Weak prediction principles
  We prove the consistency of the failure of the weak diamond $\Phi_\lambda$ at strongly inaccessible cardinals. On the other hand, we show that the very weak diamond $\Psi_\lambda$ is equivalent to the statement $2^{<\lambda}<2^\lambda$ and hence holds at every strongly inaccessible cardinal. 
Omer Ben-Neria, Shimon Garti and Yair Hayut
math.LO