1. Propositional Logic
In this lecture we solve a standard propositional problem of circuit equivalence and a puzzle about common knowledge.
Problems and puzzles crystallize larger issues,
eg., consistency and co-ordination 1

2. Problem: Logic Circuits
not i2 or o1
Are these two circuits equivalent ? i1
and
not o1 i2
not 2

3. Problem: Common Knowledge
A husband unfaithful to his wife will be shot by her on the day she finds out.
Day 1: Three wives are told that at least one of their husbands is unfaithful. None knows whether her husband is unfaithful but each knows about the others.
Day 2: The three wives meet again and learn that no husband has been shot.
Day 3: The three wives meet again and learn that no husband has been shot. That night there is shooting.
How many husbands were unfaithful ? 3

4. Truth Tables P Q not(P) P and Q P or Q P  Q P  Q t t f t t t t t f f4

5. Inconsistencies
P and not(P)
P and not(P) t t f f f t f f 5

6. Contingencies (P or Q)  P (P or Q)  P (P or Q)  P (P or Q)  P t tf t t t t t
(P or Q)  P
(P or Q)  P f t f f f f t f f t 6

7. Tautologies P and (P  Q)  Q P and (P  Q)  Q P and (P  Q)  Qf f t f t f t t
P and (P  Q)  Q
P and (P  Q)  Q f f t t f f f f t t f f t t 7

8. Clicker Question 1 Which of the following is a contingency? (a) P  P
(b) Q or (not(Q)
(c) P  Q
(d) (P and Q) and (not(Q)) 8

9. Solution: Logic Circuits (1)
not(i1) or i2
not(i1 and not(i2))
equivalent ? i1
not i2 or o1
equivalent ? i1
and
not o1 i2
not 9

10. Solution: Logic Circuits (2)
not(i1)
not(i2)
not(i1) or i2
i1 and not(i2)
not(i1 and not(i2)) t t f f t f t t f f t f t f f t t f t f t f f t t t f t 10

11. Solution: Logic Circuits (2)
not(i1) or i2
equivalent
not(i1 and not(i2))
i2 equivalent not(not(i2))
not(i1) or not(not(i2))
equivalent
not(i1 and not(i2))
not(i2) in both sides (call it B)
not(i1) or not(B)
equivalent
not(i1 and B)
Here we are performing a proof using equivalences 11

12. Problem: Common Knowledge
A husband unfaithful to his wife will be shot by her on the day she finds out.
Day 1: Three wives are told that at least one of their husbands is unfaithful. None knows whether her husband is unfaithful but each knows about the others.
Day 2: The three wives meet again and learn that no husband has been shot.
Day 3: The three wives meet again and learn that no husband has been shot. That night there is shooting.
How many husbands were unfaithful ? 12

13. Solution: Common Knowledge
cheat1
cheat2
cheat3 t t t
BANG! t t f
X – day 3 t f t
X – day 3 t f f
X – day 2 f t t
X – day 3 f t f
X – day 2 f f t
X – day 2 f f f
X – day 1
(See notes for full truth table) 13

14. Clicker Question 2
Same setting as before, but now 5 wives are told about infidelity on Day 1. And there is shooting at the end of Day 1! How many straying husbands?
(a) 1
(b) 2
(c) 4
(d) 5 14

15. Real Common Knowledge Process Process Process Process Process Process15

16. Things to Practice
Writing English statements in propositional logic (exercises in tutorials).
Being able to work out the truth of any propositional expression using truth tables (exercises in tutorials).
Understanding why some expressions are equivalent (see equivalences in notes). 16

17. Recreational Reading Raymond Smullyan,
“What is the Name of This Book? The Riddle of Dracula and Other Logical Puzzles”
&
“Alice in Puzzle-Land” 17