CSCI 2400/SENG 1010
Solutions for practice questions set 1016

1. Define suitable predicates and convert the following into first-order logic. What are the axioms? What needs to be proved? Give a logical proof of the following, showing the rule of inference that is used at each step. "All men are mortal. Socrates is a man. Therefore, socrates is mortal."

   Predicates:

   mortal(x): x is mortal.

   man(x): x is a man.

   Axioms:

   1. ∀x(man(x) → mortal(x))
   2. man(Socrates)

   Proof:

   1. man(Socrates) → mortal(Socrates) [Universal instantiation of (1)]
   2. mortal(Socrates) [Modus ponens of (2) and (A)]
   
   
   
2. For each of the following arguments, say which rule(s) of inference is (are) used.
   
   
   1. Linda, a student in this class, owns a red convertible. Everyone who owns a red convertible has gotten at least one speeding ticket. Therefore, someone in this class has gotten a speeding ticket.

      Predicates:

      student(x): x is a student in this class.

      convertible(x): x owns a red convertible.

      ticket(x): x has gotten a speeding ticket.

      Axioms:

      1. student(Linda)
      2. convertible(Linda)
      3. ∀x(convertible(x) → ticket(x))

      Proof:

      1. convertible(Linda) → ticket(Linda) [Universal instantiation of (3)]
      2. ticket(Linda) [Modus ponens of (A)]
      3. student(Linda) ∧ ticket(Linda) [Conjunction of (1) and (B)]
      4. ∃x(student(x) ∧ ticket(x)) [Existential generalization of (C)]
      
      
      
   2. There is someone in this class who has been to Paris. Everyone who goes to Paris visits the Louvre. Therefore, someone in this class has visited the Louvre.

      Predicates:

      student(x): x is in this class.

      paris(x): x has been to Paris.

      louvre(x): x has visited the louvre.

      Axioms:

      1. ∃x(student(x) ∧ paris(x))
      2. ∀x(paris(x) → louvre(x))

      Proof:

      1. student(c) ∧ paris(c) [Existential instantiation of (1)]
      2. student(c) [Simplification of (A)]
      3. paris(c) [Simplification of (A)]
      4. paris(c) → louvre(c) [Universal instantiation of (2)]
      5. louvre(c) [Modus ponens of (C) and (D)]
      6. student(c) ∧ louvre(c) [Conjunction of (B) and (E)]
      7. ∃x(student(x) ∧ louvre(x)) [Existential generalization of (F)]
      
      
      
3. For each of the following arguments, say whether the argument is valid.
   
   
   1. Everyone enrolled in the university has lived in a dormatory. Mia has never lived in a dormatory. Therefore, Mia is not enrolled in the university.

      Valid.

      
      
      
   2. A convertible car is fun to drive. Isaac's car is not a convertible. Therefore, Isaac's car is not fun to drive.

      Not valid.

      
      
      
   3. Quincy likes all action movies. Quincy likes the movie Eight Men Out. Therefore, Eight Men Out is an action movie.

      Not valid.

      
      
      
   4. All lobstermen set at least a dozen traps. Hamilton is a lobsterman. Therefore, Hamilton sets at least a dozen traps.

      Valid.