Looking at the rules of projective planes the rule indicates:
There exists a set of four points, no three of which belong to the same line.
But I'm wondering why should there be a set of 4 points? I'm asking what could have gone wrong if we had this rule instead:
There exists 3 points not all on a same line.
This rule is meant to exclude degenerate cases like the near pencil.
http://mathworld.wolfram.com/Near-Pencil.html
The nearpencil consists of $n+1$ points $O,A_1,\dots,A_n$. The lines are given by the sets $\{A_1,\dots,A_n\}$, $\{O,A_1\}$, $\{O,A_2\}$, ..., $\{O,A_n\}$.
One can think of such a near pencil as a projective space of dimension 1 (a line) and a projective space of dimension 0 (a point) where all possible connections between both are made.