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.
$\forall x,y (P(x) \land P(y) \implies \exists z ( L(z) \land I(x,z) \land I(y,z)))$
is the part of the axiom concerning the existence of a line that joins every two points. I’ll let you figure out how to formalize the uniqueness of that line (Hint: what must happen if we pick two lines joining the same pair of points?)