Solutions to Starred Exercises

Chapter 11a: One Sided Truth Trees




1.(b)  -(C & D)        open
       -(C v D)
           ---
           -C
           -D
          /  \
        -C   -D

2.(b)  W -> D
           -W
        --D
          ---
           D
          / \
        -W   D

3.(b)     B -> W
         -B -> A
         -(W v A)
             ---
              -W
              -A
             /  \
           -B    W
          /  \   *
        --B   A
         *    *

5.(b)             R v -R           valid
             R -> (F & S)
            -R -> -(F v S)
              --(F & -S)
                   ---
                F & -S
                    F
                    -S
               /          \
             -R       F & S
          /      \        F
       --R  -(F v S)    S
        *        -F       *
                 -S
                 *
Note that premise one is superfluous. Section 12.1 sheds light on this matter.
7.         -B        valid
       B v M
           -M
          ---
          /  \
         B    M
         *    *

11.              -W                 invalid
     --{[(W & K) & P] -> E}
                 ---
        [(W & K) & P] -> E
              /        \
    -[(W & K) & P]    E
         /       \
   -(W & K)    -P
       /  \
     -W   -K
The double conjunction in the conclusion may be grouped differently.

Tree construction can stop after the first right branch is complete.

12.(b)       P <-> W          valid
             -(P & W)
            --(P v W)
                 ---
              P v W
            /          \
           P           -P
           W           -W
          / \        /    \
        -P  -W     -P      -W
         *   *    /  \    /  \
                 P    W  P    W
                 *    *  *    *

Chapter 11a: One Sided Truth Trees