In 1986, Flagg and Friedman \cite{ff} gave an elegant alternative proof of
the faithfulness of G\"{o}del (or Rasiowa-Sikorski) translation (⋅)□
of Heyting arithmetic HA to Shapiro's epistemic arithmetic EA. In
\S 2, we shall prove the faithfulness of (⋅)□ without using
stability, by introducing another translation from an epistemic system to
corresponding intuitionistic system which we shall call \it the modified
Rasiowa-Sikorski translation\rm . That is, this introduction of the new
translation simplifies the original Flagg and Friedman's proof. In \S 3, we
shall give some applications of the modified one for the disjunction property
(DP) and the numerical existence property (NEP) of
Heyting arithmetic. In \S 4, we shall show that epistemic Markov's rule
EMR in EA is proved via HA. So EA⊢EMR and HA⊢MR are equivalent. In \S 5, we
shall give some relations among the translations treated in the previous
sections. In \S 6, we shall give an alternative proof of Glivenko's theorem. In
\S 7, we shall propose several(modal-)epistemic versions of Markov's rule for
Horsten's modal-epistemic arithmetic MEA. And, as in \S 4, we shall study
some meta-implications among those versions of Markov's rules in MEA and
one in HA. Friedman and Sheard gave a modal analogue FS (i.e.
Theorem in \cite{fs}) of Friedman's theorem F (i.e. Theorem 1 in
\cite {friedman}): \it Any recursively enumerable extension of HA which
has DP also has NPE\rm . In \S 8, we shall give a proof
of our \it Fundamental Conjecture \rm FC proposed in Inou\'{e}
\cite{ino90a} as follows: FC:FS⟹F. This is a new type of proofs. In \S 9, I
shall give discussions.Comment: 33 page