1. Lecture Notes 8
CS1502

2. Example Proof
A  (B  C)
(A  B)  (A  C)

4. Valid Argument Valid Argument P1 P2 … Pn Q
Q is a tautological (logical) consequence of P1, P2, …, Pn
(P1  P2  …  Pn) Q is a tautology (logical necessity). NEW IDEA
Valid Argument

5. Example Show P is a tautological consequence of (P  Q).
Methods of attack:
Boole
Show P is a tautological consequence of (P  Q).
Show (P  Q)  P is a tautology.
Fitch
Show (P  Q) is a valid argument P

6. Tautological Consequence

7. Tautology

8. Using Fitch

9. Example Show P is not a tautological consequence of (P  Q).
Method of attack:
Boole
Show P is not a tautological consequence of (P  Q).
Show (P  Q)  P is not a tautology.
Build a world
Show (P  Q) is an invalid argument P

10. Not a Tautological Consequence

11. Not a Tautology

12. Build a World
Let P be assigned true and Q false. (P  Q) is true while P is false.
conclusion
premises

13. Example
Show the following argument is valid. Cube(b) (Cube(c)  Cube(b)) Cube(c)

14. Logical Consequence

15. Logical Necessity
Every non-spurious row is true! In fact, every row is true, so a Tautology!!

16. Fitch

17. Non-consequence
Show the following argument is invalid. Cube(a)  Cube(b) (Cube(c)  Cube(b)) Cube(c)

18. Counterexample

19. Inference Patterns
Modus Ponens P  Q P Q

20. Tautological Consequence

21. Tautology

22.  Elimination
P  Q … P … Q
 Elim

23.  Introduction
P … Q P  Q
 Intro

24.  Elimination
P  Q … P … Q
 Elim

25.  Introduction
P … Q Q … P P  Q
 Intro

26. Inference Patterns
Modus Tollens P  Q Q P

27. Modus Tollens