TITLE: Two versions of "absolutely ccc"
QUESTION [17 upvotes]: I have recently been slogging my way through Shelah's "Large continuum, oracles".  Essentially from the start there has been a question needling me which I cannot seem to answer.

In the paper, Shelah says that a forcing notion $\mathcal{P}$ is absolutely ccc if it remains ccc after forcing with any ccc notion.
Elsewhere, I have seen it defined that a forcing notion $\mathcal{P}$ is absolutely ccc if it remains ccc after any forcing.  (This would be indestructibly ccc from Bartoszyński-Judah.)

Any forcing having the Knaster property is absolutely ccc (in the strong sense), and MA$_{\aleph_1}$ implies that all ccc forcings have the Knaster property.  Thus, it is consistent that the two are equivalent.
Do these two versions of absolute ccc-ness provably coincide?

REPLY [7 votes]: Bartoszyński-Judah's definition is NOT what you say, it is in fact the same as Shelah's notion.  In Set Theory: On the Structure of the Real Line, they define "indestructibly c.c.c." on page 177:

"Recall that a forcing notion $\mathcal{P}$ is indestructibly ccc if for every forcing notion $\mathcal{Q}$ satisfying ccc, $V^\mathcal{Q} \vDash$ "$\mathcal{P}^V$ satisfies ccc."
 
Then they prove (theorem 3.5.26) that if $\mathcal{P}$ has the Knaster property, then it is indestructibly ccc, per this definition.  (One can also show that having the Knaster property is itself indestructible by ccc forcing.)
I have not seen elsewhere the claim that the Knaster property implies that the ccc is indestructible by any forcing.  Do you have a proof of this?
Tangentially, while I was looking in Bartoszyński and Judah's book, I found a notable mistake.  They claim (lemma 1.5.14) that $\mathcal{P} * \dot{\mathcal{Q}}$ has precaliber $\aleph_1$ iff $\mathcal{P}$ has precaliber $\aleph_1$ and $\Vdash_{\mathcal{P}} \dot{\mathcal{Q}}$ has precaliber $\aleph_1$.  This is false.  As noted in the comments on the original post, MA$_{\aleph_1}$ implies the ccc is equivalent to having precaliber $\aleph_1$.  So assume MA$_{\aleph_1}$, and let $\mathcal{C}$ be Cohen forcing, and $\dot{\mathcal{T}}$ be a $\mathcal{C}$-name for a Suslin tree (by Shelah).  So in $V$, the iteration $\mathcal{C} * \dot{\mathcal{T}}$ has precaliber $\aleph_1$, but $\Vdash_{\mathcal{P}} \dot{\mathcal{T}}$ does not have precaliber $\aleph_1$.