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)

Category theory has become central to certain aspects of theoretical physics. Bain has recently argued that this has significance for ontic structural realism. We argue against this claim. In so doing, we uncover two pervasive forms of category-theoretic generalization. We call these ‘generalization by duality’ and ‘generalization by categorifying physical processes’. We describe in detail how these arise, and explain their significance using detailed examples. We show that their significance is two-fold: the articulation of high-level physical concepts, and the generation (...) of new models. 1 Introduction2 Categories and Structuralism 2.1 Categories: abstract and concrete 2.2 Structuralism: simple and ontic3 Bain’s Two Strategies 3.1 A first strategy for defending Objectless 3.2 A second strategy for defending Objectless4 Two Forms of Categorical Generalization 4.1 Generalization by duality 4.2 Generalization by categorification 4.3 The role of category theory in physics5 Conclusion. (shrink)

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.

One of the central problems in the philosophy of psychology is an updated version of the old mind-body problem: how levels of theories in the behavioral and brain sciences relate to one another. Many contemporary philosophers of mind believe that cognitive-psychological theories are not reducible to neurological theories. However, this antireductionism has not spawned a revival of dualism. Instead, most nonreductive physicalists prefer the idea of a one-way dependence of the mental on the physical.In Psychoneural Reduction, John Bickle presents a (...) new type of reductionism, one that is stronger than one-way dependency yet sidesteps the arguments that sank classical reductionism. Although he makes some concessions to classical antireductionism, he argues for a relationship between psychology and neurobiology that shares some of the key aims, features, and consequences of classical reductionism. Parts of Bickle's "new wave" reductionism have emerged piecemeal over the past two decades; this is his first comprehensive statement and defense of it to appear. (shrink)

According to the semantic view of scientific theories, theories are classes of models. I show that this view -- if taken seriously as a formal explication -- leads to absurdities. In particular, this view equates theories that are truly distinct, and it distinguishes theories that are truly equivalent. Furthermore, the semantic view lacks the resources to explicate interesting theoretical relations, such as embeddability of one theory into another. The untenability of the semantic view -- as currently formulated -- threatens to (...) undermine scientific structuralism. (shrink)

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

v. 1. Language.--v. 2. Mythical thought.--v. 3. The phenomenology of knowledge.--v. 4. The metaphysics of symbolic forms.

I review some recent work on applications of category theory to questions concerning theoretical structure and theoretical equivalence of classical field theories, including Newtonian gravitation, general relativity, and Yang-Mills theories.

As a prolegomenon to understanding the sense in which dualities are theoretical equivalences, we investigate the intuitive `equivalence' of hyper-regular Lagrangian and Hamiltonian classical mechanics. We show that the symplectification of these theories provides a sense in which they are isomorphic, and mutually and canonically definable through an analog of `common definitional extension'.

I explain the difficulty of making various concepts of and relating to probability precise, rigorous and physically significant when attempting to apply them in reasoning about objects living in infinite-dimensional spaces, working through many examples from cosmology. I focus on the relation of topological to measure-theoretic notions of and relating to probability, how they diverge in unpleasant ways in the infinite-dimensional case, and are even difficult to work with on their own. Even in cases where an appropriate family of spacetimes (...) is finite-dimensional, and so admits a measure of the relevant sort, however, it is always the case that the family is not a compact topological space, and so does not admit a physically significant, well behaved probability measure. Problems of a different but still deeply troubling sort plague arguments about likelihood in that context, which I also discuss. I conclude that most standard forms of argument used in cosmology to estimate the likelihood of the occurrence of various properties or behaviors of spacetimes have serious mathematical, physical and conceptual problems. (shrink)

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.

From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categories of syntax and semantics, mediated by homming into a common dualizing object, in this (...) case 2.In the present work, we generalize the entire arrangement from propositional to first-order logic, using a representation result of Butz and Moerdijk. Boolean algebras are replaced by Boolean categories presented by theories in first-order logic, and spaces of models are replaced by topological groupoids of models and their isomorphisms. A duality between the resulting categories of syntax and semantics, expressed primarily in the form of a contravariant adjunction, is established by homming into a common dualizing object, now Sets, regarded once as a boolean category, and once as a groupoid equipped with an intrinsic topology.The overall framework of our investigation is provided by topos theory. Direct proofs of the main results are given, but the specialist will recognize toposophical ideas in the background. Indeed, the duality between syntax and semantics is really a manifestation of that between algebra and geometry in the two directions of the geometric morphisms that lurk behind our formal theory. Along the way, we give an elementary proof of Butz and Moerdijkʼs result in logical terms. (shrink)

Of the various notions of reduction in the logical literature, relative interpretability in the sense of Tarskiet al. [6] appears to be the central one. In the present note, this syntactic notion is characterized semantically, through the existence of a suitable reduction functor on models. The latter mathematical condition itself suggests a natural generalization, whose syntactic equivalent turns out to be a notion of interpretability quite close to that of Ershov [1], Szczerba [5] and Gaifman [2].