How do I construct a proof for an argument that stands alone? This one has me scratching my head:
$$(A \lor \lnot B) \implies C$$
$$C \iff (D \land\lnot D)$$
$$B \implies  A$$
$$∴ E$$
How am I supposed to reach the conclusion when E is not mentioned in any other premise?
Thanks :)
 A: This is called "Principle of Explosion". It means that you can prove anything from a contradiction.
Notice that you have that $C$ is a contradiction in your premises, because it is equivalent to the contradiction $(D \wedge \neg D)$.
$(A \vee \neg B)$ is the same as $B\rightarrow A$, so looking at the premises, you have that $B\rightarrow A$ leads to a contradiction. In a proof system, these premises will allow you to infer $C$ from the truth of $B\rightarrow A$ 
Since $(A \vee \neg B)\iff (B\rightarrow A)$, replacing gives you $(B\rightarrow A)\rightarrow C$. 
Both
$B\rightarrow A$ 
and 
$(B\rightarrow A)\rightarrow C$
leads to
$C$.
That is to say, with such set of premises you have that $C$ is true, and from $C$, by the "Principle of Explosion", you can prove anything. $E$ is supposed to stand for such arbitrary formula that you can prove.
just one small observation: you would need to prove that $B\rightarrow A$ is derivable from $(A \vee \neg B)$ (or vice versa) to formally do this, but let's just assume you already have this easy-to-check fact.
A: The answer really depends on what rules of inference you have in your proof system.
If you are able to use reductio ad absurdum (i.e. proof by contradiction), then just assume $\neg E$ and find a contradiction to conclude that $E$ must be true.
Another possibility is to use the rule of disjunction introduction (a.k.a. addition), which states that from $P$ you can infer $P \vee Q$ (where $Q$ is any statement). In your particular example, you can derive $D \wedge \neg D$, from which you can derive $D$ and $\neg D$ separately. Then use disjunction introduction to introduce $D \vee E$. Now, since you have $\neg D$, you can immediately derive $E$.
A: Here is the question:

How am I supposed to reach the conclusion when E is not mentioned in any other premise?

If $E$ is not mentioned in the premises and that is what you need to derive, the premises are likely inconsistent, that is, you can derive a contradiction from them and from that derive anything, in particular $E$.  
Note that it is possible to derive a contradiction from the second premise. Here is a proof using natural deduction to show how that could be done:

To use second premise to derive the contradiction $D \land \neg D$, I need to first derive $C$. I use the law of the excluded middle (LEM) to do that considering the two cases $B$ and $\neg B$. That allows me to derive the contradiction (absurdity: ⊥) and then eliminate that by deriving anything I want. Since I want $E$, I derive $E$ on line 16.

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/
