Abstract
In this paper it is argued that qualitative theories (Q-theories) can be used to describe the statistical structure of cross classified populations and that the notion of verisimilitude provides an appropriate tool for measuring the statistical adequacy of Q-theories. First of all, a short outline of the post-Popperian approaches to verisimilitude and of the related verisimilitudinarian non-falsificationist methodologies (VNF-methodologies) is given. Secondly, the notion of Q-theory is explicated, and the qualitative verisimilitude of Q-theories is defined. Afterwards, appropriate measures for the statistical verisimilitude of Q-theories are introduced, so to obtain a clear formulation of the intuitive idea that the statistical truth about cross classified populations can be approached by falsified Q-theories. Finally, it is argued that some basic intuitions underlying VNF-methodologies are shared by the so-called prediction logic, developed by the statisticians and social scientists David K. Hildebrand, James D. Laing and Howard Rosenthal.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
I am very grateful to Theo Kuipers for his remarks on previous versions of this paper.
As far as we know, the first explicit suggestion that an adequate analysis of confirmation should include strictly non-falsificationist principles has been made by Cohen (1973). According to Cohen any satisfactory analysis of confirmation must be capable of solving the so-called paradox of anomaly, where “anomaly” refers to a fact described by a statement e, regarded as observationally true, which conflicts with a theory h. According to Cohen (ib., p. 79) one can solve the paradox of anomaly only by “allowing a high level of support to some hypotheses even on the basis of observable evidence that contradicts them”. While this non-falsificationist condition is satisfied by Cohen’s concept of inductive support (1977, 1989), it is satisfied neither by Popper’s methodology nor by the standard versions of Bayesian epistemology. In fact, in presence of a falsifying evidence e, both corroboration of a hypothesis h (in Popper’s sense) and its posterior probability (i.e., its Bayesian confirmation) collapse to their minimum value.
Lakatos (ib., pp. 384–385) argues that “theories rarely pass severe tests of their new content with flying colours; even some of the best theories may never get ‘strictly corroborated’. But such theories, even if in a strict sense they fail all their tests, may have something of their excess empirical content, namely some weaker but still interesting consequences, corroborated [...]”. In some cases it happens that “[t]heories, while failing all their tests quantitatively, frequently pass some of them ‘qualitatively’ [...]” (italics added). In such cases, a theory is refuted-and-corroborated by the same experimental test, which is failed quantitatively but passed qualitatively; this means that the theory is corroborated by the falsifying result of the test. This feature of Lakatos’ corroboration clearly reveals its strictly non-falsificationist nature. As an historical example of quantitatively-refuted-and-qualitatively-corroborated theories, Lakatos (ib., p. 385) mentions Bohr’s first theories of the hydrogen atom; indeed, they “were immediately falsified by the fact that spectral lines were multiplets, but the subsequent discovery of Layman’s, Brackett’s and Pfund’s series corroborated, and indeed, strictly corroborated, the weaker but still novel, previously undreamt of, consequence of Bohr’s theory that there are spectral lines in the immediate neighbourhood of predicted wavelengths”.
Lakatos (1974, p. 269, note 117) claims that “the methodology of scientific research programmes is better suited for approximating the truth in our actual universe that any other methodology”.
In 1986 and 1987, the first decade of intensive work on the post-Popperian theories of verisimilitude was summarized in the first book-length expositions of the so-called similarity approach to verisimilitude by Oddie (1986), Niiniluoto (1987) and Kuipers (1987). An excellent survey of the modern history of verisimilitude, started with Karl Popper’s definition of verisimilitude in 1960, is provided by Niiniluoto (1998).
The most articulated versions of VNF-methodologies have been developed by Niiniluoto (1987) and Kuipers (2000). In particular, the so-called evaluation methodology, proposed by Kuipers (2000), where (also falsified) theories are judged in terms of their successes and problems, is explicitly motivated by Lakatos’ idea that theory evaluation has a primarily comparative character, leaving falsified theories in the game as long as there are no better alternatives.
The formulation of principles of this kind can be seen as a response to the late Lakatos’ challenge to identify inductive principles linking corroboration and verisimilitude in the appropriate way.
The subscripts “ c” in “x c ” and “ r” in “y r ” stay for “column” and “row”, w.r.t. the tables used to represent cross classifications; see Table 1 below.
It should be noted that, if “the truth” about D is given by the trivial constituent C T , then the only true Q-theory is the tautological Q-theory G T while any non-tautological Q-theory is not only false, but completely false.
More precisely, d(C *,C) should be defined as an increasing function of \({|C^{-}_{*} \cap C^+|}\) and |C + * ∩ C −| and a decreasing function of |C − * ∩ C −| and |C + * ∩ C +|. This intuition can be specified in different ways. Several of the distance measures d(C *,C) proposed in the literature are particular cases of a parametric class of distance measures introduced by Festa (1987, p. 158).
For instance, it is easy to check that, if C T is “the truth” and the similarity function s α, β, γ,δ = 1 − d α, β, γ,δ is used, then the most verisimilar Q-theory is the tautological Q-theory G T .
The “dot” in the symbol P 1. indicates that the variable X whose index j is omitted has been ignored.
It should be noted that any true theory is s-true, while the vice versa does not hold.
If one does not like the notion of statistically true Q-theory and finds that “G is s-true” is a problematic statement, one could introduce the “statistical” statement G° ≡ “The Q-predicates in G − are P-rare”, and interpret “G is s-true” as a handful reformulation of “G° is true”.
It should be recalled that, if the commonness function com t is adopted, then com t (Q ij ) ≡ P ij [see (13)], so that any P-positive Q-predicate belongs to the set P + of the P-common Q-predicates. It follows that G − ∩ P + includes all the P-positive Q-predicates in G −. This implies that \({{\sum\limits_{Q_{{ij}} \in G^{-} \cap P^{+}} {P_{{ij}}}} = {\sum\limits_{Q_{{ij}} \in G^{-}} {P_{{ij}}}}.}\)
The final version of the present paper was sent to Prof. Rosenthal who made several stimulating comments and suggestions for further research. He was also so kind to provide interesting information about the circumstances in which prediction logic was developed. David K. Hildebrand, James D. Laing and Howard Rosenthal began their work in the late 1960s when Rosenthal and Laing were assistant professors at Carnegie Mellon University and Hildebrand, whose doctorate is from Carnegie Mellon, returned for a year as a visiting assistant professor. Their collaboration was truly one of equals, so that the order of the authors in the titles of their joint papers depends just on the circumstance that, in economics and political science, senior authors are listed in strict alphabetical order. Rosenthal agrees that there is little reference to prediction logic after the 1970s and suggests a couple of convincing reasons for this. First, perhaps as a result of the controversy with Leo Goodman and William Kruskal (see below, note 19), the measures of predictive power introduced within prediction logic never made it into standard statistical packages used in those years, such as SPSS. Second, social science has been more concerned with the development of sophisticated measures and tests of statistical association between qualitative variables than with the evaluation of specific qualitative theories (see below, in Sect. 5.1). However, the ideas of prediction logic have been used in some biological sciences, such as zoology and health science. Indeed the little pamphlet Analysis of Ordinal Data (Hildebrand et al. 1977) continues to sell a few hundred copies every year and has sold over 40,000 since 1977. Unfortunately, Hildebrand passed away on July 13th, 1999 after a long struggle with cancer. He was 59 years old, and had spent 34 of those years at the University of Pennsylvania where he served as chairman of the statistics department. Laing moved to Penn in 1976. He is now fully retired and lives with his wife on a boat which cruises between North Carolina and southern waters. Rosenthal moved from Carnegie Mellon to Princeton in 1993 and retired in 2005. Afterwards he accepted a one semester a year appointment at New York University. My deepest thanks to Prof. Rosenthal for sharing this information with me.
The issues (1) and (2) correspond to the logical and methodological problem of verisimilitude, respectively (see the end of Sect. 2).
Recall that rule U is used in a condition where the X-state of the individuals in D is unknown.
Among other things, the conceptual relations between our measures of qualitative verisimilitude and the approaches developed by Tuomela (1978) and Niiniluoto (1987) might be explored. Moreover, the cognitive significance of the commonness functions underlying our measures of statistical verisimilitude may be better understood and, presumably, alternative commonness functions may be introduced.
Such analysis might profit a lot both from the work by Hildebrand et al. (1976b, Chap. 6 and 7) and from the analysis of different notions of estimated verisimilitude that has been made in the last 30 years by the scholars concerned with the development of VNF-methodologies.
References
Cohen LJ (1973) The paradox of anomaly. In: Bogdan RJ, Niiniluoto I (eds) Logic, language, and probability. Reidel, Dordrecht, pp 78–82
Cohen LJ (1977) The probable and the provable. Oxford University Press, Oxford
Cohen LJ (1989) An introduction to the philosophy of induction and probability. Clarendon, Oxford
Coleman JS (1964) Introduction to mathematical sociology. Free Press, New York
Costner HL (1965) Criteria for measures of association. Am Soc Rev 30:341–353
Duverger M (1954) Political parties. Wiley, New York
Festa R (1987) Theory of similarity, similarity of theories, and verisimilitude. In: Kuipers TK (ed) What is closer-to-the-truth?. Rodopi, Amsterdam, pp 145–176
Festa R (2007) Verisimilitude, qualitative theories, and statistical inferences. In: Sintonen M, Pihlström S, Raatikainen P (eds) Festschrift for Ilkka Niiniluoto (forthcoming)
Goodman LA, Kruskal W (1974a) Empirical valuation of formal theory. J Math Soc 3:187–196
Goodman LA, Kruskal W (1974b) More about empirical valuation of formal theory. J Math Soc 3:211–213
Hildebrand DK, Laing JD, Rosenthal H (1974a) Prediction logic: a method for empirical evaluation of formal theory. J Math Soc 3:163–185
Hildebrand DK, Laing JD, Rosenthal H (1974b). Prediction logic and quasi-independence in empirical evaluation of formal theory. J Math Soc 3:197–209
Hildebrand DK, Laing JD, Rosenthal H (1975) A prediction logic approach to causal models of qualitative variates. In: Heise DR (ed) (1976) Sociological methodology. JosseyBass, San Francisco, pp 146–175
Hildebrand DK, Laing JD, Rosenthal H (1976a) Prediction analysis in political research. Am Polit Rev 70:509–535
Hildebrand DK, Laing JD, Rosenthal H (1976b) Prediction analysis of cross classification. Wiley, New York
Hildebrand DK, Laing JD, Rosenthal H (1977) Analysis of ordinal data. Sage Publication, Newbury Park
Homans GC (1961) Social behavior: its elementary forms. Harcourt Brace Jovanovich, New York
Kuipers TAF (ed) (1987) What is closer-to-the-truth? Rodopi, Amsterdam
Kuipers TAF (2000) From instrumentalism to constructive realism. Kluwer, Dordrecht
Kuipers TAF (2001) Structures in science. Kluwer, Dordrecht
Lakatos I (1968) Changes in the problem of inductive logic. In: Lakatos I (ed) The problem of inductive logic. NorthHolland, Amsterdam, pp 315–417
Lakatos I (1970) Falsificationism and the methodology of scientific research programmes. In: Lakatos I, Musgrave A (eds) Criticism and the growth of knowledge. Cambridge University Press, Cambridge, pp 91–196. Reprinted in: Lakatos I (1978) The methodology of scientific research programmes, edited by Worrall J, Currie G. Cambridge University Press, Cambridge, pp 8–101
Lakatos I (1974) Popper on demarcation and induction. In: Schilpp PA (ed) The philosophy of Karl Popper. Open Court, La Salle, pp 241–273
Lipset SM (1960) Political man: the social bases of politics. Doubleday, Garden City
Miller D (1974) Popper’s qualitative theory of verisimilitude. Br J Philos Sci 25:166–177
Niiniluoto I (1987) Truthlikeness. Reidel, Dordrecht
Niiniluoto I (1989) Corroboration, verisimilitude, and the success of science. In: Gavroglu K, Goudaroulis Y, Nicolacopoulos P (eds) Imre Lakatos and theories of scientific change. Reidel, Dordrecht, pp 229–243
Niiniluoto I (1998) Verisimilitude: the third period. Br J Philos Sci 49:11–29
Oddie G (1986) Likeness to the truth. Reidel, Dordrecht
Popper KR (1959) The logic of scientific discovery, Hutchinson, London. (1st edn, 1934: Logik der Forschung, Wien)
Popper KR (1963) Conjectures and refutations. Routledge and Kegan Paul, London
Popper KR (1972) Objective knowledge. Clarendon Press, Oxford
Tichý P (1974) On Popper’s definition of verisimilitude. Br J Philos Sci 25:155–160
Tuomela R (1978) Verisimilitude and theory-distance. Synthese 38:213–246
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Festa, R. Verisimilitude, cross classification and prediction logic. Approaching the statistical truth by falsified qualitative theories. Mind & Society 6, 91–114 (2007). https://doi.org/10.1007/s11299-006-0022-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11299-006-0022-2