# Towards a General Theory of Identifiability

Chapter

## Abstract

What is identifiability, anyway, and what does it have to do with definitions and definability? The basic intuitive idea is clear. A concept (say, a one-place predicate *P*) occurring in a theory *T*[*P*] is *definable* on the basis of this theory iff the theory determines the interpretation of *P* as soon as the interpretations of the other concepts occurring in *T*[*P*] are fixed. More explicit, this is what, the definability of *P* on the basis of *T*[*P*] means.

## Keywords

Left Column Interpolation Theorem Interrogative Model Universal Reading Inquiry Pertain
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## Notes

- 1.For a more complete treatment of the subject, see V. Rantala’s
*Aspects of definability*, Acta Philosophica Fennica, Vol.**29**, No. 2 (1977), North Holland, Amsterdam.Google Scholar - 2.See Cheng Hsiao, ‘Identification’, in Z. Griliches and M.D. Intriligator (eds.),
*Handbook of Econometrics*, Vol.**1**, Ch. 4, North Holland, Amsterdam, 1983;Google Scholar - 2a.F. Fisher,
*The Identification Problem in Econometrics*, McGraw-Hill, New York, 1966.Google Scholar - 3.See H. A. Simon, ‘The Axiomatization of Physical Theories’,
*Philosophy of Science*, Vol.**37**, No. 1 (1970), pp. 16–26.CrossRefGoogle Scholar - 4.See H. A. Simon,
*op. cit*;‘The Axioms of Newtonian Mechanics’,*Philosophical Magazine*, Series 7, Vol.**33**(1947), pp. 888–905; ‘The Axiomatization of Classical Mechanics’,*Philosophy of Science*, Vol.**21**(1954), pp. 340–343; ‘Definable Terms and Primitives in Axiom Systems’, in L. Henkin, P. Suppes and A. Tarski (eds.),*The Axiomatic Method*, North-Holland, Amsterdam, 1959; J. C. C. McKinsey, A. C. Sugar and P. Suppes, ‘Axiomatic Foundations of Classical Particle Mechanics’,*Journal of Rational Mechanics and Analysis*, Vol.**2**, pp. 253–272;Google Scholar - 4a.J. C. C. McKinsey and Patrick Suppes, ‘Transformations of Systems of Classical Particle Mechanics’,
*ibid.*, pp. 273–289;Google Scholar - 4b.M. Jammer,
*Concepts of Mass in Classical and Modern Physics*, Harvard U.P., Cambridge, 1961, especially ch. 9.Google Scholar - 5.E. Mach,
*Die Mechanik in ihrer Entwicklung*, F. A. Brockhaus, Leipzig, 1883.Google Scholar - 6.See, e.g., J. Hintikka, ‘The Logic of Science as Model-Oriented Logic’, in P. Asquith and P. Kitcher (eds.),
*PSA 1984*, Vol.**1**, Philosophy of Science Association, East Lansing, MI, 1984, pp. 177–185; ‘Knowledge Representation and the Interrogative Approach to Inquiry’, in M. Clay and K. Lehrer (eds.);*Knowledge and Skepticism*, Westview Press, Boulder, CO., 1989; ‘What Is the Logic of Experimental Inquiry?’,*Synthese*vol.**74**(1988), pp. 173–188.Google Scholar - 7.See E. W. Beth, ‘Semantic Entailment and Formal Derivability’,
*Mededelingen ven der Koninklijke Nederlandse Akademie van Wetenschappen*, Afdeling Letterkunde, N.R., Vol.**18**, No. 13 (1955), Amsterdam. This technique is formally speaking but a mirror image of a version of a Gentzen-type sequent calculus.Google Scholar - 8.There are of course other types of interrogative inquiry in which the aim of the game is not to prove a predetermined conclusion but to answer a question. I shall return to this matter in sec. 14 below.Google Scholar
- 9.For these results and for their background see, e.g., J. Barwise (eds.),
*Handbook of Mathematical Logic*, Part D, North Holland, Amsterdam, 1977.Google Scholar - 10.See my paper, ‘Knowledge Representation and the Interrogative Model of Inquiry’, in Marjorie Clay and Keith Lehrer (eds.),
*Knowledge and Skepticism*, Westview Press, Boulder, Colorado, 1989, pp. 155–183.Google Scholar - 11.W. Craig, ‘Three Uses of the Herbrand-Gentzen Theorem in Relating Model Theory to Proof Theory’,
*Journal of Symbolic Logic*, Vol.**22**(1957), pp. 269–285.CrossRefGoogle Scholar - 12.E. W. Beth, ‘On Padua’s Method in the Theory of Definition’,
*Indagationes Mathematicae*, Vol.**15**(1953), pp. 330–339.Google Scholar - 13.L. Svenonius, ‘A Theorem About Permutation in Models’,
*Theoria*, Vol.**25**(1959), pp. 173–178.CrossRefGoogle Scholar - 14.E. W. Beth,
*op. cit.*Google Scholar - 15.This theorem was first formulated in Jaakko Hintikka and Stephen Harris, ‘On the Logic of Interrogative Inquiry’, in A. Fine and J. Leplin (eds.),
*PSA 1988*, Vol.**1**, Philosophy of Science Association, East Lansing, MI, 1988, pp. 233–240.Google Scholar - 16.See C. C. Chang, ‘Some New Results in Definability’,
*Bulletin of the American Mathematic Society*, Vol.**70**(1964), pp. 808–813;CrossRefGoogle Scholar - 16a.M. Makkai, ‘A Generalization of a Theorem of E. W. Beth’,
*Acta Math. Acad. Sci. Hungar.*, Vol.**15**(1964), pp. 227–235.CrossRefGoogle Scholar - 17.For this issue, see, e.g., J. Giedymin, ‘Logical Comparability and Conceptual Disparity between Newtonian and Relativistic Mechanics’,
*British Journal for the Philosophy of Science*, Vol.**24**(1973), pp. 270–276;CrossRefGoogle Scholar - 17a.David Pearce,
*Roads to Commensurability*, D. Reidel, Dordrecht, 1987, pp. 154–158.CrossRefGoogle Scholar - 18.See, e.g., R. L. Causey, ‘Theory and Observation’, in P. Asquith and H. Kyburg, Jr. (eds.),
*Current Research in the Philosophy of Science*, Philosophy of Science Association, East Lansing, MI, 1979, pp. 187–206;Google Scholar - 18a.R. Tuomela,
*Theoretical Concepts*, Springer-Verlag, Wien & New York, 1973.CrossRefGoogle Scholar - 19.See here J. Hintikka and P. Sibelius, ‘Identification and Heisenbergian Uncertainty’, forthcoming.Google Scholar
- 20.Notice that in the principal wh-question to be answered through inquiry the whquestion is in effect given what I have called the universal reading, while the “small” questions through which it is answered is given an existential reading. (See here Jaakko Hintikka,
*The Semantics of Questions and the Questions of Semantics, Acta Philosophica Fennica*, Vol.**28**, No. 4, Societas Philosophica Fennica, Helsinki, 1976, especially ch. 4.) This is as it ought to be. The Inquirer is trying to extract the maximal information from the questioning procedure, and hence prefers the universal reading of whquestions. Nature is operating with the contrary purpose, and hence chooses the reading of wh-questions on which they have the least informative answers.Google Scholar - 21.In writing this paper, I have profited greatly from cooperation with Stephen Harris.Google Scholar

## Copyright information

© Springer Science+Business Media Dordrecht 1999