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MORITA EQUIVALENCE

Published online by Cambridge University Press:  29 July 2016

THOMAS WILLIAM BARRETT  and
HANS HALVORSON
THOMAS WILLIAM BARRETT*
Affiliation:
Department of Philosophy, Princeton University
HANS HALVORSON*
Affiliation:
Department of Philosophy, Princeton University
*
*DEPARTMENT OF PHILOSOPHY PRINCETON UNIVERSITY PRINCETON, NJ, USA E-mail: thomaswb@princeton.edu
† DEPARTMENT OF PHILOSOPHY PRINCETON UNIVERSITY PRINCETON, NJ, USA E-mail: hhalvors@princeton.edu
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Abstract

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.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 9 , Issue 3 , September 2016 , pp. 556 - 582
Copyright
Copyright © Association for Symbolic Logic 2016 

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References

BIBLIOGRAPHY

Adámek, J., Sobral, M., & Sousa, L. (2006). Morita equivalence of many-sorted algebraic theories. Journal of Algebra, 297(2), 361–371.CrossRefGoogle Scholar
Andréka, H., Madarász, J. X., & Németi, I. (2005). Mutual definability does not imply definitional equivalence, a simple example. Mathematical Logic Quarterly, 51(6), 591–597.CrossRefGoogle Scholar
Andréka, H., Madaraśz, J. X., & Németi, I. (2008). Defining new universes in many-sorted logic. Mathematical Institute of the Hungarian Academy of Sciences, Budapest, 93.Google Scholar
Artigue, M., Isambert, E., Perrin, M., & Zalc, A. (1978). Some remarks on bicommutability. Fundamenta Mathematicae, 101(3), 207–226.CrossRefGoogle Scholar
Awodey, S. (2010). Category Theory. Oxford: Oxford University Press.Google Scholar
Awodey, S. & Forssell, H. (2010). First-order logical duality. Manuscript.Google Scholar
Barrett, T. W. (2015). On the structure of classical mechanics. The British Journal for the Philosophy of Science, 66(4), 801–828.CrossRefGoogle Scholar
Barrett, T. W. & Halvorson, H. (2015a). Glymour and Quine on theoretical equivalence. the Journal of Philosophical Logic. Forthcoming.Google Scholar
Barrett, T. W. & Halvorson, H. (2015b). Quine’s conjecture on many-sorted logic. Manuscript. URL: http://philsci-archive.pitt.edu/11660/.Google Scholar
Borceux, F. (1994). Handbook of Categorical Algebra, Vol. 1. Cambridge: Cambridge University Press.Google Scholar
Coffey, K. (2014). Theoretical equivalence as interpretative equivalence. The British Journal for the Philosophy of Science, 65(4), 821–844.CrossRefGoogle Scholar
Curiel, E. (2014). Classical mechanics is Lagrangian; it is not Hamiltonian. The British Journal for the Philosophy of Science, 65(2), 269–321.CrossRefGoogle Scholar
de Bouvére, K. L. (1965). Synonymous theories. Symposium on the Theory of Models. Amsterdam: North-Holland Publishing Company, pp. 402–406.Google Scholar
Dukarm, J. J. (1988). Morita equivalence of algebraic theories. Colloquium Mathematicae, 55(1), 11–17.CrossRefGoogle Scholar
Eilenberg, S. & Mac Lane, S. (1942). Group extensions and homology. Annals of Mathematics, 757–831.CrossRefGoogle Scholar
Eilenberg, S. & Mac Lane, S. (1945). General theory of natural equivalences. Transactions of the American Mathematical Society, 58(2), 231–294.CrossRefGoogle Scholar
Freyd, P. (1964). Abelian categories. New York: Harper and Row.Google Scholar
Friedman, H. M. & Visser, A. (2014). When bi-interpretability implies synonymy. Logic Group Preprint Series, 320, 1–19.Google Scholar
Glymour, C. (1971). Theoretical realism and theoretical equivalence. PSA 1970. Springer, pp. 275–288.CrossRefGoogle Scholar
Glymour, C. (1977). The epistemology of geometry. Noûs, 11, 227–251.CrossRefGoogle Scholar
Glymour, C. (1980). Theory and Evidence. Princeton, NJ: Princeton University Press.Google Scholar
Glymour, C. (2013). Theoretical equivalence and the semantic view of theories. Philosophy of Science, 80(2), 286–297.CrossRefGoogle Scholar
Halvorson, H. (2011). Natural structures on state space. Manuscript.Google Scholar
Halvorson, H. (2012). What scientific theories could not be. Philosophy of Science, 79(2), 183–206.CrossRefGoogle Scholar
Halvorson, H. (2013). The semantic view, if plausible, is syntactic. Philosophy of Science, 80(3), 475–478.CrossRefGoogle Scholar
Halvorson, H. (2015). Scientific theories. Oxford Handbooks Online. Oxford, UK: Oxford University Press, ForthcomingGoogle Scholar
Hodges, W. (2008). Model Theory. Cambridge University Press.Google Scholar
Johnstone, P. T. (2003). Sketches of an Elephant. Oxford: Oxford University Press.Google Scholar
Kanger, S. (1968). Equivalent theories. Theoria, 34(1), 1–6.CrossRefGoogle Scholar
Knox, E. (2014). Newtonian spacetime structure in light of the equivalence principle. The British Journal for the Philosophy of Science, 65(4), 863–880.CrossRefGoogle Scholar
Kuratowski, K. (1966). Topology, Vol. 1. Warsaw and New York: Academic Press.Google Scholar
Mac Lane, S. (1948). Groups, categories, and duality. Proceedings of the National Academy of Sciences, 34(6), 263–267.CrossRefGoogle Scholar
Mac Lane, S. (1971). Categories for the working mathematician. New York: Springer.CrossRefGoogle Scholar
Makkai, M. (1991). Duality and definability in first order logic, Vol. 503. Providence, RI: American Mathematical Society.Google Scholar
Manzano, M. (1996). Extensions of First Order Logic. Cambridge University Press.Google Scholar
Mere, M. C. & Veloso, P. (1992). On extensions by sorts. Monografias em Ciências da Computaçao, DI, PUC-Rio, 38, 92.Google Scholar
Montague, R. (1957). Contributions to the Axiomatic Foundations of Set Theory. PhD thesis, University of California, Berkeley.Google Scholar
Nestruev, J. (2002). Smooth Manifolds and Observables. New York: Springer.Google Scholar
North, J. (2009). The ‘structure’ of physics: A case study. The Journal of Philosophy, 106, 57–88.CrossRefGoogle Scholar
Pelletier, F. J. & Urquhart, A. (2003). Synonymous logics. Journal of Philosophical Logic, 32(3), 259–285.CrossRefGoogle Scholar
Pinter, C. C. (1978). Properties preserved under definitional equivalence and interpretations. Mathematical Logic Quarterly, 24(31–36), 481–488.CrossRefGoogle Scholar
Quine, W. V. O. (1975). On empirically equivalent systems of the world. Erkenntnis, 9(3), 313–328.CrossRefGoogle Scholar
Rosenstock, S., Barrett, T. W., & Weatherall, J. O. (2015). On Einstein algebras and relativistic spacetimes. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 52, 309–316.CrossRefGoogle Scholar
Schmidt, A. (1951). Die Zulässigkeit der Behandlung mehrsortiger Theorien mittels der üblichen einsortigen Prädikatenlogik. Mathematische Annalen, 123(1), 187–200.CrossRefGoogle Scholar
Schwabhäuser, W., Szmielew, W., & Tarski, A. (1983). Metamathematische Methoden in der Geometrie. New York: Springer.CrossRefGoogle Scholar
Shoenfield, J. (1967). Mathematical Logic. Addison-Wesley.Google Scholar
Sklar, L. (1982). Saving the noumena. Philosophical Topics, 13(1), 89–110.CrossRefGoogle Scholar
Swanson, N. & Halvorson, H. (2012). On North’s ‘The structure of physics’. Manuscript.Google Scholar
Szczerba, L. (1977). Interpretability of elementary theories. Logic, Foundations of Mathematics, and Computability Theory. Amsterdam: Springer.Google Scholar
Tsementzis, D. (2015). A syntactic characterization of Morita equivalence. Manuscript.Google Scholar
van Benthem, J. and Pearce, D. (1984). A mathematical characterization of interpretation between theories. Studia Logica, 43(3), 295–303.CrossRefGoogle Scholar
van Fraassen, B. C. (2014). One or two gentle remarks about Hans Halvorson’s critique of the semantic view. Philosophy of Science, 81(2), 276–283.CrossRefGoogle Scholar
Weatherall, J. O. (2015a). Are Newtonian gravitation and geometrized Newtonian gravitation theoretically equivalent? Erkenntnis. Forthcoming.CrossRefGoogle Scholar
Weatherall, J. O. (2015b). Categories and the foundations of classical field theories. In Landry, E., editor, Categories for the Working Philosopher. Oxford University Press. Forthcoming.Google Scholar
Weatherall, J. O. (2015c). Understanding gauge. Philosophy of Science. Forthcoming.Google Scholar