Fitch-Style Proof

Question

Hi I'm having trouble solving a Fitch Style Proof and I was hoping someone would be able to help me.

Premises:

$A \land (B \lor C)$
$B \to D$
$C \to E$

Goal: $\neg E \to D$

Thank You

Answer 1

To begin with: you have $\neg E$, and combine this with $C \to E$, then you have: $\neg C$, but $B \lor C$ must be true, so this yields $B$, and with $B \to D$, apply Modus Ponen: we have: $D$.

Answer 2

Do a proof by contradiction. Assume $\neg E \wedge \neg D$. Then we have from $A \wedge (B \vee C)$ that $B \vee C$. If $B$, then $D$, which contradicts $\neg D$. If $C$ then $E$ which contradicts $\neg E$.

Answer 3

You should be able to transform the following in a formal proof.

Assume $\neg E$.

Prove $B\lor \neg B$ with the intent to use $\lor$-$\text{Elim}$.

If $B$ holds, then use $\to$-$\text{Elim}$ on the premise $B\to D$ to conclude $D$.

Suppose $\neg B$ holds. Use $\land$-$\text{Elim}$ on the first premise to get $B\lor C$. You will want to use $\lor$-$\text{Elim}$ on $B\lor C$. If $B$ holds you can get $D$ in two ways, choose one of them. If $C$ holds, eliminate $\to$ on the premise $C\to E$ to get $E$ and a consequently a contradiction, thus getting $D$ with $\bot$-$\text{Elim}$. Eliminating the disjunction $B\lor C$ gives you $D$.

Eliminating the disjunction $B\lor \neg B$ yields $D$.

Finish off.

You can find the proof hidden in the grey area below.