I understand with truth tables the Conditional Law: $[P \Longrightarrow Q] \equiv [\lnot P \vee Q]$. However, what's the intuition or a natural motivation? Source 1, all but intuitive, now appears as an Answer beneath. Retrospectively, cogent and instructive were Cameron Buie's and Prof Scott's Answer.
Source 2: P46 of How to Prove It by Daniel Velleman :
Let $P$ be the statement “You will neglect your homework” and $Q$ be “You’ll fail the course.”
Then “You won’t neglect your homework, or you’ll fail the course.” = $\lnot P \vee Q$.
But what message is the teacher trying to convey with this statement?
Clearly the intended message is “If you neglect your homework, then you’ll fail the course,” or in other words $P \rightarrow Q.$ Thus, in this example, the statements $\lnot P \vee Q$ and $ P \rightarrow Q $ seem to mean the same thing.
Based upon “If you neglect your homework, then you’ll fail the course,”
could I not aver its "intended meaning" as
$P \vee \lnot Q $ = "You neglect your homework or you won't fail the course"?
This would fail to motivate the Conditional Law.
Source 3 : Ex 2.16 on P48 of Gary Chartrand et al's Mathematical Proofs, 2nd Ed:
"If you earn an A on the final exam, then you will receive an A for the.final grade."
We need know only that the student did not receive an A on the final exam or the student received an A as a final grade to see that she kept her promise.
I can't twig this due to its terseness. Could the details please be explicated?