On Generalization of Definitional Equivalence to Languages with Non-Disjoint Signatures

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 non-disjoint signatures

On Generalization of Definitional Equivalence to Languages with Non-Disjoint Signatures

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 non-disjoint signatures

On Generalization of Definitional Equivalence to Languages with Non-Disjoint Signatures

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 non-disjoint signatures

On Generalization of Definitional Equivalence to Languages with Non-Disjoint Signatures

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 non-disjoint signatures

Distances between formal theories

In the literature, there have been several methods and definitions for working out if two theories are "equivalent" (essentially the same) or not. In this article, we do something subtler. We provide means to measure distances (and explore connections) between formal theories. We define two main notions for such distances. A natural definition is that of axiomatic distance, but we argue that it might be of limited interest. The more interesting and widely applicable definition is that of conceptual distance which measures the minimum number of concepts that separate two theories. For instance, we use conceptual distance to show that relativistic and classical kinematics are distinguished by one concept only. We also develop further notions of distance, and we include a number of suggestions for applying and extending our project. We end with a philosophical discussion of the significance of these approaches

On variable non-dependence of first-order formulas

In this paper, we introduce a concept of non-dependence of variables in formulas. A formula in first-order logic is non-dependent of a variable if the truth value of this formula does not depend on the value of that variable. This variable non-dependence can be subject to constraints on the value of some variables which appear in the formula, these constraints are expressed by another first-order formula. After investigating its basic properties, we apply this concept to simplify convoluted formulas by bringing out and discarding redundant nested quantifiers. Such convoluted formulas typically appear when one uses a translation function interpreting a theory into another

Distances Between Formal Theories

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. © 2019 BMJ Publishing Group. All rights reserved

Distances between formal theories

In the literature, there have been several methods and definitions for working out if two theories are "equivalent" (essentially the same) or not. In this article, we do something subtler. We provide means to measure distances (and explore connections) between formal theories. We define two main notions for such distances. A natural definition is that of axiomatic distance, but we argue that it might be of limited interest. The more interesting and widely applicable definition is that of conceptual distance which measures the minimum number of concepts that separate two theories. For instance, we use conceptual distance to show that relativistic and classical kinematics are distinguished by one concept only. We also develop further notions of distance, and we include a number of suggestions for applying and extending our project. We end with a philosophical discussion of the significance of these approaches