  We formalize the notion of Herbrand Consistency in an appropriate way for
bounded arithmetics, and show the existence of a finite fragment of ${\rm
I\Delta_0}$ whose Herbrand Consistency is not provable in the thoery ${\rm
I\Delta_0}$. We also show the existence of an ${\rm I\Delta_0}-$derivable
$\Pi_1-$sentence such that ${\rm I\Delta_0}$ cannot prove its Herbrand
Consistency.
