  We define a countable antichain condition (ccc) property for partial
orderings, weaker than precalibre-$\aleph_1$, and show that Martin's axiom
restricted to the class of partial orderings that have the property does not
imply Martin's axiom for $\sigma$-linked partial orderings. This answers an old
question of the first author about the relative strength of Martin's axiom for
$\sigma$-centered partial orderings together with the assertion that every
Aronszajn tree is special. We also answer a question of J. Steprans and S.
Watson (1988) by showing that, by a forcing that preserves cardinals, one can
destroy the precalibre-$\aleph_1$ property of a partial ordering while
preserving its ccc-ness.
