80-211                                                                                                             Spring 2003

Note:  These are the answers to some of the questions in the book and some answers to problems not in the book, most of these were not problems you had to do for homework, but are similar.

Problems from pages 27-28:

1-a                   P |- Qà (P & Q)

1          (1)        P                                              A

2          (2)        Q                                             A

1,2       (3)        P & Q                                      1,2 &-I

1          (4)        Qà (P & Q)                           2,3 CP

1-f                    (Pà R) & (Qà R) |- (P v Q)à R

1          (1)        (Pà R) & (Qà R)                  A

2          (2)        P v Q                                       A

3          (3)        P                                              A

1          (4)        Pà R                                      1, &-E

1,3       (5)        R                                              3,4 MPP

6          (6)        Q                                             A

1          (7)        Qà R                                      1, &-E

1, 6      (8)        R                                              6,7 MPP

1,2       (9)        R                                              2, 3, 5, 6, 8 v-E

1          (10)      (P v Q)à R                             2, 9 CP

1-g                   Pà Q, Rà S |- (P & R) à (Q & S)

1          (1)        Pà Q                                      A

2          (2)        Rà S                                      A

3          (3)        P & R                                      A

3          (4)        P                                              3, &-E

1, 3      (5)        Q                                             1, 4 MPP

3          (6)        R                                              3, &-E

2, 3      (7)        S                                              2, 6 MPP

1, 2, 3 (8)        Q & S                                      5, 7 &-I

1, 2      (9)        (P & R) à (Q & S)                 3, 8 &-I

1-i                    Pà (Q & R) |- (Pà Q) & (Pà R)

1          (1)        Pà (Q & R)                            A

2          (2)        P                                              A

1, 2      (3)        Q & R                                      1,2 MPP

1, 2      (4)        Q                                             3, &-E

1          (5)        Pà Q                                      2, 4 CP

1, 2      (6)        R                                              3, &-E

1          (7)        Pà R                                      2, 6 CP

1          (8)        (Pà Q) & (Pà R)                  5, 7 &-I

Problems on page 33:

1-c                   P « Q |- ~ P « ~Q

1          (1)        P « Q                                     A

1          (2)        (Pà Q) & (Qà P)                  1, def «

1          (3)        Pà Q                                      2, &-E

1          (4)        Qà P                                      2, &-E

5          (5)        ~Q                                           A

1, 5      (6)        ~P                                            3, 5 MTT

1          (7)        ~Qà ~P                                 5,6 CP

8          (8)        ~P                                            A

1, 8      (9)        ~Q                                           4, 8 MTT

1          (10)      ~Pà ~Q                                 8, 9 CP

1          (11)      (~Pà ~Q) & (~Qà ~P)         7, 10 &-I

1          (12)      ~P « ~Q                                11, def «

1-e                   (P v Q) « P |- Qà P

1          (1)        (P v Q) « P                                        A

1          (2)        ((P v Q)à P) & (Pà (P v Q))             1, def «

3          (3)        Q                                                         A

3          (4)        P v Q                                                   3, v-I

1          (5)        (P v Q)à P                                         2, &-E

1, 3      (6)        P                                                          4,5 MPP

1          (7)        Qà P                                                  3, 6 CP

Problems on page 41:

1-c                   P & (Q v R) -||- (P & Q) v (P & R)

Part a:              P & (Q v R) |- (P & Q) v (P & R)

1          (1)        P & (Q v R)                             A

1          (2)        P                                              1, &-E

1          (3)        Q v R                                       1, &-E

4          (4)        Q                                             A

1, 4      (5)        P & Q                                      2, 4 &-I

1, 4      (6)        (P & Q) v (P & R)                   5, v-I

7          (7)        R                                              A

1, 7      (8)        P & R                                      2, 7 &-I

1, 7      (9)        (P & Q) v (P & R)                   8, v-I

1          (10)      (P & Q) v (P & R)                   3, 4, 6, 7, 9 v-E

Part b:              (P & Q) v (P & R) |- P & (Q v R)

1          (1)        (P & Q) v (P & R)                   A

2          (2)        P & Q                                      A

2          (3)        P                                              2, &-E

2          (4)        Q                                             2, &-E

2          (5)        Q v R                                       4, v-I

2          (6)        P & (Q v R)                             3, 5 &-I

7          (7)        P & R                                      A

7          (8)        P                                              7, &-E

7          (9)        R                                              7, &-E

7          (10)      Q v R                                       9, v-I

7          (11)      P & (Q v R)                             8, 10 &-I

1          (12)      P & (Q v R)                             1, 2, 6, 7, 11, v-E

1-f                    ~ (P v Q) -||- ~P & ~Q

Part a:              ~ (P v Q) |- ~P & ~Q

1          (1)        ~ (P v Q)                                 A

2          (2)        P                                              A

2          (3)        P v Q                                       2, v-I

1, 2      (4)        ~ (P v Q) & (P v Q)                 1, 3 &-I

1          (5)        ~P                                            2, 4 RAA

6          (6)        Q                                             A

6          (7)        P v Q                                       6, v-I

1, 6      (8)        ~ (P v Q) & (P v Q)                 1, 7 &-I

1          (9)        ~Q                                           6, 8 RAA

1          (10)      ~P & ~Q                                 5, 9 &-I

Part b:              ~P & ~Q |- ~ (P v Q)

1          (1)        ~P & ~Q                                 A

2          (2)        P v Q                                       A

3          (3)        P                                              A

1          (4)        ~P                                            1, &-E

1, 3      (5)        P & ~P                                                3, 4 &-I

3          (6)        ~ (~P & ~Q)                            1, 5 RAA

7          (7)        Q                                             A

1          (8)        ~Q                                           1, &-E

1, 7      (9)        Q & ~Q                                   7, 8 &-I

7          (10)      ~ (~P & ~Q)                            1, 9 RAA

1, 2      (11)      ~ (~P & ~Q)                            2, 3, 6, 7, 10, v-E

1, 2      (12)      ~ (~P & ~Q) & (~P & ~Q)     1, 11 &-I

1          (13)      ~ (P v Q)

Note: The following 4 proofs use derived rules.

|- (P®Q) v (Q®P)

(1)        P v ~P                                      TI 44

2          (2)        P                                              A

2          (3)        Q®P                                       2 SI 50

2          (4)        (P®Q) v (Q®P)                      3, v-I

5          (5)        ~P                                            A

5          (6)        P®Q                                       5 SI 51

5          (7)        (P®Q) v (Q®P)                                  6, v-I

(8)        (P®Q) v (Q®P)                      1, 2, 4, 5, 7 v-E

~(P v Q) v ~(~P v ~Q) |- P«Q

1          (1)        ~(P v Q) v ~(~P v ~Q)              A

2          (2)        ~(P v Q)                                   A

2          (3)        ~P & ~ Q                                 2 SI DeMorgan

2          (4)        ~P                                            3, &-E

2          (5)        ~Q                                           3, &-E

2          (6)        P®Q                                       4 SI 51

2          (7)        Q®P                                       5 SI(S) 51

2          (8)        (P®Q)  & (Q®P)                    6, 7 &-I

9          (9)        ~(~P v ~Q)                               A

9          (10)      P & Q                                      9 SI DeMorgan (prob 1-h)

9          (11)      P                                              10 &-E

9          (12)      Q                                             10 &-E

9          (13)      Q®P                                       11 SI-50

9          (14)      P®Q                                       12 SI(S)-50

9          (15)      (P®Q)  & (Q®P)                    13, 14 &-I

1          (16)      (P®Q)  & (Q®P)                    1, 2, 8, 9, 15 v-E

1          (17)      P«Q                                       16 « Def

P v Q, (Q®R) & (~P v S), (Q & R)®T |- T v S

1          (1)        P v Q                                       A

2          (2)        (Q®R) & (~P v S)                   A

3          (3)        (Q & R)®T                             A

4          (4)        P                                              A

2          (5)        ~P v S                                      2, &-E

4          (6)        ~~P                                          4, DN

2, 4       (7)        S                                              5, 6 MTP (aka disjunctive syllogism)

2, 4       (8)        T v S                                        7, v-I

9          (9)        Q                                             A

2          (10)      Q®R                                       2, &-E

2, 9       (11)      R                                              9, 10 MPP

2, 9       (12)      Q & R                                      9, 11 &-I

2, 3, 9   (13)      T                                              3, 12 MPP

2, 3, 9   (14)      T v S                                        13, v-I

1, 2, 3   (15)      T v S                                        1, 4, 8, 9, 14 v-E