The first (left) part of Law 3 is the same as the end of Step 4, and the second (right) part of Step 3 is the same as the beginning of step 4. So, in Step 4, Law 3 is applied right-to-left.
Ok so we are allowed to repeat the same step multiple times ...
So now we have:
Step 1: ¬p ∨ q = Law 3
Step 2: q ∨¬p = ...
Step 3: ¬(¬q) ∨ ¬p = Law 2
Step 4: ¬q ⇒ ¬p = Law 3
So now we only need to find out Step 2 ...
Step 2: q ∨¬p = ...
and it looks like that Law 3 is used again here ... is that possible?
That is your problem.
Unless you get a whole book and start from the beginning and make sure you understand each and every chapter ,then you might have a chance to realise what you are doing and the meaning behind it .
Taking for example the classic book "introduction to logic" by Irving Copi you will find that the problem you are trying to solve is in chapter 9.
How about the 8 chapters before??
Back on topic :-)
So far we still have:
Step 1: ¬p ∨ q = Law 3
Step 2: q ∨ ¬p = ...
Step 3: ¬(¬q) ∨ ¬p = Law 2
Step 4: ¬q ⇒ ¬p = ...
The Laws that we can choose from:
Law 1: (a ∨ b) ⇔ (b ∨ a)
(commutativity of ∨)
Law 2: ¬(¬a) ⇔ a
(law of negation)
Law 3: (a ⇒ b) ⇔ (¬a ∨ b)
(law of implication)
Law 4: (a ⇔ b) ⇔ ((a ⇒ b) ∧ (b ⇒ a))
(law of equivalence)
Explanation of the steps:
So with step 1 it was obvious that Step 1 (¬p ∨ q) belonged to Law 3 (a ⇒ b) ⇔ (¬a ∨ b)
Because we see that "¬p" corrosponds with "¬a" and "q" corrosponds with "b"
And with step 3 ( ¬(¬q) ∨ ¬p ) it was also clear that law 2 (a ⇒ b) ⇔ (¬a ∨ b) is the lucky one.
Becasue when we replace the "a" with "¬q" and "b" with "¬p" the result will be (¬q ⇒ ¬p) ⇔ (¬¬q ∨ ¬p)
So the only steps we have left to solve is Step 2 and Step 4...
And the Laws that we have not used yet are Law 1 and Law 4
This is correct until now right?
I see now ...
Step 1 = Law 3
Step 2 = Law 1
Step 3 = Law 2
Step 4 = Law 3
You know what my big problem is ...
That is that I was in the understanding that we should use all given laws only 1 time ... and now that Law3 is used twice and Law 4 is not used at all I realise that this is not needed...
We just use the Laws acording to needs ... and this can well be that a Law is used multiple times...
Thanks for your patience with this one :-)