challenge problems

(advanced material) Your attempted solutions must be your own work: you may not work on these problems with anyone else. (Don't work on 2 or 3 yet: I may change them!)

  1. For the following argument: (i) Identify the premise and conclusion indicators; (ii) Identify the premises, conclusion, and sub-conclusions; (iii) Identify the component statements; (iv) Build the argument diagram.
  • Honesty is a virtue, and virtuous behaviour transcends the boundary between public and private life. Therefore, if one is a liar in his private life, he will also be a liar in his public life. For this reason, one should never vote for a political candidate who has been known to be deceptive in his personal life because one should never vote for someone who will lie in public office.
  • Your attempted solution is due in your tutorial in week 4 (January 23-27).
  1. Consider the two statements (1)  (A v B) → C and (2) (A → C) v (B → C). One of these two statements entails the other, but not the other way round. Determine which entails which by giving a valid proof of one from the other.
  • Full credit for a correct proof in 6-lines; partial credit for longer correct proofs. Your attempted solution is due in your tutorial in week 5 (January 30-February 3).
  1. (a) Prove the validity of the abstract argument:

     A & B, A v C  ∴  (A → B) &  (C → A)

    (Hint: the second premise is a distraction!)

  1. (b)—alternative to (a). Use the instructions on p. 122 of your textbook to analyze and diagram the following natural argument:
  • Suppose we do not reduce carbon emissions on the planet. Then the global temperature will rise by two degrees celsius. But in that case many natural ecosystems will be destroyed. The destruction of many natural ecosystems would result in a decrease in biodiversity and an increase in natural disasters. Therefore, if we don't reduce carbon emissions, there will be an increase in natural disasters.
  • Your attempted solution to one of these two problems is due in your tutorial in week 6 (February 6-10).
  1. Construct a truth table for Peirce’s Arrow Operator ↓ (see pp. 185, 187 for the definition). Find an expression in terms of this operator for p ↔ q.
  • Your attempted solution is due in your tutorial in week 7 (February 13-17).
  1. Determine, by the method of Truth Trees, whether the following sequent is valid or invalid:
  • [(A → B) ↔ ¬C] ⊢ [¬(A → B) ↔ C]
  • Full credit for a correct proof in a 6-line truth tree; partial credit for longer correct proofs. Due in tutorial in week 9 (November 9-13).
  • Your attempted solution is due in your tutorial in week 8 (February 27-March 3).
  1. Using predicate logic, demonstrate that the following argument is formally valid by treating the conclusion as the negation of an A-statement:
  • All PHILOSOPHERS are ABSENT-minded. Socrates is GREEK, although he is now DEAD. No one who is dead is absent-minded. Therefore it is false that all Greeks are philosophers.
  • Why is the argument not convincing? Your attempted solution is due in your tutorial in week 9 (March 6-10).
  1. Assuming every predicate occurs exactly twice in the argument, what conclusion can be validly inferred from the following premises?
  • (1) All planets that do not have liquid WATER lie OUTSIDE the asteroid belt. (2) No RINGED planets are in the habitable ZONE. (3) Only planets in the habitable ZONE have liquid WATER. (4) Some INHABITED planets are not OUTSIDE the asteroid belt. [UD: planets]
  • Demonstrate this using a Carroll diagram for 5 predicates, and verify by giving a proof of your conclusion from these premises using predicate logic.
  • Your attempted solution is due in your tutorial in week 10 (March 13-17).
  1. Using predicate logic, prove that the following argument is penevalid:
  • All HUMANS are GREAT apes. All GREAT apes are ANIMALS. All HUMANS have highly developed BRAINS. All those with highly developed BRAINS have COMPLEX language. There does not exist a being that it is either IMMORTAL or OMNISCIENT. All who are not omniscient have LIMITED knowledge. Therefore, some animals have complex language, but also have limited knowledge.
  • Your attempted solution is due in your tutorial in week 11 (March 20-24).

© Richard T. W. Arthur 2016