Assuming
ω
\omega
is the only measurable cardinal, we prove: (i) Let
∼
\sim
be an equivalence relation such that
∼
=
≡
L
\sim \, = \,{ \equiv _L}
for some logic
L
⩽
L
∗
L \leqslant {L^{\ast }}
satisfying Robinson’s consistency theorem (with
L
∗
{L^{\ast }}
arbitrary); then there exists a strongest logic
L
+
⩽
L
∗
{L^ + } \leqslant {L^{\ast }}
such that
∼
=
≡
L
+
\sim \, = \,{ \equiv _{{L^ + }}}
; in addition,
L
+
{L^ + }
is countably compact if
∼
≠
≅
\sim \, \ne \, \cong
. (ii) Let
∼
˙
\dot \sim
be an equivalence relation such that
∼
=
≡
L
0
\sim \, = \,{ \equiv _{{L^0}}}
for some logic
L
0
{L^0}
satisfying Robinson’s consistency theorem and whose sentences of any type
τ
\tau
are (up to equivalence) equinumerous with some cardinal
κ
τ
{\kappa _\tau }
; then
L
0
{L^0}
is the unique logic
L
L
such that
∼
=
≡
L
\sim \, = \,{ \equiv _L}
; furthermore,
L
0
{L^0}
is compact and obeys Craig’s interpolation theorem. We finally give an algebraic characterization of those equivalence relations
∼
\sim
which are equal to
≡
L
{ \equiv _L}
for some compact logic
L
L
obeying Craig’s interpolation theorem and whose sentences are equinumerous with some cardinal.