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DISTANCES BETWEEN FORMAL THEORIES

Published online by Cambridge University Press:  04 October 2019

MOHAMED KHALED ,
GERGELY SZÉKELY ,
KOEN LEFEVER  and
MICHÈLE FRIEND
MOHAMED KHALED*
Affiliation:
Faculty of Engineering and Natural Sciences, Bahçeşehir University
GERGELY SZÉKELY*
Affiliation:
Alfréd Rényi Institute of Mathematics, and Department of Natural Sciences, National University of Public Service
KOEN LEFEVER*
Affiliation:
Centre for Logic and Philosophy of Science, Vrije Universiteit Brussel
MICHÈLE FRIEND*
Affiliation:
George Washington University
*
*FACULTY OF ENGINEERING AND NATURAL SCIENCES BAHÇEŞEHIR UNIVERSITY ISTANBUL, TURKEY E-mail: [email protected]
**SET THEORY, LOGIC AND TOPOLOGY ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS BUDAPEST, HUNGARY E-mail: [email protected]
†CENTRE FOR LOGIC AND PHILOSOPHY OF SCIENCE VRIJE UNIVERSITEIT BRUSSEL BRUSSELS, BELGIUM E-mail: [email protected]
‡DEPARTMENT OF PHILOSOPHY GEORGE WASHINGTON UNIVERSITY WASHINGTON, DC, USA and UNIVERSITÉ DE LILLE LILLE, FRANCE E-mail: [email protected]
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Abstract

In the literature, there have been several methods and definitions for working out whether two theories are “equivalent” (essentially the same) or not. In this article, we do something subtler. We provide a means to measure distances (and explore connections) 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 concepts that distinguish two theories. For instance, we use conceptual distance to show that relativistic and classical kinematics are distinguished by one concept only.

Keywords

Primary 03B9903C0703A10Secondary 03G9903B80network of theoriesdegrees of nonequivalenceconceptual distancerelativistic and classical kinematics
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 3 , September 2020 , pp. 633 - 654
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Copyright
Copyright © Association for Symbolic Logic 2019 

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References

BIBLIOGRAPHY

[1]Andréka, H., Comer, S. D., Madarász, J. X., Németi, I., & Ahmed, T. S. (2009). Epimorphisms in cylindric algebras and definability in finite variable logic. Algebra Universalis, 61 (3), 261–282.CrossRefGoogle Scholar
[2]Andréka, H., Madarász, J. X., & Németi, I. (2004). Logical analysis of relativity theories. In Hendricks, V., Neuhaus, F., Pedersen, S. A., Scheffler, U., and Wansing, H., editors. First-order Logic Revisited. Berlin: Logos Verlag, pp. 7–36.Google Scholar
[3]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
[4]Andréka, H., Madarász, J. X., Németi, I., & Székely, G. (2012). A logic road from special relativity to general relativity. Synthese, 186(3), 633–649.CrossRefGoogle Scholar
[5]Andréka, H., Madarász, J. X., Németi, I., with contributions from: Andai, A., Sági, G., Sain, I., & Tőke, C. (2002). On the Logical Structure of Relativity Theories. Research report, Alfréd Rényi Institute of Mathematics, Budapest: Hungarian Academy of Sciences. Available at: https://old.renyi.hu//pub/ algebraic-logic/Contents.html.Google Scholar
[6]Andréka, H. & Németi, I. (2017). How many varieties of cylindric algebras are there. Transactions of the American Mathematical Society, 369, 8903–8937.CrossRefGoogle Scholar
[7]Andréka, H., van Benthem, J., & Németi, I. (1998). Modal languages and bounded fragments of predicate logic. Journal of philosophical logic, 27(3), 217–274.CrossRefGoogle Scholar
[8]Baltag, A. & Smets, S. (2005). Complete axiomatizations for quantum actions. International Journal of Theoretical Physics, 44(12), 2267–2282.CrossRefGoogle Scholar
[9]Banerjee, A. & Khaled, M. (2018). First order logic without equality on relativized semantics. Annals of Pure and Applied Logic, 169(11), 1227–1242.CrossRefGoogle Scholar
[10]Barrett, T. W. & Halvorson, H. (2016). Glymour and quine on theoretical equivalence. Journal of Philosophical Logic, 45(5), 467–483.CrossRefGoogle Scholar
[11]Carathéodory, C. (1909). Untersuchungen über die grundlagen der thermodynamik. Mathematische Annalen, 64, 355–386.CrossRefGoogle Scholar
[12]Chang, H. (2012). Is Water H2O ? Evidence, Realism and Pluralism. Dordrecht: Springer.CrossRefGoogle Scholar
[13]Cooper, J. L. B. (1967). The foundations of thermodynamics. Journal of Mathematical Analysis and Applications, 17, 172–193.CrossRefGoogle Scholar
[14]Fine, K. (1985). Natural deduction and arbitrary objects. Journal of Philosophical Logic, 14(1), 57–107.CrossRefGoogle Scholar
[15]Floyd, R. W. (1967). Assingining meanings to programs. Proceedings of Symposia in Applied Mathematics, 19, 19–32.CrossRefGoogle Scholar
[16]Goodman, N. (1943). On the simplicity of ideas. The Journal of Symbolic Logic, 8(4), 107–121.CrossRefGoogle Scholar
[17]Henkin, L. (1950). The Completeness of Formal Systems. Ph.D. Thesis, Princeton University.Google Scholar
[18]Henkin, L., Monk, J., & Tarski, A. (1971). Cylindric Algebras Part I. Amsterdam: North-Holland.Google Scholar
[19]Henkin, L., Monk, J., & Tarski, A. (1985). Cylindric Algebras Part II. Amsterdam: North-Holland.Google Scholar
[20]Hoare, C. A. R. (1969). An axiomatic basis for computer programming. Communications of the ACM, 12(10), 576–583.CrossRefGoogle Scholar
[21]Japaridze, G. & Jongh, D. (1998). The logic of provability. In Buss, S. R., editor. Handbook of Proof Theory, Vol. 137. Amsterdam: Elsevier, pp. 475–550.CrossRefGoogle Scholar
[22]Khaled, M. (2019). The finitely axiomatizable complete theories of nonassociative arrow frames. Advances in Mathematics, 346, 194–218.CrossRefGoogle Scholar
[23]Krause, D. & Arenhart, J. R. B. (2017). The Logical Foundations of Scientific Theories: Languages, Structures, and Models. New York: Routledge.Google Scholar
[24]Lefever, K. (2017). Using Logical Interpretation and Definitional Equivalence to Compare Classical Kinematics and Special Relativity Theory. Ph.D. Thesis, Vrije Universiteit Brussel.Google Scholar
[25]Lefever, K. & Székely, G. (2018a). Comparing classical and relativistic kinematics in first-order-logic. Logique et Analyse, 61(241), 57–117.Google Scholar
[26]Lefever, K. & Székely, G. (2018b). On generalization of definitional equivalence to nondisjoint languages. Journal of Philosophical Logic, 48(4), 709–729.CrossRefGoogle Scholar
[27]Lieb, E. H. & Yngvason, J. (2000). A fresh look at entropy and the second law of thermodynamics. Physics Today, 53(4), 32–37.CrossRefGoogle Scholar
[28]Meyer, A. R. & Halpern, J. Y. (1982). Axiomatic definitions of programming languages: A theoretical assessment. Journal of the Association for Computing Machinery, 29(2), 555–576.CrossRefGoogle Scholar
[29]Pinter, C. C. (1978). Properties preserved under definitional equivalence and interpretations. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 24, 481–488.CrossRefGoogle Scholar
[30]Ryll-Nardzewski, C. (1952). The role of the axiom of induction in elementary arithmetic. Fundamenta Mathematicae, 39, 239–263.CrossRefGoogle Scholar
[31]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, Vol. 26. Wellesley, MA: ASL, A.K. Peters, Ltd., pp. 284–341.Google Scholar