I am trying to understand what “$p$ implies $q$” means. I read that $p$ is a sufficient condition for $q$, and $q$ is a necessary condition for $p$.
Further from Wikipedia,
A necessary condition of a statement must be satisfied for the
statement to be true. Formally, a statement $P$ is a necessary condition
of a statement $Q$ if $Q$ implies $P,\quad (Q \Rightarrow P)$.A sufficient condition is one that, if satisfied, assures the
statement's truth. Formally, a statement $P$ is a sufficient condition
of a statement $Q$ if $P$ implies $Q,\quad (P \Rightarrow Q)$.
Now what I am stuck with is that if $P$ is not satisfied will the condition still always be true?
Best Answer
This is a simple matter answered by the truth table of $\Rightarrow$:
$$\begin{array}{ c | c || c | } P & Q & P\Rightarrow Q \\ \hline \text T & \text T & \text T \\ \text T & \text F & \text F \\ \text F & \text T & \text T \\ \text F & \text F & \text T \end{array}$$
This shows that when $P$ is false, the implication is true. Note that this is the definition of the table, there is no need to prove it. This is how $\Rightarrow$ is defined to work.
As an example, here is one:
$$\textbf{If it is raining then there are clouds in the sky}$$
In this case $P=$It is raining, and $Q=$There are clouds in the sky. Note that $P$ is sufficient to conclude $Q$, and $Q$ is necessary for $P$. There is no rain without clouds, and if there are no clouds then there cannot be any rain.
However, note that $P$ is not necessary for $Q$. There could be light clouds without any rain, and there could be clouds of snow and blizzard (which is technically not rain).