Abstract
Earlier chapters deal with abductive inferences to explanations which are deductive or inductive-probabilistic. This more or less standard account has so far ignored the fact that explanatory and predictive success in science is often approximate. Therefore, the analysis of abduction should cover also approximate explanations, which is illustrated by Newton’s explanation of Kepler’s harmonic law (Sect. 8.1). The notions of approximate truth (closeness to being true), verisimilitude (closeness to complete qualitative or quantitative truth) and legisimilitude (closeness to the true law) are defined in Sect. 8.2. This leads us to generalize Peirce’s model of abduction to cases where the conclusion states that the best theory is truthlike or approximately true, with illustrations from idealized theories and models (Sect. 8.3). In a comparative formulation, if theory Y is a better explanation of the available evidence E than theory X, then conclude for the time being that Y is more truthlike than X. To justify such abduction, we need a method of estimating degrees of truthlikeness by their expected values. Another tool is the notion of probable approximate truth. Then, in order to answer to Laudan’s challenge, the probabilistic link between empirical success and truth has to be replaced with a fallible bridge from the approximate empirical success of a theory to its truthlikeness (Sect. 8.4). Section 8.5 gives some remarks on abductive belief revision, which is related to cases where the evidence is conflict with the theory. This theme extends Aliseda’s way of linking belief revision models with abductive reasoning.
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Notes
- 1.
Sloughter (1996) argues that plausibility in Peirce’s probabilistic abduction (see e.g. (1.9)) can be measured by the significance level in R. A. Fisher’s theory of statistics. Significance level tells how unlikely the evidence must be for the null hypothesis to be rejected, so that in a sense the improbability of data given a null hypothesis measures their discrepancy. This idea is related to the likelihood principle (7.3). But Slaughter does not discuss the controversial aspects of Fisher’s “fiducial” inference, which attempts to reach something like doxastic probabilities without Bayesian priors.
- 2.
- 3.
Rott (1989) compares several ways of analyzing approximations and idealizations in the Kepler-Newton case .
- 4.
For details, see Niiniluoto (1987), Ch. 11.
- 5.
Alternatively the target could be the true state description in a monadic language L, where a state description specifies which Q-predicate each individual in L satisfies. All sentences of L have a normal form as a disjunction of state descriptions. For the treatment of truthlikeness of singular statements along these lines, see Niiniluoto (1987), Ch. 8.
- 6.
This measure can be applied also to nomic monadic constituents , which claim that some Q-predicates are physically possible and some physically impossible. In the definition of legisimilitude , it is natural to take the true nomic constituent as the target. This is in harmony with the modal treatment of quantitative laws which express the physically possible combinations of quantitative properties (see Niiniluoto 1987, 95, 112). The theories of Kuipers (2000) can be interpreted as nomic constituents (see Niiniluoto 1987, 381).
- 7.
- 8.
Rowbottom (2015) argues that scientific progress is possible in the absence of increasing verisimilitude . He asks us to imagine that the scientists in a specific area of physics have found the maximally verisimilar theory C*. Then it may seem that no more progress is possible, but yet this general true theory could be used for further predictions and applications. One reply to this argument is that predictions from C* constitute new cognitive problems for the scientists. Moreover, on the basis of conceptual pluralism , in Rowbottom’s thought experiment it would still be possible for the physicists to achieve further progress by extending their conceptual framework and replacing the target C* with a stronger truth in order to find a still deeper truths about their research domain.
- 9.
Chang Liu (1999) has made the pertinent observation that this treatment of singular approximation is “flat” in the sense that it assumes a global metric in a uniform state space Q and thus does take into account the influence of true laws on the structure of Q. For example, a claim that the velocity of light slightly exceeds the value 300.000 km/s cannot be approximately true, since it is physically impossible by the theory of relativity. As a possible reply, one could argue that the state space Q should be a neutral framework for comparing various kinds of hypotheses independently of assumed background knowledge. As false hypotheses can be truthlike, why could not physically impossible claims be approximately true? Perhaps a more interesting line of thought is suggested by Liu’s another remark: if e.g. a linear law L is assumed to hold in R2, the singular truth x* is a point on L, and the hypothesis y lies outside L, then the distance of y from x* should depend also on its closeness to L. This idea would allow that physical possibilities influence the metric structure of the state space in the same way as the distribution of masses generate non-Euclidean geometries of physical space in relativity theory. In Liu’s example, the circles of Euclidean metrics around the point x* would be replaced by ellipses with L as its major axis.
- 10.
Cf. the discussion of approximate predictions in Niiniluoto (1999a), 195–196.
- 11.
In this case, the better theory has larger expected predictive success, if the values of argument x are chosen randomly. Cf. the Expected Success principle in Kuipers (2000) , 310.
- 12.
This reformulation allows us to re-evaluate Duhem’s and Popper’s criticism of induction: Newton’s laws are not derived from Kepler’s laws by induction but by the modified model of abduction.
- 13.
- 14.
Twelve volumes on idealization have appeared in the book series Poznan Studies in the Philosophy of Science between 1990 and 2005.
- 15.
Independently of my proposal, originally presented in 1983, Michael Shaffer has treated idealization statements as counterfactuals. See Shaffer (2007).
- 16.
Idealized theories are sometimes called “models”, so that one may consider explanation by idealized models. (Magnani and Bertolotti (2017) includes several survey articles on models and modeling.) Such models may be either sets of mathematical equations or structures which are interpretations of theoretical statements. In the latter sense, models may be compared to real systems by similarity metrics which are special cases of truthlikeness measures. For example, Weisberg (2013) uses the same method of “feature matching” as Cevolani et al. (2013) . Cf. also Niiniluoto (1987), 335–338.
- 17.
- 18.
See the survey in Niiniluoto (2018).
- 19.
- 20.
In Sect. 9.2 we shall return to a specific form of downward inference, the no miracle argument for the success of science.
- 21.
The standard Bayesian trick to allow that P(E/H) > 0 is to add to regression models H a random error with a Gaussian probability distribution (cf. (4.5)). See Bandyopadhyay and Boik (1999) .
- 22.
The function PA can be used to explicate the notion of “probable approximate correct” or PAC-learning in machine learning (see Niiniluoto 2005b).
- 23.
Our reply is thus different from Abner Shimony’s (1970) “tempered personalism”, which argues that all “seriously proposed” hypotheses should have non-zero probabilities.
- 24.
Zamora Bonilla’s (1996) alternative solution is to measure directly the distance between a theory and empirical laws, so that for such evidence-relative notion of truthlikenesss he does not need the objective notions of truth and truthlikeness at all. Schippers (2015) develops a “two-sided” coherence measure (cf. Sect. 6.4) for the distance between theory and evidence .
- 25.
Popper’s proposal of interpreting his degrees of corroboration - variants of the relevance measures of confirmation (see Sect. 6.2) – as indicators of verisimilitude (see Popper 1972, 103) fails precisely on this point, since such degrees have the value zero as soon as the hypothesis is falsified by evidence.
- 26.
Such approximate steps may occur in inter-theoretic reduction, so that their probabilistic analysis should be complemented by the notion of PA. See e.g. the notion of “analogy ” in Dizadji-Bahmani et al. (2011) .
- 27.
The Bayesian decision method of point and interval estimation by minimizing expected loss can be reinterpreted in terms of maximizing expected truthlikeness , but then the sum (25) has to be replaced by an integral (see Niiniluoto 1987, 426–441).
- 28.
Similar arguments can be applied to existential-universal generalizations which make some positive claims and some negative claims about the Q-predicates , but may leave some Q-predicates with question marks (see Niiniluoto 1987, 337; 2005a, 269). Such statements, which include constituents as a special case, are called c-theories by Cevolani et al. (2013) .
- 29.
The convergence results (36) and (37) can be relativized to a true background theory B, so that ver(H/E&B) converges to the truth given ES and B, but as fallibilists we cannot be completely certain that B is true.
- 30.
Connections between theoretical and observational predicates have been called “reduction sentences” (Carnap), “rules of correspondence” (Nagel) , and “bridge principles” (Hempel) (see Niiniluoto 1999a, 112). For example, in Wilson’s cloud chamber a curved trail of ionized gas particles is a manifestation of the negative electric charge of an electron.
- 31.
This definition of contraction uses an idea proposed by Adam Grove .
- 32.
However, Isaac Levi’s work on belief revision is based on his treatment of truth and information as epistemic utilities (see Sect. 7.1), but Levi explicitly denies the relevance of truthlikeness considerations.
- 33.
- 34.
See papers in the special issue “Belief Revision Aiming at Truth Approximation” of Erkenntnis 75:3 (2011). See also Cevolani et al. (2013).
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Niiniluoto, I. (2018). Abduction and Truthlikeness. In: Truth-Seeking by Abduction. Synthese Library, vol 400. Springer, Cham. https://doi.org/10.1007/978-3-319-99157-3_8
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