Comment by danbruc

The key correctness insight is this: any two majorities of nodes must overlap in at least one node. So between any two consecutive global state changes [...] at least one node participated in both. This single overlapping node carries forward the knowledge of what was previously committed, preventing conflicts and ensuring consistency.

There is another side to this, it must not be possible for two »majorities« to coexist, otherwise they could independently move on in case of a split cluster. This also rules out allowing consensus by majority in addition to majority by a bloc. In the seven node example, there could be a { 1, 2, 3 } and { 4, 5, 6, 7 } split, the first partition being a bloc and the second one being a majority but not containing a bloc.