Understanding the rules of inference I'm struggling in my discrete mathematics class and need some clarification on a topic.
Recently we learned the truth table for P -&gt; Q.
P   Q  (P → Q)
T   T   T
T   F   F
F   F   T
F   T   T

After that we learned about rules of inference - these are used to prove an argument to be true or false. However, I don't truly understand why these are necessary.
For example, one of the rules in Modus Ponens, which states this:
assume P → Q is true.
if p is true,
then Q is true as well.

When would that rule of inference ever be necessary in a real world application? Why would we need to assume a hypothesis such as P → Q to be true? What is the point of these &quot;rules of inference&quot; in a real world setting, or perhaps does anyone have a real world example where it's useful? (I'd like to know why these laws have a use so an example would be greatly appreciated. Something besides P= I ate my dog. and Q = there is a blizzard outside. I think my textbook is trying to confuse me with its examples...)
I apologize if this seems like an ignorant question - it's because I am ignorant and am seeking knowledge so I can dig myself out of this hole of confusion.
EDIT: I should probably add that my textbook uses weird example like p=the moon is made of cheese - assume it to be true. So I'm wondering how these rules are actually applicable when we are talking about things that are actually used in the real world - not moons made of cheese. I am a software engineering student, so I'm sure there's a use for this in coding and I just don't know what it is yet.
Here is a real-world example to consider: &quot;If it is raining, then it is cloudy.&quot;
This does not mean that rain causes cloudiness. It means only that, at the moment, it is not the case that it is raining and not cloudy. It permits all other combinations of truth values:

It can be raining and cloudy

It can be cloudy and not raining

It can be not raining and not cloudy


Hence the entries in the truth table.

The use of bizarre falsehoods is often used in logic textbooks to demonstrate the principle of vacuous truth. In general, it states that for any propositions $A$ and $B$, we have the tautology:
$$A \implies (\neg A \implies B)$$
In words, all things follow from a falsehood. Sort of. If $A$ is true, then the implication $\neg A \implies B$ is also true (vacuously so). Fortunately, we cannot infer anything about the truth value of $B$ from this implication since the antecedent ($\neg A$) is assumed to be false.
To use a popular, bizarre example: &quot;If pigs could fly then $X$.&quot;
Since pigs cannot fly, this implication will be true no matter what proposition we may substitute for $X$, be it true OR false. Of course, we cannot infer anything about the truth value of $X$ without more information.
Here is an example where you use Modus Ponens.
You know that every differentiable function at $ x=a$ is continuous at $ x =a$.
So, if $ g$ is special function differentiable at $ x=a$, you are Sure that $ g $ is continuous at $ x= a$.
Every man is mortal.
You are a man.
So, be sure, you are mortal.
My remark:
In mathematics, an implication never begins by a False statement.