Logic

Hi,

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 |- ¬A 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!

Thanks.
Kit

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 |- ¬A 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 ¬A v B is true as well.

Hi,

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...

Quote:

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Originally Posted by kit

Hi,

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 |- ¬A 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!

Thanks.
Kit

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to your second question you can use this process:
1)A->B
2)A
3)B 1,2 M.P
4)Bv~A 3,Add.

about your first question:
1) XvY
2)X->Z
3)Y->Z
4)~X->Y 1,m.c
5)~X->Z 4,3 h.s
6)~Z->~X 2,trans
7)~Z->Z 6,5 h.s
8)ZvZ 7,m.c
9)Z 8, idemp.

that's all there is to it.