# Formal proof (using Fitch Format) with an OR

• Jun 27th 2010, 09:27 PM
devouredelysium
Formal proof (using Fitch Format) with an OR
I want to prove that

A /\ (B V C) => (A /\ B) V (A /\ C)

using the Fitch system.

I have done the following:

1. A /\ (B V C)
------
2. A /\ Elim : 1
3 B V C /\ Elim: 1
4. B
-----
5. A /\ B /\ Intro: 2, 4

6. C
-----
7. A /\ C /\ Intro: 2, 6

8. (A /\ B) V (A /\ C) V Intro : 3, 4-5, 6-7

Can I do step 8)? Or am I cheating here? How should I do this?

From what I've read on Logic and Proof, it gave me the impression that you must always conclude the same thing in all "paths" of the OR. I am not doing that, so that is why I'm a bit confused on how to approach this.

Thanks
• Jun 28th 2010, 02:12 AM
Ackbeet
I'm assuming 4-5 is a subproof, as is 6-7. You should use some sort of indentation to do Fitch subproofs. I realize it's difficult here. I use multiple periods with spaces in-between (see here for an example) to indicate indentation.

My thoughts:
1. The rule you're citing for Step 8 is incorrect. It should be OR elimination (which is the "By Cases" inference rule).
2. Step 8 is not technically valid as yet, but your proof can be made valid by just including two more lines inserted in the right place. Hint: try OR intro.
• Jun 28th 2010, 03:03 PM
devouredelysium
OK. I think I did it. Here is another similar one:

I have:
1. B /\ C
2. ~B V ~C V D
-----
2. B
3. C

4. ~B
-----
5. _|_

6. ~C
-----
7. _|_

8. D
-----
(here I want to be able to conclude D, but I can't with V Elim as all "paths" from the V should return the same thing! How am I to prove this?)

From the 2 initial premisses, I want to prove that D follows from them. Am I taking a wrong approach here?
• Jun 28th 2010, 06:01 PM
Ackbeet
I'm totally lost. What are you trying to prove?
• Jun 29th 2010, 01:00 AM
devouredelysium
that D follows from B /\ C and ~B V ~C V D

edit: I actually did it. All I had to do was assume ~D and then do an OR Elim, which would yield a contradiction, thus proving D.

Could you take a look at this twin thread, http://www.mathhelpforum.com/math-he...tch-v-b-b.html ? Thanks
• Jun 29th 2010, 03:06 AM
Ackbeet
Yes, you can do that. Also don't forget that you can derive anything from a contradiction. That is most useful in cases like this.