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.

I review the philosophical literature on the question of when two physical theories are equivalent. This includes a discussion of empirical equivalence, which is often taken to be necessary, and sometimes taken to be sufficient, for theoretical equivalence; and “interpretational” equivalence, which is the idea that two theories are equivalent just in case they have the same interpretation. It also includes a discussion of several formal notions of equivalence that have been considered in the recent philosophical literature, including (generalized) definitional (...) equivalence and categorical equivalence. The article concludes with a brief discussion of the relationship between equivalence and duality. The article is in two parts; this is Part 1, which addresses empirical equivalence and definitional equivalence. (shrink)

In his book Philosophy of Logic, Putnam (1971) presents a short argument which reads like—and indeed, can be reconstructed as—a formal proof that a nominalistic physics is impossible. The aim of this paper is to examine Putnam’s proof and show that it is not compelling. The precise way in which the proof fails yields insight into the relation that a nominalistic physics should bear to standard physics and into Putnam’s indispensability argument.

I review the philosophical literature on the question of when two physical theories are equivalent. This includes a discussion of empirical equivalence, which is often taken to be necessary, and sometimes taken to be sufficient, for theoretical equivalence; and “interpretational” equivalence, which is the idea that two theories are equivalent just in case they have the same interpretation. It also includes a discussion of several formal notions of equivalence that have been considered in the recent philosophical literature, including (generalized) definitional (...) equivalence and categorical equivalence. The article concludes with a brief discussion of the relationship between equivalence and duality. The article is in two parts; this is Part 2, which addresses categorical equivalence, interpretational equivalence, and duality. (shrink)

I review the philosophical literature on the question of when two physical theories are equivalent. This includes a discussion of empirical equivalence, which is often taken to be necessary, and sometimes taken to be sufficient, for theoretical equivalence; and "interpretational" equivalence, which is the idea that two theories are equivalent just in case they have the same interpretation. It also includes a discussion of several formal notions of equivalence that have been considered in the recent philosophical literature, including definitional equivalence (...) and categorical equivalence. The article concludes with a brief discussion of the relationship between equivalence and duality. (shrink)

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.

Quine often argued for a simple, untyped system of logic rather than the typed systems that were championed by Russell and Carnap, among others. He claimed that nothing important would be lost by eliminating sorts, and the result would be additional simplicity and elegance. In support of this claim, Quine conjectured that every many-sorted theory is equivalent to a single-sorted theory. We make this conjecture precise, and prove that it is true, at least according to one reasonable notion of theoretical (...) equivalence. Our clarification of Quine’s conjecture, however, exposes the shortcomings of his argument against many-sorted logic. (shrink)

Glymour and Quine propose two different formal criteria for theoretical equivalence. In this paper we examine the relationships between these criteria.

Sometimes structures or theories are formulated with different sets of primitives and yet are definitionally equivalent. In a sense, the transformations between such equivalent formulations are rather like basis transformations in linear algebra or co-ordinate transformations in geometry. Here an analogous idea is investigated. Let a relational signature \ be given. For a set \ of \-formulas, we introduce a corresponding set \ of new relation symbols and a set of explicit definitions of the \ in terms of the \. (...) This is called a definition system, denoted \. A definition system \ determines a translation function\. Any \-structure A can be uniquely definitionally expanded to a model \, called \. The reduct \ to the Q-symbols is called the definitional image\ of A. Likewise, a theory T in \ may be extended a definitional extension \; the restriction of this extension \ to \ is called the definitional image\ of T. If \ and \ are in disjoint signatures and \, we say that \ and \ are definitionally equivalent and \). Some results relating these notions are given, culminating in two characterization theorems for the definitional equivalence of structures and theories. (shrink)

In _On Empirically Equivalent Systems of the World_ from 1975, Quine formulated a thesis of underdetermination roughly to the effect that every scientific theory has an empirically equivalent but logically incompatible rival, one that cannot be discarded merely as a terminological variant of the former. For Quine, the truth of this thesis was an open question. If true, some would argue that it undermines any belief in scientific theories that is based purely on their empirical success. But despite its potential (...) significance, surprisingly little has been done by way of establishing or refuting it. My aim is to establish the thesis for as large a class of theories as possible. Under various precisifications of the concepts involved, I show that it holds for all consistent and recursively axiomatizable theories that postulate infinitely many theoretical entities. (shrink)

This paper discusses the general problem of translation functions between logics, given in axiomatic form, and in particular, the problem of determining when two such logics are "synonymous" or "translationally equivalent." We discuss a proposed formal definition of translational equivalence, show why it is reasonable, and also discuss its relation to earlier definitions in the literature. We also give a simple criterion for showing that two modal logics are not translationally equivalent, and apply this to well-known examples. Some philosophical morals (...) are drawn concerning the possibility of having two logical systems that are "empirically distinct" but are both translationally equivalent to a common logic. (shrink)

This paper began with the simple object of finding an account that allowed us to compare incompatible false theories. This we achieved with ρ. But that relation is language — or interest — dependent. ρ' is free from this limitation; though thus liberated it is perhaps rather unconcerned about what is true, and further fails to deliver certain intuitive comparisons. Whether ρ is to be preferred to ρ' or vice versa, seems to me a largely fruitless question: In fact it (...) might be that as we seek to free the idea of verisimilitude from any dependence on language, interests or problems, so we gradually collapse it to nothing more than logical power. (shrink)