Axiom:Inner Connectivity of Betweenness

Axiom
Let $\mathsf{B}$ be the relation of betweenness.

This axiom asserts that:
 * $\forall a,b,c,d: \left({\mathsf{B}abd \land \mathsf{B}acd}\right) \implies \left({\mathsf{B}abc \lor \mathsf{B}acb}\right)$

where $a, b, c, d$ are points.

Intuition
Let $abd$ and $acd$ be line segments.

They exist in one of the following configurations:

Case 1

 * Inner Connectivity of Betweenness Case1.png

Point $b$ is between $a$ and $c$.

Case 2

 * Inner Connectivity of Betweenness Case2.png

Point $c$ is between $a$ and $b$.

Note that this axiom does not assert that exactly one of the cases happens.

This axiom still holds in the degenerate cases where the points are not distinct.

For example, if we are dealing with exactly three points, this axiom could be interpreted as "three points on a line are between each other".

Also see

 * Outer Connectivity of Betweenness