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ON MORITA EQUIVALENCE AND INTERPRETABILITY

Published online by Cambridge University Press:  13 August 2019

PAUL ANH MCELDOWNEY
PAUL ANH MCELDOWNEY*
Affiliation:
Department of Philosophy, University of Notre Dame
*
*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF NOTRE DAME 100 MALLOY HALL NOTRE DAME, INDIANA 46556, USA E-mail: pmceldow@nd.edu or pmceldow@gmail.com
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Abstract

In a recent article, Barrett & Halvorson (2016) define a notion of equivalence for first-order theories, which they call “Morita equivalence.” To argue that Morita equivalence is a reasonable measure of “theoretical equivalence,” they make use of the claim that Morita extensions “say no more” than the theories they are extending. The goal of this article is to challenge this central claim by raising objections to their argument for it and by showing why there is good reason to think that the claim itself is false. In light of these criticisms, this article develops a natural way for the advocate of Morita equivalence to respond. I prove that this response makes her criterion a special case of bi-interpretability, an already well-established barometer of theoretical equivalence. I conclude by providing reasons why the advocate of Morita equivalence should opt for a notion of theoretical equivalence that is defined in terms of interpretability rather than Morita extensions.

Keywords

03A1003F2500A3003A05Morita equivalenceinterpretabilitymany-sorted logictheoretical equivalence
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 2 , June 2020 , pp. 388 - 415
Copyright
Copyright © Association for Symbolic Logic 2019 

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References

BIBLIOGRAPHY

Barrett, T. W. & Halvorson, H. (2016). Morita equivalence. The Review of Symbolic Logic, 9(3), 556–582.CrossRefGoogle Scholar
Barrett, T. W. & Halvorson, H. (2017a). From geometry to conceptual relativity. Erkenntnis, 82(5), 1043–1063.CrossRefGoogle Scholar
Barrett, T. W. & Halvorson, H. (2017b). Quine’s conjecture on many-sorted logic. Synthese, 194(9), 3563–3582.CrossRefGoogle Scholar
Button, T. & Walsh, S. (2018). Philosophy and Model Theory. Oxford: Oxford University Press.CrossRefGoogle 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. In Buck, R. C., and Cohen, R. S., editors. PSA: Proceedings of the Biennial Meeting Of The Philosophy Of Science Association, Vol. 1970. Dordrecht: Springer, pp. 275–288.Google Scholar
Hodges, W. (1993). Model Theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Hodges, W. (2008). Tarski’s theory of definition. In Patterson, D., editor. New Essays On Tarski, Chapter 5. Oxford: Oxford University Press, pp. 94–132.CrossRefGoogle Scholar
Kaye, R. & Wong, T. L. (2007). On interpretations of arithmetic and set theory. Notre Dame Journal of Formal Logic, 48(4), 497–510.CrossRefGoogle Scholar
Makkai, M. & Reyes, G. E. (1967). First Order Categorical Logic. Berlin: Springer-Verlag.Google Scholar
Marker, D. (2002). Model Theory: An Introduction. New York: Springer Verlag.Google Scholar
Moosa, R. (2005). The model theory of compact complex spaces. In Matthias, B., Friedman, S.-D., and Krajícek, J., editors. Logic Colloquium ’01. Lecture Notes in Logic, Vol. 20. Wellesley: A K Peters, Ltd./CRC Press, pp. 317–349.Google Scholar
Niebergall, K.-G. (2013). On the logic of reducibility: Axioms and examples. Erkenntis, 53(1), 27–61.CrossRefGoogle Scholar
Slaman, T. A. (2008). Global properties of the Turing degrees and the Turing jump. Computational Prospects of Infinity. Part I. Tutorials, 14, 83–101.CrossRefGoogle Scholar
Suppes, P. (1957). Introduction to Logic. New York: Van Nostrand Heinhold Company.Google Scholar
Szczerba, L. W. (1975). Interpretability of elementary theories. In Butts, R. E. and Hintikka, J., editors. Logic, Foundations of Mathematics, and Computability Theory. Dordrecht: D. Reidel Publishing Company, pp. 129–145.Google 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
Visser, A. (2006). Categories of theories and interpretations. In Enayat, A., Kalantari, I., and Moniri, M., editors. Logic in Tehran: Proceedings of the Workshop and Conference on Logic, Algebra, and Arithmetic, held October 18–22, 2003, Lecture Notes in Logic, Volume 26. Cambridge: Cambridge University Press, pp. 284–341.Google Scholar
Visser, A. (2015). Extension & interpretability. Logic Group preprint series, volume 329. Available at https://dspace.library.uu.nl/handle/1874/327588.Google Scholar
Walsh, S. (2014). Logicism, interpretability, and knowledge of arithmetic. The Review of Symbolic Logic, 7(1), 84–119.CrossRefGoogle Scholar