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On Identifiability in Extended Domains

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Part of the The University of Western Ontario Series in Philosophy of Science book series (WONS,volume 11)

Abstract

For the sake of the present discussion, by definability I shall understand what is usually called the explicit (or syntactic) definability; by identifiability — the implicit (or semantic) definability. I shall restrict myself to considering empirical theories which can be formalized in first-order logic. As it is known, with regard to such theories these two concepts coincide. A term t is definable in a theory T if it is identifiable in it. That is to say, a definition of t is a theorem of T if the interpretation of the remaining terms in any model of T fixes uniquely the interpretation of t. That is why all cases of nonidentifiability considered thus far are always connected with some kind of non-definability. A concept is not identifiable only if it is not definable; e.g. if it is governed by some postulate weaker than an explicit definition (a conditional, or piecewise, or partial definition — to mention the typical cases). In what follows, I wish to call attention to some other kind of non-identifiability and its characteristic sources. Non-identifiability of this kind is not connected with non-definability. It appears as a result of some looser conception of a theory’s interpretation. On the standard approach, any extension of a given language has as its semantic counterpart a suitable expansion of the language’s structures. But there seem to be cases where an extension of language is connected not only with expanding its structures, but also with extending their domains. This seems to be the case when an observational language is being extended by introducing into it certain types of theoretical terms. In such cases the correspondence between definability and identifiability mentioned above does not apply any longer. Being definable in a given theory, a theoretical concept is not identifiable in the theory’s models — if these are structures with extended domains.

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Bibliography

  • Przelgcki, M.: 1976, ‘Interpretation of Theoretical Terms: In Defence of an Empiricist Dogma’ in M. Przelgcki et al. (eds.), Formal Methods in the Methodology of Empirical Sciences, D. Reidel, Dordrecht — Ossolineum, Wroclaw.

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  • Winnie, J. A.: 1967, ‘The Implicit Definition of Theoretical Terms’, The British Journal for the Philosophy of Science 18, 223–229.

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© 1977 Springer Science+Business Media Dordrecht

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Przełęcki, M. (1977). On Identifiability in Extended Domains. In: Butts, R.E., Hintikka, J. (eds) Basic Problems in Methodology and Linguistics. The University of Western Ontario Series in Philosophy of Science, vol 11. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0837-1_6

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  • DOI: https://doi.org/10.1007/978-94-017-0837-1_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-8350-0

  • Online ISBN: 978-94-017-0837-1

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