There are 2 issues. First, the actual situation is an infinite dimension vector space, so it is not clear that that result applies. Second, the idea was to take advantage of properties of projection in order to converge to the projection as a series of local moves, and I don't see a way to do that with a simplex.

The goal was to use this projection and apply it to the point that assigns 1/2 to all statements, in order to get a coherent probability distribution on sentences and a simple procedure that converges to it, so that we can look at the properties of this distribution.

It also could have given an answer to the question: How should I correct if I observe that my probability assignment is incoherent.