Do inflation expectations matter for inflation?

Jeremy Rudd in this Federal Reserve working paper questions the foundations of monetary policy (HR Prof Sashi Sivramkrishna). Rudd says concept of inflation expectations is based on shaky foundations:

Economists and economic policymakers believe that households’ and firms’ expectations of future inflation are a key determinant of actual inflation. A review of the relevant theoretical and empirical literature suggests that this belief rests on extremely shaky foundations, and a case is made that adhering to it uncritically could easily lead to serious policy errors.

He looks at papers of Friedman, Phelps, Lucas and so on which show inflation expectations are central to inflation and thus to monetary policy. But finds them inadequate and wanting…

Need to read this carefully.