## φ-Definability in α structures

At the start of §2.2 Reduction and Determination, Hellman introduces the notion of definability:

An $n$-place predicate $\textup{P}$ is definable in terms of a vocabulary $\phi$ in $\alpha$ structures if, and only if, there is a finite sentence $\textup{A}$ containing no nonlogical terms not in $\phi$ and with occurrences of $n$ distinct variables, $x_{1},\dots, x_{n}$, such that every structure in $\alpha$ models $(\forall x_{1})\dots (\forall x_{n}) (\textup{P}x_{1} \dots x_{n} \leftrightarrow \textup{A})$.

$\textup{P}$ is not just coextensive with a term in the $\phi$ vocabulary.  What the claim of definability says is that there is a term, $\textup{A}$, in the $\phi$ vocabulary such that the $n$-place predicate $\textup{P}$ is coextensive with it in every structure in $\alpha$.

Hellman notes that to say that a term is definable in a given vocabulary is not to say that the term is synonymous with a term in that vocabulary.  While the notion of definability can be spelled out explicitly, the notion of synonymy cannot.  Also, the coextensiveness at issue here involves definability of terms in $\phi$ terms over the structures in $\alpha$ and since each structure $\mathfrak{A}$ in $\alpha$ is a model of the laws of science,  Hellman claims that definability is a kind of lawlike coextensiveness between terms terms.

From here, Hellman will go on to discuss reducibility that holds when all the terms in the vocabulary being reduced are definable in the reducing vocabulary.