Chapter XVII in [[Barwise, Feferman - Model-theoretic logics]]
# Abstract
Simply put, an abstract logic is determined by two predicates of set theory, " $x \in \mathscr{L}$ " and " $y \vDash_{\mathscr{L}} x$." The general problem to be considered in this chapter is as follows: What can we say of the model-theoretic properties of $\mathscr{L}$ if we known how the predicates " $x \in \mathscr{L}$ " and " $y \vDash_{\mathscr{L}} x$ " behave as predicates of set theory?
# 1. Model-Theoretic Definability Criteria
## 1.1. Adequacy to Truth
>[!note] Notation
> For any set $a$, let $\mu_a(z)$ be the following (possibly) infinitary formula in the vocabulary $\tau_{\text {set }}=\{\in\}$ :
> $
> \mu_a(x)=\forall y\left(y \in x \leftrightarrow \bigvee_{b \in a} \mu_b(y)\right)
> $
> Now, let
> $
> \pi_a(x)=\mu_a(x) \wedge \bigwedge_{b \in \operatorname{TC}(a)} \exists y \mu_b(y) .
> $
>[!note] Remark
>Intuitively $\mu_{a}(x)$ means that "$x$ has the same structure as $a
quot; as long as all required elements are present. $\pi_{a}(x)$ says that this is in fact the case.
>Formally: if $\mathfrak{B}=(B, E)$ is a model of the axiom of extensionality, $\mathfrak{B}_0$ the well-founded part of $\mathfrak{B}$, and $\mathfrak{A}$ the transitive collapse of $\mathfrak{B}_0$ via $i: \mathfrak{B}_0 \rightarrow \mathfrak{U}$, then:
> $\mathfrak{B} \vDash \pi_a(x) \quad$ if and only if $\quad x \in B_0, a \in A$ and $i(x)=a$.
>[!info] Definition
>We say that *the syntax of $\mathscr{L}$ is represented on $A$* if $A$ is a transitive set closed under primitive recursive set functions, $\mathscr{L}(\tau) \subseteq A$ for all $\tau$ considered, and
> $
> \operatorname{Mod}\left(\pi_a\right) \in \mathrm{EC}_{\mathscr{L}\left[\tau_{\mathrm{set}}\right]} \text { for } \quad a \in A
> $
>[!info] Definition
>We say that a logic $\mathscr{L}$ is *adequate to truth in* a logic $\mathscr{L}^{\prime}$ if for every $\tau$ there is $\tau^{+}=\left[\tau, \tau_{\text {set }}, \mathrm{Th}, \tau^{\prime}\right]$ and $\theta \in \mathscr{L}\left[\tau^{+}\right]$ such that for every $\mathfrak{M} \in \operatorname{Str}[\tau]$, the following conditions hold:
> (AT1) $\left(\mathfrak{M}, \mathfrak{A}', \mathrm{Th}_{\mathscr{L}'}(\mathfrak{M}), \mathfrak{N}\right) \vDash_{\mathscr{L}} \theta$ for some $\mathfrak{N}$.
> (AT2) If $(\mathfrak{M}, \mathfrak{B}, T, \mathfrak{N}) \vDash_{\mathscr{L}} \theta \wedge \pi_{\varphi}(b)$, then $b \in T$ if and only if $\mathfrak{M} \vDash_{\mathscr{L}^{\prime}} \varphi$, whenever $\varphi \in A^{\prime}$ and $b \in B$.
>
>Usually $\theta$ will be of the form $\eta \wedge \forall x(\operatorname{Th}(x) \leftrightarrow \exists s S(x, s))$ where $S\in \tau'$ (the satisfaction predicate) and $\eta$ an $\mathscr{L}$-sentence (in the language containing $S$) embodying the truth definition of $\vDash_{\mathscr{L}'}$
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