When and how can I change the conclusion in rules of inference

Question

I have seen people changing the conclusion of the examples in the rules of inference. My question is when I'm able to do that? If I have to prove P and Q which one should I choose?

Examples 1: premises { p → q, p ∨ q } conclusion/prove q ∨ r But they are making it, just prove q but they add ¬r in the premises so we have

Final: Premises { p → q, p ∨ q, ¬r } conslusion q

Example 2:

Premises { p ∨ q, ¬q ∨ r, r → s} conclusion: ¬p → s

Final: they change it to { p ∨ q, ¬q ∨ r, r → s, ¬p } conclusion s

Can I simplify this every time?

Answer 1

Here is a proof of the first example:

Note that it doesn't matter if one adds $\neg R$ to the premises. One can still derive $Q$ or $Q \lor R$ with or without that premise.

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Here is a proof of the second example:

If one also assumed $\neg P$ as a premise the proof might look like this:

In this example, unlike the first example, one could make use of the additional premise. However, what one is asked to prove has changed as well. It is no longer a conditional, $\neg P \to S$, but the proposition $S$.

Regardless, the two examples do not illustrate a similar pattern. Given premises and a desired goal, one should use only defined inference rules to reach the goal from the premises.

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Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

Answer 2

Both examples above are simple application of the Deduction Theorem :

in order to prove $Γ ⊢ p → q$, prove $Γ,p ⊢ q$.

Also the first one is based on the DT, with the "trick" of using the Material Implication rule :

$q∨r$ is equivalent to : $¬ r → q$.

Thus, the proof is :

1) $p → q, p ∨ q, ¬r \vdash q$ --- given

2) $p → q, p ∨ q \vdash ¬r \to q$ --- from 1) by DT

3) $p → q, p ∨ q \vdash r \lor q$ --- from 2) by Material Implication.