What Do Symmetries Tell Us About Structure?

Mathematicians, physicists, and philosophers of physics often look to the symmetries of an object for insight into the structure and constitution of the object. My aim in this paper is to explain why this practice is successful. In order to do so, I present a collection of results that are closely related to (and in a sense, generalizations of) Beth’s and Svenonius’ theorems

Are Newtonian Gravitation and Geometrized Newtonian Gravitation Theoretically Equivalent?

I argue that a criterion of theoretical equivalence due to Clark Glymour [Nous 11(3), 227-251 (1977)] does not capture an important sense in which two theories may be equivalent. I then motivate and state an alternative criterion that does capture the sense of equivalence I have in mind. The principal claim of the paper is that relative to this second criterion, the answer to the question posed in the title is "yes", at least on one natural understanding of Newtonian gravitation.Comment: 27 page

Mutual Translatability, Equivalence, and the Structure of Theories

This paper presents a simple pair of first-order theories that are not definitionally (nor Morita) equivalent, yet are mutually conservatively translatable and mutually 'surjectively' translatable. We use these results to clarify the overall geography of standards of equivalence and to show that the structural commitments that theories make behave in a more subtle manner than has been recognized

Extension, Translation, and the Cantor-Bernstein Property

The purpose of this paper is to examine in detail a particularly interesting pair of first-order theories. In addition to clarifying the overall geography of notions of equivalence between theories, this simple example yields two surprising conclusions about the relationships that theories might bear to one another. In brief, we see that theories lack both the Cantor-Bernstein and co-Cantor-Bernstein properties

Testing definitional equivalence of theories via automorphism groups

Two first-order logic theories are definitionally equivalent if and only if there is a bijection between their model classes that preserves isomorphisms and ultraproducts (Theorem 2). This is a variant of a prior theorem of van Benthem and Pearce. In Example 2, uncountably many pairs of definitionally inequivalent theories are given such that their model categories are concretely isomorphic via bijections that preserve ultraproducts in the model categories up to isomorphism. Based on these results, we settle several conjectures of Barrett, Glymour and Halvorson

Morita Equivalence

Logicians and philosophers of science have proposed various formal criteria for theoretical equivalence. In this paper, we examine two such proposals: definitional equivalence and categorical equivalence. In order to show precisely how these two well-known criteria are related to one another, we investigate an intermediate criterion called Morita equivalence.Comment: 30 page

Bases for Structures and Theories II

In Part I of this paper, I assumed we begin with a (relational) signature $P = \{P_i\}$ and the corresponding language $L_P$, and introduced the following notions: a \emph{definition system} $d_{\Phi}$ for a set of new predicate symbols $Q_i$, given by a set $\Phi = \{\phi_i\}$ of defining $L_P$-formulas (these definitions have the form: $\forall \overline{x}(Q_i(x) \iff \phi_i)$); a corresponding \emph{translation function} $\tau_{\Phi}: L_Q \to L_P$; the corresponding \emph{definitional image operator} $D_{\Phi}$, applicable to $L_P$-structures and $L_P$-theories; and the notion of \emph{definitional equivalence} itself: for structures $A + d_{\Phi} \equiv B + d_{\Theta}$; for theories, $T_1 + d_{\Phi} \equiv T_2 + d_{\Theta}$. Some results relating these notions were given, ending with two characterizations for definitional equivalence. In this second part, we explain the notion of a \emph{representation basis}. Suppose a set $\Phi = \{\phi_i\}$ of $L_P$-formulas is given, and $\Theta = \{\theta_i\}$ is a set of $L_Q$-formulas. Then the original set $\Phi$ is called a \emph{representation basis} for an $L_P$-structure $A$ with inverse $\Theta$ iff an inverse explicit definition \forall \x(P_i(\overline{x}) \iff \theta_i) is true in $A + d_{\Phi}$, for each $P_i$. Similarly, the set $\Phi$ is called a \emph{representation basis} for a $L_P$-theory $T$ with inverse $\Theta$ iff each explicit definition $\forall \overline{x}(P_i(\overline{x}) \iff \theta_i)$ is provable in $T + d_{\Phi}$. Some results about representation bases, the mappings they induce and their relationship with the notion of definitional equivalence are given. In particular, we show that $T_1$ (in $L_P$) is definitionally equivalent to $T_2$ (in $L_Q$), with respect to $\Phi$ and $\Theta$, if and only if $\Phi$ is a \emph{representation basis} for $T_1$ with inverse $\Theta$ and $T_2 \equiv D_{\Phi}T_1$

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