Using a Hilbert system to show the correctness of a few logical arguments

Question

I have system:

$(A1)\qquad A → (B → A)$

$(A2)\qquad (A → B) → ((A → (B → C)) → (A → C))$

$(A3)\qquad (A → B) → ((A → \neg B) → \neg A)$

$(A4)\qquad \neg \neg A → A $

$(MP)\qquad \frac{A \quad A → B}{A} $

I would like to show proof for:

a) Premise 1: p
   Premise 2: p → (q → r)
   Premise 3: p → q
   Conclusion: r

b) Premise: q → r
   Conclusion: p → (q → r)

c) Premise 1: p → q
   Premise 2: ¬q
   Conclusion: ¬p

I can show proof for a) like so:

1 p → (q → r) premise
2 p → q premise
3 p premise
4 q → r (MP) 1, 3
5 q (MP) 2, 3
6 r (MP) 4, 5

However with b) and c) I am not sure how to proceed, thanking you sincerely for you guidance.

Answer 1

Your proof for $a)$ is correct.

For $b)$ you just need the first axiom and modus ponens. The shortest proof is three rows long (one of them for the premise, the last one for the conclusion).

For $c)$ I reasoned similarly to what I did in this answer. A proof starts like this:

1. $p\to q\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(Premise)}$
2. $\neg q\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(Premise)}$
3. $(p\to q)\to ((p\to \neg q)\to \neg p)\,\,\,\,\text{(A3)}$
4. $\neg q\to (p\to \neg q)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(A1)}$

Can you finish?

Answer 2

For b):

1) $(q \to r) \to (p \to (q \to r))$ --- axiom A1

2) $q \to r$ --- premise

3) $p \to (q \to r)$ --- from 1) and 2) by mp.

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For c) you need some ausiliary results, like the Deduction Theorem : if $\Gamma \cup \{ A \} \vdash B$, then $\Gamma \vdash A \to B$, easily provable with axioms (A1) and (A2), and the derived rule of Syllogism : $A \to B, B \to C \vdash A \to C$, provable by modus ponens twice and the Deduction Th.