Show: $\lnot F \rightarrow G, F \rightarrow H \vdash G ∨ H$

Question

I'm having trouble attempting to prove the above argument. What assumption of $F$ can be made in order to obtain either $H$ or $G$ and still be properly discharged?

Everything I've tried so far gives me an assumption of $F$ that I'm not able to discharge...

Answer 1

Hint:

This could be one approach of the proof:

See if you can fill the missing steps

$$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}} \fitch{~1.~~\neg F\to G\\~2.~~F\to H}{\fitch{~3.~~\neg(F\lor\neg F)}{\fitch{~4.~~F}{~5.~~F\lor\neg F&\lor\text{Intro}~4\\~6.~~\bot&\bot\text{Intro}~3,5}\\~7.\\\fitch{~8.~~\neg F}{~9.~~F\lor\neg F&\lor\text{Intro}~8\\~10.~~\bot&\bot\text{Intro}~3,9}\\~11.\\~12.}\\~13.\\\fitch{~14.~~F}{~15.~~H&\to\text{Elim}~2,14\\~16.~~G\lor H&\lor\text{Intro}~15}\\\fitch{~17.~~\neg F}{~18.~~G&\to\text{Elim}~1,17\\~19.~~G\lor H&\lor\text{Intro}~18}\\~20.}$$

Answer 2

Assuming you are working with a fairly standard natural deduction system with subproofs, you have two main strategies here:

First, do what Max does. This is the 'intuitive' approach ... the 'cleaner' of the two.

Second, do a Proof by Contradiction directly on the Conclusion, i.e. assume $\neg (G \lor H)$. That typically leads to a contradiction a little more quickly ... but it isn't as intuitive:

Answer 3

Thanks to Manx and Bram28 for the help. I'm not sure why but I just had particular difficulties with assuming $¬(F\lor \lnot F)$ and deriving $(F\lor \lnot F)$.

Here is my answer