  The use of null hypotheses (in a statistical sense) is common in hard
sciences but not in theoretical linguistics. Here the null hypothesis that the
low frequency of syntactic dependency crossings is expected by an arbitrary
ordering of words is rejected. It is shown that this would require star
dependency structures, which are both unrealistic and too restrictive. The
hypothesis of the limited resources of the human brain is revisited. Stronger
null hypotheses taking into account actual dependency lengths for the
likelihood of crossings are presented. Those hypotheses suggests that crossings
are likely to reduce when dependencies are shortened. A hypothesis based on
pressure to reduce dependency lengths is more parsimonious than a principle of
minimization of crossings or a grammatical ban that is totally dissociated from
the general and non-linguistic principle of economy.
