Infinitary logics $\mathcal{L}_{\infty\omega}$ is an extension of first-order logics such that
- $\bigvee\Phi \in \mathcal{L}_{\infty\omega}$ if $\Phi$ is a set of $\mathcal{L}_{\infty\omega}$-sentences
- $\bigwedge\Phi \in \mathcal{L}_{\infty\omega}$ if $\Phi$ is a set of $\mathcal{L}_{\infty\omega}$-sentences
with the obvious semantics of an infinite disjunction or conjunction.
Since other than "set" there is no restriction, infinitary logic is very powerful. Consider, that every Turing Machine $\mathfrak{T}$ is identified by a first-order sentence $\varphi_{\mathfrak T}$.
Then
$$ \bigvee_{\mathfrak T \in H} \varphi_{\mathfrak T},$$
where $H$ is the Halting Problem, is undecidable.
Now I have this definition of abstract logic:
A abstract logic is a pair $(\mathcal{L},\models_{\mathcal{L}})$ consisting of a set of $\mathcal{L}$-sentences for each signature $\tau$ and a mapping which associates a property $\mathcal P_\varphi$ of $\tau$-structures with each $\varphi \in \mathcal{L}[\tau]$.
My question: $\mathcal{L}_{\infty\omega}$ is an abstract logic?
I think yes, since $\mathcal{L}[\tau]$ is a set (right?) and there properties $\mathcal{P}_\varphi$ (not necessarly decidable) for each $\varphi \in \mathcal{L}[\tau]$.
I feel really on shaky ground with this stuff. Is it correct? Any ideas?