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Johan van Benthem on Logic and Information Dynamics pp 143–172Cite as

Comparing Theories: The Dynamics of Changing Vocabulary

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Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 5))

Abstract

There are several first-order logic (FOL) axiomatizations of special relativity theory in the literature, all looking different but claiming to axiomatize the same physical theory. In this chapter, we elaborate a comparison between these FOL theories for special relativity. We do this in the framework of mathematical logic. For this comparison, we use a version of definability theory in which new entities can also be defined besides new relations over already available entities. In particular, we build an interpretation (in Alfred Tarski’s sense) of the reference-frame oriented theory \({\textsc {SpecRel}}\) developed in the Budapest Logic Group into the observationally oriented Signalling theory of James Ax published in Foundations of Physics. This interpretation provides \({\textsc {SpecRel}}\) with an operational/experimental semantics. Then we make precise, “quantitative” comparisons between these two theories via using the notion of definitional equivalence. This is an application of mathematical logic to the philosophy of science and physics in the spirit of Johan van Benthem’s work.

Research supported by the Hungarian grant for basic research OTKA No K81188.

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Notes

  1. 1.

    A similar investigation for Newtonian gravitation, but not in the framework of mathematical logic, can be found in [45].

  2. 2.

    Other names for vocabulary are signature and set of nonlogical constants.

  3. 3.

    In the formulas, the scope of a quantifier is till the end of the formula if not indicated otherwise. Lower case Roman and Greek letters denote variables of sorts \({\small \mathsf{{Par}}}\) and \({\small \mathsf{{Sig}}}\), respectively. Instead of conjunction \(\wedge \) we will simply write a comma.

  4. 4.

    Or, if we are content with more approximate measurements, we can imagine all this happening on a big lake covered with smooth ice (but then we have to take space to be 2-dimensional).

  5. 5.

    Similar observations apply to a slight variant \({\textsc {SpecRel}}_0+{\textsc {Compl}}\) of \({\textsc {SpecRel}}\) in place of \({\textsc {SigTh}}\) (cf. Theorem 6.2 in Sect. 6.7). This can be extended to the Newtonian theory in [2, Sect. 4.1, p. 423].

  6. 6.

    The easiest way of making this precise is that there are fields with no automorphisms at all, e.g., the field of real numbers, and this means that the structure \(\langle {\small \mathsf{{Par}}},\pi _P,{\textsc {EFd}}\rangle \) will have no automorphism, either.

  7. 7.

    We note that \({\textsc {SpecRel}}\) can be interpreted in \({\textsc {SigTh}}\) in the way that we interpret \({\textsc {SpecRel}}\) in the field \({\small \mathsf Q},+,\star \).

References

  1. Andréka H, Madarász JX, Németi I (2001) Defining new universes in many-sorted logic. Mathematical Institute of the Hungarian Academy of Sciences, Budapest, p 93 (Preprint)

    Google Scholar 

  2. Andréka H, Madarász JX, Németi I (2002) On the logical structure of relativity theories. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest. Research report, 5 July 2002, with contributions from Andai A, Sági G, Sain I, Tőke Cs, 1312 pp. http://www.math-inst.hu/pub/algebraic-logic/Contents.html

  3. Andréka H, Madarász JX, Németi I (2007) Logic of space-time and relativity theory. In: Aiello M, Pratt-Hartmann I, van Benthem J (eds) Handbook of spatial logics. Springer, Berlin, pp 607–711

    Google Scholar 

  4. Andréka H, Madarász JX, Németi I, Németi P, Székely G (2011) Vienna circle and logical analysis of relativity theory. In: Máte A, Rédei M, Stadler F (eds) The Vienna circle in Hungary (Der Wiener Kreis in Ungarn). Veroffentlichungen des Instituts Wiener Kreis, Band 16, Springer, New York, pp. 247–268

    Google Scholar 

  5. Andréka H, Madarász JX, Németi I, Székely G (2012) A logic road from special relativity to general relativity. Synthese 186(3):633–649

    Article  Google Scholar 

  6. Ax J (1978) The elementary foundations of spacetime. Found Phys 8(7/8):507–546

    Article  Google Scholar 

  7. Balzer W, Moulines U, Sneed JD (1987) An architectonic for science. The structuralist program. D. Reidel Publishing Company, Dordrecht

    Book  Google Scholar 

  8. van Benthem J (1982) The logical study of science. Synthese 51:431–472

    Article  Google Scholar 

  9. van Benthem J (2012) The logic of empirical theories revisited. Synthese 186(2):775–792

    Article  Google Scholar 

  10. van Benthem J, Pearce D (1984) A mathematical characterization of interpretation between theories. Studia Logica 43(3):295–303

    Article  Google Scholar 

  11. Burstall R, Goguen J (1977) Putting theories together to make specifications. In: Proceeding of IJCAI’77 (Proceedings of the 5th international joint conference on artificial intelligence), vol 2. Morgan Kaufmann Publishers, San Francisco, pp 1045–1058

    Google Scholar 

  12. Carnap R (1928) Die Logische Aufbau der Welt. Felix Meiner, Leipzig

    Google Scholar 

  13. Friedman M (1983) Foundations of space-time theories: Relativistic physics and philosophy of science. Princeton University Press, Princeton

    Google Scholar 

  14. Friedman H (2004) On foundational thinking 1. FOM (foundations of mathematics) posting. http://www.cs.nyu.edu/pipermail/fom/. Posted 20 Jan 2004

  15. Friedman H (2007) Interpretations of set theory in discrete mathematics and informal thin- king. Lectures 1–3, Nineteenth annual Tarski lectures, Berkeley. http://u.osu.edu/friedman.8/

  16. Gärdenfors P, Zenker F (2013) Theory change as dimensional change: conceptual spaces applied to the dynamics of empirical theories. Synthese 190:1039–1058

    Article  Google Scholar 

  17. Goldblatt R (1987) Orthogonality and spacetime geometry. Springer, Berlin

    Google Scholar 

  18. Harnik V (2011) Model theory versus categorical logic: two approaches to pretopos completion (a.k.a. \(T^{eq}\)). In: Centre de Recherches Mathématiques CRM proceedings and lecture notes, vol 53. American Mathematical Society, pp 79–106

    Google Scholar 

  19. Hodges W (1993) Model theory. Cambridge University Press, Cambridge

    Google Scholar 

  20. Hoffman B (2013) A logical treatment of special relativity, with and without faster-than-light observers. BA Thesis, Lewis and Clark College, Oregon, 63pp. arXiv:1306.6004 [math.LO]

  21. Konev B, Lutz C, Ponomaryov D, Wolter F (2010) Decomposing description logic ontologies. In: Proceedings of 12th conference on the principles of knowledge representation and reasoning, Association for the advancement of artificial intelligence, pp 236–246

    Google Scholar 

  22. Lutz C, Wolter F (2009) Mathematical logic for life science ontologies. In: Ono H, Kanazawa M, de Queiroz R (eds) Proceedings of WOLLIC-2009, LNAI 5514. Springer, pp 37–47

    Google Scholar 

  23. Madarász, JX (2002) Logic and relativity (in the light of definability theory). PhD Dissertation, Eötvös Loránd University. http://www.math-inst.hu/pub/algebraic-logic/diszi.pdf

  24. Madarász JX, Székely G (2013) Special relativity over the field of rational numbers. Int J Theor Phys 52(5):1706–1718

    Article  Google Scholar 

  25. Makkai M (1985) Ultraproducts and categorical logic, vol 1130. Methods in mathematical logic, Springer LNM, Berlin, pp 222–309

    Google Scholar 

  26. Makkai M (1993) Duality and definability in first order logic, vol 503. Memoirs of the American Mathematical Society, Providence

    Google Scholar 

  27. Makkai M, Reyes G (1977) First order categorical logic. Lecture Notes in Mathematics, vol 611. Springer, Berlin

    Google Scholar 

  28. Pambuccian V (2004/05) Elementary axiomatizations of projective space and of its associated Grassman space. Note de Matematica 24(1):129–141

    Google Scholar 

  29. Pambuccian V (2005) Groups and plane geometry. Studia Logica 81:387–398

    Article  Google Scholar 

  30. Pambuccian V (2007) Alexandrov-Zeeman type theorems expressed in terms of definability. Aequationes Math 74:249–261

    Article  Google Scholar 

  31. Previale F (1969) Rappresentabilit\(\grave{a}\) ed equipollenza di teorie assomatiche i. Ann Scuola Norm Sup Pisa 23(3):635–655

    Google Scholar 

  32. Schelb U (2000) Characterizability of free motion in special relativity. Found Phys 30(6):867–892

    Google Scholar 

  33. Schutz JW (1997) Independent axioms for Minkowski space-time. Longman, London

    Google Scholar 

  34. Suppes P (1959) Axioms for relativistic kinematics with or without parity. In: Henkin L, Tarski A, Suppes P (eds) Symposium on the axiomatic method with special reference to physics, North Holland, pp 297–307

    Google Scholar 

  35. Suppes P (1972) Some open problems in the philosophy of space and time. Synthese 24:298–316

    Article  Google Scholar 

  36. Szabó LE (2009) Empirical foundation of space and time. In: Suárez M, Dorato MM, Rédei M (eds) EPSA07: launch of the European philosophy of science association, Springer. http://phil.elte.hu/leszabo/Preprints/LESzabo-madrid2007-preprint.pdf

  37. Szczerba LW (1977) Interpretability of elementary theories. In: Butts RE, Hintikka J (eds) Logic, foundations of mathematics and computability theory (Proceeding of fifth international congress of logic, methodology and philosophical of science, University of Western Ontario, London), Part I. Reidel, Dordrecht, pp 129–145

    Google Scholar 

  38. Székely G (2009) First-order logic investigation of relativity theory with an emphasis on accelerated observers. PhD Dissertation, Eötvös Lorand University, Faculty of Sciences, Institute of Mathematics, Budapest, 150pp. ArXiv:1005.0973[gr-qc]

  39. Székely G (2010) A geometrical characterization of the twin paradox and its variants. Studia Logica 95(1–2):161–182

    Article  Google Scholar 

  40. Tarski A (1936) Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica 1:152–278

    Google Scholar 

  41. Tarski A, Mostowski A, Robinson RM (1953) Undecidable theories. North-Holland, Amsterdam

    Google Scholar 

  42. Tarski A (1959) What is elementary geometry? In: Henkin L, Suppes P, Tarski A (eds) The axiomatic Method with Special Reference to Geometry and Phsics. North-Holland, Amserdam, pp 16–29

    Google Scholar 

  43. Tarski A, Givant SR (1999) Tarski’s system of geometry. Bull Symbolic Logic 5(2):175–214

    Article  Google Scholar 

  44. Taylor EF, Wheeler JA (1963) Spacetime physics. Freeman, San Francisco

    Google Scholar 

  45. Weatherall JO (2011) Are Newtonian gravitation and geometrized Newtonian gravitation theoretically equivalent? (Unpublished manuscript)

    Google Scholar 

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Authors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, Budapest, 1053, Hungary

    Hajnal Andréka

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, Budapest, 1053, Hungary

    István Németi

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Correspondence to Hajnal Andréka .

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Editors and Affiliations

  1. University of Amsterdam Inst Logic,Language & Computation, Amsterdam, The Netherlands

    Alexandru Baltag

  2. Amsterdam, The Netherlands

    Sonja Smets

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Andréka, H., Németi, I. (2014). Comparing Theories: The Dynamics of Changing Vocabulary. In: Baltag, A., Smets, S. (eds) Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-06025-5_6

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