# Thread: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

1. ## In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B?

I believe I need to assume A and do something Elim V, but I am really stumped as to what to do exactly. Any help would be appreciated.

2. ## Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

Is a truth table not an acceptable proof for this? That's what I would use.

3. ## Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

Originally Posted by CDs
In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B?
What do you mean by "corresponds to"?

To prove (~A v B) -> (A -> B), assume ~A v B and A and do Elim V on ~A v B. If ~A, then you get get a contradiction with A, so you can derive B by ex falso. Otherwise, you already have B.

To prove the converse implication is a little more difficult and requires double negation elimination.

4. ## Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

Originally Posted by CDs
In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B?

I believe I need to assume A and do something Elim V, but I am really stumped as to what to do exactly. Any help would be appreciated.
Fitch-Style Proof Builder

5. ## Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

This is what I've done so far. It's how to get from step 5 to 6 that confuses me.

6. ## Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

Nevermind, I figured it out. And I even know how to do the reverse. Thanks a lot guys!

7. ## Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

So, could you say how you derived B in step 6 for everybody's benefit?

8. ## Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

Fitch says that I've proved what I tried to prove, yet upon more reflection I'm worried it might due to a bug in the program (view steps 6-9). Could somebody confirm if I did this properly?

9. ## Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

Usually natural deduction is presented with the following two rules:

$\displaystyle \frac{\bot}{A}\bot\ \mbox{Elim}$ and $\displaystyle \frac{\neg\neg A}{A}\mbox{DNE}$ (double-negation elimination, called $\displaystyle \neg$ Elim in your program) for all formulas $\displaystyle A$

The rule $\displaystyle \bot$ Elim can be derived from DNE, which is what you did in your derivation:

Code:
  ~B    Assumption
...
_|_   _|_ Intro
~~B     ~ Intro
B       DNE
Here the assumption ~B can be vacuous, in which case you derive B just from $\displaystyle \bot$.

The reason to keep these two rules separate is that the logic without DNE is quite interesting on its own.

Which program do you use to create Fitch-style derivations?

10. ## Re: In formal Logic (Fitch Format), how do I prove ~A v B corresponds to A -> B

The program is called Fitch by a CD called Language, Proof and Logic.