Abstract
In the process of scientific discovery, knowledge ampliation is pursued by means of non-deductive inferences. When ampliative reasoning is performed, probabilities cannot be assigned objectively. One of the reasons is that we face the problem of the unconceived alternatives: we are unable to explore the space of all the possible alternatives to a given hypothesis, because we do not know how this space is shaped. So, if we want to adequately account for the process of knowledge ampliation, we need to develop an account of the process of scientific discovery which is not exclusively based on probability calculus. We argue that the analytic view of the method of science advocated by Cellucci is interestingly suited to this goal, since it rests on the concept of plausibility. In this perspective, in order to account for how probabilities are in fact assigned in uncertain contexts and knowledge ampliation is really pursued, we have to take into account plausibility-based considerations.
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Notes
‘Uncertainty’ has different usages, at least an informal and a technical usage, and this may give rise to some misunderstandings. In order to make clearer what we mean by ‘uncertainty’, it may be useful to compare this concept with the concept of ‘risk’. As Hansson states, in decision theory, “a decision is said to be made ‘under risk’ if the relevant probabilities are available and ‘under uncertainty’ if they are unavailable or only partially available. […]. Although this distinction between risk and uncertainty is decision-theoretically useful,” it is not completely satisfactory from an epistemological point of view, since “only very rarely are probabilities known with certainty. Strictly speaking, the only clear-cut cases of ‘risk’ (known probabilities) seem to be idealized textbook cases that refer to devices such as dice or coins that are supposed to be known with certainty to be fair. In real-life situations, even if we act upon a determinate probability estimate, we are not fully certain that this estimate is exactly correct, hence there is uncertainty” (Hansson 2014, Sect. 2). In this view, one deals with ‘uncertainty’ whenever one cannot make any probability estimate of the possible outcomes or one cannot know with certainty whether one’s probability estimates are correct.
On certainty and the difficulty of giving an account of certainty which is compatible with a fallibilistic view of knowledge, see Reed (2011, especially Sect. 2).
Cf. Cellucci (2017, p. 142): “Methods can be divided into algorithmic and heuristic. An algorithmic method is a method that guarantees to always produce a correct solution to a problem. Conversely, a heuristic method is a method that does not guarantee to always produce a correct solution to a problem.”
For a detailed explanation of why mathematical logic failed to be the logic of justification, see Cellucci (2015).
Cf. e.g. Howson and Urbach (2006, p. 79): “A nontrivial conclusion (one which is not itself a theorem of logic) of a deductively valid inference depends on at least one nontrivial premise. Similarly, a nontrivial conclusion (one which is not a theorem of probability) of a valid probabilistic argument depends on one or more nontrivial probabilistic premises. And just as the logical axioms in Hilbert-style axiomatisations of classical logic are regarded as empty of factual content because they are universally valid, so is the same true of the probability axioms in the view we have been advocating. They too are logical axioms, empty of factual content because universally valid. Putting all this together, we can derive a probabilistic analogue of the celebrated conservation result of deductive logic, that valid deductive inference does not beget new factual content, but merely transforms or diminishes the content already existing in the premises. So too here: valid probabilistic inference does not beget new content, merely transforming or diminishing it in the passage from premises to conclusion.”
By underlying these points, we do not mean to deny the theoretical relevance of both formal approaches to confirmation and probabilistic approaches to discovery. We only mean to stress that some relevant theoretical insights might be provided by considering the role played by non-formalizable components of reasoning.
We are dealing here with an inference to the best explanation (Lipton 2004; see below, Sect. 7.2). In this kind of inference, some criterion is needed in order to rank different explanations and select the one that displays the highest score, i.e. the best explanation. Here we consider the case that probability is an adequate criterion to rank rival hypotheses. In this case, the best hypothesis is the one which displays the highest score with respect to probability, and so the most probable hypothesis is regarded as the true one. We will not enter here the debate over the validity of the inference to the best explanation, nor the debate over the difficulty of ranking different hypotheses which are able to explain the same set of empirical phenomena but differ under some other theoretical respect, see e.g. Tulodziecki (2012). We just wish to stress that we are not focusing here on the epistemic and pragmatic difficulties that one has to face when one attempts to effectively determine the score of each hypothesis. We concede, for argument’s sake, that hypotheses can be effectively ranked, for instance by performing reliable empirical tests. Rather, here we wish to focus on and appraise the theoretical consequences of accepting the claim that probability is objective, and it can be used to rank hypotheses. We thank an anonymous reviewer for pressing us to clarify this point.
This issue is strictly related to the ‘bad lot argument’ developed by van Fraassen (1989) against the inference to the best explanation. According to van Fraassen, we “can watch no contest of the theories we have so painfully struggled to formulate, with those no one has proposed. So our selection may well be the best of a bad lot. […]. For me to take it that the best of set X will be more likely than not, requires a prior belief that the truth is already more likely to be found in X, than not” (van Fraassen 1989, p. 143). This objection is analogous to the ‘problem of the unconceived alternatives’ raised by Stanford (2006) against scientific realism. In both cases, the point is that we cannot exclude that there may be unconceived alternatives to the hypotheses we are evaluating. If probability is objective, probability values are assigned to each possible outcome, no one excluded, so it can be claimed that our ‘lot’ of hypotheses is not a bad lot, since no hypothesis is left out. In this case, we can claim that there cannot be unconceived alternatives, and the true hypothesis is among our set of hypotheses.
It may be objected that deductive arguments are at least ampliative in an epistemic sense, because otherwise we should say that we learn nothing in mathematics beyond what we already knew by knowing the premises of a proof. We agree that mathematical proofs can be regarded as ampliative in this epistemic sense, since we are not deductively omniscient. But epistemic ampliation is not equivalent to knowledge ampliation, see Cellucci (2017, Sect. 12.7), and here we are focusing on the latter. If mathematical proofs rest exclusively on deductions, a mathematical theorem “asserts nothing that is objectively or theoretically new as compared with the postulates from which it is derived, although its content may well be psychologically new in the sense that we were not aware of its being implicitly contained in the postulates” (Hempel 1945, p. 9). Deduction is extremely useful because it “discloses what assertions are concealed in a given set of premises, and it makes us realize to what we committed ourselves in accepting those premises.” Nevertheless, “none of the results obtained by this technique ever goes by one iota beyond the information already contained in the initial assumptions” (Ibidem). So, if deduction is ampliative in a merely epistemic sense, it is unable to account for how we reach mathematical knowledge which cannot even in principle be deductively derived from what we already know.
On the concept of ‘plausibility’, see also Gettys and Fisher (1979); Agassi (2014). These views of plausibility differ from Cellucci’s view under several respects. Nevertheless, all these conceptions of plausibility share some common features, namely the idea that plausibility assessment cannot be reduced to probability calculus, and that plausibility assessment is crucial for ampliating knowledge.
For a more detailed treatment of this issue, see Cellucci (2017, Chap. 9), where plausibility is confronted with related (but distinct) concepts, such as truth, probability, and warranted assertibility.
See Cellucci (2013, Chap. 20). An example of plausible hypotheses that have zero probability are all the plausible hypotheses derived by an Induction from a Single Case (ISC). On the classical concept of probability as the ratio between favorable and possible cases, a conclusion obtained by (ISC) has zero probability when the number of possible cases is infinite. An example of implausible hypotheses that have non-zero probability are those implausible hypotheses obtained by Induction from Multiple Cases (IMC). Consider the hypothesis that all swans are white. Until the end of the seventeenth century, “all swans observed were white. From this, by (IMC), it was inferred that all swans are white. But in 1697 black swans were discovered in Western Australia.” Since then, the hypothesis that all swans are white is highly implausible. But, this contrasts with the fact that, “on the classical concept of probability, a conclusion obtained by (IMC) has non-zero probability when the number of possible cases is not infinite,” and such “is the case of the hypothesis that all swans are white” (Cellucci 2013, p. 335).
On the relevance of the Dutch Books Arguments in the debate over the rationality requirement advocated by the Bayesians, see Vineberg (2016).
It is worth clarifying that we are not claiming that one should not rely on estimated probabilities for practical purposes, nor are we denying that those who rely on estimated probabilities for practical purposes are successful in dealing with the world. Here we are focusing on the epistemological analysis of the process of scientific discovery, which is a distinct issue. That estimated probabilities may well be adequate for dealing with every-day activities does not impinge on the theoretical issue of assessing whether scientific knowledge is certain. Indeed, the consequences of not considering some unknown possibilities may well be negligible in some contexts. Humans built ships long before coming to understand on scientific grounds why they were able to build ships that floated. Nevertheless, their practical ability in building ships did not make their pre-scientific beliefs about the reasons why they were able to build ships that floated more justified. The point is that practical success is not able to justify by itself our beliefs about the reasons why we succeeded in practical activities. As regard estimated probabilities, in many cases, even “if a decision problem is treated as a decision ‘under risk’, this does not mean that the decision in question is made under conditions of completely known probabilities. Rather, it means that a choice has been made to simplify the description of this decision problem by treating it as a case of known probabilities. This is often a highly useful idealization in decision theory” (Hansson 2014, Sect. 2). But the fact that we can deal with some practical affairs to a satisfactory extent by assuming that our probabilities are objective does not mean that our probabilities are really objective, and we really know for certain that there cannot be unconceived alternatives. We wish to thank an anonymous reviewer for urging us to clarify this point.
Many replies have been elaborated in the last decade to address Stanford’s instrumentalist challenge to scientific realism. See e.g. Magnus (2006, 2010); Saatsi et al. (2009); Ruhmkorff (2011); Devitt (2011); see Saatsi et al. (2009) for Stanford’s rejoinder to some criticisms; see Rowbottom (2016) and Wray (2016) for interesting extensions of Stanford’s line of reasoning.
See above, fn. 9.
If rival hypotheses are not only assumed to be mutually exclusive but also jointly exhaustive, “then one’s model will represent a situation in which one knows that one of these competing hypotheses must be true. In such a case, there is no need to include a ‘catch-all’ hypothesis to represent all unimagined hypotheses,” but “there are many contexts in which it is not known with certainty that the true hypothesis is one of those considered” (Schupbach 2011, p. 119). The problem in these cases is that we cannot demonstrate that there cannot be other unimagined hypotheses. “Such scenarios correspond to van Fraassen’s best of a bad lot objection as well as what Kyle Stanford (2006) calls ‘the problem of unconceived alternatives’” (Schupbach 2011, p. 119, fn. 2).
Cf. Stanford (2006, p. 42): “There is simply no way to assign an absolute probability or level of confirmation to the theory without solving the problem of estimating the likelihood conferred on the evidence by a catch-all hypothesis of unknown content and constitution.”
Cf. Rowbottom (2016, p. 3): “Why be confident that the confirmation value of any given theory […] would not change drastically if all the unconceived alternatives were appreciated? What licenses inferring absolute confirmation values from relative confirmation values?”
See above, fn. 9.
We say ‘at least’ because it can be argued that plausibility-based considerations play an indispensable role also in those cases in which the space of the possible alternatives is known with certainty, because the preliminary judgement that we do know the shape of the possible alternatives with certainty is better accounted for in terms of plausibility-based considerations, rather than in terms of probability-based considerations. We have no space here to further develop this argument.
Cf. e.g. Howson and Urbach (2006, p. 9): “the ideal of total objectivity is unattainable and […] classical methods, which pose as guardians of that ideal, actually violate it at every turn; virtually none of those methods can be applied without a generous helping of personal judgment and arbitrary assumption.”
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Fabio Sterpetti and Marta Bertolaso declare that they have no conflict of interest.
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The article was jointly developed by both authors and the thesis commonly shared. Fabio Sterpetti was mainly responsible for the writing of Sects. 1, 3, 5, 7, and 7.1, Marta Bertolaso was mainly responsible for the writing of Sects. 2, 4, 6, 7.2, and 8.
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Sterpetti, F., Bertolaso, M. The Pursuit of Knowledge and the Problem of the Unconceived Alternatives. Topoi 39, 881–892 (2020). https://doi.org/10.1007/s11245-018-9551-7
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DOI: https://doi.org/10.1007/s11245-018-9551-7