SOLUTION: Use conditional proof: 1. G ⊃ (E ⊃ N) 2. H ⊃ (∼N ⊃ E) / G ⊃ (H ⊃ N)

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Question 1169574: Use conditional proof:

1. G ⊃ (E ⊃ N)
2. H ⊃ (∼N ⊃ E) / G ⊃ (H ⊃ N)
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Here's one way to do the derivation. I'm using arrow symbols in place of horseshoe symbols.

The idea is to assume G is the case and show it leads to (H -> N).
NumberStatementLines UsedReason
1G -> (E -> N)
2H -> (~N -> E)
:.G -> (H -> N)
3GAssumption for Conditional Proof
4E -> N1,3Modus Ponens
5(H & ~N) -> E2Exportation
6(H & ~N) -> N5,4Hypothetical Syllogism
7H -> (~N -> N)6Exportation
8H -> (~~N v N)7Material Implication
9H -> (N v N)8Double Negation
10H -> N9Tautology
11G -> (H -> N)3-10Conditional Proof