Systems Between First-Order and Second-Order Logics by S. Shapiro (in Handbook of Philosophical Logic, vol 1, ed 2) gives a definition similar to the following:
Let $L(Q)$ denote the logic that augments First-Order Logic with a new quantifier $Qx$ such that $Qx(\phi)$ holds just in case there are infinitely many $x$ such that $\phi$.
I'd like to understand why $L(Q)$ has the downward Lowenheim-Skolem property. My current understanding is around here: Weak Second-Order Logic includes $L(Q)$ (in the sense that WSOL if capable of distinguishing every model that $L(Q)$ can distinguish), WSOL is $w-$projective, and the downward (though not upward) Lowenheim-Skolem property holds for $w-$logic. It follows that $L(Q)$ has the downward LS property as well. So the Lowenheim number of $L(Q)$ would be $\aleph_0$, right? Or is my reasoning flawed here?
Instead of relying on appeals to relations between $L(Q)$ and other extensions of $FOL$, would a direct proof of the downward LS theorem for $L(Q)$ work if it were analogous to the standard proof of the LS theorem for $FOL$? Nothing pops out at me immediately as for why the traditional argument wouldn't work, but I'm nevertheless unsure.
Thanks!
I am not sure about your approach through second order logic, but it is not hard to directly prove that the Lowenheim number of $L(Q)$ is $\aleph_1$.
Firstly note that $Qx (x = x)$ has a model of cardinality $\aleph_1$ but no countable model. Therefore the Lowenheim number is at least $\aleph_1$.
On the other hand given a structure $M$ and $A \subseteq M$ one can add witnesses to quantifiers ($\aleph_1$-witnesses for $Qx \phi(x)$) and iterate to obtain a model $N \supseteq A$ which is $L(Q)$-elementary inside $M$. More precisely define $A_0 = A$ and $A_{n+1}$ from $A_n$ as follows. Enumerate all $L(Q)$ formulas $(\phi_i(x, \bar a_i) : \bar a_i \in A_n)$. Now for each $i$ if $M \models Qx \phi_i(x, \bar a_i)$, then add $\aleph_1$-many solutions to $A_n$ otherwise if $M \models \exists x \phi_i(x, \bar a_n)$, then add a single solution to $A_n$. Call the resulting set $A_{n+1}$. Then $|A_{n+1}| \le \max(A_n, \aleph_1)$. Finally define $N = \bigcup_{n < \omega} A_n$. Now $N$ is an $L(Q)$-elementary substructure of $M$ (induction on complexity).
Edit: the above argument assumes that $Qx \phi(x)$ is interpreted as "there are uncountably many solutions to $\phi(x)$". This is more common in model theory, however the question interprets $Qx \phi(x)$ as "there are infinitely many solutions to $\phi(x)$". In this case the same argument shows that the Lowenheim number is $\aleph_0$ (just substitute $\aleph_1$ with $\aleph_0$).