Faithful and Full Translations Between Logics

In computer science, an important application of translations includes comparing the expressive power among logics to achieve reasoning tasks of a logic in another defined logic. General properties of translations found in the literature do not give a comprehensive study on semantically translations and the preservation of the unsatisfiability. To preserve the satisfiability and the unsatisfiability of formulas, the definition of faithful and full translation is given, in this paper, and connections between the faithful and full translation, and other definitions of translations in the literature are discussed. Some properties of logics, such as the soundness, the completeness, the decidability, the compactness, the logical equivalence of formulas and the elementary equivalence of models, which are characterized by the existence of faithful and full translations between logics are also studied. By definition of faithful and full translation, the concept of synonymous logics is introduced and the proof for the equivalence of the synonymous relation is also given.