A logic $L$ in a propositional language $\mathfrak{L}$ is said to be structural if $\Gamma\vdash_{L}\phi$, then $\sigma({\Gamma})\vdash_{L}\sigma({\phi})$ for each $\mathfrak{L}$ substitution $\sigma$ (an $\mathfrak{L}$ substitution being an endomorphism of $Fm_{\mathfrak{L}}$ and, the absolutely free algebra generated by $X$, say $X$ is countably infinite for this case. And $\Gamma\cup\{\phi\}\subseteq{Fm_{\mathfrak{L}}}$). How does this extend to the predicate case, i.e. a language $(\mathfrak{L},\mathfrak{P})$?
In http://www.carlesnoguera.cat/files/ch2.pdf (pg 182) the word logic is used in the same sense for predicate language (pg 177, 178 also carry relevant definitions) but it is not mentioned how the above property should be viewed. Basically what is the analouge of an $\mathfrak{L}$ substitution?