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We prove that any Boolean algebra with the subsequential completenessproperty contains an independent family of size continuum. This improves aresult of Argyros from the 80ties which asserted the existence of anuncountable independent family. In fact we prove it for a bigger class ofBoolean algebras satisfying much weaker properties. It follows that the Stonespaces of all such Boolean algebras contains a copy of the Cech-Stonecompactification of the integers and the Banach space of contnuous functions onthem has $l_\infty$ as a quotient. Connections with the Grothendieck propertyin Banach spaces are discussed.

Independent families in Boolean algebras with some separation properties