Reading in a book on logic I found some examples of structures and languages in first order logic:

The language $L_{geom}$ of basic plane geometry has two 1-place relation symbols $P$

and $L$ for “point” and “line”, and a 2-place relation symbol $I$ for “point $x$ lies on

line $y$”. Examples of axioms are:

- $forall x(P(x) ∨ L(x))$
- $forall x¬(P (x) ∧ L(x))$

these axioms mean: everything is either a point or a line; and

not both.

how to express this axiom in $L_{geom}$: for every two points there is a unique line they lie on.