Help with propositional logic

Question

Hi all this is for a homework where we just started learning logic and I am not very familiar with propositional logic.

So we have two problems:

To show a proof of the Sherlock Holmes syllogism using Quine's method

My solution is:

The syllogism is (a V b) -> ~a -> b

We begin by splitting on a.

1)Substituting phi[a:=⊤]

(a V b) -> ~a -> b = (⊤ V b) -> ~⊤ -> b = ⊤-> ⊥ -> b = ⊤ -> ⊤
The last line is a tautological

2)Substituting phi[a:=⊥]. (a V b) -> ~a -> b = (⊥ V b) -> ~⊥ -> b = b -> ⊥ -> b = b -> b The last line is tautological

Am I on the right track here? Can the proof be further improved upon? It would be great to get some feedback!

The other problem involves giving a natural deduction proof of (α→β)→(β→γ)→(α→γ) What is natural deduction? can anyone provide a concise example of how can we prove a similar expression using natural deduction so I can work my way with the hw.

Hope this post is in abiding with the community standards.

Thanks!