On p127 in §3. Extensions by Definitions in VIII Syntactic Interpretations and Normal Forms In Ebbinghaus' Mathematical Logic: $S$ is a (non-logical) symbol set
Let $S$ be given, and let $s$ be a relation symbol, a function symbol or a con- stant with $s \notin S$. Furthermore, let $\Phi \subset L_0^S$ and let $\delta_s$ be an $S$-definition of $s$ in $\Phi$. We define, in the obvious way, the associated syntactic interpretation $I$ of $S' := S \cup \{s\}$ in $S$ to be the identity on the symbols from $S$ and
$$ (I(S') =)\phi_{S'}(v_0) := v_0 == v_0, \quad I(s) := \phi_s $$
(where $\phi_s$ is as in Definition 3.1 on p126). So $\Phi_I$ (If I am correct $\Phi_I$ is induced from $I$ but not from $\Phi$, c.f. p120) is logically equivalent to
- the empty set of sentences, if $s$ is a relation symbol,
- $\{\forall v_0, ... \forall v_{n-1} \exists^{=1} v_n \phi_f( v_0, ... ,v_{n-1}, v_n)\}$, if $s$ is an $n$-ary function symbol $f$,
- $\{\exists^{=1} v_0 \phi_c(v_0)\}$, if $s$ is a constant $c$.
Therefore, we have for every $S$-structure $\mathfrak{A}$ with $\mathfrak{A} \models \Phi$:
(*) $\mathfrak{A} \models \Phi_I$
(**) $(\mathfrak{A}, s^A) \models \Phi$ iff $\mathfrak{A}^{-I} = (\mathfrak{A}, s^A) $.
If I am correct, $\Phi$ and $\delta_s$ induce $I$, and then $I$ induces $\Phi_I$. What relation and difference are between $\Phi$ and $\Phi_I$, both of which are sets of $S$-sentences? Does (*) show a relation between the two?
In (**), what does $s^A$ mean? What is $A$ ? (I guess that because $\mathfrak{A}^{-I} |_S = \mathfrak{A}$ (c.f. p121), $s^A$ may mean $I(s)$, but I am not sure what $A$ and $s^A$ mean. I doubt $A$ is the domain of a structure, specifically $I$, because $I$ is a syntactic interpretation, which I am not sure is an interpretation (and therefore has a domain), and whose domain if it has one is unclear to me and at least not chosen as some $$ on p120. )
Thanks.
It means the interpretation (pg. 28). $A$ is the underlying set and sometimes it’s identified with the whole structure, like how we refer to a group $G$ (which should include its composition function, plus inverse and identity by some definitions) but also use that as the underlying set too, such as when one writes “$g\in G$“.
On pg 29, $s^A$ is another way of writing $a(s)$, where $a$ is the mapping in a structure.