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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. |
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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. |
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The purported fact that geometric theories formulated in terms of points and geometric theories formulated in terms of lines are “equally correct” is often invoked in arguments for conceptual relativity, in particular by Putnam and Goodman. We discuss a few notions of equivalence between first-order theories, and we then demonstrate a precise sense in which this purported fact is true. We argue, however, that this fact does not undermine metaphysical realism. |
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For simplicity, most of the literature introduces the concept of definitional equivalence only for disjoint languages. In a recent paper, Barrett and Halvorson introduce a straightforward generalization to non-disjoint languages 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 another formalization of definitional equivalence due to Andréka and Németi which is equivalent to the Barrett–Halvorson (...) |
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According to Stewart Shapiro's coherence principle, structures exist whenever they can be coherently described. I argue that Shapiro's attempts to justify this principle are circular, as he relies on criticisms of modal nominalism which presuppose the coherence principle. I argue further that when the coherence principle is not presupposed, his reasoning more strongly supports modal nominalism than ante rem structuralism. |
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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 (...) |
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Glymour and Quine propose two different formal criteria for theoretical equivalence. In this paper we examine the relationships between these criteria. |
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In the literature, there have been several methods and definitions for working out whether two theories are “equivalent” or not. In this article, we do something subtler. We provide a means to measure distances 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 (...) |
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We characterize Morita equivalence of theories in the sense of Johnstone in terms of a new syntactic notion of a common definitional extension developed by Barrett and Halvorson for cartesian, regular, coherent, geometric and first-order theories. This provides a purely syntactic characterization of the relation between two theories that have equivalent categories of models naturally in any Grothendieck topos. |
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