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Glymour and Quine on Theoretical Equivalence

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Abstract

Glymour (1970, 1977, 1980) and Quine (1975) propose two different formal criteria for theoretical equivalence. In this paper we examine the relationships between these criteria.

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Notes

  1. For example, [4, 12, 21, 28], and [2] discuss whether Hamiltonian and Lagrangian mechanics are theoretically equivalent. [9, 31], and [19] discuss standard Newtonian gravitation and geometrized Newtonian gravitation. Rosenstock et al. [25] consider general relativity and the theory of Einstein algebras. And [11, 1315, 27, 29], and [3] discuss more general issues about theoretical equivalence.

  2. The reader is encouraged to consult [16] for details and notation.

  3. As is evident from the work of [5, 20, 26], and [18], logicians were familiar with definitional equivalence before the 1970s. But Glymour was the first to introduce the notion into philosophy of science.

  4. For example, see [16] Section 2.6, [1, 5, 6, 18, 22, 23] for some results.

  5. Quine explains his proposal as follows: “By a reconstrual of the predicates of our language, accordingly, let me mean any mapping of our lexicon of predicates into our open sentences (n-place predicates to n-variable sentences). […] I propose to individuate theories thus: two formulations express the same theory if they are empirically equivalent and there is a reconstrual of predicates that transforms the one theory into a logical equivalent of the other” [24, 320].

  6. As of June 30, 2015, according to scholar.google.com, there have been no technical investigations of Quine’s proposal.

  7. Since the theories in these examples only use predicate symbols, these problems will stand regardless of how one extends Quine’s original proposal to theories in arbitrary signatures.

  8. Coffey [3] argues that symmetry is not necessarily a good feature for a proposed criterion for theoretical equivalence to have. But both Coffey and Quine suggest that Quine equivalence is an equivalence relation. We have shown here that this is not the case. If one were to “symmetrize” Quine equivalence, the problem posed by Example 4 would be avoided, but the more pressing problem posed by Example 3 would still stand.

  9. Knox [19] and [3] make this same remark.

  10. This map is defined in a perfectly analogous manner to the map between Σ-formulas and \(\widehat {\Sigma }\)-formulas described above. Indeed, one can easily verify that the map between Σ and \(\widehat {\Sigma }\) formulas is a reconstrual in this extended sense.

  11. Glymour remarks that definitional equivalence “guarantees that all and only theorems of [T 1] are translated as theorems of [ T 2], and conversely” [8, 279]. Here we provide a strengthening of Glymour’s remark. Theorems 1 and 2 make precise a sense in which this requirement is no stronger and no weaker than definitional equivalence. Friedman and Visser [6] and [23] state these two results, but do not provide proofs. Ingredients for proofs using tools of category theory are contained in [30 ]. Further ingredients are contained in [ 17 ]. Pelletier and Urquhart [ 22] provide proofs for the special case of propositional logic. Here we extend the results to full first-order logic using only elementary methods.

  12. One can compare this with [7].

  13. See [23, 2.4] for a similar lemma.

  14. This material is based upon work supported by the National Science Foundation under Grant No. DGE 1148900.

References

  1. Andréka, H., Madarász, J.X., & Németi, I. (2005). Mutual definability does not imply definitional equivalence, a simple example. Mathematical Logic Quarterly.

  2. Barrett, T.W. (2014). On the structure of classical mechanics. Forthcoming in The British Journal for the Philosophy of Science.

  3. Coffey, K. (2014). Theoretical equivalence as interpretive equivalence. The British Journal for the Philosophy of Science.

  4. Curiel, E. (2014). Classical mechanics is Lagrangian; it is not Hamiltonian. The British Journal for the Philosophy of Science.

  5. de Bouvére, K.L. (1965). Synonymous theories. In Symposium on the theory of models. North-holland publishing company.

  6. Friedman, H., & Visser, A. (2014). When bi-interpretability implies synonymy. Manuscript.

  7. Gajda, A., Krynicki, M., & Szczerba, L. (1987). A note on syntactical and semantical functions. Studia Logica.

  8. Glymour, C. (1970). Theoretical realism and theoretical equivalence. In PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association.

  9. Glymour, C. (1977). The epistemology of geometry. Nous.

  10. Glymour, C. (1980). Theory and Evidence. Princeton.

  11. Glymour, C. (2013). Theoretical equivalence and the semantic view of theories. Philosophy of Science.

  12. Halvorson, H. (2011). Natural structures on state space. Manuscript.

  13. Halvorson, H. (2012). What scientific theories could not be. Philosophy of Science.

  14. Halvorson, H. (2013). The semantic view, if plausible, is syntactic. Philosophy of Science.

  15. Halvorson, H. (2015). Scientific theories. Forthcoming in Oxford Handbooks Online.

  16. Hodges, W. (2008). Model Theory. Cambridge University Press.

  17. Hoehnke, H. (1966). Zur Strukturgleichheit axiomatischer Klassen. Mathematical Logic Quarterly.

  18. Kanger, S. (1968). Equivalent theories. Theoria.

  19. Knox, E. (2013). Newtonian spacetime structure in light of the equivalence principle. The British Journal for the Philosophy of Science.

  20. Montague, R. (1957). Contributions to the axiomatic foundations of set theory. PhD thesis, University of California, Berkeley.

  21. North, J. (2009). The ‘structure’ of physics: A case study. Journal of Philosophy.

  22. Pelletier, F.J., & Urquhart, A. (2003). Synonymous logics. Journal of Philosophical Logic.

  23. Pinter, C.C. (1978). Properties preserved under definitional equivalence and interpretations. Mathematical Logic Quarterly.

  24. Quine, W.V.O. (1975). On empirically equivalent systems of the world. Erkenntnis.

  25. Rosenstock, S., Barrett, T.W., & Weatherall, J.O. (2015). On Einstein algebras and relativistic spacetimes. Manuscript.

  26. Shoenfield, J. (1967). Mathematical Logic. Addison-Wesley.

  27. Sklar, L. (1982). Saving the noumena. Philosophical Topics.

  28. Swanson, N., & Halvorson, H. (2012). On North’s ‘The structure of physics’. Manuscript.

  29. van Fraassen, B.C. (2014). One or two gentle remarks about Hans Halvorson’s critique of the semantic view. Manuscript.

  30. Visser, A. (2006). Categories of theories and interpretations. In Logic in Tehran. Proceedings of the workshop and conference on Logic, Algebra and Arithmetic, held October 18–22, 2003. ASL.

  31. Weatherall, J.O. (2015). Are Newtonian gravitation and geometrized newtonian gravitation theoretically equivalent? Manuscript.

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Correspondence to Thomas William Barrett.

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Thanks to JB Manchak, Jim Weatherall, Jeff Barrett, Albert Visser, Neil Dewar, and an anonymous referee for comments.

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Barrett, T.W., Halvorson, H. Glymour and Quine on Theoretical Equivalence. J Philos Logic 45, 467–483 (2016). https://doi.org/10.1007/s10992-015-9382-6

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