I.   Derivations for the following tautologies:

           A.   P v ~P

               1.       P                       Assumption
               2.       P                       1, R
               3.   P -> P                      1-2, CP
               4.   ~P v P                      3, Impl.
               5.   P v ~P                      4, Comm.

           B.   ~(P & ~P)

               1.       P                       Assumption
               2.       P                       1, R
               3.   P -> P                      1-2, CP
               4.   ~P v P                      3, Impl.
               5.   ~P v ~~P                    4, DN
               6.   ~(P & ~P)                   5, DM

           C.   P <-> (~P -> P) (Note: pay attention to this one. It will
                form the basis for the next lesson.)

               1.       P                       Assumption
               2.           ~P                  Assumption
               3.           P                   1, R
               4.       ~P -> P                 2-3, CP
               5.   P -> (~P -> P)              1-4, CP
               6.       ~P -> P                 Assumption
               7.       ~~P v P                 6, Impl.
               8.       P v P                   7, DN
               9.       P                       8, Taut.
              10.   (~P -> P) -> P              6-9, CP
              11.   [P -> (~P -> P)] & [(~P -> P) -> P]
                                                5, 10, Conj.
              12.   P <-> (~P -> P)             11, Equiv.

              or

               1.       P                       Assumption
               2.       P v P                   1, Taut.
               3.       ~~P v P                 2, DN
               4.       ~P -> P                 3, Impl.
               5.   P -> (~P -> P)              1-4, CP
               6.       ~P -> P                 Assumption
               7.       ~~P v P                 6, Impl.
               8.       P v P                   7, DN
               9.       P                       8, Taut.
              10.   (~P -> P) -> P              6-9, CP
              11.   [P -> (~P -> P)] & [(~P -> P) -> P]
                                                5, 10, Conj.
              12.   P <-> (~P -> P)             11, Equiv.

      II.   Without using Repetition (R), derive P -> P.

               1.       P                       Assumption
               2.       P & P                   1, Taut.
               3.       P                       2, Simp.
               4.   P -> P                      1-3, CP

               or

               1.       P                       Assumption
               2.       P v P                   1, Taut.
               3.       P                       2, Taut.
               4.   P -> P                      1-3, CP

     III.   Without using Material Implication (Impl.), derive (~P v Q) ->
            (P -> Q). (Hint: Use one conditional proof inside another.)

               1.       ~P v Q                  Assumption
               2.           P                   Assumption
               3.           ~~P                 2, DN
               4.           Q                   1, 3, DS
               5.       P -> Q                  2-4, CP
               6.   (~P v Q) -> (P -> Q)        1-5, CP

      IV.   Without using Exportation (Exp.), derive [(P & Q) -> R] -> [P
            -> (Q -> R)].

               1.       (P & Q) -> R            Assumption
               2.           P                   Assumption
               3.               Q               Assumption
               4.               P & Q           2, 3, Conj.
               5.               R               1, 4, MP
               6.           Q -> R              3-5, CP
               7.       P -> (Q -> R)           2-6, CP
               8.   [(P & Q) -> R] -> [P -> (Q -> R)]
                                                1-7, CP

       V.   Without using Hypothetical Syllogism (HS), derive [(P -> Q) &
            (Q -> R)] -> (P -> R).

               1.       (P -> Q) & (Q -> R)     Assumption
               2.           P                   Assumption
               3.           P -> Q              1, Simp.
               4.           Q                   2, 3, MP
               5.           (Q -> R) & (P -> Q) 1, Comm.
               6.           Q -> R              5, Simp.
               7.           R                   4, 6, MP
               8.       P -> R                  2-7, CP
               9.   [(P -> Q) & (Q -> R)] -> (P -> R)
                                                1-8, CP