SOLUTION: 55. 1. (L ≡ N) ⊃ C 2. (L ≡ N) v (P ⊃ ~E) 3. ~E ⊃ C 4. ~C /~P 56. 1. ~N ⊃ [(B ⊃ D) ⊃ (N v ~E)] 2. (B ⊃ E) ⊃ ~N 3. B ⊃ D 4. D ⊃ E /

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Question 1194113: 55.
1. (L ≡ N) ⊃ C
2. (L ≡ N) v (P ⊃ ~E)
3. ~E ⊃ C
4. ~C /~P
56.
1. ~N ⊃ [(B ⊃ D) ⊃ (N v ~E)]
2. (B ⊃ E) ⊃ ~N
3. B ⊃ D
4. D ⊃ E /~D
5. ~E 1,3, MD
N v ~ E 2,5 Add
57.
1. G ⊃ [~O ⊃ (G ⊃ D)]
2. O v G
3. ~O /D
Answer by math_tutor2020(3838) (Show Source):
You can put this solution on YOUR website!

Please only post one problem at a time.

I'll do problem 55 to get you started.

Number Statement Line(s) Used Reason
1 (L = N) -> C
2 (L = N) v (P -> ~E)
3 ~E -> C
4 ~C
:. ~P
5 ~~E 3, 4 Modus Tollens
6 ~(L = N) 1, 4 Modus Tollens
7 P -> ~E 2, 6 Disjunctive Syllogism
8 ~P 7, 5 Modus Tollens

I used a standard equal sign in place of a triple equal sign. I used arrows in place of the horseshoe symbols.

For more information, check out the various rules of inference and rules of replacement as shown in the link below.
https://logiccurriculum.com/2019/02/09/rules-for-proofs/