On the Elusive Formalisation of the Risky Condition for Hypothesis Testing

ABSTRACT

In this paper, we examine possible formalisations of the riskiness condition for hypothesis testing. First, we informally introduce derivability and riskiness as testing conditions together with the corresponding arguments for refutation and confirmation. Then, we distinguish two different senses of confirmation and focus our discussion on one of them with the aid of a historical example. In the remaining sections, we offer a brief overview of the main references to the risky condition in the literature and scrutinise different options for formally capturing riskiness; we show why none of them works. We conclude with some remarks about the relation between derivability and riskiness and claim that riskiness essentially involves a contextual, pragmatic component that eludes a complete formal reconstruction.

Notes

1 Here, we do not need to assume that the prediction P is strictly speaking a purely observational claim, nor that there is a sharp observational/theoretical divide. The only requirement is that there be a relevant contrast between the hypothesis and the prediction with regard to empirical assessment such that an experimental determination of whether the latter occurs is substantially more direct.

2 For an explanation of what the auxiliaries or the initial conditions are, see, for instance, Giere (1979, 87–88).

3 Actually, one may not know it after the test either, since it is possible that P fails, in which case knowing that the material conditional is true would require us to know that something in H∧A∧IC failed, which we may not know—of course we know this if we know that P failed and we also know that the conditional is true.

4 We thank Carl Hoefer for this observation.

5 Tycho’s was not the only geocentric system predicting the phases of Venus. Although not as well known, Paul Wittich’s system (inspired by Martianus Capella’s work, and most probably inspiring Tycho’s system), in which Mercury and Venus orbit around the Sun, and the Sun, the Moon and the other planets orbit around the Earth at the centre, also predicts the phases of Venus.

6 Notice, in this respect, the trivial logical fact that, for every hypothesis, H, and prediction, P, made by H, there conceptually exist infinitely many other incompatible, alternative hypotheses that make the same prediction. In addition, in most cases it is possible to concoct alternative—often quite artificial—hypotheses with the same prediction.

7 With this caveat, we only aim to show that Popper’s measure of the severity of a test captures one aspect of riskiness, not that it is equivalent to this latter condition. Moreover, given Popper’s explicit anti-inductivism, we are perfectly aware that he would not endorse an argument like (CONF). However, it remains an interesting question whether Popper’s ‘severity of a test’—or his closely related ‘degree of corroboration’—should be considered pace the author an inductive measure (see Díez 2010 for a discussion of this issue).

8 Mayo (1996, ch. 6) addresses the issue of alternative hypotheses, but in the context of the general problem of underdetermination of theory by data, explicitly referring to all conceptually possible alternatives, not only to the contextually considered rivals, which differs from the problem we deal with here.

9 In general, the argument ‘(i) not-H → very likely not-P, (ii) P, THEREFORE (iii) H’ is neither deductively nor inductively valid. It is not deductively valid since the second premise is not the negation of the consequent of the first premise (the occurrence of P is compatible with its unlikelihood). And it is not inductively valid either, since the two premises may be true due to the actual occurrence of a very unlikely P, which tells us nothing about the possibly completely unrelated H (as in the (Fresnel) case above).

10 Giere claims that the conclusion in point is not literally ‘H’, but ‘approximately H’. This is not intended to substitute the inductive argument for a deductive one, but to qualify the conclusion of the inductive argument. The reasons Giere provides for such a move are worth considering; but since nothing of what follows hinges on them, we will not discuss them here.

11 Cf Salmon (1990, 49).

12 We should add that Pr(P) is low, in order to capture the idea that P is unlikely on its own, but since nothing in what follows hinges on this, we omit it for the sake of simplicity.

13 Notice that we are not seeking a probabilistic explication of causation, in which case Pr (P|H) >>Pr (P|¬H) would not be a candidate for well-known reasons related with the existence of co-effects; we are just looking for non-contingent evidential relations that may perfectly well include common cause co-effects of the barometer/storm kind.

14 V. Cruppi, personal communication.

15 For the sake of simplicity, in what follows we talk of H predicting P; but as explained before, this could be relaxed to just a positive relation in the sense of condition (RISK*ii), which is compatible with, but may be weaker than, strict prediction.

16 Notice that to conclude that two hypotheses H₁ and H₂ cannot both satisfy (2*ii#), we have only assumed that they are incompatible and some axioms/theorems of probability theory, such as Bayes’ theorem.

Additional information

Funding

This work has been supported by Ministerio de Ciencia Tecnología y Telecomunicaciones [grant number PID2020-115114GB-I00].