Dgdrxrt

This one has me scratching my head:

$$(A lor lnot B) implies C$$
$$C iff (D landlnot D)$$
$$B implies A$$
$$∴ E$$

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

Thanks :)

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 $Brightarrow A$, so looking at the premises, you have that $Brightarrow A$ leads to a contradiction. In a proof system, these premises will allow you to infer $C$ from the truth of $Brightarrow A$

Since $(A vee neg B)iff (Brightarrow A)$, replacing gives you $(Brightarrow A)rightarrow C$.

Both

$Brightarrow A$

and

$(Brightarrow A)rightarrow C$

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 $Brightarrow A$ 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.

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 $E$.