The Poisson's ratio $\nu$ is always less than $0.5$.
A traction force ($\Delta L >0$) can cause an increase in volume, while a compression force ($\Delta L <0$) can only decrease the volume.
$$\Delta V \approx V \frac{\Delta L}{L}(1-2\nu)$$
I'm totally ok with these facts but I don't understand which one follows the other.
Is $\nu<0.5$ a consequence of an empirical observation that $\Delta V<0$ with traction, or viceversa?
Yes, $ν$ being always less than 0.5 would be a consequence of $ΔV$ always being negative with traction.
Would be, if it were true, which it is not. Citing the Wikipedia article,
The Poisson's ratio of a stable, isotropic, linear elastic material cannot be less than −1.0 or greater than 0.5 (...) Some anisotropic materials (...) have one or more Poisson's ratios above 0.5 in some directions.
(emphasis mine).
Granular matter also can be in peculiar states. If you've ever walked on wet sand at low tide, you may have noticed sand dries up around your footsteps: the compression disrupts the close packing of sand grains, locally spacing them out (to the point where water flows away) and increasing the volume.
