are knights (because at this stage we already know is a knave) • In particular, is a knight • Exactly one of them is a knight Contradiction! So can only be a knave. : None of us is a knight. : At least three of us are knights. : At most three of us are knights. : Exactly five of us are knights. : Exactly two of us are knights. : Exactly one of us is a knight. Knights always tell the truth. Knaves always lie. • What we know at this stage: • is a knave

27. Knights and Knaves Suppose is a knight • At least three of them can be knights • At least three people among are knights • Either or must be a knight (by pigeon-hole principle) • Exactly one(or two) of them is a knight Contradiction! So can only be a knave. : None of us is a knight. : At least three of us are knights. : At most three of us are knights. : Exactly five of us are knights. : Exactly two of us are knights. : Exactly one of us is a knight. Knights always tell the truth. Knaves always lie. • What we know at this stage: • is a knave • is a knave

28. Knights and Knaves At this stage we already identify three people as knaves • There can be at most three of them are knights • is a knight : None of us is a knight. : At least three of us are knights. : At most three of us are knights. : Exactly five of us are knights. : Exactly two of us are knights. : Exactly one of us is a knight. Knights always tell the truth. Knaves always lie. • What we know at this stage: • is a knave • is a knave • is a knave

29. Knights and Knaves Suppose is a knight • Exactly one of them is a knight Contradiction! (because both are knights in this case) Therefore must be a knave. : None of us is a knight. : At least three of us are knights. : At most three of us are knights. : Exactly five of us are knights. : Exactly two of us are knights. : Exactly one of us is a knight. Knights always tell the truth. Knaves always lie. • What we know at this stage: • is a knave • is a knave • is a knight • is a knave

30. Knights and Knaves Suppose is a knave • is the only knight • There is exactly one knight • is a knight Contradiction! Therefore must be a knight. : None of us is a knight. : At least three of us are knights. : At most three of us are knights. : Exactly five of us are knights. : Exactly two of us are knights. : Exactly one of us is a knight. Knights always tell the truth. Knaves always lie. • What we know at this stage: • is a knave • is a knave • is a knight • is a knave • is a knave

31. Knights and Knaves Therefore: • is a knave • is a knave • is a knight • is a knave • is a knight • is a knave : None of us is a knight. : At least three of us are knights. : At most three of us are knights. : Exactly five of us are knights. : Exactly two of us are knights. : Exactly one of us is a knight. Knights always tell the truth. Knaves always lie. • What we know at this stage: • is a knave • is a knave • is a knight • is a knave • is a knave

32. Summary • How to write logical formula from a truth table? • How to check whether two logical formulas are equivalent? • How to simplify logical formula using rules including (but not limited to) De Morgan’s law? • What are conditional statements and contrapositive? • How to check whether an argument is valid or not?

33. Thank You!