SOLUTION: INSTRUCTIONS: Use natural deduction to derive the conclusion in each problem. Prove this using natural deduction. NOTE: Use * for dot, v for wedge, ~ for tilde, = for tri

Question 1179695: INSTRUCTIONS: Use natural deduction to derive the conclusion in each problem.
Prove this using natural deduction.

NOTE: Use * for dot, v for wedge, ~ for tilde, = for triple bar (or copy and paste ≡), and > for horseshoe (or copy and paste ⊃ )

1. M ⊃ (∼B ⊃ J)
2. B ⊃ (~M * ~M)
3. ∼J / ~M
Answer by RBryant(14) (Show Source): You can put this solution on YOUR website!



Here is a proof using the Introduction and Elimination Rules of Natural Deduction:

1. M ⊃ (∼B ⊃ J)	 
2. B ⊃ (~M * ~M)	 
3. ∼J	/ ∴  ~M 
----------------------------------------
4.      M            Assumption
5.      ~B ⊃ J       1,4                 ⊃E
6.           B            Assumption      
7.      ~M & ~M       2,6                 &I
8.      ~M            7                   &E
9.      M & ~M        4,8                 &I
10.      ~B           6-9                 ~I
11.     B & ~B        6,10                &I
12. ~M                4-11                ~I
                                                 QED

HERE is a Proof using Copi Rules:

1. M ⊃ (∼B ⊃ J)	 
2. B ⊃ (~M * ~M)	 
3. ∼J	/ ∴  ~M 
----------------------------------------
4. B ⊃ (~M & ~M)                    2      Tautology
5. ~(~B ⊃ J) ⊃ ~M                   4      Transposition
6. (~B ⊃ J) v ~M                    5       Material Implication
7. ~ ~B v J v ~M                     6      Material Implication
8. B v J v ~M                        7      Double Negation
9. (B v ~M) v J                      8      Association
10. B v ~M                           9      Disjunctive Syllogism
11. ~B ⊃ ~M                         10      Material Implication                           
12. M ⊃ B                           11      Transposition
13. M ⊃ ~M                          4,12    Hypothetical Syllogism
14. ~M v ~M                          13      Material Implication
15. ~M                               14      Tautology
                                                                     QED

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