Question

Is it consistent that every non-trivial $\aleph_1-$closed forcing notion of size $2^{\aleph_0}$ (which is trivially the minimal size of such forcing notions) collapses some cardinals? (we can ask the same question for larger cardinals.)

Remark: This statement follows trivially from "Foreman`s maximality principle" which states that every non-trivial forcing notion either adds a real or collapses some cardinals. As far as I know the consistency of this principle is unknown.

Answer 1

I can't answer your question, but I can give a simpler sounding formulation that might be helpful. Analyze the question in two cases.

Case 1: The continuum hypothesis holds
In this case, the statement is false, because any $<\aleph_1$ -closed forcing of size $\aleph_1$ cannot collapse cardinals. The forcing to add a cohen subset to $\aleph_1$ is a nontrivial example of such a forcing.

Case 2: The continuum hypothesis fails
In this case, it is a theorem that every $<\aleph_1$-closed forcing notion which collapses a cardinal collapses the continuum. (See this question, referenced by Joel in the comments.) But every such forcing is equivalent to the canonical collapse forcing to collapse the continuum to $\aleph_1$. The most general version of this latter theorem that I know of (although the degree of generality makes it hard to follow) can be found in Handbook of Boolean Algebras, Volume 2, Corollary 1.15.

So really, your question boils down to whether every $<\aleph_1$-closed forcing of size continuum is isomorphic to Coll$(\aleph_1, c)$ This sounds strange to me, but I can't prove it's false, and if Foreman is entertaining it, who am I to judge it?