I don't know if this the place to post this question, but I guess I can always try.
My questions are regarding Logic.
1) I'm trying to prove 'proof by cases' - X v Y, X -> Z, Y -> Z |- Z
I'm trying to prove it using the natural rules, but am having problems!
2) Using proof by cases, I want to prove the implication rule: A -> B |- ČA v B
Either that, or by using natural rules.
This is really starting to annoy me - as I have spent a long time working on this!
Any help / advice would be helpful!
I think that 'proof by cases' refers to the process of writing down all the possible assignments of truth values to X,Y,Z,... which make all the statements preceding the |- true; then you check that for every such assignment the statement folliwing the |- is true as well. So for A -> B |- ČA v B you find that the truth values (A,B) = (T,T), (F,T), (F,F) make A->B true and for every one of these assignments ČA v B is true as well.
Thanks for that - I actually want to show this without using truth tables.
So if X v Y, X -> Z, Y -> Z |- Z
Would it be ok to say the following:
1) X assume
2) X-> Z ->I, 1, and X->Z
3) Z -> E, 2
4) Y assyne
5) Y -> Z ->I, 4, Y->Z
6) Z -> E, 5
7) Z 3, 6.
Would this work?
For the implication rule - I'm really stuck on how to do that one...
to your second question you can use this process:
3)B 1,2 M.P
about your first question:
5)~X->Z 4,3 h.s
7)~Z->Z 6,5 h.s
9)Z 8, idemp.
that's all there is to it.