I'm afraid this question might be too localized, but I have no better place to ask it:

In the section Classes which interpret any structure of his *Model Theory* Hodges shows how each $L$-structure $B$ can be converted into a graph $A = \Delta(B)$, $L$ a first order language with finite signature.

He associates every element of $B$ with a so-called 5-tagged element $a$ of $A$. Likewise he associates the *i*-th symbol of the signature $S_i$ with a (*i*+6)-tagged element of $A$. For further details see Hodges, p. 228 ff.

The only question I have at this point is: Why isn't it enough to associate all the relation symbols with different 6-tagged (and 6-tagged *only*) elements of $A$? Why is it necessary - for his argument's sake - to distinguish them further by their "tag count"?

`$B\models\forall x\,P(x)\land\neg\forall x\,Q(x)$`

, we’d better make sure that the gadgets which are used in $A$ to read off the values of $P$ and $Q$ cannot be swapped by an automorphism, as otherwise we couldn’t interpret $\forall x\,P(x)$ and $\forall x\,Q(x)$ by a different formula. – Emil Jeřábek Jun 9 '11 at 18:03