Mathematical Logic - 2015
Propositional Logic: exercises
Fausto Giunchiglia and Mattia Fumagalli (ref. Chiara Ghidini slides “PL Formalization” and Enzo Maltese sleides “PL Exercises”)
Expansion Rules of Propositional Tableau
2
Tableaux: exercise 1
Is the the following formula satisfiable?
((P  Q)  P)
(P  Q)
P
P
P
Q
X
3
Tableaux: Exercise 2
Is the following formula valid? (and satisifiable?)
((P  Q)  P)
(P  Q)
P
P
X
4
Q
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
(P  Q)
X
P
X
5
Q
X
R
X
Tableaux: Exercise 4
Is the following argumentation valid?
P  Q, P  Q ⊨ P
PQ
P  Q
P
P
P
X
6
Q
P
X
Q
X
Tableaux: Exercise 5
Check the model for the following formula:
(P  Q  P  Q)
(P  Q)
(P  Q)
P
P
Q
Q
X
I(P) = True; I(Q) = False
I(P) = False; I(Q) = True
7
P
Q
X
Tableaux: Exercise 6
Is the following argumentation valid?
P, (P  Q) ⊨ P ⟷ Q
P
PQ
(P ⟷ Q)
P
X
Q
P
Q
X
8
Q
P
X
Reduction in CNF
9
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)
10
SYNTAX :: MODELING A PROBLEM :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING :: SOME NICE PROBLEMS
Convert a formula in CNF
 (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
11