In modal logic, what is 'classical entailment'? I am familiar with the concepts of 'global entailment' and 'local entailment', and the distinction between them, but I am not familiar with 'classical entailment'. I also do not know a good textual reference for it, but I heard of it during a modal logic course lecture.
Similarly to how global entailment and local entailment are analogized to $∀x,P(x) → ∀x,Q(x)$ and $∀x(P(x) → Q(x))$, respectfully, what would 'classical entailment' be analogized to? How is classical entailment formally defined?

Update: I have met with my professor and apparently 'classical entailment' is defined as follows: where $p⊨_Cq$ notates that that $p$ classically entails $q$, the formal definition of classical entailment is
$p⊨_C q≔((p→q)=T)$
 A: I wish to unfold the question a bit further for those interested in the concepts involved. I believe also it is a good practice not to leave the answer part blank as possible.
The parallelism drawn between local and global entailments in modal logic and the statements $\forall x(P(x)\rightarrow Q(x))$ and $\forall xP(x)\rightarrow\forall xQ(x)$ is appropriate. As noted by the OP, by "classical entailment", the notion of logical consequence that we are familiar with from the standard first-order predicate logic must be meant. These may be said to be some extensional images of intensional matters.
Let us move towards the definitions of local and global entailments reviewing the context.
In the possible worlds semantics for modal logic, frames act as domain discourse. A frame $\mathcal{F}$, denoted by a pair $(W, R)$, is a relational structure composed of a set of worlds $W$ and a binary accessibility relation $R$ on $W$. The relation $R$ specifies for each world $w\in W$ which worlds $w'$ in $W$ are relevant to (viz., accessible from) $w$. Thus, if the state of affairs represented by $w'$ does not differ from $w$ in the right respect, $wRw'$ does not hold.
A model for a basic language of modal language (notice that not of a formula as we are familiar with from model theory) consists of a frame $(W, R)$ and a valuation function $V$
$$\mathcal{M} = \langle W, R, V\rangle$$
$V$ assigns to each atomic proposition $p$ a set of possible worlds $V(p)\subseteq W$ at which $p$ is true. Given a set of atomic propositions, every model $\mathcal{M}$ is said to be based on a frame $\mathcal{F}$. Reciprocally, a frame $\mathcal{F}$ induces a class of models $\mathsf{Mod}$.
I ought to be remark that, studying these concepts, it is a worthwhile exercise to play with diagrams like the following (from the Open Logic Project, which offers excellent sources for free):

We need to define what it is for a formula $\phi$ to be true at a world $w$ in a model $\mathcal{M}$. We denote truth at a world by
$$\mathcal{M}, w\Vdash\phi$$
and define it in the inductive fashion we are familiar from model theory:
If $\phi$ is an atomic proposition $p$, then $\mathcal{M}, w\Vdash p\iff w\in V(p)$
If $\phi\equiv\neg\psi$, then $\mathcal{M}, w\Vdash\phi\iff\mathcal{M}, w\nVdash\psi$
If $\phi\equiv(\psi_{1}\wedge\psi_{2})$, then $\mathcal{M}, w\Vdash\phi\iff\mathcal{M}, w\Vdash\psi_{1}\text{ and }\mathcal{M}, w\Vdash\psi_{2}$
If $\phi\equiv(\psi_{1}\vee\psi_{2})$, then $\mathcal{M}, w\Vdash\phi\iff\mathcal{M}, w\Vdash\psi_{1}\text{ or }\mathcal{M}, w\Vdash\psi_{2}$
If $\phi\equiv(\psi_{1}\rightarrow\psi_{2})$, then $\mathcal{M}, w\Vdash\phi\iff\mathcal{M}, w\nVdash\psi_{1}\text{ or }\mathcal{M}, w\Vdash\psi_{2}$
If $\phi\equiv\Box\psi$, then $\mathcal{M}, w\Vdash\phi\iff\forall w′\in W,\;\mathcal{M}, w′\Vdash\psi\text{ such that }wRw′$
If $\phi\equiv\Diamond\psi$, then $\mathcal{M}, w\Vdash\phi\iff\exists w′\in W,\;\mathcal{M}, w′\Vdash\psi\text{ such that }wRw′$
Now, we can define what it is to be true in a model $\mathcal{M}$ for a formula $\phi$. We denote truth in a model by
$$\mathcal{M}\Vdash\phi$$
and define it as
$$\mathcal{M}\Vdash\phi\iff\forall w\in W,\;\mathcal{M}, w\Vdash\phi$$
Suppose we have a class of models $\mathsf{Mod}=\{\mathcal{M}_{1}, \mathcal{M}_{2},\ldots\}$. We can define validity of a formula $\phi$ with respect to $\mathsf{Mod}$ as
$$\phi\text{ is valid in }\mathsf{Mod}\iff\forall\mathcal{M}\in\mathsf{Mod},\; \mathcal{M}\Vdash\phi$$
If $\phi$ is valid in $\mathsf{Mod}$, we denote this by $\mathsf{Mod}\vDash\phi$. We write only $\vDash\phi$ to indicate that $\phi$ is valid in the class of all models. We can also define validity of a formula $\phi$ in a frame $\mathcal{F}$. We denote this by $\mathcal{F}\vDash\phi$ and define it as truth in all the models $\mathcal{M}$ in the class of models $\mathsf{Mod}$ based on (viz., induced by) $\mathcal{F}$ and denote it by $\mathsf{Mod}\vDash\phi$.
So far, we have recorded the definitions of fundamental notions. Having grasped them, we can talk about local and global entailments (logical consequences). The conceptions fork into two groups: One is grounded in models, the other one in frames. First, we shall take up the definitions by models, thereafter those by frames.
For a model $\mathcal{M}$ and a set of formulas $\Gamma\cup\{\phi\}$, we say that $\Gamma$ locally entails $\phi$ in $\mathcal{M}$, or $\phi$ is a local semantic
consequence of $\Gamma$ in $\mathcal{M}$, when we have for any $w\in W$ (of $\mathcal{M}$)
$$\mathcal{M}, w\Vdash\Gamma\implies\mathcal{M}, w\Vdash\phi$$
We say that $\Gamma$ globally entails $\phi$ in $\mathcal{M}$, or $\phi$ is a global semantic consequence of $\Gamma$ in $\mathcal{M}$, when we have
$$\mathcal{M}\Vdash\Gamma\implies\mathcal{M}\Vdash\phi$$
Notice that, for local entailment, we employ the notion of truth at a world, and for global entailment, the notion of truth in a model. We denote these conceptions of local and global entailments by $\Gamma\vDash^{l}_{\mathcal{M}}\phi$ and $\Gamma\vDash^{g}_{\mathcal{M}}\phi$, respectively; usually the superscript $l$ is dropped.
Alternatively, the local and global entailments can be defined with respect to frames. Given a class of frames $\mathsf{F}$, for any $\mathcal{F}\in\mathsf{FR}$ and for any model $\mathcal{M}$ based on $\mathcal{F}$, we say that $\Gamma$ locally entails $\phi$ in $\mathcal{M}$, or $\phi$ is a local semantic consequence of $\Gamma$, when we have for any $w\in W$ (of $\mathcal{M}$)
$$\mathcal{M}, w\Vdash\Gamma\implies\mathcal{M}, w\Vdash\phi$$
We say that $\Gamma$ globally entails $\phi$ in $\mathcal{M}$, or $\phi$ is a global semantic consequence of $\Gamma$, when we have
$$\mathcal{M}\Vdash\Gamma\implies\mathcal{M}\Vdash\phi$$
In this case, we denote by $\Gamma\vDash^{l}_{\mathsf{F}}\phi$ and $\Gamma\vDash^{g}_{\mathsf{F}}\phi$, respectively. The superscript $l$ is usually dropped also here.
