Relationships in Truth Statements

Exercise

Fill in the blanks with the valid form of inference that is being used and the lines the inference follows from. Note: the conclusion is written to the right of the last premise, following the "/∴" symbols.

Example 1:

1. M ⊃ ~N
2. M
3. H ⊃ N /∴ ~H
4. ~N Modus ponens, 1, 2
5. ~H Modus tollens, 3, 4
   
   

Example 2:

1. A v B
2. C ⊃ D
3. A ⊃ C
4. ~D /∴ B
5. A ⊃ D Hypothetical syllogism, 3, 2
6. ~A Modus tollens, 5, 4
7. B Disjunctive syllogism, 1, 6

1. 
   
   1. A ⋅ C /∴ (A v E) ⋅ (C v D)
   2. A _________________
   3. C _________________
   4. A v E _________________
   5. C v D _________________
   6. (A v E) ⋅ (C v D) ______________
2. 
   
   1. A ⊃ (B ⊃ D)
   2. ~D
   3. D v A /∴ ~B
   4. A _________________
   5. B ⊃ D _________________
   6. ~B _________________
3. 
   
   1. A ⊃ ~B
   2. A v C
   3. ~~B ⋅ D /∴ C
   4. ~~B _________________
   5. ~A _________________
   6. C _________________
4. 
   
   1. A ⊃ B
   2. A ⋅ ~D
   3. B ⊃ C /∴ C ⋅ ~D
   4. A _________________
   5. A ⊃ C _________________
   6. C _________________
   7. ~D _________________
   8. C ⋅ ~D _________________
5. 
   
   1. C
   2. A ⊃ B
   3. C ⊃ D
   4. D ⊃ E /∴ E v B
   5. C ⊃ E _________________
   6. C v A _________________
   7. E v B _________________
6. 
   
   1. (A v M) ⊃ R
   2. (L ⊃ R) ⋅ ~R
   3. ~(C ⋅ D) v (A v M) /∴ ~(C ⋅ D)
   4. ~R _______________
   5. ~(A v M) _______________
   6. ~(C ⋅ D) _______________
7. 
   
   1. (H ⋅ K) ⊃ L
   2. ~R ⋅ K
   3. K ⊃ (H v R) /∴ L
   4. K _________________
   5. H v R _________________
   6. ~R _________________
   7. H _________________
   8. H ⋅ K _________________
   9. L _________________
8. 
   
   1. C ⊃ B
   2. ~D ⋅ ~B
   3. (A ⊃ (B ⊃ C)) v D
   4. A v C /∴ B ⊃ C
   5. ~D _________________
   6. A ⊃ (B ⊃ C) _____________
   7. ~B _________________
   8. ~C __________________
   9. A __________________
   10. B ⊃ C __________________
   11. (B ⊃ C) v B __________________