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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, 13–15, 27, 29], and [3] discuss more general issues about theoretical equivalence.
The reader is encouraged to consult [16] for details and notation.
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].
As of June 30, 2015, according to scholar.google.com, there have been no technical investigations of Quine’s proposal.
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.
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.
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.
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.
One can compare this with [7].
See [23, 2.4] for a similar lemma.
⋆ This material is based upon work supported by the National Science Foundation under Grant No. DGE 1148900.
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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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DOI: https://doi.org/10.1007/s10992-015-9382-6