1. CS 1502 Formal Methods in Computer Science
Lecture Notes 11

2. Example Infer A  C from A  B and B  C.
A B B  C A  C This argument is known as the Transitivity of the Biconditional.

3. Using Resolution A  B is (A  B)  (B  A)
B  C is (B  C)  (C  B)
(A  C) is (A  C )  (A  C)
{A, B} {B,A}
{B, C} {C, B}
{A, C} {A, C}
Resolution Proof: In Lecture

4. Using Fitch

5. Law of Excluded Middle
P  P
A Tautology

6. Exercise 6.33 (in the pdf solution)

7. Exercise 6.33 (shorter version)

8. Using DeMorgan’s, with Taut Con doing the work for you

9. Using DeMorgan’s, but
we do the work…(turns
out to be redundant)

10. Law of Excluded Middle P  P Use with V-Elim in Proofs!
To introduce it:
Use Taut Con, if the rules allow
Otherwise, insert proof 6.33

11. Exercise 8.53 (on LPL Web site)
Note:
Fitch
lines
after
line 4
And 11
were
eaten by
adobe.
They
should be
there.

12. Example
Prove this argument is valid from no premises (P  Q) (P  Q)
Logical truth

13. Example

14. Prove this argument
Horned(unicorn)  (Elusive(unicorn)  Dangerous(unicorn)) (Elusive(unicorn)  Mythical(unicorn))  Rare(unicorn) Mammal(unicorn)  Rare(unicorn) Horned(unicorn)   Mammal(unicorn)
Proof is on the next slide; Go through it for practice on your own.

15. 1. Horned(unicorn)  (Elusive(unicorn)  Dangerous(unicorn) 2
1. Horned(unicorn)  (Elusive(unicorn)  Dangerous(unicorn) (Elusive(unicorn)  Mythical(unicorn))  Rare(unicorn) Mammal(unicorn)  Rare(unicorn) Horned(unicorn) Elusive(unicorn)  Dangerous(unicorn)  Elim 1, Elusive(unicorn)  Elim Elusive(unicorn)  Mythical(unicorn)  Intro Rare(unicorn)  Elim 2, Mammal(unicorn) Rare(unicorn)  Elim 3,   Intro 8,  Mammal(unicorn)  Intro Horned(unicorn)   Mammal(unicorn) Intro 4-12

16. Informal Proof Example
Prove there exists irrational numbers b and c such that bc is rational. Proof: Consider  = 22. This number is either rational or irrational. If  is rational we are finished since b = c = 2 . Assume  is irrational. Consider   2 = 2. Again we are finished since b =  and c = 2.

17. English Translation EGs P  Q
If P then Q (If you are human then you are a mammal)
P implies Q (Being a human implies being a mammal)
If P, Q (If you are human, you are a mammal)
P only if Q (You’ll live a long time only if you eat veggies)
P is sufficient for Q (Knowing you’re living a long time is sufficient to know you eat veggies)
Q is necessary for P (Eating veggies is necessary to live a long time)
Q if P (You are a mammal if you are human)

18. P  Q Home(max)  Library(claire) Large(b)  Cube(b)
If not P then Q If Max is not home, then Claire is at the library
If b is not large, then it is a cube
Unless P, Q Unless Max is at home, Claire is at the library
Unless b is large, b is a cube
Q, unless P
Claire is at the library unless Max is home
B is a cube unless b is large
Why not  for last two? (section 7.3 and lecture)

19. Lecture: look at related questions on Assignment 3, in 7.12 and 7.15