1.1 Induction

Induction is a species of inference. It is distinguished from practical reason as it involves the reasoned adoption of new beliefs. It is distinguished from deduction in that the new beliefs are not logical consequences of the old. Induction is closely associated with scientific reasoning, what Hume called ‘inferences from experience’ (Reference Hume and Beauchamp1999/1748, sec. 4.21): making predictions and reasoning from evidence to hypothesis.

Consider what happened when scientists adopted Hutton and Lyell’s Uniformitarian approach to geology (Lyell Reference Lyell1830). Uniformitarians argued that the same geological processes (erosion, deposition, lithification, orogeny, etc.,) active today are also those responsible for shaping the landscape of the Earth throughout time. Ripple marks in an exposed vertical rock layer, far from the sea, demand an explanation. The Uniformitarian explains them as arising from the operation, over geological timescales, of familiar processes: postulating an ancient shallow sea in which the ripple marks were formed, the covering of those rippled layers by subsequent sediments, the gradual folding and uplift of the resulting sedimentary rock into a mountain range, and the erosion of that range over millions of years to reveal the ripples again to the eye (Drexel et al. Reference Drexel, Preiss and Parker2012/1993, pp. 171–197).

In postulating this continuity between present and past geological processes, Uniformitarians endorse a canonical form of inductive inference. They reason from evidence of observed geological activity to predictions concerning the outcomes of unobserved geological activity. (Here, and throughout, evidence is treated as propositional – that volcanic rocks intruded into older sedimentary rocks is evidence, not the rocks themselves.) While most geological inference concerns what our evidence reveals about events of the distant past, Uniformitarianism also supports inference from past to future, perhaps a more traditional conception of inductive inference. This type of inference is at least parallel to inverse inference in statistics, reasoning from features of a known sample to those of the larger population from which it is drawn.

If accepted, Uniformitarianism would vindicate many inverse inferences. Its characteristic thesis is, in effect, the application to geology of what has come to be known as the Uniformity Principle:

But what might justify the acceptance of Uniformitarianism itself? This appears to involve another scientific inference schema. Uniformitarians were opposed by Catastrophists (Gohau Reference Gohau1990 pp. 139–150), who proposed that various geological irregularities (discontinuities in the fossil record, unconformities in rock strata) were to be explained by currently unattested processes that dramatically transformed the earth over very short timescales. (Some were keen to link these ideas to flood narratives in various religious traditions.) Principal advantages of Uniformitarianism over Catastrophism include its simplicity and avoidance of ad hoc bespoke geological processes. Uniformitarianism is supported over Catastrophism, on the basis of the agreed geological evidence, by an inference to the best explanation (IBE) that takes these features to be truth-conducive virtues (Harman Reference Harman1965; Henderson Reference Henderson2014; Lipton Reference Lipton2004). That is: the simpler and less ad hoc theory is regarded as better than its rivals, and indeed, good enough to meet a threshold for acceptance. The inference is motivated by the natural thought that a false theory would not be able to give such a simple and satisfactory account of the evidence. Darwin says as much in his description of this inference schema:

As Darwin notes, IBE and inverse inference are both exemplified in ordinary reasoning as well as scientific contexts. We conclude that the last evening train will depart late, on the basis that it’s always been late. We may also conclude in turn that something is a little different about the population of passengers who tend to take the last train, perhaps hypothesising that they are particularly prone to the sorts of behaviours that cause delays.

While there are interesting special features of inverse inference as opposed to the inference to the best explanation more generally, both exhibit an ‘ampliative’ character. In the example previously discussed, the Uniformitarian hypothesis is not a logical consequence of the given geological evidence, because Catastrophism is consistent with the evidence but incompatible with Uniformitarianism. Logical deduction may exclude hypotheses that are inconsistent with the evidence, but mere consistency does not favour one among the hypotheses consistent with the evidence. By contrast, reasonable inductive inference can lead us to favour one of many coherent hypotheses, as in the examples discussed previously. It is coherent to suppose the future quite unlike the past; yet we habitually infer that the future will broadly resemble the past. As such, inductive inference ‘goes beyond’ the evidence.

The central normative question about induction arises at this point. What, if anything, makes it reasonable to go beyond the evidence, by accepting one of many coherent hypotheses? Or, from the other end of the process, how was ‘our hard-won factual knowledge … secured by any process of demonstrably sound reasoning’ (Howson Reference Howson2000, p. 1)? So stated, this is an enormously challenging question. It involves the identification, characterisation, and justification of inductive methods for forming beliefs in response to empirical evidence. A complete account of such methods will tackle many of the central issues in epistemology and philosophy of science, going well beyond the scope of this Element.

1.2 Inductive Reason and Evidential Support

My focus is narrower. Any inference is a ‘reasoned change in view’ (Harman Reference Harman1986, p. 5). A theory of inference should then provide an account of the reasons involved, and what changes one should make to one’s beliefs in light of them.

Consider deductive inference. The reasons in question are grounded in relations of implication: that $P$ logically implies $P\vee Q$ , for example. These provide reasons such as this: ‘the fact that one’s view logically implies $P$ can be a reason to accept $P$ ’ (Harman Reference Harman1986, p. 11). This may be a reason, but it is hardly a decisive one; it depends what else one believes and the other reasons one has. So any account of deductive change in view should not entail that one must come to believe $P$ when one’s view logically implies $P$ . Sometimes one should revise one’s view to remove the implication; there are some formal theories of how to do that (Alchourrón et al. Reference Alchourrón, Gärdenfors and Makinson1985).

It is rather tricky to state principles of belief revision that exceptionlessly cover all the possibly reasonable ways of responding to reasons provided by deductive logic (Harman Reference Harman1986, chapter 2). Principles of reasoned change in view hold in general only ceteris paribus. Principles of belief revision involve balancing various reasons one has – one has a reason to avoid unmanageable clutter in one’s beliefs, but also a reason to have opinions about the many and varied things one might encounter in one’s environment, and these reasons may push in opposite directions. Finding the resultant force of various component epistemic reasons to rationalise a particular change in view is not an easy matter, and the reasons provided by logical implications do not trump other reasons. This makes inference quite unlike implication: while $Q$ is implied by $P$ of necessity if it is implied at all, $Q$ may reasonably be inferred from $P$ in some circumstances while not in others.

A general theory of theoretical reasoning, of how to respond to one’s epistemic reasons, will include consideration of all the reasons one might have: those provided by deductive logic, testimony (Fricker Reference Fricker1995; Lackey Reference Lackey2008), memory (Fernández Reference Fernández2015; Lackey Reference Lackey2005), as well as empirical evidence and explanatory considerations of the sort considered in Section 1.1. There is nothing especially distinctively inductive about theoretical reason in general. But an appropriate topic for us, starting from an interest in scientific reasoning, is to focus on these ‘inductive reasons’ provided by evidence. This would involve principles about reasons that might parallel the defeasible reasons provided by logical implication. For example, IBE looks like it might be captured by something like this principle: that $P$ is the best explanation of some evidence you possess can be a reason to accept $P$ (Harman Reference Harman1986, chapter 7). There will be parallel principles about the epistemic force of simplicity, strength, coherence, and so on.

The diversity of such principles may seem at first glance to be unmanageable. But in fact each may fall under a plausible general schema:

1.3 Absolute and Incremental Support

The Evidential Support schema stands in need of clarification, as there are both absolute and incremental relations of evidential support, both of which can provide reasons to believe. (While our schema is ambiguous, it seems true on either disambiguation.) Absolute evidential support exists when a particular hypothesis is on balance overall most favoured by the total evidence. Incremental evidential support – what is often termed confirmation – exists when some evidence is in favour of a hypothesis. These notions come apart. Quite how they diverge will depend in part on the precise account of evidential support on offer (I’ll follow this up in Section 3.1). But even before a precise model is on the table, there are intuitive cases where some evidence favours a hypothesis without absolutely supporting it; and there are cases where a hypothesis is on balance most supported while being disfavoured by some evidence:

• Many incompatible hypotheses can be supported by the evidence; at most one can be absolutely supported by the evidence. Suppose we are searching for a bushwalker, lost on a plateau. Lost walkers tend to follow drainage basins downstream (Koester Reference Koester2008); we see there are four possible creeks they might have followed. Ground-based searchers have excluded one of them. This evidence favours (supports) all three remaining hypotheses about where the hiker might be without necessarily favouring any one in particular.

• If we have only a small sample, the evidence may favour a hypothesis without being sufficient to absolutely support it overall. A small survey yields evidence that all smokers in the sample have impaired lung capacity, and this evidence supports the hypothesis that all smokers have impaired lung capacity. However, the small sample size precludes that evidence being sufficient to warrant an inference to that conclusion.

• A positive result from a reliable medical diagnostic is evidence in favour of the hypothesis that one has the disease. But even reliable diagnostics are imperfect, giving rise to ‘false positives’ – cases where the test is positive but where the subject doesn’t in fact have the disease. If a disease is rare in the population, a good diagnostic test can favour the hypothesis that one has the illness without making it overall credible – it may just be rendered more credible than it would have been in the absence of that evidence. (This is an example of the relevance of the ‘base rate’; see Section 3.3.)

• This is a good example of the way that a hypothesis can be supported overall while being disconfirmed. A positive test result makes it slightly less incredible that one has a rare disease, but not sufficiently so to make that hypothesis acceptable.

Both absolute and incremental supports provide epistemic reasons. Neither suffices to determine the outcome of inductive reasoning: even if $H$ is overall most credible in light of evidence, there may be reasons to suspend judgement, or to change one’s mind about other things. A belief in $H$ needn’t result from the acquisition of evidence that overall supports $H$ .

The project that now comes into view is to offer a general characterisation of absolute and incremental evidential support, as relations that may obtain between any hypothesis and any body of evidence such that had it ‘actually obtained, [it] would constitute favourable evidence for’ that hypothesis (Hempel Reference Hempel1945a, p. 2). (Notice Hempel’s formulation exhibits the same ambiguity in ‘evidence for’.) When applied in scientific practice, perhaps the vital questions concern which among the hypotheses that are presently live for us are supported by the actual evidence. But the theory of evidential support is more general.

1.4 Inductive Logic

Deductive logic is the study of consequence relations. Those consequence relations underlie the coherence and completeness of various states of belief, and those features yield some special cases of evidential support, for example, that if the evidence logically entails $P$ , then $P$ is supported by the evidence, or if $Q$ is inconsistent with the evidence, then $Q$ is not supported by the evidence. So the relation of evidential support should have deductive consequence as a special case. This is captured in this widely accepted condition:

1.5 Formality and Background Knowledge

Early proponents of inductive logic pursued the project in an ambitious and literal fashion, attempting to pin down, using formal resources alone, a notion of support of a conclusion by some premises that would generalise the notion of logical consequence. Conclusiveness is all-or-nothing, but inconclusive evidence supports hypotheses to varying degrees, so any such inductive logic must say something about which invalid argument forms are nevertheless ‘logically confirming’ in virtue of the logical properties of their constituents. Let us for now introduce the (non-standard) symbol ‘ $⫦$ ’ to denote this (still hypothetical) relation of logical confirmation; $\Gamma ⫦\varphi$ says that the premises $\text{Γ}$ logically confirm $\varphi$ .

Any premises should logically confirm any of their logical consequences: if $\text{Γ}\models \varphi$ then $\Gamma ⫦\varphi$ . But that is where the similarity to classical deductive logic ceases. For one thing, even if $\Gamma ⫦\varphi$ , it need not be that $\Gamma ,\psi ⫦\varphi$ . Some premises about balls drawn from an opaque bag known to contain 100 balls might be ‘ball 1 is red’, ‘ball 2 is red’, …, ‘ball 99 is red’. Suppose these premises logically confirm the conclusion that all the balls in the bag are red. But add the premise ‘ball 100 is black’ to the others, and that conclusion is definitely not supported. This feature of a consequence relation is known as non-monotonicity (Hawthorne Reference Hawthorne and Zalta2021, section 2.2; Straßer Reference Straßerforthcoming). Another disanalogy is that while logical consequence is transitive (if $\varphi \models \psi$ and $\psi \models \chi$ , then $\varphi \models \chi$ ), logical confirmation is not. Shogenji (Reference Shogenji2003, p. 613) gives the following example: that X is an academic philosopher confirms that X has a PhD; that X has a PhD confirms that X is well-paid; but it does not follow that X’s being an academic philosopher confirms that X is well-paid.

These features are logically unusual, but arguably must be captured by any adequate inductive logic. They are already suggestive of the thought that logical features of sentences are not going to match up with what we want from an inductive logic. This suggestion is correct. Consider this principle, which is about the most plausible example of purely logical confirmation: that instances confirm generalisations:

1.6 Hume’s Problem

In the foregoing example, the adoption of a particular stochastic model is itself likely the upshot of some inductive reasoning. That the trials are IID is a hypothesis justified on the basis of evidence (perhaps evidence about outcome frequencies or about the mechanisms involved in drawing from and mixing up the urn). The evidence itself only yields inductive reasons in the context of some inductive framework.

Hume’s infamous ‘problem of induction’ can be seen to rest on an early recognition of this point (Henderson Reference Henderson, Zalta and Nodelman2022). Hume recognised that our inductive habits rested on the pattern of causal relationships evident in our experience, and the ‘supposition’ that future patterns ‘will be conformable to the past’ (Hume Reference Hume and Beauchamp1999/1748, para. 4.19):

Hume frames the question as one about the reasons evidence about the past provides for beliefs about the future course of nature. He doesn’t explicitly invoke evidential support. But his target, that the past provides a ‘rule for the future’, is fairly clearly a claim about evidential support for hypotheses about the future by evidence about the past. His aim, in my terminology, is to reveal what reasons there are for us to adopt an inductive framework $ℱ$ in which the past provides a rule for the future. His worry is one of circularity: that any reasons to adopt framework $ℱ$ are themselves only reasons on the presupposition of $ℱ$ . For those are reasons drawn from our evidence about the past, and our views about how that evidence bears on hypotheses are likewise grounded in how past evidence bore on past hypotheses. When Hume emphasises the possibility ‘that the course of nature may change’, he is not merely pointing out a difference between induction and deduction. He is pointing out that our grasp on relations of evidential support is itself dependent on our evidential history. More concisely: any general hypotheses about the nature and structure of evidential support we adopt can only be for reasons that presuppose those same hypotheses about evidential support (Howson Reference Howson2000, pp. 10–15).

1.7 The Justification of Induction and the Structure of Evidential Support

Hume’s resolution of his problem is characteristically bold. He claims that our inductive inferences are but ‘a species of natural instincts, which no reasoning or process of the thought and understanding is able, either to produce, or to prevent’ (Hume Reference Hume and Beauchamp1999/1748, para. 5.8). As I would approach the question, Hume here acknowledges the dependence of evidential support on a background inductive framework, and suggests that our adoption of such a framework is not itself the product of reasons and hence need not wait on a further prior inductive framework.

What does Hume’s naturalistic explanation of our inductive habits tell us about evidential support? In one way, it merely underscores the result from Section 1.5 that evidential support must be understood as a three-place relation between evidence, hypotheses, and an explicit relativisation to some background inductive framework or assumptions. The fact that the same evidence can bear differently on hypotheses in different worldly circumstances, while those worldly circumstances may not themselves be known to us, suggests that it may not generally be possible to justify the adoption of any such framework.

We don’t need to follow Hume in his scepticism about inductive frameworks. It might well be that we can, ultimately, find a justification for accepting a particular conception of evidential support. For example, perhaps some conception of how evidence bears on hypotheses is justified by default, or we might have justification for some inductive assumptions ‘without being in a position to cite anything that could count as ampliative, non-question-begging evidence for those beliefs’ (Pryor Reference Pryor2000, p. 520). Perhaps we acquire our inductive habits of thought by testimony, deference, or immersion in a scientific community, and justification comes subsequent to the fruits of our epistemic endeavours. Perhaps we are justified simply because ‘the world is so constituted that inductive arguments lead on the whole to true opinions’ (Ramsey Reference Ramsey and Mellor1990, p. 93). These would all be ways in which an inductive framework could be adopted without internalistic justification of the sort Hume appears to seek.

However such justification might be acquired, it is fruitful to separate the question of how to justify a particular view about the bearing of evidence on hypotheses from the question of the structure of evidential support. The latter issue can be pursued to a significant extent by characterising the three-way connections between evidence, hypotheses, and standards of evidential support, without taking a stand on what the actual evidence is, what the actually live hypotheses are, or what the actual standards of evidential support are. Because this investigation concerns the abstract structural features of this three-place relation ‘ $E$ supports $H$ relative to inductive framework $F$ ’, it is a deserving bearer of the label ‘inductive logic’. (Though to illustrate its significance, it will be helpful to give at least some examples of how the relata can be concretely filled in.)

That is how I will pursue inductive logic in this Element. In particular, I will explore Bayesianism, a particularly influential and fruitful approach to the formal features of evidential support. Bayesians propose that an inductive framework can be represented by a probability function, which captures both any relevant background information and encodes (in its conditional probabilities) a conception of absolute evidential relevance. A Bayesian conception of inductive frameworks is presented in Section 2, where roughly the following account is offered: the evidence absolutely supports $H$ , relative to some probability model, just when $H$ is more probable than not, conditional on $E$ . That same model also allows us to represent a comparative notion of confirmation: to regard $E$ as confirming $H$ is to regard $H$ as more probable given $E$ than otherwise. The Bayesian theory of confirmation and its consequences is the subject of Section 3. I will return to the issue Hume raises, of how to justify inductive assumptions, in Section 3.7. We delve further into whether there are significant prior constraints on what sorts of probability models we may permissibly adopt in Section 4, including some consideration of Goodman’s (Reference Goodman1954) ‘new riddle’ of induction (Section 4.4).

2.1 Prospects and Perspectives

The prospects for a hypothesis are – not wholly metaphorically – what is in view in light of some body of evidence. (Note the related but distinct usage by Kahneman & Tversky (Reference Kahneman and Tversky1979).) A hypothesis which is supported by some potential evidence has an improved prospect of turning out to be true when that potential evidence is gathered. What is visually in prospect for you depends on where you are and on the direction you are facing: your perspective. Likewise, what is epistemically in prospect depends on your epistemic perspective: where you find yourself evidentially as well as on the significance of that position in the space of possibilities.

Camp characterises occupying an epistemic perspective as having ‘an open-ended disposition to characterise: to encounter, interpret, and respond to some parts of the world in certain ways’ (Reference Camp and Grimm2019, p. 24). That is what it is to occupy a perspective. A perspective itself need not be embodied, existing independently of any agent occupying it, but it must suffice to characterise such dispositions. To fulfil that role, I suggest that each epistemic perspective must involve the following:

1. 1. A representation of a space of possibilities;

2. 2. A representation of the total evidence currently on hand; and

3. 3. Some policy, or set of standards (Schoenfield Reference Schoenfield2012, p. 199), that capture, numerically, the significance of potential evidence for various entertainable hypotheses.

Let me expand on these components. ‘A space of possibilities’ captures all the ways that things might have turned out and might yet turn out, at least according to that perspective. It is natural to take this space of possibilities to have the structure of a field of propositions (Eagle Reference Eagle2011, pp. 1–3; Hájek & Hitchcock Reference Hájek, Hitchcock, Hájek and Hitchcock2017, p. 17). That is, for each outcome that might occur, according to that perspective, there is a proposition in the field to the effect that the outcome comes to pass; there is a trivial outcome, represented by a trivially true proposition (i.e., that something or other comes to pass); and whenever the field of propositions includes $P$ and $Q$ , it also includes $\neg P$ , $\left(P\vee Q\right)$ , and $\left(P\wedge Q\right)$ . A maximally specific possibility will be represented by a logically complex proposition that fixes, for each possible outcome, whether or not it occurs. This will play the role of a ‘possible world’ from that perspective. I will often work with collections of propositions $\left\{P_1,\dots ,P_n\right\}$ such that no two of them are true in a single possible world, and at least one is true in every possible world. Such a collection is called a partition. The set of all possible worlds is a partition, but so are various more coarse-grained divisions of the space of apparent possibilities, for example, a complete set of possible experimental outcomes.

I will not distinguish propositions true of exactly the same possibilities; depending on how possibilities themselves are individuated, this may or may not mean identifying with one another those propositions that have, necessarily, the same truth value (Stalnaker Reference Stalnaker1984, p. 2). These possibilities thus need not be ‘metaphysically possible’, nor must each metaphysical possibility be among those represented in a perspective – they are potential epistemic (or perhaps doxastic) possibilities (Titelbaum Reference Titelbaum2022, pp. 27–38).

It must be acknowledged that different perspectives needn’t agree on how to analyse any given proposition into more specific outcomes. There is no guarantee that all perspectives will agree on what the most fine-grained possibilities are, or on which propositions are possible. I will assume that there are perspectives capable of representing any hypotheses or pieces of evidence we might need to consider. Even so, some perspectives – something like those Savage called ‘small worlds’ (Savage Reference Savage1954, p. 16) – might be very limited, able to represent only a narrow range of hypotheses and evidence. (We might consider the possible outcomes of an experiment, and the hypotheses on which they bear, without including any possibilities in which the experiment was never performed at all.) The question of what happens when an agent manages to expand their conception of what is possible is philosophically very rich and much discussed – sometimes under the label of ‘partition sensitivity’ – but lies beyond the scope of this Element (Paul Reference Paul2014; Pettigrew Reference Pettigrew2020a).

An epistemic perspective must also represent a body of total evidence. Evidence constrains which possibilities in the space of possibilities are left open and which are excluded. Because evidence is propositional, evidence narrows down a location in the space of possibilities by excluding those possibilities in which propositions inconsistent with the evidence are true (Stalnaker Reference Stalnaker1984, p. 120). So an epistemic perspective can represent the current total evidence by indicating in some way a region of the space of possibilities which, from that perspective, might be actual – those consistent with the evidence.

Finally, and most importantly for inductive logic, an epistemic perspective must represent in some way the bearing of the evidence on the remaining live hypotheses. It must provide a way of discriminating among all those hypotheses not ruled out by the evidence. I will assume that this discrimination is effected numerically; so an epistemic perspective assigns numbers to hypotheses that somehow reflect their support by the evidence. As Horwich puts it, ‘our inductive practice may be represented by a function which specifies, for any evidential circumstance, the permissible degrees of belief in any statement’ (Reference Horwich1982, p. 79). Such a function will be an epistemic perspective, noting that it is not to be identified with one’s actual attitudes (one’s ‘degrees of belief’, in Horwich’s treatment), but circumscribes permissible attitudes.

An epistemic perspective thus represents a possible way of ‘proportion[ing] belief to the evidence’ (Hume Reference Hume and Beauchamp1999/1748, sec. 10.4). A proportioning will in general assign different numbers to hypotheses, even those it regards as on balance supported by the evidence. That evidential support varies in extent is fairly commonsensical. Suppose that my current total evidence supports the claim that it is raining outside my office right now, and that it also supports the claim that the weather next weekend will be very hot. It would be foolish to deny that it may support the former to a considerably greater extent than it supports the latter. That this variability in evidential support can be quantified numerically is a reasonable starting point.Footnote ¹

Someone’s actual attitudes may not correspond to any epistemic perspective. For example, they may have incoherent or indeterminate attitudes to a hypothesis, precluding the assignment of a single number as the degree to which the evidence supports it. Perhaps at best an imprecise range of attitudes could be ascribed to them (Jeffrey Reference 88Jeffrey and Earman1983a, pp. 139–140). And an epistemic perspective need not correspond to anyone’s actual state of belief or knowledge. But an epistemic perspective does appear to be something a rational agent could have as an ideal for a coherent body of epistemic attitudes. To put it in explicitly normative terms, an epistemic perspective has a structure that any reasonable total system of epistemic attitudes must approximate, and provides a regulative ideal for such attitudes (see Section 2.8). (If there are reasonable yet indeterminate systems of epistemic attitudes, then perhaps the regulative norm should be that any such state can be precisified into one or more determinate epistemic perspectives.) Epistemic perspectives rationalise the states of belief and knowledge of any agents who occupy them.

Given this, one way of describing the present project is that we begin from the assumption that all epistemic perspectives are rational, and use that fact to establish the properties that distinguish epistemic perspectives from other purported evaluations of the prospects of hypotheses.

2.2 Prospects and Bets

If an epistemic perspective represents the prospects of hypotheses, it must be subject to certain norms about the evaluation of prospects. One way to bring out these norms is to consider the ideal evaluation of bets (de Finetti Reference de Finetti, Smokler and Kyburg1964/1937; Pettigrew Reference Pettigrew2020b; Ramsey Reference Ramsey and Mellor1990). This is not because one is likely to be involved in ‘betting on theories’ (Maher Reference Maher1993). Rather, among the dispositions involved in adopting a perspective are dispositions to evaluate, as favourable or otherwise, wagers on hypotheses about how things are, given an exogenously settled assignment of values to outcomes. If there are constraints on the acceptability of evaluations, those constraints may reveal structural features of epistemic perspectives that rationalise those evaluations.

At first glance, it might seem that we should assign prospects to hypotheses by trying to elicit betting behaviour and infer an epistemic perspective from it. Witness Kyburg’s insistence that ‘how seriously someone believes what he says he believes’ is elicited by inviting him ‘to put his money where his mouth is’ (Reference Kyburg and Henry1983, p. 64). This is unlikely to succeed, given the confounding factors in real-life gambling behaviour, which is psychologically and logistically complicated, not to say morally fraught. The declining marginal utility of money (Pettigrew Reference Pettigrew2020b, pp. 17–19), the difficulty of finding parties and counterparties to every possible bet, the fact that some outcomes will be resolved only after the agent’s lifespan if ever, the fact of risk-averse and risk-seeking individuals (Buchak Reference Buchak2013; Kahneman & Tversky Reference Kahneman and Tversky1979), all make it difficult to draw direct conclusions about the prospects of $P$ from an agent’s (un)willingness to bet on $P$ . It is complicated to deduce an agent’s own mental state from the betting prices they offer (Bradley & Leitgeb Reference Bradley and Leitgeb2006), let alone an epistemic perspective legitimising their attitudes.

Nevertheless, behind any reasoned decision to bet lies an epistemic perspective: some evaluation of the prospects of the propositions one is betting on, given the evidence one has (Howson & Urbach Reference Howson and Urbach1993, pp. 76–77). That the prospects of a proposition are decent can go some way towards justifying taking a bet, without constituting, or being constituted by, anyone’s actual willingness to bet. Doubtless that epistemic perspective justifies other dispositions too – perhaps dispositions to assert, or to employ hypotheses in scientific explanation. But betting has a neat connection to evaluation which makes these dispositions of particular interest for us here, despite the suspicion that assertion and explanation are of greater importance.

A bet on a proposition $H$ is a right to receive $s$ units of value if $H$ turns out to be true, and nothing otherwise. The price of the bet is the value $x$ that is assigned to that right. (So we can imagine a bet on a coin toss as a right to receive $10 if the coin lands tails, and a bookmaker could decide to price this right at $7.) For each bet, there is a counterbet, which is a bet against $H$ that yields $s$ units if $H$ turns out to be false and which costs $s-x$ units. The payout $s$ is the total staked, the sum of the units of value contributed jointly by the bet and counterbet, $s=x+\left(s-x\right)$ . (In the example, the bookmaker is implicitly taking the counterbet, putting up $3 for the right to receive $10 if the coin does not land tails.) For simplicity, let us restrict attention to bets where the total stake $s$ is 1 unit. We can treat a counterbet against $H$ at $s-x$ as equivalent to a bet on $\neg H$ at that same price. This is justified by this basic principle of equivalence: any bets which have the same payoffs in the same circumstances must rationally be assigned the same price. A bet against $H$ that costs $s-x$ to yield $s$ pays off when $H$ is false; as does a bet on $\neg H$ with the same cost and yield.Footnote ²

Real agents only enter into bets they regard as favourable. (We have not assumed that stakes are monetary; an agent who loves the thrill of gambling may find a bet favourable while it is unfavourable in purely monetary terms.) A bet on $H$ is favourable, intuitively, if the prospect of gaining the payout $s$ more than compensates for the risk of losing what you have staked $x$ . We have assumed that an epistemic perspective represents prospects numerically. Suppose that relative to a given perspective the prospect of $H$ is $p$ . A bet is favourable if the payout, weighted by the prospect of getting the payout, exceeds the cost: $ps>x$ . A bet is unfavourable if $ps<x$ ; a case where the prospect of gain is outweighed by the risk of loss. An agent is epistemically justified in entering into a bet only if the bet is favourable-from-their-perspective. (This is a necessary condition for all-things-considered justification for a bet, but is not sufficient.)

What if the prospect of gain and the risk of loss are exactly balanced according to some perspective? Such a bet is neutrally priced. The potential bettor lacks positive reason to enter into a neutrally priced bet; the prospect of gain is too little. But nor is the bet unfavourable; the prospect of loss isn’t sufficiently great to motivate a favourable evaluation of the counterbet. It would not be reasonable to prefer to accept a neutrally-priced bet or its counterbet over the status quo, given the prospects involved. Thus a neutrally-priced bet will represent an accurate evaluation of the prospects of $H$ according to an epistemic perspective: it is a bet on $H$ that is calibrated to the degree to which the evidence supports $H$ . I will say that the numerical prospect of $H$ , revealed in a neutrally priced bet on $H$ from a particular perspective, is the degree of support that perspective provides to $H$ .

2.3 Evaluating Prospects

To assign a neutral price to a bet would involve fixing on some particular epistemic perspective. The general structural principles on epistemic perspectives we are concerned with can be uncovered without making assumptions about neutral prices except that they exist. We uncover those general structural constraints – constraints that, it will turn out, suffice to ensure neutral prices are mathematically like probabilities (Section 2.4) – by noting that unless certain constraints are placed on the numerical evaluation of prospects, certain bets (and packages of bets) seem to yield an incoherent evaluation of their value. Presented one way, the bets have a certain neutral price; presented another way, they are assigned another non-neutral price. So a single neutral price doesn’t exist unless we constrain the acceptable numerical representation of evidential support in some way.

In particular, suppose an epistemic perspective could assign a prospect $-\delta$ to a proposition $H$ , where $\delta >0$ . Such a perspective entails that the neutral price for a bet on $H$ that pays 1 unit is $-\delta$ . But at that price, there is an advantage to purchasing the bet – the price is negative, so, win or lose, represents an advantage over the status quo. A perspective that assigns a negative number as the degree to which the evidence supports $H$ leads to a situation where the agent cannot assign a coherent neutral price. Their assessment of the prospects of $H$ leads them to be indifferent to accepting this bet, while an assessment of the possible payoff of the bet should make them positively disposed to accept it. So this epistemic perspective turns out to be unable to assign a single value to the bet. To assign a neutral price to a bet, no perspective could assign a degree of support less than zero. That is, every epistemic perspective must satisfy:

Normality

The degree of support provided by any body of evidence for a trivial hypothesis (necessary in light of that evidence) is 1.

Finally, suppose that there are two hypotheses, $H$ and $H\prime$ , such that the truth of either one excludes the truth of the other, according to some epistemic perspective. Suppose an epistemic perspective assigns a degree of support $\alpha$ to $H$ , and degree of support $\beta$ to $\mathrm{H\prime }$ , and a degree of support $\delta$ to their disjunction $H\vee H\prime$ , but where $\delta \ne \alpha +\beta$ . The neutral prices for bets on these propositions $H$ , $H\prime$ , and $H\vee H\prime$ are fixed by those degrees of support. Consider the book of bets consisting of bets on $H$ and $H\prime$ that each pay 1 unit; its payoffs are depicted in Table 1.

Table 1 Payoffs for the book of bets on $H$ and $H\text{'}$ .

$H$	$H\prime$	Bet on $H$	Bet on $H\prime$	Total payoff	$H\vee H\prime$
T	F	$1-\alpha$	$-\beta$	$1-\left(\alpha +\beta \right)$	T
F	T	$-\alpha$	$1-\beta$	$1-\left(\alpha +\beta \right)$	T
F	F	$-\alpha$	$-\beta$	$-\left(\alpha +\beta \right)$	F

It is easy to see from the table that where $H\vee H\prime$ is true (the top two possibilities), the payoff is $1-\left(\alpha +\beta \right)$ , and when $H\vee H\prime$ is false, the payoff is $-\left(\alpha +\beta \right)$ . But the neutral price for a bet on $H\vee H\prime$ is $\delta$ , leading to a different payoff of $1-\delta$ when true, and $-\delta$ when false. So the book of bets pays out the same amount, in the same circumstances, as the individual bet on the disjunction – but has a different neutral price. Clearly, this either violates the equivalence requirement to assign the same neutral price to bets that pay off in the same circumstances, or violates the requirement on epistemic perspectives that we assign a single neutral price to a given bet. Hence:

2.4 Probabilities and Degrees of Support

The argument of the previous section reached the conclusion that epistemic perspectives must satisfy Non-negativity, Normality, and Additivity to assign a single neutral price for a bet on each proposition in their scope. A numerical function from propositions which satisfies these constraints is, mathematically speaking, a probability function (Eagle Reference Eagle2011, pp. 1–4; Hájek & Hitchcock Reference Hájek, Hitchcock, Hájek and Hitchcock2017; Kolmogorov Reference Kolmogorov and Morrison1956/1933). So it turns out that an epistemic perspective assigns probabilistic prospects to every proposition in its field (Section 2.1).

Satisfying these constraints is also sufficient for justifying a single neutral price (Kemeny Reference Kemeny1955; Lehman Reference Lehman1955). Epistemic perspectives can accordingly be represented as evidential probability functions, assigning numbers to hypotheses, representing degrees of support, in light of the total evidence to which a perspective is committed. In line with Section 2.1, the total evidence according to an evidential probability function comprises those propositions which are assigned probability 1. This usage might be slightly revisionary, liberalising our conception of evidence so that it includes at a given moment any proposition on which we rely in evaluating other claims with which we are confronted. ‘Taking the evidence into consideration’ doesn’t mean only taking things you happen to have learned empirically into consideration.

The argument in Section 2.3 showed that epistemic perspectives have the formal features common to all probability functions. Some have suggested that epistemic perspectives might obey tighter constraints. (I discuss these proposed constraints in the Online Appendix B.) For example, some have suggested that the evidential probabilities representing them should be regular (assigning zero probability only to impossibilities, or perhaps only to contradictions). More have suggested that evidential probabilities should obey a stronger Additivity constraint, countable Additivity. I do not insist that these are features of every epistemic perspective, though they may be features of some.

This approach to evidential probability has precedent in the literature. As argued in Section 2.1, epistemic perspectives rationalise the dispositions of their occupants to judge hypotheses. Thus we agree in part with Williamson (Reference Williamson2000), who suggests that evidential probability represents the degree to which ‘the evidence tells for or against the hypothesis’ (Reference Williamson2000, p. 209), rather than reflecting anyone’s actual credences. Williamson’s own view is that evidential probability should reflect ‘something like the intrinsic plausibility of hypotheses prior to investigation’ (Reference Williamson2000, p. 211). On the present view, this only captures some epistemic perspectives, omitting those one can come to occupy after some investigation. Climenhaga puts it well: ‘the distinctive claims of the degree-of-support interpretation [of evidential probability] are that probabilities are mind-independent relations between propositions and that probabilities constrain rational degrees of belief’ (Climenhaga Reference Climenhaga2024, p. 3).

The tradition of objective Bayesianism that Climenhaga and Williamson represent goes back at least to Keynes (Reference Keynes1921), Johnson (Reference Johnson1932), and Carnap (Reference Carnap1962). It has however almost invariably been accompanied with an additional and not necessarily welcome commitment to there being such a thing as ‘the degree to which evidence supports a hypothesis’. While each epistemic perspective on my view articulates a conception of how evidence bears on hypotheses, there is no commitment to the existence of a ‘best’ perspective. Perspectives – for all I’ve established so far – might disagree with one another on the space of possibilities, on the background evidence they incorporate, or on the significance they attach to it. We return to the question of whether there is a unique best perspective in Sections 2.8 and 4. For now, I would wish to distance the view defended here both from the ‘subjective’ Bayesian, who identifies epistemic perspectives with the actual degrees of belief of individual agents, and from the objective Bayesian who accepts a unique best perspective. The view I’m defending could be called non-subjective Bayesianism.

2.5 Probabilities and the Dutch Book Argument

In Section 2.3, I presented a version of what is known as the Dutch Book Argument (DBA) for the conclusion that degrees of support must be probabilities (Eagle Reference Eagle2011, pp. 28–32; Hájek Reference Hájek, Anand, Pattanaik and Puppe2008; Howson & Urbach Reference Howson and Urbach1993, pp. 75–81; Pettigrew Reference Pettigrew2020b). The earliest versions of this argument were due to Ramsey (Reference Ramsey and Mellor1990) and de Finetti (Reference de Finetti, Smokler and Kyburg1964/1937); many variants have been offered since.

The typical DBA looks rather different from the version I have presented, however. The typical version applies to credences (degrees of belief), not to epistemic perspectives (degrees of support). A more important difference is the following. I argued that non-probabilistic assignments of numbers to hypotheses fail to reflect the prospects of those hypotheses (see also Armendt Reference Armendt, Hull, Forbes and Okruhlik1992, p. 218; Christensen Reference Christensen2004, p. 121; Titelbaum Reference Titelbaum2022, section 9.2.1). By contrast, the standard DBA argues that non-probabilistic credences are practically unreasonable because they subject those who act on such credences to a sure loss: ‘Dutch Book arguments evaluate the rationality of credences by looking at the quality of the choices that they do or should lead us to make’ (Pettigrew Reference Pettigrew2020b, p. 1). There is some connection with choices in the argument in Section 2.3, because an epistemic perspective is supposed to justify a neutral price for a bet, which may partly rationalise choices to bet at non-neutral prices. But the standard DBA relies on a much stronger link with choice, namely, that non-probabilistic credences rationally require the agent to commit to a package of bets that guarantees a sure loss.

These features of the DBA I’ve offered may help defuse some challenges that face more standard versions. The standard version is accused of being unrealistic (because real agents don’t have sufficiently many determinate betting preferences for the argument to succeed), or because probabilism is descriptively inaccurate of real agents (Kyburg Reference Kyburg and Henry1978). Defenders of the standard DBA have tried to fend off the accusation that it requires us to postulate a plethora of bookies wandering around looking to exploit incoherent agents.

Better still, however, to focus not on the action of betting, but on inputs to the justification of betting, namely, on the evaluation of prospects. In this I follow other presentations of ‘de-pragmatised’ DBAs (Armendt Reference Armendt1993; Earman Reference Earman1992, p. 42; Howson Reference Howson2000, pp. 124–134; Skyrms Reference Skyrms1984, p. 22; though see Maher Reference Maher1997 for a dissenting voice), all of whom emphasise that non-probabilistic credences involve an inconsistent evaluation of the very same options. Nevertheless, even these authors err in focussing on credences. It may well be that there are good reasons for an agent to evaluate options in an inconsistent way, perhaps because of their practical situation. A risk-averse agent whose neutral price for a bet on a fair coin landing heads is $0.49$ , and likewise for a bet on it landing tails, inconsistently values the status quo, neutrally valuing the tautology $H\vee \neg H$ at $0.98$ . If there is pragmatic encroachment on belief, such grounds for incoherent credence might be widespread (Kim Reference Kim2017; though see Jackson Reference Jackson2019 for an argument that credence isn’t encroachable to the same extent as belief). So I think it preferable to focus on the epistemic perspectives that rationalise credence, rather than on realised credences themselves. (How do epistemic perspectives bear on credences? I say a little about that in Section 2.8.)

It is not mandatory to make use of justifications of betting to reveal the structure of rational prospects, though it is a deservedly popular approach. I make use of it here because of its accessibility, but also because it gets at something essential to belief, namely, as Stalnaker puts it, that beliefs are ‘representational mental states [and] should be understood primarily in terms of the role that they play in the characterisation and explanation of action’ (Stalnaker Reference Stalnaker1984, p. 4). Betting actions are best explained as normed by appropriately structured epistemic perspectives, and these must be probabilistic if they are to perform this normative function.

A more general approach that still begins with preference over practical options, and thus shares with the DBA an unavoidably decision-theoretic cast, proceeds via representation theorems. A representation theorem begins with some axioms taken to govern rational preference and shows that sufficiently rich rational preferences over options can be represented as if they involved the maximisation of probabilistic-expected value. Most representation theorems are framed explicitly as accounting for subjective credences (Jeffrey Reference Jeffrey1983b; Maher Reference Maher1993; Meacham & Weisberg Reference Weisberg, Gabbay, Hartmann and Woods2011; Neth Reference Neth2025; Savage Reference Savage1972; Titelbaum Reference Titelbaum2022, pp. 285–309), but can be understood equally as offering norms on those credences, captured by properties of epistemic perspectives. In this case, the idea would be that an epistemic perspective would manifest in ideally rational preference, and must have a probabilistic structure to do so. This kind of approach has no prospect of being ‘de-pragmatised’, as substantive assumptions about utility must be made.

Another recently popular argument aims to offer purely epistemic grounds for probabilism, on the basis that probabilistic epistemic perspectives are closer to representing how things are – more accurate – than non-probabilistic ways of representing an ideal epistemic state, come what may (Joyce Reference Joyce1998; Pettigrew Reference Pettigrew2016; Titelbaum Reference Titelbaum2022, pp. 338–361). I discuss this approach in Online Appendix C.

2.6 Conditional Prospects

Frequently, one doesn’t want to know only the present prospects of a proposition, the degree of support one’s current evidence provides. One wants also to know what bearing hypothetical evidence has on hypotheses one is concerned with: what are the prospects of $H$ , given $E$ , where $E$ is not (yet) in evidence? For instance, we might be interested in the support for the hypothesis that some particular die that we haven’t yet tossed is fair, given the further potential evidence that it lands ‘ $⚅$ ’ on 100 consecutive rolls. Presumably that number will be different from the support provided by the current background evidence for its fairness, or from the support provided by the potential evidence that it lands a random mixture of ‘ $⚀$ ’, ‘ $⚁$ ’, …, ‘ $⚅$ ’ in roughly equal proportions in 100 rolls. The conditional prospect of $H$ given $E$ relative to some evidential probability $\text{Pr}$ is written $\text{Pr}\left(H\mid E\right)$ .

What is the prospect that both $H$ and $E$ turn out true? Intuitively, it is the proportion among cases where $E$ turns out true that are also cases where $H$ turns out true. Since $H$ may depend in some way on $E$ , this second quantity involves the conditional prospect of $H$ given $E$ . That is, the prospect of the conjunction of $H$ and $E$ is the prospect of $H$ turning out true given $E$ , weighted by the prospect of $E$ turning out true in the first place. The low prospect of my going to the beach and getting sunburnt is the prospect that I get sunburnt given I go to the beach (high), weighted by the prospect that I go to the beach (low). This is captured in this rule connecting conditional and unconditional prospects:Footnote ³

Total Probability

$\text{Pr}\left(H\right)=\text{Pr}\left(H\mid J_1\right)\text{Pr}\left(J_1\right)+\cdots +\text{Pr}\left(H\mid J_n\right)\text{Pr}\left(J_n\right).$

Another elementary consequence of Product is that $\text{Pr}\left(H\mid E\right)\text{Pr}\left(E\right)=\text{Pr}\left(E\mid H\right)\text{Pr}\left(H\right)$ . When $\text{Pr}\left(E\right)>0$ , this can be rearranged into a theorem that has a prominence belying its obvious proof:

Bayes

$\text{Pr}\left(H\mid E\right)=\frac{\text{Pr}\left(E\mid H\right)\text{Pr}\left(H\right)}{\text{Pr}\left(E\right)}.$

Applying the theorem of Total probability to ‘ $\text{Pr}\left(E\right)$ ’ given a partition including $H$ (e.g., $\left\{H,H_1,\dots ,H_n\right\}$ ), we can reformulate Bayes’ theorem:

$\text{Pr}\left(H\mid E\right)=\frac{\text{Pr}\left(E\mid H\right)\text{Pr}\left(H\right)}{\text{Pr}\left(E\mid H\right)\text{Pr}\left(H\right)+\text{Pr}\left(E\mid H_1\right)\text{Pr}\left(H_1\right)+\cdots \text{Pr}\left(E\mid H_n\right)\text{Pr}\left(H_n\right)}.$

Bayes’ theorem is important because it shows how conditional probabilities $\text{Pr}\left(H\mid E\right)$ are fixed by other quantities we often know the value of:

• The prior probability of the hypothesis $H$ , $\text{Pr}\left(H\right)$ ;

• The likelihood of the evidence given the hypothesis, $\text{Pr}\left(E\mid H\right)$ ;

• The probability of the evidence, $\text{Pr}\left(E\right)$ , which can also be expressed as the expected likelihood of the evidence, given hypotheses spanning the space of possibilities.

The quantity $\text{Pr}\left(H\mid E\right)$ is often called the posterior probability of the hypothesis.

Here ‘prior’ and ‘posterior’ do not refer to temporal priority; $\text{Pr}\left(H\mid E\right)$ is evaluated at the same time, and using the same evidential probability $\text{Pr}$ , as $\text{Pr}\left(H\right)$ . There is a very common idea that if one is using a probability function to represent an agent’s state of mind at some time, and that agent was to learn exactly $E$ , then the agent’s new state of mind should be ${\text{Pr}}_E$ . The rule of Conditionalization says that the relation between this new mental state, taking $E$ into account, and the old mental state, is this: for any $H$ , ${\text{Pr}}_E\left(H\right)=\text{Pr}\left(H\mid E\right)$ ; that is, that the new attitudes should be the old conditional attitudes. In light of this, the genuinely posterior state of mind ${\text{Pr}}_E\left(H\right)$ is equal to a pre-existing conditional attitude, which (by unfortunate extension) is also called the posterior. Conditionalization allows quite radical changes between successive unconditional probabilities, but requires ‘rigidity’: agreement on conditional probabilities between existing and revised assignments (Weisberg Reference Weisberg2009, pp. 14–16).

There is considerable controversy, not unrelated to the earlier discussion of reasoning in Section 1.2, around whether Conditionalization is required as a condition of rationality (Hedden Reference Hedden2015, pp. 29–44; Lewis Reference Lewis1999; Talbott Reference Talbott1991; van Fraassen Reference van Fraassen1989, pp. 160–182; Williamson Reference Williamson2000, pp. 213–221). My goal is to offer a theory of inductive reasons – not a theory of what one ought to do with them – so I will not pursue this issue (though see Section 3.1). (I will retain the entrenched terms ‘prior’ and ‘posterior’; it would risk unintelligibility to refrain from using them.) Still, the view adopted here that inductive reasons are grounded in a single epistemic perspective fits naturally with conceptions of rationality that take it to be principally a feature of a single ‘time slice’ of an individual (Hedden Reference Hedden2015, pp. 6–9, 28–55). What matters for rationality is that at each moment a Bayesian agent is probabilistically coherent, adopting a single epistemic perspective that matches one’s total evidence. It may not be irrational to have radical discontinuities between successive coherent states of opinion, and there may be no general rule on offer to govern how newly acquired evidence should prompt revisions in the epistemic perspective one comes to adopt, as long as one is rational before and after the revision.

2.7 (Degree of) Evidential Support

With conditional degrees of support, I have finally reached the third key role of epistemic perspectives, namely, capturing the significance of potential evidence for various hypotheses (Section 2.1). That is, an epistemic perspective serves as an inductive framework (Section 1.4), which therefore can be represented by an evidential probability function. Given an epistemic perspective, an unconditional probability already discriminates between hypotheses in light of the background evidence held fixed. Conditional probabilities add a new aspect to that evaluation: they give us some sense of the support various hypotheses would receive from propositions perhaps not yet in evidence, given that we retain the same inductive framework (i.e., standards of evidential significance). So conditional and unconditional probabilities together articulate an ‘epistemic standard’ that might be deployed in the evaluation of hypotheses (Schoenfield Reference Schoenfield2012, p. 199). That is, we may finally offer an account of the degree of evidential support and overall evidential support (Section 1.3), relative to an inductive framework:

2.8 The Plurality of Epistemic Perspectives

Previously I have only supposed that the numerical prospects assigned to propositions given an epistemic perspective must satisfy the probability axioms. Very many functions do this. So if epistemic perspectives are to justify actual credences (Section 2.2), we shall need some way of deciding which epistemic perspective(s) to adopt as normative for our own individual credences, and what exactly adopting a perspective amounts to.

The second question is easier than the first. To adopt an epistemic perspective is to take its verdicts on support as your own. If $\text{Cr}$ is your credence function, representing your degrees of belief, then you’ve adopted an epistemic perspective $\text{Pr}$ iff for any $X$ , $\text{Cr}\left(X\right)=\text{Pr}\left(X\right)$ . The question of whether you can adopt an epistemic perspective is another matter. Perhaps epistemic perspectives are too cognitively demanding for us. In that case, adopting an epistemic perspective is an idealisation of rationality, rather than a prescriptive norm (Carr Reference Carr2021; Staffel Reference Staffel2020).

Return to the first question. You ought not to adopt a perspective that disagrees with you on your total evidence. There are many perspectives meeting this requirement. Can we identify further constraints that allow us to answer our first question as follows: one is doxastically rational in adopting $\text{Pr}$ iff $\text{Pr}$ is the uniquely permitted epistemic perspective consistent with your total evidence?

Consider an expert weather forecaster, to whom one might defer completely about a limited subject matter, or an omniscient being, to whom one ought to defer completely about everything. One ought, as a seeker after truth, to adopt an (globally or locally) expert epistemic perspective – if only one knew what it was! The literature on deference and the epistemology of chance provides a way of proceeding: one’s attitude should be one’s expectation of the expert’s attitude (Elga Reference Elga2007, pp. 478–480; Gaifman Reference Gaifman, Skyrms and Harper1988; Hall Reference Hall2004, pp. 100–101; Titelbaum Reference Titelbaum2022, pp. 133–145). An expected value is a probability-weighted mean value of a quantity. In the much discussed case of chance (Ismael Reference Ismael2008; Joyce Reference Joyce2007; Lewis Reference Lewis1986a), chance deference principles entail (given Total probability from Section 2.6) that one’s current attitude to a chancy proposition $X$ should be its expected chance, where each possible value $x_i$ of $X$ ’s chance is weighted by your judgement of the probability of the hypothesis that $x_i$ turns out to be the actual chance of $X$ .

Since the probabilities of the hypotheses about chance are just as much a matter of epistemic perspective as any other contingent theoretical proposition, it is not plausible that chance deference principles narrow down the class of admissible epistemic perspectives to a uniquely privileged perspective. But deference to chance provides a model for how we might respond to that variety of admissible epistemic perspectives: namely, we ought to set our credence in some hypothesis $H$ equal to the expected support $H$ receives from the various admissible epistemic perspectives under consideration, weighted by our judgements about which epistemic perspectives will turn out to capture the degree to which our evidence supports $H$ .

Mathematically, let there be various epistemic perspectives ${\text{Pr}}_i$ that assign degrees of support $p_i$ to $H$ , and $\text{Cr}$ be our actual credence or degree of belief. Next assume that the possible answers to the question what is the correct degree of support the evidence assigns to $H$ ? form a partition. Let $D_i$ be the proposition that ${\text{Pr}}_i$ captures the correct degrees of support for $H$ in assigning it $p_i$ . Then Total probability will get us this:

$\text{Cr}\left(H\right)=\sum _i\text{Cr}\left(H\mid D_i\right)\text{Cr}\left(D_i\right),$

and the epistemic deference principle that $\text{Cr}\left(H\mid D_i\right)=p_i$ then yields

$\text{Cr}\left(H\right)=\sum _i{p}_i\text{Cr}\left(D_i\right).$

This kind of approach works well if there is a single right degree of support or unique evidential probability given a body of evidence – perhaps something like Williamson’s ‘intrinsic plausibility of hypotheses’ (Section 2.4) conditional on our total evidence – and our goal in setting our credences is to best approximate this ideal. Some recent formulations of deference principles for rational belief in fact seem to build in a presupposition of this sort of Uniqueness (Meacham Reference Meacham2019, p. 717; cf. Greco & Hedden Reference Greco and Hedden2016). The question of whether Uniqueness is true of evidential probability is the focus of Section 4.

This argument for how to set credences breaks down if there are multiple epistemic perspectives consistent with the evidence. Each embodies some standards for evaluating how the evidence supports hypotheses. There may be no sense in which they are rival accounts of ‘the’ evidential support relation, and we need not be required to see them as forming a partition of ways this relation could turn out. Their relation to one another may be more like the relation among epistemic perspectives that reflect different bodies of total evidence. There is no one perspective that reflects the evidence one uniquely should have, rather there are many perspectives that may reflect the various bodies of evidence one might have. The question what is the correct degree to which the evidence supports $H$ may be in no better shape than the question what is the correct body of total evidence to draw on when evaluating $H$ . You draw on the evidence you find yourself having; likewise, perhaps you ought simply to draw on the epistemic standards to which you find yourself subscribing. If that is the case, it is harder to justify adopting as your credence the weighted mean of these various epistemic perspectives. (We can only rationalise expert deference if there is an expert, if there is a fact of the matter about getting the degree of support right that transcends each individual evidential probability function; and in this scenario, there is no single fact of the matter about this.)

Without a unique degree of support, there may be no way to recommend a unique attitude in view of the plurality of epistemic perspectives. That will be bad news for a restrictive epistemology, which aims to tell us what we must do. But if epistemology is permissive – in the business of telling us what we may do – then a natural idea suggests itself: adopt an epistemic perspective consistent with your total evidence and which reflects your standards for epistemic evaluation. As van Fraassen puts it: ‘rationality is only bridled irrationality … what it is rational to believe includes anything that one is not rationally compelled to disbelieve’ (van Fraassen Reference van Fraassen1989, pp. 171–172). Epistemic rationality requires us to evaluate the prospects of hypotheses. In the absence of constraints that delimit those prospects uniquely, we do what we must by means of what we may: adopt some permissible epistemic perspective. Compare: in unrestricted sections of the Bundesautobahn system, there is no speed limit. While cars must travel at some speed, drivers achieve this requirement by choosing one of the many permissible speeds. This suggests a picture of rational credence in the presence of multiple potentially normative epistemic perspectives:

3.1 Incremental Confirmation Defined

A theory of evidential support aims to capture a number of widely shared beliefs about what evidence does for us. For example: that unexpected or novel predictions should favour a hypothesis more than banal or familiar predictions; or that simpler hypotheses, other things being equal, are favoured by the evidence. Behind these beliefs seems to be a persistent metaphor: evidence as a mass stuff, something that can be gathered and, once collected, weigh in favour of a hypothesis and against its rivals. The incremental notion of confirmation aims to capture the idea that each new piece of evidence adds some weight to the scales (or takes some off).

The resultant of all of these pieces of evidence will be a particular degree of evidential support (relative to some background conception of what supports what, of course). Given my conception of degree of support (Section 2.7), it is natural to take the component weight of individual pieces of evidence to be associated with the changes in overall degree of support those pieces of evidence produce against a fixed background. If there is some positive weight $E$ contributes to $H$ , we say $E$ confirms $H$ . (Whether this positive weight can be measured is discussed in Section 3.6.) Confirmation is a three-place relation between some piece of potential evidence $E$ , some hypothesis $H$ , and a background probability model. This last is assumed to represent an epistemic perspective and hence to capture both background knowledge and a specific inductive framework, that is, a conception of evidential relevance (Fitelson Reference Fitelson, Pfeifer and Sarkar2005, p. 391).

Biased Die

Suppose we have background information that leaves open the hypotheses that a die is biased towards ‘ $⚅$ ’ (the chance of ‘ $⚅$ ’ being $1/3$ ), or that it is fair, and regards these hypotheses as equally supported and exhaustive, so $\text{Pr}\left(\text{fair}\right)=\text{Pr}\left(\text{‘}⚅\text{’-biased}\right)$ . No outcomes are yet in evidence. Against this background, consider the proposition that the die lands ‘ $⚅$ ’ 5 times consecutively. That proposition confirms the hypothesis that the die is biased, and disconfirms the hypothesis that the die is fair.

Can Bayesian confirmation theory capture this basic observation about confirmation? Eventually, we might use a theory of confirmation to decide complex cases about which our judgements are unclear, but as usual with a philosophical explication, we need reassurance that it gets the basics right, typically by checking that it reproduces obvious facts about the target relation. In this case, we begin with an appeal to the theorem of Total probability to calculate the prior of $E$ :

$\begin{array}{ccc}\text{Pr}\left(E\right)& =& \text{Pr}\left(E\mid \text{fair}\right)\text{Pr}\left(\text{fair}\right)+\text{Pr}\left(E\mid \text{‘}⚅\text{’–biased}\right)\text{Pr}\left(\text{‘}⚅\text{’–biased}\right)\\ & =& \frac{1}{6^5}\cdot \frac{1}{2}+\frac{1}{3^5}\cdot \frac{1}{2}\\ & =& \frac{1}{15,552}+\frac{1}{486}\approx \mathrm{0.002122.}\end{array}$

Plug this into Bayes’ theorem (Section 2.6), and we obtain

$\begin{array}{cccc}\text{Pr}\left(\text{fair}\mid E\right)& \approx 0.0606\cdot \text{Pr}\left(\text{fair}\right)& <& \text{Pr}\left(\text{fair}\right);\\ \text{Pr}\left(\text{‘}⚅\text{’–biased}\mid E\right)& \approx 1.9394\cdot \text{Pr}\left(\text{‘}⚅\text{’–biased}\right)& >& \text{Pr}\left(\text{‘}⚅\text{’–biased}\right).\end{array}$

So here this proposition confirms the hypothesis of bias, and disconfirms the hypothesis of fairness.

The same sort of reasoning can be applied in less idealised examples. Consider Howson and Urbach’s account of Babbage’s investigation into the origins of tables of logarithms:

Intransitivity

There is a $\text{Pr}$ and $A,B,C$ such that $\text{Pr}\left(B\mid A\right)>\text{Pr}\left(B\right)$ , $\text{Pr}\left(C\mid B\right)>\text{Pr}\left(C\right)$ , but $\text{Pr}\left(C\mid A\right)\le \text{Pr}\left(C\right)$ . Consider a dice rolling case; let $A$ be ‘ $⚀$ ’; let $B$ be the disjunction that the die came up odd, that is, ‘ $⚀\vee ⚂\vee ⚄$ ’ ; let $C$ be the disjunction ‘ $⚂\vee ⚄$ ’. The standard die-rolling $\text{Pr}$ gives us $\text{Pr}\left(A\right)=1/6$ , $\text{Pr}\left(B\right)=1/2$ , $\text{Pr}\left(C\right)=1/3$ . The relevant conditional probabilities are $\text{Pr}\left(B\mid A\right)=1$ (so $A$ confirms $B$ ); $\text{Pr}\left(C\mid B\right)=2/3$ (so $B$ confirms $C$ ); but $\text{Pr}\left(C\mid A\right)=0$ , so $A$ conclusively disconfirms $C$ .

Many Bayesians are tempted to import their views on diachronic rationality into confirmation theory. From this perspective, ‘confirmation’ is a thing that happens to a theory when new evidence arrives; the theory is confirmed or disconfirmed as its probability shifts around over time. Most Bayesians accept updating upon receipt of new evidence $E$ goes by Conditionalization, adopting one’s old conditional credences given $E$ as one’s new ‘unconditional’ credences. In this case, $\text{Pr}\left(H\mid E\right)$ represents the new posterior probability of $H$ ; $H$ is confirmed by $E$ if it has come to be more probable once news of $E$ is in. So one sometimes sees Bayesians present confirmation as essentially diachronic:

an experience provides evidence that confirms a hypothesis, for that scientist, if … this evidence ‘boosts’ the scientist’s credence in the hypothesis.

(Eagle Reference Eagle2011, p. 210)

At the core of modern Bayesianism is a rule for changing the subjective probabilities assigned to hypotheses in the light of new evidence. … where $\text{Cr}(\cdot )$ is your subjective probability distribution before observing $E$ and ${\text{Cr}}^{+}(\cdot )$ is your subjective probability distribution after observing $E$ , …

${\text{Cr}}^{+}\left(H\right)=\frac{\text{Cr}\left(E\mid H\right)}{\text{Cr}\left(E\right)}\text{Cr}\left(H\right).$

More or less anyone who counts themselves a proponent of BCT thinks that this rule is the rule that governs the way that scientists’ opinions should change in the light of new evidence.

(Strevens Reference Strevens2004, p. 369; see also Strevens Reference Strevens2006, section 5.1)

But this really runs together two completely separate issues: whether Conditionalization is the right updating rule, and under what synchronic circumstances does one proposition support another. (Recall here Sections 1.2 and 2.6.) It is perfectly possible to endorse the Bayesian account of evidential support while rejecting Conditionalization. Better, then, to interpret the confirmation inequality $\text{Pr}\left(H\mid E\right)>\text{Pr}\left(H\right)$ as telling us the current significance of $E$ for $H$ from the single perspective of $\text{Pr}$ . That view of evidential significance that may or may not be preserved across successive perspectives, pre- and post-acquisition of $E$ – if the agent adopts a non-rigid update rule, and does not conditionalize on $E$ , they need not in general retain their old view of the bearing of $E$ on $H$ .

3.2 Old Evidence

The foregoing bears on a problem which has been seen by many as a serious challenge to the Bayesian account of confirmation, Glymour’s problem of old evidence

${\text{Pr}}_t\left(T\mid E\right)=\frac{{\text{Pr}}_t\left(T\right)\times {\text{Pr}}_t\left(E\mid T\right)}{{\text{Pr}}_t\left(E\right)}={\text{Pr}}_t\left(T\right).$

As many have noted, the problem is not so much the antiquity of the evidence, as the fact that evidence and hypothesis seem to come in the wrong temporal order. If the age of the evidence were the only problem, we could solve the problem by ‘rolling back’ to an earlier state of knowledge in which the crucial evidence isn’t included – this seems to be what Howson and Urbach have in mind when they say ‘ $\text{Pr}\left(H\right)$ measures your belief in a hypothesis when you do not know the evidence’ (Reference Howson and Urbach1993, p. 117, my emphasis). But Climenhaga (Reference Climenhaga2024, section 3) sets up a simple example in which a piece of evidence is acquired, then some crucial information about the probability distribution over hypotheses is acquired, and then a posterior probability over hypotheses is calculated. Roll back the credence to before the acquisition of the evidence, and one also loses the distributional information. There was never, in his case, a state of belief that represented the background against which this evidence is confirmatory in the way it appears to be.

Introduced like that, old evidence is not a problem for the Bayesian view I have presented, which involves no ‘kinematic’/diachronic element. Though I follow the Bayesian literature in talking of confirmation of hypotheses by evidence, and use the suggestive variable ‘ $E$ ’, I explicitly reject the idea that confirmation occurs when evidence is newly acquired (Carnap Reference Carnap1962, p. 468; Hempel Reference Hempel1945a, section 6). There is no sense in which the evidence considered with respect to confirmation has to be collected at all. Recall my discussion in Section 2.1; an epistemic perspective is associated with some body of total evidence, some body of propositions (it turned out) that are assigned probability 1 by the perspective. Nothing temporal is involved in this characterisation. Utilise an epistemic perspective including $E$ as evidence, and it won’t confirm anything; utilise an epistemic perspective not including $E$ , and it may well have confirmatory power. This is independent of when $E$ is gathered, or even if it is gathered. The incremental confirmation relation informs us of the evidential bearing of one proposition on another, relative to an epistemic perspective; neither needs to be ‘evidence’ in a folk or philosophical sense for this evidential bearing to obtain. Rather, $E$ is some claim that might bear on $H$ – perhaps $H$ predicts it, or some rival of $H$ predicts it – and we wish to evaluate its significance, relative to some perspective embodying some appropriate principles of evidential bearing.

To do this, of course, one must make use of an epistemic perspective according to which there is some bearing of $E$ on $H$ . Glymour’s argument certainly emphasises that a perspective that assigns probability 1 to $E$ is not appropriate for this purpose. Nor for that matter is one that assigns probability 1 to $H$ , which would then be incapable of being confirmed. Nothing in the framework I’ve presented requires us to make such inappropriate choices; but nothing tells us which choices to make, either. So the synchronic problem of old evidence is to give defensible guidance about which epistemic perspective we ought to consider when we evaluate confirmation of hypotheses by claims which are already in evidence for us.

One obvious candidate is clearly excluded because it will simply reinscribe the problem of old evidence. This is the proposal that we ought to evaluate claims of confirmation relative to an epistemic perspective we’ve adopted. My account of adoption in Section 2.8 guaranteed that anything in evidence for us is assigned probability 1 by any adoptable perspective. Hence the problem of old evidence shows that even at a fixed point in time there is no single body of background evidence: the background evidence relevant to the adoption of an epistemic perspective is the evidence possessed by the adopting agent, which may be different than the body of evidence against which that very same agent assessed claims of confirmation. A theory of confirmation is quite distinct from a theory of individual belief, as Glymour (Reference Glymour1981, p. 74) pointed out; this unworkable proposal would collapse them.

Some suggest simply removing $E$ from the background evidence, and evaluating all confirmation claims relative to an epistemic perspective that treats that background (mutilated, from our perspective) as its total evidence:

The proposal is an example of a more general class of counterfactual theories, those that evaluate the confirmatoriness of $E$ with respect to how much $E$ ‘would increase’ the credibility of $H$ against a counterfactual background, what we would have known had we not known $E$ . Howson’s approach is, in effect, that we would have known everything but $E$ . Howson’s suggestion seems to give two sorts of incorrect predictions about confirmation.

The first is this. Sometimes a generalisation is confirmed by its instances (Section 3.5), and, in the Baconian fashion (Bacon Reference Bacon, Jardine and Silverthorne2000/1620), may only be proposed after diligent apian collection of facts. We know that there is a point at which confirmatory returns to repeated experiments must diminish to zero (Howson & Urbach Reference Howson and Urbach1993, p. 120). In such a case, subtracting any particular instance from background knowledge leaves many similar instances, and no confirmation by that instance. The same is true, clearly, for every instance – so no instance confirms. Howson seems blasé about this (Reference Howson1991, p. 550). Yet intuitively each instance may be highly confirmatory were none of the similar instances present, and our instincts in particular cases go with this latter observation. In this case, the counterfactual about what the Baconian would have believed had they not believed $E$ seems to give us $K\backslash \left\{E\right\}$ , as Howson claims, but that seems to be the wrong body of background evidence to use in evaluating confirmation.

In that case, the counterfactual yields the ‘wrong’ body of background evidence to assess confirmation. There are other cases in which Howson’s counterfactual account yields the right background, but predicts the wrong epistemic perspective: it not only subtracts $E$ from the background knowledge, it also shifts us to an epistemic perspective in which evaluations of the bearing of $E$ on $H$ are different. This is the second sort of incorrect prediction Howson’s account makes. Maher gives this example:

What we want is something like this: a surgical modification of our current adopted epistemic perspective that preserves our dispositions to evaluate the bearing of $E$ on $H$ , while removing $E$ and evidence substantially similar to $E$ to predict the right judgements about confirmation. That suggests the following proposal (Eells & Fitelson Reference Eells and Fitelson2000, pp. 667–669; Jeffrey Reference Jeffrey2008, pp. 44–47; Meacham Reference Meacham2016, pp. 461–462):

3.3 Base Rates

Some students of the scientific method analyse the cases from Section 3.1 slightly differently. They are sympathetic to the idea that probability is central to evidential support. But they are suspicious of the apparent arbitrariness that goes into selecting a particular epistemic perspective to assign prior probabilities to evidence and hypothesis (Royall Reference Royall1997, p. xiii), and unconvinced by any of the proposals I will discuss later (Section 4) to allow the background evidence to constrain that selection uniquely. However these philosophers are impressed with the apparent objectivity of the likelihoods that play such an important role in confirmation. They want to rest their whole account of confirmation on likelihoods, and set aside prior probabilities of evidence and hypothesis as much as they can. I take a closer look at these Likelihoodists (Milne Reference Milne1996; Royall Reference Royall1997; Sober Reference Sober1994) in Online Appendix D. The evidential probability framework ameliorates to a certain extent this Likelihoodist charge of ‘subjectivity’: there is nothing subjective about confirmation relative to a probability model.

This Likelihoodist impulse should probably be resisted because there are cases where the prior probability of hypotheses seems to be confirmationally significant. Let’s consider a simplified example.

Mammogram

Mary, a woman aged 47, attends her annual mammogram. Mary has no other symptoms, but while she is waiting for the initial results, she worries: what if there is something abnormal on the scan? Won’t that be evidence of cancer?

Mary has some reason to worry, but perhaps not as much as we might antecedently think. Suppose we consider an epistemic perspective that incorporates information about the error rates for mammograms for women in Mary’s situation. Table 2 shows some broadly indicative aggregate statistics about mammogram results, though the actual data is rather complex.

Table 2 Aggregate test results for 1,682,504 mammograms 2007–2013 (Lehman et al. Reference Lehman, Arao and Sprague2017, p. 53, table 2).

	Normal mammogram	Abnormal mammogram	Total
No cancer	1,486,553 (TN)	186,139 (FP)	1,672,692 ( $\neg C$ )
Cancer	1,283 (FN)	8,529 (TP)	9,812 ( $C$ )
Total	1,487,836 ( $\neg A$ )	194,668 ( $A$ )	1,682,504

The true positive rate, otherwise known as the sensitivity of a test, is the likelihood of an abnormal mammogram given cancer. Given these statistics,

$\text{Pr}\left(A\mid C\right)=\frac{\text{TP}}{\text{TP}+\text{FN}}=\frac{8529}{9812}=\mathrm{0.869.}$

The true negative rate, or specificity, is the likelihood of a normal mammogram given that the test subject doesn’t have cancer:

$\text{Pr}\left(\neg A\mid \neg C\right)=\frac{\text{TN}}{\text{TN}+\text{FP}}=\frac{1,486,553}{1,672,692}=\mathrm{0.889.}$

Accordingly, the likelihood of an abnormal mammogram given no cancer is $\text{Pr}\left(A\mid \neg C\right)=1–\text{Pr}\left(\neg A\mid \neg C\right)=0.111$ (cf. Hendrick & Helvie Reference Hendrick and Helvie2011, p. W113).

These likelihoods tell us about the test: how reliable is it? Because of how rare cancer actually is, the reliability of the test is not especially informative for the patient. We can see this in the overall support that an abnormal test result provides to the hypothesis of cancer. On the data in Table 2, the cancer rate among all screenings is $\text{Pr}\left(C\right)=9,812/1,682,504\approx 0.006$ . This is the base rate. Bayes’ theorem tells us that the predictive value of a positive test result is not the sensitivity of the test, but rather the probability of cancer given an abnormal result:

$\begin{array}{cc}\text{Pr}\left(C\mid A\right)& =\frac{\text{Pr}\left(A\mid C\right)\text{Pr}\left(C\right)}{\text{Pr}\left(A\mid C\right)\text{Pr}\left(C\right)+\text{Pr}\left(\neg A\mid C\right)\text{Pr}\left(\neg C\right)}\\ & =\frac{0.869\cdot 0.006}{0.869\cdot 0.006+0.111\cdot 0.994}\approx \mathrm{0.045.}\end{array}$

In effect, this posterior probability balances the sensitivity of the test against the number of opportunities for even a reliable test to go wrong, indicated by the high base rate for no cancer. So while we see incremental confirmation of cancer from the abnormal result, we see that the degree to which the total evidence supports a cancer diagnosis is very low.

The Bayesian approach makes it easy to incorporate the evidential relevance of the base rate. In my framing, the base rate provides a constraint on the bearing of statistical evidence about test reliability. In a world where the base rate of $H$ is low, a positive result on a sensitive test is good evidence for $H$ . In a world where the base rate of $H$ is high, that same positive result need not be good evidence for $H$ . The Bayesian approach incorporates both a story about confirmation, and a story about how evidential relevance is fixed by background perspective, and cases where base rates are significant show that understanding the impact of evidence needs both. Accounts which neglect the base rate – including Likelihoodism, significance testing, and frequentist statistics – do not provide an account of evidential relevance, and suffer in comparison to the unified picture provided by Bayesianism.

3.4 The Scientific Method

That Bayesian confirmation theory can reproduce judgements about evidential support in particular cases is promising. That it has a natural and unified account of the relevance of base rates is also promising. But we also have an existing body of principles and heuristics that together form a proto-theory of evidential support, embedded in the practice of science. The success of science indicates some value to these truisms comprising the ‘scientific method’, so Bayesian accounts of evidential support are vindicated when they are able to reconstruct and systematise ideal scientific practice. As Earman notes, ‘an adequate account of confirmation is not under obligation to give an unqualified endorsement to all such truisms’ (Earman Reference Earman1992, p. 77), but ideally, it should explain the success of those we should endorse. For reasons of space, my discussion here will be partial. The topic of Bayesian philosophy of science is treated extensively elsewhere (Earman Reference Earman1992, pp. 63–86; Horwich Reference Horwich1982, pp. 100–130; Howson & Urbach Reference Howson and Urbach1993, pp. 117–164; Schupbach Reference Schupbach2022).

Consider the role of refutation in scientific inference. When a theory makes a determinate prediction (relative as always to background assumptions) which is not borne out in experiment, that is often taken to decisively undermine the prospects of that theory. For example, the simplest aether theory of the propagation of light was decisively undermined by the outcome of the Michelson–Morley experiment, which did not observe the predicted difference in the speed of light in perpendicular directions, while background assumptions excluded rival aether-preserving explanations.Footnote ⁵ This sort of case is central enough to scientific practice that Popper (Reference Popper1959) was able to make it the centrepiece of his ‘falsificationist’ approach to theory choice. The correctness of this judgement is supported by a Bayesian model. When $H$ determinately predicts $E$ , relative to background assumptions, that is reflected in the likelihood $\text{Pr}\left(\neg E\mid H\right)=0$ (there is no prospect of $E$ ’s falsity). In that case,

$\text{Pr}\left(H\mid \neg E\right)=\text{Pr}\left(\neg E\mid H\right)\frac{\text{Pr}\left(H\right)}{\text{Pr}\left(\neg E\right)}=0.$

So falsifying evidence conclusively undermines a hypothesis. The displayed equation shows that, in general, the degree of support of a hypothesis by evidence is proportional to increasing likelihood, the limit case being where $H$ entails $E$ .

There is an asymmetry here (which Popper’s view may mesh with), in that evidence that a theory predicts we won’t see provides conclusive disconfirmation, while evidence that a theory predicts we will see does not yield conclusive confirmation. ( $H$ is conclusively confirmed by $E$ only when $E$ excludes $\neg H$ , relative to background knowledge.) But in those cases where a hypothesis predicts some proposition, so that $\text{Pr}\left(E\mid H\right)=1$ , we see a degree of support of $H$ by $E$ that is equal to $\frac{\text{Pr}\left(H\right)}{\text{Pr}\left(E\right)}$ . Firstly, note that if we hold the prior probability of $H$ fixed, the more improbable $E$ is, the more support it provides for $H$ . This gives us the value of surprising evidence: other things being equal, antecedently unexpected evidence has greater evidential impact for the hypotheses that predict it than evidence we’d expect anyway.Footnote ⁶

Secondly, note that $\text{Pr}\left(H\mid E\right)=\frac{\text{Pr}\left(H\right)}{\text{Pr}\left(E\right)}$ entails $\text{Pr}\left(H\mid E\right)>\text{Pr}\left(H\right)$ : entailed evidence invariably confirms, just as the Entailment condition would have it (Section 1.4). This prompts us to briefly revisit Hempel’s theory of confirmation (Section 1.5). Hempel endorsed this principle, which superficially seems akin to the Entailment Principle:

Special Consequence

If an observation report confirms a hypothesis $H$ , then it also confirms every consequence of $H$ . (Hempel Reference Hempel1945b, p. 103).

This principle has intuitive counterexamples, and a Bayesian account of confirmation explains why. Here is an example:

There is a jar of ten marbles in front of me, five of which were made in Canada and five of which were made in the United States. Of the five marbles made in Canada, four are white and one is red. Of the five marbles made in the United States, all five are red. I am blindfolded, and a friend picks a marble at random from the bag and calls the selected marble ‘marble $X$ ’. He tells me that marble $X$ is red; let $E$ be ‘ $X$ is red’, $H_1$ be ‘ $X$ is the (unique) red marble from Canada’, and $H_2$ be ‘ $X$ is from Canada’.

(Kotzen Reference Kotzen2012, p. 63)

$H_2$ is a logical consequence of $H_1$ . The intuitive verdict is that while $E$ supports $H_1$ (along with every other hypothesis that the marble is one of the red ones), it doesn’t support $H_2$ (since the typical Canadian marble isn’t red). A Bayesian treatment endorses these verdicts. The evidence $E$ supports $H_1$ : $\text{Pr}\left(H_1\right)=1/10<1/6=\text{Pr}\left(H_1\mid E\right)$ . But $E$ disconfirms $H_2$ : $\text{Pr}\left(H_2\right)=1/2>1/6=\text{Pr}\left(H_2\mid E\right)$ . (This case also provides another counterexample to Hempel’s Consistency Condition: $E$ confirms both $H_1$ and $\neg H_2$ , even though they are inconsistent with one another.)

It appears to be a methodological rule that, other things being equal, the more diverse the sources of evidence for one’s theory, the more strongly confirmed that theory is. This can be captured in this maxim: A theory which makes predictions in a number of disparate and seemingly unconnected areas is more confirmed by that evidence than is a theory which is confirmed by predictions only about a narrow and circumscribed range. This maxim is also part of the grounds for recommending random sampling in population inference. The Bayesian insight is that diverse evidence is not internally correlated (Howson & Urbach Reference Howson and Urbach1993, p. 160; Steel Reference Steel1996, pp. 667–668). If, for example, the hypothesis is that all swans are white, then swans collected from different countries would, if white, provide better evidence for the hypothesis than swans collected from the same pond, as we know that if one swan on a pond is white, it is much more likely to be related to other swans in its pond, and those are more likely therefore to be white. If the hypothesis is false, correlations between diverse evidence are more coincidental than correlations between similar evidence. (From a falsificationist perspective, diverse predictions pose a more severe test to the proposal that our hypothesis is false.)

Again focussing on the case where hypothesis predicts evidence with (near) certainty, if the evidence is diverse, it consists of at least two propositions, $E_1$ and $E_2$ , such that truth of one is not positively relevant to the truth of the other, if the hypothesis in question is false. (If it is true, then the evidence is all true, so correlated.) So $E_1$ and $E_2$ are diverse relative to $H$ iff the likelihood $\text{Pr}\left(E_1\wedge E_2\mid \neg H\right)$ is low, or at least if it is not greater than the product of the individual likelihoods $\text{Pr}\left(E_1\mid \neg H\right)\text{Pr}\left(E_2\mid \neg H\right)$ .

The likelihood ratio $\frac{\text{Pr}\left(E_1\wedge E_2\mid \neg H\right)}{\text{Pr}\left(E_1\wedge E_2\mid H\right)}$ features in this formulation of Bayes’ theorem:

$\text{Pr}\left(H\mid E\right)=\frac{\text{Pr}\left(H\right)}{\text{Pr}\left(H\right)+\frac{\text{Pr}\left(E_1\wedge E_2\mid \neg H\right)}{\text{Pr}\left(E_1\wedge E_2\mid H\right)}\text{Pr}\left(\neg H\right)}.$

If the hypothesis $H$ predicts both $E_1$ and $E_2$ , then the likelihood $\text{Pr}\left(E_1\wedge E_2\mid H\right)$ is close to one. The likelihood ratio is therefore close to $\text{Pr}\left(E_1\wedge E_2\mid \neg H\right)$ . Substitute this in:

$\text{Pr}\left(H\mid E_1\wedge E_2\right)\approx \frac{\text{Pr}\left(H\right)}{\text{Pr}\left(H\right)+\text{Pr}\left(E_1\wedge E_2\mid \neg H\right)\text{Pr}\left(\neg H\right)}.$

But if $E_1$ and $E_2$ are diverse (uncorrelated) evidence, then so long as neither is certain given $\neg H$ , this guarantees that the term $\text{Pr}\left(E_1\wedge E_2\mid \neg H\right)<1$ , and hence that $\text{Pr}\left(H\mid E_1\wedge E_2\right)>\text{Pr}\left(H\right)$ .

Moreover, the more surprising each piece of independent evidence is, and the more we have, the more confirmatory diverse evidence is. As we consider additional pieces of diverse evidence,

$\underset{i\to \infty }{\text{lim}}\text{Pr}\left(E_1\wedge E_2\wedge \dots \wedge E_i\mid \neg H\right)=0,$

hence $\text{Pr}\left(H\mid E_1\wedge E_2\wedge \dots \wedge E_i\right)$ tends to 1. This result requires independent evidence; correlated evidence doesn’t have increasing confirmatory impact the more of it one collects.

We must bear in mind as always that judgements of diversity are relative to theoretical background:

the notion of variety of evidence has to be relativized to the background assumptions $K$ , but there is no more than good scientific common sense here, since, for example, before the scientific revolution the motions of the celestial bodies seemed to belong to a different variety than the motions of terrestrial projectiles, whereas after Newton they seem like peas in a pod. (Earman Reference Earman1992, p. 79)

Here is another methodological rule: other things being equal, science prefers naturally arising theories to ad hoc ones designed to predict the same evidence. Suppose a hypothesis, springing unbidden to the scientific mind, entails a certain piece of evidence; and another hypothesis is then designed to mimic the success of the first theory, entailing the evidence by construction. An example is provided by van Fraassen:

It is part of [Newton’s] theory that there is such a thing as Absolute Space, that absolute motion is motion relative to Absolute Space, … . He offered in addition the hypothesis (his term) that the centre of gravity of the solar system is at rest in Absolute Space. But as he himself noted, the appearances would be no different if that centre were in any other state of constant relative motion. This is the case for two reasons: differences between true motions are not changed if we add a constant factor to all velocities; and force is related to changes in motion (accelerations) and to motion directly.

(van Fraassen Reference van Fraassen1980, p. 46)

Consider Newton’s theory $N$ , and the constructed alternative $N+\overrightarrow{v}$ , that the centre of gravity of the solar system has constant absolute velocity $\overrightarrow{v}$ . These theories will make the same empirical predictions, so from the point of view of evidence they are indistinguishable. Yet one might think, Newton’s theory is clearly to be preferred to each of the arbitrary variants. (Perhaps an even better theory is neo-Newtonian, doing away with absolute space altogether.)

This sort of case has been raised as an objection to Likelihoodism (cf. Norton Reference Norton, Bandyopadhyay and Forster2011, pp. 420–22). Considering only the likelihoods of hypotheses, what resources does the Likelihoodist have to explain our distaste for ad hoc hypotheses? $N$ and $N+\overrightarrow{v}$ both entail the evidence, so the likelihoods are the same, and there the Likelihoodist account stops. To explain this methodological preference, we cannot appeal to the likelihoods alone, but must also appeal to the disparity in prior probability between the antecedently plausible Newtonian theory $N$ and the antecedently implausible $N+\overrightarrow{v}$ . One might offer all sorts of explanations for why this prior disparity exists – perhaps Newton’s theory is simpler, more natural, and less arbitrary – but that it exists and drives judgements of confirmation is undeniable. The Bayesian view of ad hoc theories then is that they may have some credibility, and may be supported by evidence, but that generally the fact that they are cooked up to preserve the empirical predictions prompts people to assign them low probability:

people often respond immediately with incredulity, even derision, on first hearing certain ad hoc hypotheses. … it is … likely that they are reacting to what they see as the utter implausibility of the hypothesis.

(Howson & Urbach Reference Howson and Urbach1993, p. 158)

There is one kind of case that may trouble the Bayesian: when the ad hoc hypothesis is cooked up to be entailed by the original hypothesis. Suppose $N^{†}$ is stipulated to be the theory, ‘ $N$ or the empirical appearances are just as if $N$ ’. Any evidence $E$ entailed by $N$ is also entailed by $N^{†}$ ; since $N$ entails $N^{†}$ , $\text{Pr}\left(N^{†}\right)\ge \text{Pr}\left(N\right)$ . So we can’t appeal to the implausibility of ad hoc rivals to explain our decided preference for $N$ ; if $N$ is probable enough to be believed, so is $N^{†}$ . In this case, we might need to appeal to another broadly Gricean principle: that our conversational contributions be as informative as they can be, subject to other conversational norms. $N$ is more informative than $N^{†}$ . Perhaps our preference for $N$ is about what we should say we believe, more than about what is credible.