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We make use of some observations on the core model, for example assuming $V=L[ E ]$, and that there is no inner model with a Woodin cardinal, and $M$ is aninner model with the same cardinals as $V$, then $V=M$. We conclude in thislatter situation that "$x=\mathcal{P} ( y )$" is $\Sigma_{1} ( Card )$ where$Card$ is a predicate true of just the infinite cardinals. It is known thatthis implies the validities of second order logic are reducible to $V_I$ theset of validities of the H\"artig quantifier logic. We draw some furtherconclusions on the L\"owenheim number, $\ell_{I}$ of the latter logic: that ifno $L[E]$ model has a cardinal strong up to an $\aleph$-fixed point, and$\ell_{I}$ is less than the least weakly inaccessible $\delta$, then (i)$\ell_I$ is a limit of measurable cardinals of $K$; (ii) the Weak CoveringLemma holds at $\delta$.

When cardinals determine the power set: inner models and H\"artig quantifier logic