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BEYOND LINGUISTIC INTERPRETATION IN THEORY COMPARISON

Part of: Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  21 December 2023

TOBY MEADOWS
TOBY MEADOWS*
Affiliation:
DEPARTMENT OF LOGIC AND PHILOSOPHY OF SCIENCE UNIVERSITY OF CALIFORNIA, IRVINE IRVINE, CA 92617 USA
*
E-mail: meadowst@uci.edu
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Abstract

This paper assembles a unifying framework encompassing a wide variety of mathematical instruments used to compare different theories. The main theme will be the idea that theory comparison techniques are most easily grasped and organized through the lens of category theory. The paper develops a table of different equivalence relations between theories and then answers many of the questions about how those equivalence relations are themselves related to each other. We show that Morita equivalence fits into this framework and provide answers to questions left open in Barrett and Halvorson [4]. We conclude by setting up a diagram of known relationships and leave open some questions for future work.

Keywords

theoretical equivalencecategory theoryset theorymodel theory

MSC classification

Primary: 03A05: Philosophical and critical
Type
Research Article
Information
The Review of Symbolic Logic , Volume 17 , Issue 3 , September 2024 , pp. 819 - 859
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

BIBLIOGRAPHY

Andréka, H., Madarász, J., Németi, I., & Székely, G. (2023). Testing definitional equivalence of theories via automorphism groups. The Review of Symbolic Logic, 1–22. https://doi.org/10.1017/S1755020323000242 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
Awodey, S. (2006). Category Theory. Oxford: Clarendon Press.CrossRefGoogle Scholar
Barrett, T. W., & Halvorson, H. (2016). Morita equivalence. Review of Symbolic Logic, 9(3), 556–582. https://doi.org/10.1017/S1755020316000186 CrossRefGoogle Scholar
Bourbaki, N. (1972). Univers. In Artin, M., Grothendieck, A., and Verdier, J.-L., editors. Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie Des Topos et Cohomologie étale Des schémas - (SGA 4) - Vol. 1 (Lecture Notes in Mathematics 269). Berlin: Springer, pp. 185–217.Google Scholar
Button, T., & Walsh, S. P. (2018). Philosophy and Model Theory. Oxford: Oxford University Press.CrossRefGoogle Scholar
Chang, C. C., & Keisler, H. J. (1973). Model Theory. Amsterdam: North-Holland.Google Scholar
Enayat, A., Schmerl, J. H., & Visser, A. (2011). $\omega$ -models of finite set theory. In Kennedy, J. and Kossak, R., editors. Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies. Lecture Notes in Logic. Cambridge: Cambridge University Press, pp. 43–65.CrossRefGoogle Scholar
Feferman, S., & Kreisel, G. (1969). Set-theoretical foundations of category theory. In Reports of the Midwest Category Seminar III, Lecture Notes in Mathematics, 106. Berlin: Springer, pp. 201–247. https://doi.org/10.1007/BFb0059148.CrossRefGoogle Scholar
Halvorson, H. (2016). Scientific theories. In Humphreys, P., editor. The Oxford Handbook of Philosophy of Science. Oxford: Oxford University Press, pp. 585–608.Google Scholar
Halvorson, H., & Tsementzis, D. (2018). Categories of scientific theories. In Landry, E., editor. Categories for the Working Philosopher. Oxford: Oxford University Press, pp. 402–429.Google Scholar
Hodges, W. (1997). A Shorter Model Theory. Cambridge: CUP.Google Scholar
Hudetz, L. (2019). Definable categorical equivalence. Philosophy of Science, 86(1), 47–75.CrossRefGoogle Scholar
Kelly, G. M., & Street, R. (1974). Review of the elements of 2-categories. In Kelly, G. M., editor. Category Seminar. Berlin: Springer, pp. 75–103.CrossRefGoogle Scholar
Kunen, K. (2006). Set Theory: An Introduction to Independence Proofs. Sydney: Elsevier.Google Scholar
Lefever, K., & Székely, G. (2019). On generalization of definitional equivalence to non-disjoint languages. Journal of Philosophical Logic, 48(4), 709–729.CrossRefGoogle Scholar
Marker, D. (2002). Model Theory: And Introduction. New York: Springer.Google Scholar
McEldowney, P. A. (2020). On Morita equivalence and interpretability. Review of Symbolic Logic, 13(2), 388–415 CrossRefGoogle Scholar
Scott, D. (1961). More on the axiom of extensionality. In Essays on the Foundations of Mathematics. Jerusalem: Magnes Press, pp. 115–131.Google Scholar
Simpson, S. G. (1999). Subsystems of Second Order Arithmetic. Berlin: Springer.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, Vol. 26. Wellesley: ASL, pp. 284–341.Google Scholar
Visser, A., & Friedman, H. M. (2014). When bi-interpretability implies synonymy. Logic Group Preprint Series, 320, 1–19.Google Scholar
Weatherall, J. O. (2019). Part 1: Theoretical equivalence in physics. Philosophy Compass, 14(5), e12592.CrossRefGoogle Scholar
Weatherall, J. O. (2021). Why not categorical equivalence? In Aladova, E., Barceló, P., van Benthem, J., Berger, G., Dannert, K. M., Dewar, N., Diaconescu, R., Düntsch, I., Dzik, W., Kurd-Misto, M. E., Formica, G., Friend, M., Goldblatt, R., Gottlob, G., Grädel, E., Hirsch, R., Hodkinson, I., Jackson, M., Jipsen, P., Maddux, R. D., Manchak, J. B., Orłowska, E., Pieris, A., Plotkin, B., Plotkin, T., Pratt, V. R., Pratt-Hartmann, I., Ahmed, T. S., Weatherall, J. O., Westerståhl, D., Wimberley, J., Wójtowicz, K., and Christian, W., editors. Hajnal Andréka and István Németi on Unity of Science: From Computing to Relativity Theory Through Algebraic Logic. Cham: Springer, pp. 427–451.CrossRefGoogle Scholar