Studying Discrete Structures, Logic, And Computability by James L. Hein, came across this proof.
On line 3, the author uses that rule to get $A v B$. Two questions arose:
On
Sure. If A is true, then A or B is absolutely true. A or ~B is true. A or (B and ~B) is true!
Why would you do this? Well, like in your sample proof that a false antecedent implies any consequent, you would need to add B somehow. Or if you had to prove $(A \vee B) \implies (C \vee D)$ given $A \implies C$ and $B \implies D$. You can do a proof by cases, but you need to turn $A \implies C$ to $A \implies (C \vee D)$ to generate the proof.
On
The accepted answer already explains the other points quite well, so let me just address the remaining question,
What would be the reasoning behind [introducing another variable, possibly one that does not occur among the premises]?
At first sight, the disjunction introduction might indeed seem not very useful: If we already know that A holds, why would we weaken and complicate that knowledge by turning it into "A or B"?
First of all, let's observe that such a disjunction introduction is sound: Suppose we have derived or are given as a premise the sentence "The gardener is the murderer", and we apply Add with a new sentence to arrive at the statement "The gardener is the murderer or Lady Smith's cat is grey". Then with the knowledge we have, if someone asks "Is at least one out of the claims true that the gardener is the murderer or Lady Smith's cat is grey?", then we can faithfully answer "yes". The addition of the second alternative does not invalidate our previous knowledge of A, it merely weakens it.
So what is the point of doing this?
While the addition of the second disjunct in the above example was indeed not very useful, in some situations it is. Suppose we know that the local swimming pool has an outdoor pool, which is always accessible from June until August, and in addition it is opened when it is more than 25°C outside. We can formalize this fact as a statement of the form
$(A \lor B) \to C$
where $A$ represents the proposition "It is between June and August, "B" stands for "It is more than 25°C", and $C$ is the consequent "The outside pool is open".
Now suppose it is an exceptionally warm May afternoon with 28°C outside and we want to know if we can get into the outside pool. With the knowledge we are given, we only have the premises that 1. $(A \lor B) \to C$ (The outdoor pool is open if it is between June and August or above 25°C). and 2. $B$ (It is above 25°C). From these two formulas alone, we can not draw any conclusions:
- (It is between June and August $\lor$ It is more than 25°C outside) $\to$ The outside pool is open
- It is more than 25°C outside
...?- The outside pool is open
Now the disjunction introduction comes in handy: By adding a second disjunct to the premise $B$ -- which we are allowed to do either to the left or the right, as $A \lor B$ is equivalent to $B \lor A$ -- we can arrive at the form of the antecedent of the implication that is necessary to derive the conclusion C, thereby completing the derivation:
- (It is between June and August $\lor$ It is more than 25°C outside) $\to$ The outside pool is open
- It is more than 25°C outside
- It is between June and August $\lor$ It is more than 25°C outside $\quad \quad$ 2, Add
- The outside pool is open $\hspace{6.5cm}$ 1, 3, MP
(where "MP" stands for "modus ponens", in natural deduction more often written as $\to E$, "implication eliminiation"., applied to the implication $(A \lor B) \to C$ on line 1 and antecedent $A \lor B$ on line 4.)
Yes. If $A$ holds in the context, then you may infer that $A\vee B$ holds (whether $B$ does or does not). After all, $A\vee B$ is the claim that at least one from $A$ or $B$ holds. So when at least one does (eg $A$), then the disjunction holds too.
PS: the rule of Addition is not about adding the whole statement. In logic proof systems, it refers to the rule by which any disjunct can be added to an existing statement. That if $A$ is derivable (or assumed), then $A\vee B$ may be inferred in the same context.
It is possible to raise an assumption (rule P) that contradicts a prior statement. You may open a sub-proof by assuming anything as needed for the sake of argument; just be sure to close the sub-proof by discharging it with a valid rule (such as rule CP, "conditional proof").
This is the Principle of Explosion, otherwise known as ex falso (sequitur) quodlibet (Latin for "from falsehood, anything follows"). If your premises contain a contradiction, you can derive anything via valid rules of derivation. Since it is valid to do so, therefore you may do so.
So you want to prove $\neg A\to(A\to B)$? Well, since that is a conditional statement, you should use a conditional proof: raise $\neg A$ as an assumption with the aim to derive $A\to B$.
So you want to prove $A\to B$, under the assumption of $\neg A$? Well, again you should use a conditional proof: raise $A$ as an assumption with the aim to derive $B$.
So you want to prove $B$ under the assumptions of $\neg A$ and $A$? Well, you have assumed $A$ so you may infer $A\vee B$ (via Addition), but you have also assumed $\neg A$, so you may further infer $B$ (via Disjunctive Syllogism).
Why now, you have successfully derived $B$ in the context of the two assumptions, $\neg A$ and $A$, so you may now discharge those assumptions to close the two conditional proofs.
Thus you have proven $\neg A\to(A\to B)$ using the rules of assumption, conditional proof, addition, and disjunctive syllogism.