Natural Deduction [Propositional Logic]

Question

I have to prove the following:

PREMISSES:

1. A IMPLIES ¬C
2. ¬ (B AND ¬A)
3. A OR B

CONCLUSION: ¬C

I've done the following so far:

^   AND
v   OR
¬   NEG
->  IMP


4. C              H
5. |  A           H
6. |  | ¬C        E-> 1,5
7. |  |  C        Reit 4
8. | ¬A           I¬ 5,6,7
9. |  B           H
10.|  |  B ^ ¬A   I^ 8,9
11.|  | ¬(B^¬A)   Reit 2
12.|  ¬B          I¬ 9,10,11
. . .

N. ¬C

I'm stuck at this point. I don't know how to continue and get an A or a B to get ¬C.

Answer 1

Combine the $\neg B$ with premise 3 to get $A$, which with premise 1 gives you $\neg C$, and thus the contradiction with $C$ you are looking for.

Answer 2

If you're using classical logic, then there's a quicker proof where you immediately use $\vee$ elimination on (3), deriving $A$ in both cases. (The case starting with $B$ requires both (2) and the double-negation law.) Then apply (1) and you're done.

But if you want to use only intuitionistic logic (which goes nicely with natural deduction), then your approach is necessary, with Bram28's advice to finish it.

Answer 3

$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}}$ Always keep your mind on the goal.

You are attempting to deduce $\neg C$ using a subproof with an assumption of $C$. The goal of this subproof is a contradiction.

The means of achieving that goal is using the third premise and $\vee$-elimination. $$\fitch{~~1.~A\to\neg C\\~~2.~\neg (B\wedge\neg A)\\~~3.~A\vee B}{\fitch{~~4.~C\hspace{12ex}\textsf{H}}{\fitch{~~5.~A\hspace{8.5ex}\textsf{H}}{~~6.~\neg C\hspace{7ex}{\to}\textsf{E}~1,8\\~~7.~\bot\hspace{8.5ex}\neg\textsf{E}~~4,6}\\~~8.~\neg A\hspace{11ex}\neg\textsf{I}~~5{-}7\\\fitch{~~9.~B\hspace{9.5ex}\textsf{H}}{10.~B\wedge\neg A\hspace{3ex}\wedge\textsf{I}~~8,9\\11.~\bot\hspace{9.5ex}\neg\textsf{E}~~10,2}\\12.~\bot\hspace{12ex}\vee\textsf{E}~~3,5{-}7,9{-}11}\\13.~\neg C\hspace{14ex}\neg\textsf{I}~4{-}12}$$