## Abstract

What sort of evidence can confer the strongest support to a hypothesis? A natural answer is that the evidence entails the hypothesis. Roush (Tracking Truth: Knowledge, Evidence, and Science, Clarendon Press, Oxford, 2005) claims that the likelihood ratio measure of degree of incremental support can deliver this intuitively natural result, and regards it as unifying “[the] account of induction and deduction in the only way that makes sense” (p. 163). In this paper, we highlight a difficulty in the treatment of this case, and question the great significance that is attached to this measure and its alleged capacity to accommodate the logicality requirement. We contrast the likelihood ratio measure with other measures (such as the Kemeny–Oppenheim measure and the difference measure), and argue that problems still emerge in light of tensions with plausible requirements for confirmation measures more generally.

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## Notes

- 1.
We will discuss these measures below.

- 2.
This requirement is introduced by drawing on Huber (2005, 2008). Informally, it requires that “confirmation measures lead one to accept the most informative among all true theories” (Brössel 2013), where informativeness of a hypothesis is characterized by the probability of its negation. (We make no commitment as to the adequacy of this account of informativeness.).

- 3.
Actually, what it takes to quantitatively generalize deduction requires a far more substantial discussion (see, e.g., Crupi 2015, and footnote 15 below). In this paper, we confine our attention to the case in which evidence entails a hypothesis.

- 4.
This makes a difference typically when the confirmation score depends on the prior probability of the hypothesis. For example, the ratio measure (P(h|e)/P(h)) satisfies (EC*), but fails to satisfy (L). Fitelson seems to have (L) in mind as the meaning of

*logicality*, since he argues that it is “intuitively clear that the strength of the support E provides for H in [the case E entails H] should not depend on how probable H is (a priori)” (2001b, p. 42). Brössel (2013, pp. 391–392) also seems to employ (L) in the proof that the requirement of logicality is inconsistent with the requirement of*continuity*.It turns out that (EC*) is consistent with all the other five requirements, including

*continuity*. This results from the fact that the difference measure (P(H|E) − P(H)) satisfies all of them. In light of this, one may want to employ (EC*) instead of (L). This may seem to be a better alternative than “give up monism of confirmation measures or […] give up the validity of one of the [other intuitive requirements]” (Brössel 2013, p. 394). In this paper, we leave open the question of which formulation is ultimately preferable. - 5.
We acknowledge an anonymous reviewer whose comments were very helpful in clarifying this point.

- 6.
Regarding this point, Roush (2005, p. 163) refers to her private communication with Fitelson without providing the details.

- 7.
The relative distance measure Z

_{P}(h, e) is defined as follows (Crupi 2015, p. 647):\({\text{Z}}_{P} \left( {{\text{h}},{\text{e}}} \right)\, = \,\begin{array}{*{20}l} {\left\{ {{\text{P}}\left( {{\text{h}}|{\text{e}}} \right)\,{-}\,{\text{P}}\left( {\text{h}} \right)} \right\}/\left\{ {1\,{-}\,{\text{P}}\left( {\text{h}} \right)} \right\}} \hfill & {{\text{if}}\,{\text{P}}\left( {{\text{h}}|{\text{e}}} \right){ \geqq }{\text{P}}\left( {\text{h}} \right);} \hfill \\ {\left\{ {{\text{P}}\left( {{\text{h}}|{\text{e}}} \right)\,{-}\,{\text{P}}\left( {\text{h}} \right)} \right\}/\left\{ {{\text{P}}\left( {\text{h}} \right)} \right\}} \hfill & {{\text{otherwise}} .} \hfill \\ \end{array}\)

Note that this measure came to be discussed in the literature relatively recently, and it is natural that Roush did not take it into consideration.

- 8.
Consider, for example, h

_{1}, h_{2}, and e such that P(h_{1}) = P(h_{2}) > P(h_{1}|e) > P(h_{2}|e). In this case, the posterior probabilities of h_{1}and h_{2}(given e) are less than the priors. Nonetheless, since P(h_{1}) = P(h_{2}), the Bayes’ theorem implies that P(e|h_{1}) > P(e|h_{2}), which means that e favors h_{1}against h_{2}. - 9.
Even if Royall’s measure were able to cover the entailment case (i.e., the case in which evidence entails one of the mutually exclusive but not jointly exhaustive hypotheses), it would not be clear whether some counterpart of (EC*) or (L) would be the desideratum in this case. For, as a measure of differentiation, it might be intuitive that the degree of differentiation that, say, the observation of Uranus makes between the hypothesis about its existence at the predicted orbit and the one about the non-existence of such a planet is greater than the degree of differentiation the same observation makes between the former hypothesis and another hypothesis about the existence of an unknown planet at a slightly different orbit than actually observed.

Another concern is that, in Royall’s measure the division by zero also appears when the evidence implies the negation of one of the hypotheses under consideration (but, note that this does not imply that the evidence entails or confirms the other hypothesis). However, it is not clear whether the degree of differentiation in this case should be the same as in the case in which one of the hypotheses is entailed by the evidence. Thus, there could be a whole different set of desiderata for measures of differentiation and it is not clear whether it is adequate to discuss it together with relevance measures. This is another reason why we focus on the latter in this paper.

- 10.
This line of response, and the one that admits infinity which we will discuss below, was suggested to us by Peter Lewis.

- 11.
This way out is available only to Roush’s measure, which involves a pair of hypotheses that are mutually exclusive and jointly exhaustive.

- 12.
Note that, in this instance, Bayes’ theorem tells us that the posterior probability of h is 1, and, equivalently, that of ¬h is null.

- 13.
We thank one of the reviewers for directing our attention to this possible interpretation of Roush.

- 14.
A further complication is required in order to apply this way out to Royall’s case, which compares hypotheses that are not necessarily jointly exhaustive. Consider, for example, hypotheses h

_{1}, h_{2}, and h_{3}, which are mutually exclusive and jointly exhaustive. Suppose that evidence e entails h_{1}, and we want to evaluate the strength of this evidence with respect to h_{2}and h_{3}. However, in this case, neither P(e|h_{2})/P(e|h_{3}) nor P(e|h_{3})/P(e|h_{2}) is defined because the hypotheses are incompatible with e. (We would like to thank one of the reviewers for noting this.) Thus, yet another disjunct would be required to treat this case, which would increase even further the ad hocness of this way out. - 15.
Moreover, as one of the reviewers noted, neither of the disjuncts is available when P(h) is extreme or P(e) is zero. While this problem is not limited to this particular definition of the LR measure, and while it is customary to examine contingent hypotheses and evidence, it would be important to consider deductive cases involving tautologies and contradictions in order to develop a confirmation theory that applies to deductive logic.

Doing this, however, requires various considerations. We will only be able to illustrate a few of them. For example, while tautologies are entailed by any propositions, should these cases be regarded as providing maximal support to tautologies? In particular, when the premises are themselves a tautology, it is odd that a proposition that lacks any empirical information confers maximal support. (We also thank an anonymous reviewer for noting this interesting point.)

As far as confirmation is concerned, intuitively, relevance measures should assign neutral values to such cases. Crupi (2015) introduces this as one of the fundamental desiderata, and, thus, (L) is limited to contingent hypotheses. This may be one point where our expectations about confirmation and deductive logic diverges.

Another concern arises regarding the so-called ‘explosion’, i.e., the derivation rule, from classical logic, according to which any proposition follows from a contradiction. Should contradictions then be regarded as conferring maximal support to any proposition? Paraconsistent logicians, who deny the validity of explosion, may want to avoid this counterintuitive consequence.

However these issues are resolved, parallel treatment of inductive and deductive inference would require additional, substantial considerations especially when it comes to inferences involving tautologies and contradictions, and the outcomes will vary depending on the system of logic one adopts.

Given this, the usual focus on contingent hypotheses and evidence may be quite reasonable. In this context, the question of the parallel treatment of induction and deduction can be put this way: whether it is possible to develop a confirmation theory in such a way that confirmation via deductive logic can be seen as a sort of ‘conservative extension’ of confirmation via induction; namely, in a way that, given relevant background knowledge, the same deductive relation (one of ‘maximal support’) appears in both of them

*as far as contingent hypotheses and evidence are concerned*.But there could be different notions of parallel treatment as well. For example, when Crupi (2015) considers what it takes “to generalize deductive logic” (pp. 645-647), he includes requirements such as

*contraposition of confirmation*, i.e., if e confirms h, then*C*_{P}(h, e) =*C*_{P}(¬e, ¬h). In contrast, according to the ‘conservative extension’ notion of the parallel treatment, the requirement would be much weaker: e maximally confirms h if and only if ¬h maximally disconfirms ¬e. The extent to which a correspondence is required between deductive and inductive inference is, then, another issue to be discussed when the generalization of deductive inference is fully considered. - 16.
When one employs (EC*) as the interpretation of ‘maximum support’, a similar problem arises if one considers

*an inconsistent pair of evidence*e and e′ such that e entails h and e′ entails ¬h. However, one may argue that confirmation measures do not need to address such an unrealistic case. - 17.
Here is the problem. It is clearly the case that 0 × 1 = 0 and 0 × 2 = 0. Therefore, 0 × 1 = 0 × 2. By dividing both terms by zero, we then conclude that 1 = 2! This argument obviously applies to any numbers, which are then provably identical to every other number!.

- 18.
We thank Kenny Easwaran, for noting this. Glass and McCartney (2015) also employs such a definition.

- 19.
We thank one of the reviewers for suggesting that we discuss this possibility.

- 20.
Thanks here to Branden Fitelson.

- 21.
In other words, one may argue that the LR measure,

*understood as an equivalence class*(based on ordinal equivalence), can handle the extreme case. - 22.
It is not clear whether this sort of way out is available to Royall’s case (i.e. whether there is a measure that is ordinally equivalent with it and that can handle the entailment case). Even if there is, our objection to appealing to such a quasi-ordinal-equivalence will apply to it as well (see the discussion below).

- 23.
One may contend that such an artificial measure should not be counted. However, an argument is needed to rule out this measure on these grounds, and the burden is on those who hold the contrastive approach of the LR measure to establish that their account does unify induction and deduction in the end.

- 24.
Our thanks go to Peter Brössel, Kenny Easwaran, Branden Fitelson, Peter Lewis, and two anonymous reviewers for this journal for extremely helpful discussions and suggestions, which led to substantial improvements. This research was partly supported by the Fulbright Doctoral Dissertation Research Grant (Grant number:15131828). The authors are grateful to the University of Miami (UM) and Harvey Siegel for accepting Yukinori Onishi as a visiting fellow at the UM Philosophy Department, which eventually made this research possible. The authors contributed equally to the work.

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Bueno, O., Onishi, Y. The Likelihood Ratio Measure and the Logicality Requirement.
*Erkenn* (2020). https://doi.org/10.1007/s10670-019-00202-6

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