One way to think about it would be to replace or with and: the continuum hypothesis can be true and false: it is a 'polycomputational object' [1].
[1] Using the concept of polycomputing from There’s Plenty of Room Right Here: Biological Systems as Evolved, Overloaded, Multi-Scale Machines: "Form and function are tightly entwined in nature, and in some cases, in robotics as well. Thus, efforts to re-shape living systems for biomedical or bioengineering purposes require prediction and control of their function at multiple scales. This is challenging for many reasons, one of which is that living systems perform multiple functions in the same place at the same time. We refer to this as 'polycomputing'—the ability of the same substrate to simultaneously compute different things, and make those computational results available to different observers.", https://www.mdpi.com/2313-7673/8/1/110
Interesting, that's not a concept I have come across before. But to be honest, I wasn't sure which conjunction to use (and, or or).
Here is Michael Levin, one of the paper's author, speaking at length about the polycomputing concept and more: "Agency, Attractors, & Observer-Dependent Computation in Biology & Beyond" [1].
It turns out the 'continuum hypothesis' can be true or it can be false. Neither contradicts standard ZFC set theory: the hypothesis is 'independent'.