  If we replace first order logic by second order logic in the original
definition of G\"odel's inner model $L$, we obtain HOD. In this paper we
consider inner models that arise if we replace first order logic by a logic
that has some, but not all, of the strength of second order logic. Typical
examples are the extensions of first order logic by generalized quantifiers,
such as the Magidor-Malitz quantifier, the cofinality quantifier, or stationary
logic. Our first set of results show that both $L$ and HOD manifest some amount
of {\em formalism freeness} in the sense that they are not very sensitive to
the choice of the underlying logic. Our second set of results shows that the
cofinality quantifier gives rise to a new robust inner model between $L$ and
HOD. We show, among other things, that assuming a proper class of Woodin
cardinals the regular cardinals $>\aleph_1$ of $V$ are weakly compact in the
inner model arising from the cofinality quantifier and the theory of that model
is (set) forcing absolute and independent of the cofinality in question. We do
not know whether this model satisfies the Continuum Hypothesis, assuming large
cardinals, but we can show, assuming three Woodin cardinals and a measurable
above them, that if the construction is relativized to a real, then on a cone
of reals the Continuum Hypothesis is true in the relativized model.
