Abstract
The problem of distance from the truth, and more generally distance between hypotheses, is considered here with respect to the case of quantitative hypotheses concerning the value of a given scientific quantity.
Our main goal consists in the explication of the concept of distance D(I, θ) between an interval hypothesis I and a point hypothesis θ. In particular, we attempt to give an axiomatic foundation of this notion on the basis of a small number of adequacy conditions.
Moreover, the distance function introduced here is employed for the reformulation of the approach to scientific inference — developed by Hintikka, Levi and other scholars — labelled “cognitive decision theory”. In this connection, we supply a concrete illustration of the rules for inductive acceptance of interval hypotheses that can be obtained on the basis of D(I, θ).
Lastly, our approach is compared with other proposals made in literature about verisimilitude and distance from the truth.
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References
Aitchison, J. and I. R. Dunsmore: 1968, ‘Linear-Loss Interval Estimation of Location and Scale Parameters’, Biometrika 55, 141–8.
Aitchison, J. and I. R. Dunsmore: 1975, Statistical Prediction Analysis, Cambridge University Press, Cambridge.
Andersson, G.: 1978, ‘The Problem of Verisimilitude’ in G. Radnitzky and G. Andersson (eds.), Progress and Rationality in Science, Reidel, Dordrecht.
Carnap, R.: 1950, Logical Foundations of Probability, The University of Chicago Press, Chicago.
Carnap, R.: 1966, ‘Probability and Content Measure’, in P. K. Feyerabend and G. Maxwell (eds.), Mind, Matter and Method, University of Minnesota Press, Minneapolis.
Carnap, R.: 1980, ‘A Basic System of Inductive Logic. Part II’, in R. Jeffrey (ed.), Studies in Inductive Logic and Probability, Vol. II, University of California Press, Berkeley.
Ferguson, T. S.: 1967, Mathematical Statistics: A Decision- Theoretic Approach, Academic Press, New York.
Hempel, C. G.: 1960, ‘Inductive Inconsistencies’, Synthese 12, 439–69, reprinted in C. G. Hempel, Aspects of Scientific Explanation, The Free Press, New York, 1965.
Hilpinen, R.: 1968, Rules of Acceptance and Inductive Logic (Acta Philosophica Fennica 21), North-Holland, Amsterdam.
Hilpinen, R.: 1972, ‘Decision-Theoretic Approaches to Rules of Acceptance’, in R. E. Olson and A. M. Paul (eds.), Contemporary Philosophy in Scandinavia, The Johns Hopkins Press, Baltimore and London.
Hintikka, J.: 1968, ‘The Varieties of Information and Scientific Explanation’ in B. van Rootselaar and J. F. Staal (eds.), Logic, Methodology and Philosophy of Science, North-Holland, Amsterdam. pp. 311–31.
Kuipers, T. K.: 1978, Studies in Inductive Probability and Rational Expectation, Reidel, Dordrecht.
Levi, I.: 1967a, Gambling with Truth, Alfred A. Knopf, New York.
Levi, I.: 1967b, ‘Information and Inference’, Synthese 17, 369–99.
Miller, D.: 1975, ‘The Accuracy of Predictions’, Synthese 30, 159–91, 207–19.
Miller, D.: 1976, ‘Verisimilitude Redeflated’, The British Journal for the Philosophy of Science 27, 363–80.
Miller, D.: 1978a, ‘On Distance from the Truth as a True Distance’, in J. Hintikka, I. Niiniluoto, and E. Saarinen (eds.), Essays on Mathematical and Philosophical Logic, Reidel, Dordrecht.
Miller, D.: 1978b, ‘The Distance between Constituents’, Synthese 38, 197–212.
Niiniluoto, I.: 1977, ‘On a K-dimensional System of Inductive Logic’ in F. Suppe and P. Asquith (eds.), PSA 1976, Philosophy of Science Association, East Lansing.
Niiniluoto, I.: 1978a, ‘Truthlikeness in First-Order Languages’, in J. Hintikka, I. Niiniluoto, and E. Saarinen (eds.), Essays on Mathematical and Philosophical Logic, Reidel, Dordrecht.
Niiniluoto, I.: 1978b, ‘Truthlikeness: Comments on Recent Discussion’, Synthese 38, 281–329.
Niiniluoto, I.: 1980, ‘Scientific Progress’, Synthese 45, 427–62.
Niiniluoto, I.: 1982, ‘Truthlikeness for Quantitative Statements’ in P. D. Asquith and T. Nickles (eds.) PSA 1982, Vol. I, Philosophy of Science Association, East Lansing.
Niiniluoto, I.: 1985, ‘Theories, Approximations, and Idealizations’, in Logic, Methodology, and Philosophy of Science VII, North-Holland, Amsterdam.
Oddie, G.: 1978, ‘Verisimilitude as Distance in Logical Space’, in I. Niiniluoto and R. Tuomela (eds.) The Logic and Epistemology of Scientific Change (Acta Philosophica Fennica 30), North-Holland, Amsterdam.
Popper, K. R.: 1963, Conjectures and Refutations: The Growth of Scientific Knowledge, Routledge and Kegan Paul, London.
Popper, K. R.: 1972, Objective Knowledge, Oxford University Press, Oxford.
Popper, K. R.: 1976, ‘A Note on Verisimilitude’, The British Journal for the Philosophy of Science 27, 147–59.
Rosenkrantz, R. D.: 1980, ‘Measuring Truthlikeness’, Synthese 45, 463–87.
Tichý, P.: 1976, ‘Verisimilitude Redefined’, The British Journal for the Philosophy of Science 25, 25–42.
Tichý, P.: 1978, ‘Verisimilitude Revisited’, Synthese 38, 175–96.
Tuomela, R.: 1978, ‘Verisimilitude and Theory-Distance’, Synthese 38, 213–46.
Wald, A.: 1950, Statistical Decision Functions, John Wiley, New York.
Zellner, A.: 1971, An Introduction to Bayesian Inference in Econometrics, John Wiley, New York.
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I am grateful for the possibility of spending a period of some months at the Department of Philosophy of University of Gröningen. I profited there a lot from the supervision by Dr. Theo Kuipers. I am also grateful to Prof. Ilkka Niiniluoto who gave rise to my interest in this kind of research during a period spent at Department of Philosophy of University of Helsinki, and who made a number of useful suggestions in the final stage of this paper. Moreover, I would like to thank Dr. Carlo Buttasi for suggesting a better formulation of the proof of (T.24).
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Festa, R. A measure for the distance between an interval hypothesis and the truth. Synthese 67, 273–320 (1986). https://doi.org/10.1007/BF00540073
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DOI: https://doi.org/10.1007/BF00540073