  The chiral properties of the pseudoscalar mesons are studied numerically on a
quenched $20^4$ lattice with the overlap fermion. We elucidate the role of the
zero modes in the meson propagators, particularly that of the pseudoscalar
meson. The non-perturbative renormalization constant $Z_A$ is determined from
the axial Ward identity and is found to be almost independent of the quark mass
for the range of quark masses we study; this implies that the $O(a^2)$ error is
small. The pion decay constant, $f_{\pi}$, is calculated from which we
determine the lattice spacing to be 0.148 fm. We look for quenched chiral log
in the pseudoscalar decay constants and the pseudoscalar masses and we find
clear evidence for its presence. The chiral log parameter $\delta$ is
determined to be in the range 0.15 -- 0.4 which is consistent with that
predicted from quenched chiral perturbation theory.
