The meaning of Implication in Logic How do I remember Implication Logic $(P \to Q)$ in simple English?
I read some sentence like


*

*If $P$ then $Q$.

*$P$ only if $Q$.

*$Q$ if $P$.


But I am unable to correlate these sentences with the following logic.
Even though the truth table is very simple, I don't want to remember it without knowing its actual meaning.
$$\begin{array}{ |c | c || c | }
  \hline
  P & Q & P\Rightarrow Q \\ \hline
  \text T & \text T & \text T \\
  \text T & \text F & \text F \\
  \text F & \text T & \text T \\
  \text F & \text F & \text T \\ \hline
\end{array}$$

I would like to share my own understanding of this.
I like to think of Implication as a Promise rather than Causality which is the natural tendency when you come across it the first time.
Example:
You have a nice kid and you make him the following promise to him:
If you get an A in your exam, then I will buy you a car.
In this case P is kid gets A in exam and Q is You buy him a car.
Now let's see how this promise holds with various values for P and Q
If P is true (Kid gets A in exam) and Q is true (You bought him car) then your promise has held and $P \Rightarrow Q$ is true.
If P is true (Kid gets A in exam) and Q is false (You didn't buy him a car) then your promise didn't hold so $P \Rightarrow Q$ is false.
If P is false (Kid didn't get A in exam) and Q is true (You bought him car) then your promise still holds and $P \Rightarrow Q$ is true and that's because you only said what will happen if he get's an A, you basically didn't say what will happen if he doesn't which could imply anything. Basically you didn't break your promise and this is the weak property which most people find confusing in implication.
If P is false (Kid didn't get A in exam) and Q is false (you didn't buy him a car) then your promise has also held and $P \Rightarrow Q$ is true because you only promised and guaranteed a car if he gets an A.
