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Introduction,Propositional Logic First Order Logic and Model Logic.
Typology: Exercises
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Anno Accademico 2013-
We thank Annapaola Marconi for her work in previous editions of this booklet.
Mathematics is the only instructional material that can be presented in an entirely undogmatic way. The Mathematical Intelligencer, v. 5, no. 2, 1983
The purpose of this booklet is to give you a number of exercises on proposi- tional, first order and modal logics to complement the topics and exercises covered during the lectures of the course on mathematical logic. The mate- rial presented here is not a direct component of the course but is offered to you as an incentive and a support to understand and master the concepts and exercises presented during the course.
Symbol Difficulty Trivial Easy Medium Difficult Very difficult
Propositional Logic
Exercise 2.2. - Let’s consider the interpretation v where v(p) = F, v(q) = T, v(r) = T. Does v satisfy the following propositional formulas?
Solution.
v satisfies 1., 3. and 4. v doesn’t satisfy 2.
2.2 Truth Tables
Exercise 2.3. - Compute the truth table of (F ∨ G) ∧ ¬(F ∧ G).
Solution.
F G F ∨ G F ∧ G ¬(F ∧ G) (F ∨ G) ∧ ¬(F ∧ G) T T T T F F T F T F T T F T T F T T F F F F T F
Exercise 2.4. - Use the truth tables method to determine whether (p → q) ∨ (p → ¬q) is valid.
Solution.
p q p → q ¬q p → ¬q (p → q) ∨ (p → ¬q) T T T F F T T F F T T T F T T F T T F F T T T T The formula is valid since it is satisfied by every interpretation.
Exercise 2.5. - Use the truth tables method to determine whether (¬p∨q)∧(q → ¬r∧¬p)∧(p∨r) (denoted with ϕ) is satisfiable.
Solution.
p q r ¬p ∨ q ¬r ∧ ¬p q → ¬r ∧ ¬p (p ∨ r) ϕ T T T T F F T F T T F T F F T F T F T F F T T F T F F F F T T F F T T T F F T F F T F T T T F F F F T T F T T T F F F T T T F F There exists an interpretation satisfying ϕ, thus ϕ is satisfiable.
2.2 Truth Tables
Exercise 2.9. Use the truth table method to verify whether the following formulas are valid, satisfiable or unsatisfiable:
Propositional Logic
Exercise 2.10. Use the truth table method to verify whether the following logical consequences and equivalences are correct:
2.3 Propositional Formalization
Exercise 2.11. - Let’s consider a propositional language where
Formalize the following sentences:
Propositional Logic
Exercise 2.13. - Let A =“Aldo is Italian” and B =“Bob is English”. Formalize the following sentences:
Solution.
Exercise 2.14. Angelo, Bruno and Carlo are three students that took the Logic exam. Let’s consider a propositional language where
Formalize the following sentences:
2.3 Propositional Formalization
Exercise 2.15. - Let’s consider a propositional langiage where
Formalize the following sentences:
2.3 Propositional Formalization
Exercise 2.17. - Socrate says:
“If I’m guilty, I must be punished; I’m guilty. Thus I must be punished.”
Is the argument logically correct?
Solution. The argument is logically correct: if p means “I’m guilty” and q means “I must be punished”, then: (p → q) ∧ p |= q (modus ponens)
Exercise 2.18. - Socrate says:
“If I’m guilty, I must be punished; I’m not guilty. Thus I must not be punished.”
Is the argument logically correct?
Propositional Logic
Solution. The argument is not logically correct: (p → q) ∧ ¬p 2 ¬q
Exercise 2.19. Socrate says:
“If I’m guilty, I must be punished; I must not be punished. Thus I’m not guilty.”
Is the argument logically correct?
Exercise 2.20. Socrate says:
“If I’m guilty, I must be punished; I must be punished. Thus I’m guilty.”
Is the argument logically correct?
Exercise 2.21. Formalize the following arguments and verify whether they are correct: