Decidable first order logics

The logic $\mathcal L(T)$ of arbitrary first order theory $T$ is the set of predicate formulae, provable in $T$ under every interpretation into the language of $T$. It is proved, that for the theory of equation and the theory of dense linear order without minimal and maximal elements $\mathcal L(T)$ is decidable, but can not be axiomatized by any set of schemes with restricted arity. On the other hand, for most of the expressively strong theories $\mathcal L(T)$ turn out to be undecidable.