For simplicity, most of the literature introduces the concept of definitional equivalence only to languages with disjoint signatures. In a recent paper, Barrett and Halvorson introduce a straightforward generalization to languages with non-disjoint signatures and they show that their generalization is not equivalent to intertranslatability in general. In this paper, we show that their generalization is not transitive and hence it is not an equivalence relation. Then we introduce the Andréka and Németi generalization as one of the many equivalent formulations for languages with disjoint signatures. We show that the Andréka-Németi generalization is the smallest equivalence relation containing the Barrett-Halvorson generalization and it is equivalent to intertranslatability even for languages with non-disjoint signatures. Finally, we investigate which definitions for definitional equivalences remain equivalent when we generalize them for theories with nondisjoint signatures. 2. M |= (x = y)[e] holds if e(x) = e(y) holds, 3. M |= ¬ϕ[e] holds if M |= ϕ[e] does not hold, 4. M |= (ψ ∧ θ)[e] holds if both M |= ψ[e] and M |= θ[e] hold, 5. M |= ∃yψ [e] holds if there is an element b ∈ M , such that M |= ψ[e ′ ] if e ′ (y) = b and e ′ (x) = e(x) if x = y.Letx be the list of all free variables of ϕ and letā be a list of elements of M with the same number of elements asx. Then M |= ϕ[ā] iff M satsfies 6 ϕ for all (or equivalently some) evaluation e of variables for which e(x) =ā, i.e., variables inx are mapped to elements of M inā in order. In case ϕ is a sentence, its truth does not depend on evaluation of variables. So that ϕ is true in M is denoted by M |= ϕ. For theory T , M |= T abbreviates that M |= ϕ for all ϕ ∈ T .Remark 2. We will use ϕ ∨ ψ as an abbreviation for ¬(¬ϕ ∧ ¬ψ), ϕ → ψ for ¬ϕ ∨ ψ, ϕ ↔ ψ for (ϕ → ψ) ∧ (ψ → ϕ) and ∀x(ϕ) for ¬ ∃x(¬ϕ) . Definition 7. M od(T ) is the class of models of theory T , M od(T ) def = {M : M |= T }. Definition 8. Two theories T 1 and T 2 are logically equivalent, in symbols

The aim of this paper is to present a new logic-based understanding of the connection between classical kinematics and relativistic kinematics.We show that the axioms of special relativity can be interpreted in the language of classical kinematics. This means that there is a logical translation function from the language of special relativity to the language of classical kinematics which translates the axioms of special relativity into consequences of classical kinematics.We will also show that if we distinguish a class of observers (representing observers stationary with respect to the "Ether") in special relativity and exclude the non-slower-than light observers from classical kinematics by an extra axiom, then the two theories become definitionally equivalent (i.e., they become equivalent theories in the sense as the theory of lattices as algebraic structures is the same as the theory of lattices as partially ordered sets).Furthermore, we show that classical kinematics is definitionally equivalent to classical kinematics with only slower-than-light inertial observers, and hence by transitivity of definitional equivalence that special relativity theory extended with "Ether" is definitionally equivalent to classical kinematics.So within an axiomatic framework of mathematical logic, we explicitly show that the transition from classical kinematics to relativistic kinematics is the knowledge acquisition that there is no "Ether", accompanied by a redefinition of the concepts of time and space.

In the literature, there have been several methods and definitions for working out whether two theories are “equivalent” (essentially the same) or not. In this article, we do something subtler. We provide a means to measure distances (and explore connections) between formal theories. We introduce two natural notions for such distances. The first one is that of axiomatic distance, but we argue that it might be of limited interest. The more interesting and widely applicable notion is that of conceptual distance which measures the minimum number of concepts that distinguish two theories. For instance, we use conceptual distance to show that relativistic and classical kinematics are distinguished by one concept only.