  We prove that any Boolean algebra with the subsequential completeness
property contains an independent family of size continuum. This improves a
result of Argyros from the 80ties which asserted the existence of an
uncountable independent family. In fact we prove it for a bigger class of
Boolean algebras satisfying much weaker properties. It follows that the Stone
spaces of all such Boolean algebras contains a copy of the Cech-Stone
compactification of the integers and the Banach space of contnuous functions on
them has $l_\infty$ as a quotient. Connections with the Grothendieck property
in Banach spaces are discussed.
