Geodesic
Decidable first order logics
Fundamentalʹnaâ i prikladnaâ matematika, Tome 4 (1998) no. 2, pp. 733-749

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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. 
@article{FPM_1998_4_2_a18,
     author = {R. \`E. Yavorskii},
     title = {Decidable first order logics},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {733--749},
     publisher = {mathdoc},
     volume = {4},
     number = {2},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1998_4_2_a18/}
}
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R. È. Yavorskii. Decidable first order logics. Fundamentalʹnaâ i prikladnaâ matematika, Tome 4 (1998) no. 2, pp. 733-749. http://geodesic.mathdoc.fr/item/FPM_1998_4_2_a18/