Abstract
Thoroughgoing relativists typically dismiss the realist conviction that competing theories describe just one definite and mind-independent world-structure on the grounds that such theories fail to be relatively translatable even though they are equally correct. This line of argument allegedly brings relativism into direct conflict with the metaphysics of realism. I argue that this relativist line of reasoning is shaky by deriving a theorem about relativistic inquiry in formal epistemology—more specifically, in the approach Kevin Kelly has dubbed “logic of reliable inquiry”. According to the theorem, two scientists, who share some background knowledge but follow different appropriately reliable methods, will converge to relatively formally translatable competing theories, even if meaning, truth, logic and evidence are allowed to vary in time depending on each scientist’s conjectures, actions, or conceptual choices. Some final remarks on the relevance of the theorem to the incommensurability thesis that has vexed twentieth century philosophy of science are adduced.
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Notes
Hereafter I employ the term ‘framework’ as a catchall term that may cover whatever does appear as an independent variable in a relativist thesis: conceptual schemes, linguistic and logical systems, theoretical traditions, forms of experience, etc.
Quine (1975, 320), for instance, suggested that two formulations express the same theory if (a) they are empirically equivalent in the sense that they imply the same spatiotemporally pegged observation sentences under identical initial and boundary conditions and (b) there is a reconstrual of predicates (i.e., a mapping that takes, for each natural number n, n -place predicates onto n -variable open sentences) that transforms any one of them into a logical equivalent of the other.
For example, Davidson ([1974] 1984, 185) proposes to identify conceptual schemes with sets of intertranslatable languages. As is well known, however, Davidson denied the possibility of such a radical divergence.
In Davidson’s sense whereby to interpret a sentence in an object language is to provide its truth-conditions in a metalanguage.
Teller (2008) has since developed his views along the following lines. The Kuhnian theme of incommensurability threatens scientific rationality only under the assumption that science is in the “exact truth business”. But once it is realized that science is “in the business of systematically developing idealized models that are always limited in scope and never completely accurate”, one can compare alternative theoretizations, albeit in a manner relativized to the limits of human capacities as well as to human interests, both intellectual and practical. I shall not follow Teller in this wider conception of science.
Scientific theories do not treat all the aspects of the real systems they purport to study, include idealizations, tolerate inexact statements, etc.
To be more precise, at least one of the theories will be formally translatable in the other.
Here and below I write in parentheses expressions of the form ‘\(x, \ldots ,y \in \varOmega\)’ to indicate that I shall henceforth reserve the letter(s) \(x, \ldots ,y\) (on occasion, subscripted and/or superscripted) for symbolizing elements of the set \(\varOmega\).
In what follows the terms ‘scientist’, ‘method’ and ‘hypothesis generator’ may be used interchangeably insofar as formal considerations are concerned. The same holds for the expressions ‘stage of inquiry’ and ‘time’. Lastly, I shall sometimes be talking about “sentences” (maintaining the scare quotes) instead of strings when referring to elements of S.
This is a semantically and evidentially immediate world-in-itself. For other possibilities and the relevant terminology see Kelly et al. (1994, 135–136).
Here and below, \(' \ldots \Rightarrow \ldots '\) and \(' \ldots \Leftrightarrow \ldots '\) are used as abbreviations of ‘if…then…’ and ‘…if and only if…’ respectively in the metalanguage.
This corresponds to uniform identification of the complete truth according to Kelly’s (1996, 292–296) treatment of “theory discovery”.
García-Matos and Väänänen (2007, 21) dub such systems “abstract logics”. I have borrowed the term ‘model-theoretic language’ from Feferman (1974, 155) although Feferman proposed a more refined concept that explicitly covers different similarity types (signatures) and many-sorted logics. Feferman (1974, 156–157) also intimated the way in which the concept can be modified to provide for many-valued logics, Boolean-valued logics, etc.
The proofs of the two assertions are relegated to Appendix 1.
This concept of relative formal translation among theories is a generalization of the concept of relative weak representation among first-order theories explored by Schroeder-Heister and Schaefer (1989). These authors required in addition a preservation property for the logical sign of negation of the relatively weakly representable first-order languages (see condition (3.2) in Schroeder-Heister and Schaefer 1989, 137).
Therefore it would be too hasty to conclude that any two relatively formally intertranslatable theories are commensurable in the way that commensurability has been usually understood in philosophy of science. Kuhn (1982) himself insisted that translation, in the sense relevant to commensurability, must preserve taxonomy, i.e., categories and relationships between them. Likewise, on the understanding that commensurability has to do with concepts and not only with statements, the commensurability functions between first-order theories investigated by Schroeder-Heister and Schaefer (1989) represent open formulas and terms of the one theory in the other and not just sentences. See, however, also Pearce (1989).
Putnam (1983, 38–40), for instance, requires for cognitive equivalence that the two theories explain the same phenomena besides their being mutually relatively interpretable in the sense of first-order logic.
The existence of a suitable mapping is guaranteed (even for uncountable \(S\)) by the axiom of choice (see, e.g., Halmos ([1960] 1987, 59–61). \(\{ \Delta_{s} :s \in S\}\) is a non-empty collection of non-empty sets. Thus there exists a choice function \(f_{m}^{w,\gamma /\delta }\) with domain \(S\) such that for every \(s \in S\), \(f_{m}^{w,\gamma /\delta } (s) \in \Delta_{s}\). Relying on the definition of \(\Delta_{s}\), we conclude that for every \(s \in S\), \(u_{m}^{w,\gamma } (s) = u_{m}^{w,\delta } (f_{m}^{w,\gamma /\delta } (s))\), i.e., that (15) is satisfied.
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Acknowledgments
This research has been co-financed by the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: THALIS -UOA- Aspects and Prospects of Realism in the Philosophy of Science and Mathematics (APRePoSMa).
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Arageorgis, A. Relativism, translation, and the metaphysics of realism. Philos Stud 174, 659–680 (2017). https://doi.org/10.1007/s11098-016-0702-7
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DOI: https://doi.org/10.1007/s11098-016-0702-7