Conditional logical connectives, $P\to Q$ For $P \rightarrow Q$, the truth table looks like this:


*

*True P, true Q: true

*True P, false Q: false

*False P, true Q: true

*False P, false Q: true


Since this is the first time for me to study this, I'm trying to understand it using a concrete example. Let's say $P$ is "I kicked the ball" and $Q$ is "the ball moved". Then the truth table above translates to:


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*Does "I kicked the ball" and "the ball moved" support the if-then? Yes.

*Does "I kicked the ball" and "the ball did not move" support the if-then? No.

*Does "I didn't kick the ball" and "the ball moved" support the if-then? Well, at least it doesn't contradict it - so yes.

*Does "I didn't kick the ball" and "the ball did not move" support the  if-then? Well, at least it doesn't contradict it - so yes.


Is my understanding correct?
 A: That's precisely how the truth-table defines $p\to q$; your understanding is entirely spot-on with it.  
Note that in case three, perhaps a strong gust of wind moved the ball, or a toddler came along, while you were drinking water, and kicked the ball. 
The material conditional $p\to q$ is always true unless $p$ is true, but $q$ is false.

Another example some find useful:
Suppose $p$: You carry my groceries to my car.
Suppose $q:$ I pay you $\$5.$
Now we evaluate $p\to q$ as a promise to you: Under which truth value assignments did I lie (false), and which did I not break my promise/didn't lie (true).
If you carried my groceries to the car, and I paid you $\$5$, I haven't lied.
If you didn't carry my groceries to the care, and I paid you $\$5$, I haven't lied. (Perhaps I gave you $\$5$ because you declined my offer politely and explained you were already running late and had to pick up your niece from preschool, and I was impressed by your sense of responsibility.)
If you didn't carry my groceries to the car, and I didn't pay you $\$5$, I haven't lied.
But, if you did carry my groceries to the car, and I didn't pay you $\$5$, then, and only then, I certainly lied.
