Homework 4 Solutions

1. Perform truth-table tests on the following formulas, and state in each case whether the formula is tautologous, contingent, or inconsistent:
   1. P -> P

      (2 points) Tautologous:

      P | P -> P
      ----------
      T | T T  T
      F | F T  F

   2. (P v -Q) & -(-P -> -Q)

      (2 points) Inconsistent:

      P Q | (P v - Q) & -(- P -> - Q)
      -------------------------------
      T T |  T T F T  F F F T T  F T
      T F |  T T T F  F F F T T  T F
      F T |  F F F T  F T T F F  F T
      F F |  F T T F  F F T F T  T F

   3. (P & Q -> R) -> (P -> R) & (Q -> R)

      (2 points) Contingent:

      P Q R | (P & Q -> R) -> (P -> R) & (Q -> R)
      -------------------------------------------
      T T T |  T T T T  T  T   T T  T  T  T T  T
      T T F |  T T T F  F  T   T F  F  F  T F  F
      T F T |  T F F T  T  T   T T  T  T  F T  T
      T F F |  T F F T  F  F   T F  F  F  F T  F
      F T T |  F F T T  T  T   F T  T  T  T T  T
      F T F |  F F T T  F  F   F T  F  F  T F  F
      F F T |  F F F T  T  T   F T  T  T  F T  T
      F F F |  F F F T  F  T   F T  F  T  F T  F

2. Show the invalidity of the following patterns of argument by finding an assignment of truth-values to the variables such that the assumptions are all true and the conclusion is false:
   1. P & Q -> R |- P -> R

      (2 points) For the conclusion to be false, P must be true and R must be false. If R is false, then the only way the assumption P & Q -> R can be true is if the antecedent P & Q is also false; since P is true, this must mean that Q is false:

      P Q R | P & Q -> R | P -> R
      ---------------------------
      T F F | T F F T  F | T F  F

   2. P -> Q, P -> R |- Q -> R

      (2 points) As above, Q must be true and R false; if the assumptions are all to be true, then this forces P to be false as well:

      P Q R | P -> Q | P -> R | Q -> R
      --------------------------------
      F T F | F T  T | F T  F | T F  F

   3. P -> (Q -> R), Q, -R |- P

      (2 points) The conclusion is just P, so that has to be false; the assumptions will all be true if Q is true and R is false:

      P Q R | P -> (Q -> R) | Q | - R | P
      -----------------------------------
      F T F | F  T  T F  F  | T | T F | F