1. Propositional Logic: exercises
Mathematical Logic
Propositional Logic: exercises
Fausto Giunchiglia and Mattia Fumagalli (ref. Chiara Ghidini slides “PL Formalization” and Enzo Maltese sleides “PL Exercises”)

2. Expansion Rules of Propositional Tableau
(P ⟷ Q) = ((P  Q)  (Q  P))

3. Tableaux: exercise 1
Is the the following formula satisfiable? ((P  Q)  P)
(P  Q)
P P P Q X

4. Tableaux: Exercise 2
Is the following formula valid? (and satisifiable?) ((P  Q)  P)
(P  Q) P P Q X

5. Tableaux: Exercise 3
Is the following formula satisifable? ((P  Q)  (P  Q  R)  (P  R))
(P  Q)  (P  Q  R)
(P  R)
(P  Q)
(P  Q  R) P R P Q X
(P  Q) R X P Q X X

6. Tableaux: Exercise 4 Is the following argumentation valid?
P  Q, P  Q ⊨ P
P  Q
P  Q
P P
P Q X P Q X X

7. Tableaux: Exercise 5
Check the model for the following formula: (P  Q  P  Q)
(P  Q)
(P  Q) P Q P Q P Q X X
I(P) = True; I(Q) = False
I(P) = False; I(Q) = True

8. Tableaux: Exercise 6 Is the following argumentation valid?
P, (P  Q) ⊨ P ⟷ Q P
P  Q
(P ⟷ Q)
P Q X P Q Q P X X

9. Reduction in CNF

10. Convert a formula in CNF
¬(¬p  q)  (r  ¬s)
¬(¬p  q)  (¬r  ¬s)
(¬¬p  ¬q)  (¬r  ¬s)
(p  ¬q)  (¬r  ¬s) NNF
(p  ¬r  ¬s)  (¬q  ¬r  ¬s)
(C  ¬A)  ¬(B  ¬A)
(¬C  ¬A)  ¬(B  ¬A)
(¬C  ¬A)  (¬B  ¬¬A)
(¬C  ¬A)  (¬B  ¬A)

11. Convert a formula in CNF
SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS
(A  C)  (C  ¬A)
¬(A  C)  (C  ¬A)
¬(A  C)  (¬C  ¬A)
¬A  ¬C  ¬C  ¬A
¬A  ¬ C
((A → B) → A) → A
(( A  B)  A)  A
((A  B)  A)  A
((A  A)  (A  B))  A
A  (A  B)  A
(A  B)
A  B