Regarding the grandma-chicken argument, having given it some thought, I think I understand it better now. I’d explain it like this. There is a utility function u, such that all of my actions maximize Eu. Suppose that u(A) = u(B) for some two choices A, B. Then I can claim that A > B, and exhibit this preference in my choices, i.e. given a choice between A and B I would always choose A. However for every B+, such that u(B+) > u(B) I would also claim B < A < B+. This does violate continuity, however because I’m still maximizing Eu, my actions can’t be called irrational, and the function u is hardly any less useful than it would be without the violation.
Regarding the grandma-chicken argument, having given it some thought, I think I understand it better now. I’d explain it like this. There is a utility function u, such that all of my actions maximize Eu. Suppose that u(A) = u(B) for some two choices A, B. Then I can claim that A > B, and exhibit this preference in my choices, i.e. given a choice between A and B I would always choose A. However for every B+, such that u(B+) > u(B) I would also claim B < A < B+. This does violate continuity, however because I’m still maximizing Eu, my actions can’t be called irrational, and the function u is hardly any less useful than it would be without the violation.
Please see https://www.lesserwrong.com/posts/3mFmDMapHWHcbn7C6/a-simple-two-axis-model-of-subjective-states-with-possible/wpT7LwqLnzJYFMveS.