Introduction to Logic

 Use all the rules of inference (eight implication rules and ten replacement rules) to complete the proofs. Provide the justification for each step that you derive.

[18]     1. ∼ (P ⋅ Q)

            2. (P ⋅ Q) ν (R ⋅ S)                         / Q ν S

[20]    1. T ν S

           2.  ∼ T

           3. (S ν S) ⊃ (∼ P ν R)                   / ∼ R ⊃ ∼ P

[22]      1. (∼ P ν Q) ⊃ R

             2. (S ν R) ⊃ P

             3. P ⊃ Q                              / Q

[24]    1. ∼ Q

            2. R ⊃ Q

             3. ∼ S ⊃ M

             4. R ν (S ⊃ Q)                               / M ν K

[28]     1. P ⊃ (Q ν R)  

            2. (S ν T) ⊃ R

            3. ∼ Q ⋅ ∼ R                              / ∼ P ⋅ ∼ (S ν T)

[32]     1. ∼ P ⊃ (Q ν R)

            2. (S ν Q) ⊃ R

            3. ∼ R                                      / P

[34]     1. C ⊃ F

             2. A ⊃ B

             3. ∼ F ⋅ A

             4. ∼ C ⊃ (B ⊃ D )                               / B ⋅ D

[38]      1. P ⊃ (R ν S )

              2. ∼ [ (∼ P ν ∼ Q) ν (R ν ∼ L) ]                           / S