Human cognition and behavior heavily relies on the notion thatevidence (data, premises) can affect the credibility of hypotheses(theories, conclusions). This general idea seems to underlie sound andeffective inferential practices in all sorts of domains, from everydayreasoning up to the frontiers of science. Yet it is also clear that,even with extensive and truthful evidence available, drawing amistaken conclusion is more than a mere possibility. For painfullytangible examples, one only has to consider missed medical diagnoses(see Newman-Toker et al. 2022) or judicial errors (see Liebman et al.2000). The Scottish philosopher David Hume (1711–1776) isusually credited for having disclosed the theoretical roots of theseconsiderations in a particularly transparent way (see Howson 2000,Lange 2011, and Varzi 2008). In most cases of interest, Hume pointedout, many alternative candidate hypotheses remain logically compatiblewith all the relevant information at one’s disposal, so thatnone of the former can be singled out by the latter with fullcertainty. Thus, under usual circumstances, reasoning from evidencemust remain fallible.

This fundamental insight has been the source of a lasting theoreticalchallenge: if amenable to analysis, the role of evidence as supporting(or infirming) hypotheses has to be grasped by more nuanced tools thanplain logical entailment. As emphasized in a joke attributed toAmerican philosopher Morris Raphael Cohen (1880–1947), logictextbooks had to be divided in two parts: in the first part, ondeductive logic, unwarranted forms of inference (deductive fallacies)are exposed; in the second part, on inductive logic, they are endorsed(see Meehl 1990, 110). In contemporary philosophy,confirmationtheory can be roughly described as the area where efforts havebeen made to take up the challenge of defining plausible models ofnon-deductive reasoning. Its central technicalterm—confirmation—has often been used more orless interchangeably with “evidential support”,“inductive strength”, and the like. Here we will generallycomply with this liberal usage (although more subtle conceptual andterminological distinctions are sometimes drawn).

Confirmation theory has proven a rather difficult endeavour. Inprinciple, it would aim at providing understanding and guidance fortasks such as diagnosis, prediction, and learning in virtually anyarea of inquiry. Yet popular accounts of confirmation have often beentaken to run into troubles even when faced with philosophical toyexamples. Be that as it may, there is at least one real-world kind ofactivity which has remained a prevalent target and benchmark, i.e.,scientific reasoning, and especially key episodes from the history ofmodern and contemporary natural science. The motivation for this iseasily figured out. Mature sciences seem to have been uniquelyeffective in relying on observed evidence to establish extremelygeneral, powerful, and sophisticated theories. Indeed, being capableof receiving genuine support from empirical evidence is itself a verydistinctive trait of scientific hypotheses as compared to other kindsof statements. A philosophical characterization of what science iswould then seem to require an understanding of the logic ofconfirmation. And so, traditionally, confirmation theory has come tobe a central concern of philosophers of science.

In the following, major approaches to confirmation theory areoverviewed according to a classification that is relatively standard(see Earman and Salmon 1992; Norton 2005): confirmation by instances(Section 1), hypothetico-deductivism and its variants (Section 2), andprobabilistic (Bayesian) approaches (Section 3).

• 1. Confirmation by instances
  • 1.1 Hempel’s theory
  • 1.2 Two paradoxes and other difficulties
• 2. Hypothetico-deductivism
  • 2.1 HD vs. Hempelian confirmation
  • 2.2 Back to black (ravens)
  • 2.3 Underdetermination and the Duhemian challenge
  • 2.4 The extended HD menu
• 3. Bayesian confirmation theories
  • 3.1 Probabilistic confirmation asfirmness
  • 3.2 Strengths and infirmities of firmness
  • 3.3 Probabilistic relevance confirmation
  • 3.4 Differences, ratios, and partial entailment
  • 3.5 New evidence, old evidence, and total evidence
  • 3.6 Paradoxes probabilified and other elucidations
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Confirmation by instances

In a seminal essay on induction, Jean Nicod (1924) offered thefollowing important remark:

Consider the formula or the law: \(F\) entails \(G\). How can aparticular proposition, or more briefly, a fact affect itsprobability? If this fact consists of the presence of \(G\) in a caseof \(F\), it is favourable to the law […]; on the contrary, ifit consists of the absence of \(G\) in a case of \(F\), it isunfavourable to this law. (219, notation slightly adapted)

Nicod’s work was an influential source for Carl GustavHempel’s (1943, 1945) early studies in the logic ofconfirmation. In Hempel’s view, the key valid message ofNicod’s statement is that the observation report that an object\(a\) displays properties \(F\) and \(G\) (e.g., that \(a\) is a swanand is white) confirms the universal hypothesis that all \(F\)-objectsare \(G\)-objects (namely, that all swans are white). Apparently, itis by means of this kind of confirmation by instances that one canobtain supporting evidence for statements such as “sodium saltsburn yellow”, “wolves live in a pack”, or“planets move in elliptical orbits” (also see Russell1912, Ch. 6). We will now see the essential features of Hempel’sanalysis of confirmation.

1.1 Hempel’s theory

Hempel’s theory addresses the non-deductive relation ofconfirmation between evidence and hypothesis, but relies thoroughly onstandard logic for its full technical formulation. As a consequence,it also goes beyond Nicod’s idea in terms of clarity and rigor.

Let \(\bL\) be the set of the closed sentences of a first-orderlogical language \(L\) (finite, for simplicity) and consider \(h, e\in \bL\). Also let \(e\), the evidence statement, be consistent andcontain individual constants only (no quantifier), and let \(I(e)\) bethe set of all constants occurring (non-vacuously) in \(e\). So, forexample, if \(e = Qa \wedge Ra\), then \(I(e) = \{a\}\), and if \(e =Qa \wedge Qb\), then \(I(e) = \{a,b\}\). (The non-vacuity clause ismeant to ensure that if sentence \(e\) happens to be, say, \(Qa \wedgeQb \wedge (Rc \vee \neg Rc)\), then \(I(e)\) still is \(\{a, b\}\),for \(e\) does not really state anything non-trivial about theindividual denoted by \(c\). See Sprenger 2011a, 241–242.)Hempel’s theory relies on the technical construct of thedevelopment of hypothesis \(h\) for evidence \(e\), or the\(e\)-development of \(h\), indicated by \(dev_{e}(h)\). Intuitively,\(dev_{e}(h)\) is all that (and only what) \(h\) says once restrictedto the individuals mentioned (non-vacuously) in \(e\), i.e., exactlythose denoted by the elements of \(I(e)\).

The notion of the \(e\)-development of hypothesis \(h\) can be givenan entirely general and precise definition, but we’ll not needthis level of detail here. Suffice it to say that the\(e\)-development of a universally quantified material conditional\(\forall x(Fx \rightarrow Gx)\) is just as expected, that is: \(Fa\rightarrow Ga\) in case \(I(e) = \{a\}\); \((Fa \rightarrow Ga)\wedge (Fb \rightarrow Gb)\) in case \(I(e) = \{a,b\}\), and so on.Following Hempel, we will take universally quantified materialconditionals as canonical logical representations of relevanthypotheses. So, for instance, we will count a statement of the form\(\forall x(Fx \rightarrow Gx)\) as an adequate rendition of, say,“all pieces of copper conduct electricity”.

In Hempel’s theory, evidence statement \(e\) is said to confirmhypothesis \(h\) just in case it entails, not \(h\) in its fullextension, but suitableinstantiations of \(h\). Thetechnical notion of the \(e\)-development of \(h\) is devised toidentify precisely those relevant instantiations, that is, theconsequences of \(h\) as restricted to the individuals involved in\(e\). More precisely, Hempelian confirmation can be defined asfollows:

In each of clauses (i) and (ii), Hempelian confirmation(disconfirmation, respectively) is a generalization ofdirectHempelian confirmation (disconfirmation). To retrieve the latter as aspecial case of the former, one only has to posit \(s = h\) \((\negh\), respectively, for disconfirmation).

By direct Hempelian confirmation, evidence statement \(e\) that, say,object \(a\) is a white swan, \(swan(a) \wedge white(a)\), confirmshypothesis \(h\) that all swans are white, \(\forall x(swan(x)\rightarrow white(x))\), because the former entails the\(e\)-development of the latter, that is, \(swan(a) \rightarrowwhite(a)\). This is a desired result, according to Hempel’sreading of Nicod. By (indirect) Hempelian confirmation, moreover,\(swan(a) \wedge white(a)\) also confirms that a particular furtherobject \(b\) will be white, if it’s a swan, i.e., \(swan(b)\rightarrow white(b)\) (to see this, just set \(s = \forall x(swan(x)\rightarrow white(x))\)).

The second possibility considered by Nicod (“theabsence of \(G\) in a case of \(F\,\)”) can beaccounted for by Hempelian disconfirmation. For the evidence statement\(e\) that \(a\) is a non-white swan—\(swan(a) \wedge \negwhite(a)\)—entails (in fact, is identical to) the\(e\)-development of the hypothesis that there exist non-whiteswans—\(\exists x(swan(x) \wedge \neg white(x))\)—which inturn is just the negation of \(\forall x(swan(x) \rightarrowwhite(x))\). So the latter is disconfirmed by the evidence in thiscase. And finally, \(e = swan(a) \wedge \neg white(a)\) alsoHempel-disconfirms that a particular further object \(b\) will bewhite if it’s a swan, i.e., \(swan(b) \rightarrow white(b)\),because the negation of the latter, \(swan(b) \wedge \neg white(b)\),is entailed by \(s = \forall x(swan(x) \wedge \neg white(x))\) and \(e\vDash dev_{e}(s)\).

So, to sum up, we have four illustrations of how Hempel’s theoryarticulates Nicod’s basic ideas, to wit:

• (the observation report of) a white swan (directly)Hempel-confirms that all swans are white;
• (the observation report of) a white swan also Hempel-confirmsthat a further swan will be white;
• (the observation report of) a non-white swan (directly)Hempel-disconfirms that all swans are white;
• (the observation report of) a non-white swan alsoHempel-disconfirms that a further swan will be white.

1.2 Two paradoxes and other difficulties

The ravens paradox (Hempel 1937, 1945). Consider thefollowing statements:

(\(h\))

\(\forall x(raven(x) \rightarrow black(x))\), i.e., all ravens areblack;

(\(e\))

\(raven(a) \wedge black(a)\), i.e., \(a\) is a black raven;

(\(e^*\))

\(\neg black(a^*) \wedge \neg raven(a^*)\), i.e., \(a^*\) is anon-black non-raven (say, a green apple).

Is hypothesis \(h\) confirmed by \(e\) and \(e^*\) alike? That is, isthe claim that all ravens are black equally confirmed by theobservation of a black raven and by the observation of a non-blacknon-raven (e.g., a green apple)? One would want to say no, butHempel’s theory is unable to draw this distinction. Let’ssee why.

As we know, \(e\) (directly) Hempel-confirms \(h\), according toHempel’s reconstruction of Nicod. By the same token, \(e^*\)(directly) Hempel-confirms the hypothesis that all non-black objectsare non-ravens, i.e., \(h^* = \forall x(\neg black(x) \rightarrow \negraven(x))\). But \(h^* \vDash h\) (\(h\) and \(h^*\) are justlogically equivalent). So, \(e^*\) (the observation report of anon-black non-raven), like \(e\) (black raven), does (indirectly)Hempel-confirm \(h\) (all ravens are black). Indeed, as \(\negraven(a)\) entails \(raven(a) \rightarrow black(a)\), it can be shownthat \(h\) is (directly) Hempel-confirmed by the observation ofany object that is not a raven (an apple, a cat, a shoe),apparently disclosing puzzling “prospects for indoorornithology” (Goodman 1955, 71).

\(Blite\) (Goodman 1955). Consider the peculiar predicate“\(blite\)”, defined as follows: an object is blite justin case (i) it is black if examined at some moment \(t\) up to somefuture time \(T\) (say, the next expected appearance of Halley’scomet, in 2061) and (ii) it is white if possibly examined onlyafterwards. So we posit \(blite(x) \equiv (ex_{t\le T}(x) \rightarrowblack(x)) \wedge (\neg ex_{t\le T}(x) \rightarrow white(x))\). Nowconsider the following statements:

(\(h\))

\(\forall x(raven(x) \rightarrow black(x))\), i.e., all ravens areblack;

(\(h^*\))

\(\forall x(raven(x) \rightarrow blite(x))\), i.e., all ravens areblite;

(\(e\))

\(e = raven(a) \wedge ex_{t\le T}(a) \wedge black(a)\), i.e.,\(a\) is a raven observed no later than \(T\) and it is black.

Does \(e\) confirm hypotheses \(h\) and \(h^*\) alike? That is, doesthe observation of a black raven before \(T\) confirm equally theclaim that all ravens are black as the claim that all ravens areblite? Here again, one would want to say no, but Hempel’s theoryis unable to draw the distinction. For one can check that the\(e\)-developments of \(h\) and \(h^*\) are both entailed by \(e\).Thus, \(e\) (the report of a raven examined no later than \(T\) andfound to be black) does Hempel-confirm \(h^*\) (all ravens are blite)just as it confirms \(h\) (all ravens are black). Moreover, \(e\) alsoHempel-confirms the statement that a raven will be white if examinedafter \(T\), because this is a logical consequence of \(h^*\) (whichis directly Hempel-confirmed by \(e\)). And finally, suppose that\(blurple(x) \equiv (ex_{t\le T}(x) \rightarrow black(x)) \wedge (\negex_{t\le T}(x) \rightarrow purple(x)).\) We then have that the verysame evidence statement \(e\) Hempel-confirms the hypothesis that allravens are blurple, and thus also its implication that a raven will be\(purple\) if examined after \(T\)!

A seemingly obvious idea, here, is that there must be somethinginherently wrong with predicates such as \(blite\) or \(blurple\) (andperhapsnon-raven andnon-black, too) and thus aprincipled way to rule them out as “unnatural”. Then onecould restrict confirmation theory accordingly, i.e., to“natural kinds” only (see, e.g., Quine 1970). Yet thispoint turns out be very difficult to pursue coherently and it has notborne much fruit in this discussion (Rinard 2014 is a recentexception). After all, for all we know, it is a perfectly“natural” feature of a token of the “naturalkind”water that it is found in one physical state fortemperatures below 0 degrees Celsius and in an entirely differentstate for temperatures above that threshold. So why should the timethreshold \(T\) in \(blite\) or \(blurple\) be a reason to dismissthose predicates? (The water example comes from Howson 2000,31–32. See Schwartz 2011, 399 ff., for a more general assessmentof this issue.)

The above, widely known “paradoxes” then suggest thatHempel’s analysis of confirmation istoo liberal: itsanctions the existence of confirmation relations that are intuitivelyvery unsound (see Earman and Salmon 1992, 54, and Sprenger 2011a, 243,for more on this). Yet the Hempelian notion of confirmation turns outto be very restrictive, too, on other accounts. For suppose thathypothesis \(h\) and evidence \(e\) do not share any piece ofnon-logical vocabulary. \(h\) might be, say, Newton’s law ofuniversal gravitation (connecting force, distances and masses), while\(e\) could be the description of certain spots on a telescopic image.Throughout modern physics, significant relations of confirmation anddisconfirmation were taken to obtain between statements like these.Indeed, telescopic sightings have been crucial evidence forNewton’s law as applied to celestial bodies. However, as theirnon-logical vocabularies are disjoint, \(e\) and \(h\) must simply belogically independent, and so must be \(e\) and \(dev_{e}(h)\) (withvery minor caveats, this follows from Craig’s so-calledinterpolation theorem, see Craig 1957). In such circumstances, therecan be nothing but Hempel-neutrality between evidence and hypothesis.So Hempel’s original theory seems to lack the resources tocapture a key feature of inductive inference in science as well as inseveral other domains, i.e., the kind of “vertical”relationships of confirmation (and disconfirmation) between thedescription of observed phenomena and hypotheses concerning underlyingstructures, causes, and processes.

To overcome the latter difficulty, Clark Glymour (1980a) embedded arefined version of Hempelian confirmation by instances in his analysisof scientific reasoning. In Glymour’s revision, hypothesis \(h\)is confirmed by some evidence \(e\) even if appropriate auxiliaryhypotheses and assumptions must be involved for \(e\) to entail therelevant instances of \(h\). This important theoretical move turnsconfirmation into athree-place relation concerning theevidence, the target hypothesis, and (a conjunction of) auxiliaries.Originally, Glymour presented his sophisticated neo-Hempelian approachin stark contrast with the competing traditional view of so-calledhypothetico-deductivism (HD). Despite his explicitintentions, however, several commentators have pointed out that,partly because of the due recognition of the role of auxiliaryassumptions, Glymour’s proposal and HD end up being plagued bysimilar difficulties (see, e.g., Horwich 1983, Woodward 1983, andWorrall 1982). In the next section, we will discuss the HD frameworkfor confirmation and also compare it with Hempelian confirmation. Itwill thus be convenient to have a suitable extended definition of thelatter, following the remarks above. Here is one that serves ourpurposes:

One can see that in the above definition \(\alpha\) includes the\(e\)-development of further general auxiliary hypotheses (in fact,equations as applied to specific established values, in typicalexamples from Glymour 1980a), where such hypotheses are meant to beconjoined in a single statement \(k\) for convenience. This impliesthat the only terms occurring (non-vacuously) in \(\alpha\) areindividual constants already occurring (non-vacuously) in \(e\). Foran empty \(k\) (that is, tautologous: \(k = \top\)), \(\alpha\) mustbe empty too, and the original (restricted) definition of Hempelianconfirmation applies. As for the proviso that \(k \not\vDash h\), itrules out undesired cases of circularity—akin to so-called“macho” bootstrap confirmation, as discussed in Earman andGlymour 1988 (for more on Glymour’s theory and its implications,see Douven and Meijs 2006, and references therein).

2. Hypothetico-deductivism

The central idea of hypothetico-deductive (HD) confirmation can beroughly described as “deduction-in-reverse”: evidence issaid to confirm a hypothesis in case the latter, while not entailed bythe former, is able to entail it, with the help of suitable auxiliaryhypotheses and assumptions. The basic version (sometimes labelled“naïve”) of the HD notion of confirmation can bespelled out thus:

Note that clause (ii) above represents HD-disconfirmation as plainlogical inconsistency of the target hypothesis with the data (giventhe auxiliaries) (see Hempel 1945, 98).

2.1 HD vs. Hempelian confirmation

HD-confirmation and Hempelian confirmation convey different intuitions(see Huber 2008a for an original analysis). They are, in fact,distinct and strictly incompatible notions. This will be effectivelyillustrated by the consideration of the following conditions.

Entailment condition (EC)
For any \(h,e,k \in \bL\), if \(e\wedge k\) is consistent, \(e\wedge k\vDash h\) and \(k \not\vDash h\), then \(e\) confirms \(h\) relativeto \(k\).

Confirmation complementarity (CC)
For any \(h, e, k \in \bL\), \(e\) confirms \(h\) relative to \(k\) ifand only if \(e\) disconfirms \(\neg h\) relative to \(k\).

Special consequence condition (SCC)
For any \(h, e, k \in \bL\), if \(e\) confirms \(h\) relative to \(k\)and \(h\wedge k \vDash h^*\), then \(e\) confirms \(h^*\) relative to\(k\).

On the implicit proviso that \(k\) is empty (that is, tautologous: \(k= \top\)), Hempel (1943, 1945) himself had put forward (EC) and (SCC)as compelling adequacy conditions for any theory of confirmation, anddevised his own proposal accordingly. As for (CC), he took it as aplain definitional truth (1943, 127). Moreover, Hempelian confirmation(extended) satisfies all conditions above (of course, for arguments\(h\), \(e\) and \(k\) for which it is defined). HD-confirmation, onthe contrary, violates all of them. Let us briefly discuss each one inturn.

It is rather common for a theory of ampliative (non-deductive)reasoning to retain classical logical entailment as a special case (afeature sometimes called “super-classicality”; seeStrasser and Antonelli 2019). That’s essentially what (EC)implies for confirmation. Now given appropriate \(e\), \(h\) and\(k\), if \(e\wedge k\) entails \(h\), we readily get that \(e\)Hempel-confirms \(h\) relative to \(k\) in two simple steps. First,given that \(\alpha\ = dev_{e}(k)\), \(dev_{e}(e\wedge \alpha) =dev_{e}(e\wedge k)\) according to Hempel’s full definition of\(dev\) (see Hempel 1943, 131). Then because clearly \(e\wedge \alpha\vDash dev_{e}(e\wedge \alpha)\) it also follows that \(e\wedge \alpha\vDash dev_{e}(e\wedge k)\), so \(e\wedge k\) is (directly)Hempel-confirmed by \(e\) relative to \(k\) and its logicalconsequence \(h\) is likewise confirmed (indirectly). Logicalentailment is thus retained as an instance of Hempelian confirmationin a fairly straightforward way. HD-confirmation, on the contrary,does not fulfil (EC). Here is one odd example (see Sprenger 2011a,234). With \(k = \top\), just let \(e\) be the observation report thatobject \(a\) is a black swan, \(swan(a) \wedge black(a)\), and \(h\)be the hypothesis that black swans exist, \(\exists x(swan(x) \wedgeblack(x))\). Evidence \(e\) verifies \(h\) conclusively, and yet itdoes not HD-confirm it, simply because \(h \not\vDash e\). So theobservation of a black swan turns out to be HD-neutral for thehypothesis that black swans exist! The same example shows howHD-confirmation violates (CC), too. In fact, while HD-neutral for\(h\), \(e\) HD-disconfirms its negation \(\neg h\) that no swan isblack, \(\forall x(swan(x) \rightarrow \neg black(x))\), because thelatter is obviously inconsistent with (refuted by) \(e\).

The violation of (EC) and (CC) by HD-confirmation is arguably a reasonfor concern, for those conditions seem highly plausible. The specialconsequence condition (SCC), on the other hand, deserves separate andcareful consideration. As we will see later on, (SCC) is a strongconstraint, and far from sacrosanct. For now, let us point out onemajor philosophical motivation in its favor. (SCC) has often beeninvoked as a means to ensure the fulfilment of the following condition(see, e.g., Hesse 1975, 88; Horwich 1983, 57):

Predictive inference condition (PIC)
For any \(e, k \in \bL\), if \(e\) confirms \(\forall x(Fx \rightarrowGx)\) relative to \(k\), then \(e\) confirms \(F(a) \rightarrow G(a)\)relative to \(k\).

In fact, (PIC) readily follows from (SCC) and so it is satisfied byHempel’s theory. It says that, if evidence \(e\) confirms“all \(F\)s are \(G\)s”, then it also confirms that afurther object will be \(G\) if it is \(F\). Notably, this does nothold for HD-confirmation. Here is why. Given \(k = Fa\) (i.e., theassumption that \(a\) comes from the \(F\) population), we have that\(e = Ga\) HD-confirms \(h = \forall x(Fx \rightarrow Gx)\), becausethe latter entails the former (given \(k\)). (That’s the HDreconstruction of Nicod’s insight, see below.) We also have, ofcourse, that \(h\) entails \(h^* = Fb \rightarrow Gb\). And yet,contrary to (PIC), since \(h^*\) does not entail \(e\) (given \(k\)),it is not HD-confirmed by it either. The troubling conclusion is thatthe observation that a swan is white (or that a million of them are,for that matters) does not HD-confirm the prediction that a furtherswan will be found to be white.

2.2 Back to black (ravens)

One attractive feature of HD-confirmation is that it largely eludesthe ravens paradox. As the hypothesis \(h\) that all ravens are blackdoes not entail that some generally sampled object \(a\) will be ablack raven, the HD view of confirmation is not committed to theeminently Hempelian implication that \(e = raven(a) \wedge black(a)\)confirms \(h\). Likewise, \(\neg black(a) \wedge \neg raven(a)\) doesnot HD-confirm that all non-black objects are non-raven. Thederivation of the paradox, as presented above, is thus blocked.

Indeed, HD-confirmation yields a substantially different reading ofNicod’s insight as compared to Hempel’s theory (Okasha2011 has an important discussion of this distinction). Here is how itgoes.If object \(a\)is assumed to have been taken amongravens—so that, crucially, the auxiliary assumption \(k =raven(a)\) is made—and \(a\) is checked for color and found tobe black, then, yes, the latter evidence, \(black(a)\), HD-confirmsthat all ravens are black \((h)\) relative to \(k\). By the sametoken, \(\neg black(a)\) HD-disconfirms \(h\) relative to the sameassumption \(k = raven(a)\). And, again, this is as it should be, inline with Nicod’s mention of “the absence of \(G\) [here,non-black as evidence] in a case of \(F\) [here, raven as an auxiliaryassumption]”. It is also true that an object that is foundnot to be a raven HD-confirms \(h\), butonlyrelative to \(k = \neg black(a)\), that is, if \(a\) is assumed tohave been taken among non-black objects to begin with; and this seemsacceptable too (after all, while sampling from non-black objects, onemight have found the counterinstance of a raven, but didn’t).Unlike Hempel’s theory, moreover, HD-confirmation does not yieldthe debatable implication that, by itself (that is, given \(k =\top\)), the observation of a non-raven \(a\), \(\neg raven(a)\), mustconfirm \(h\).

Interestingly, the introduction of auxiliary hypotheses andassumptions shows that the issues surrounding Nicod’s remarkscan become surprisingly subtle. Consider the following statements(Maher’s 2006 example):

(\(q_1\))

\(\forall x(white(x) \rightarrow \neg black(x))\)

(\(q_2\))

\(\exists x(swan(x)) \rightarrow \exists y(swan(y) \wedgeblack(y))\)

\(q_1\) simply specifies that no object is both white and black, while\(q_2\) says that, if there are swans at all, then there also is someblack swan. Also assume, again, that \(e = swan(a) \wedgewhite(a)\). Under \(q_1\) and \(q_2\), the observation of a white swanclearlydisconfirms (indeed, refutes) the hypothesis \(h\)that all swans are white. Hempel’s theory (extended) facesdifficulties here, because for \(\alpha = dev_{e}(q_1 \wedge q_2)\) itturns out that \(e\wedge \alpha\) is inconsistent. But HD-confirmationgets this case right, thus capturing appropriate boundary conditionsfor Nicod’s generally sensible claims. For, with \(k = q_1\wedge q_2\), one has that \(h\wedge k\) is consistent and entails\(\neg e\) (for it entails that no swan exists), so that \(e\)HD-disconfirms (refutes) \(h\) relative to \(k\) (see Good 1967 foranother famous counterexample to Nicod’s condition).

HD-confirmation, however, is also known to suffer from distinctive“paradoxical” implications. One of the most frustrating issurely the following (see Osherson, Smith, and Shafir 1986, 206, forfurther specific problems).

The irrelevant conjunction paradox. Suppose that \(e\)confirms \(h\) relative to (possibly empty) \(k\). Let statement \(c\)be logically consistent with \(e\wedge h\wedge k\), but otherwiseentirely irrelevant for all of those conjuncts (perhaps belonging to acompletely separate domain of inquiry). Does \(e\) confirm \(h\wedgec\) (relative to \(k\)) as it does with \(h\)? One would want to sayno, and this implication can be suitably reconstructed inHempel’s theory. HD-confirmation, on the contrary, can not drawthis distinction: it is easy to show that, on the conditionsspecified, if the HD clause for confirmation is satisfied for \(e\)and \(h\) (given \(k\)), so it is for \(e\) and \(h\wedge c\) (given\(k\)). (This is simply because, if \(h\wedge k \vDash e\), then\(h\wedge c\wedge k \vDash e\), too, by the monotonicity of classicallogical entailment.)

Kuipers (2000, 25) suggested that one can live with the irrelevantconjunction problem because, on the conditions specified, \(e\) wouldstill not HD-confirm \(c\) alone (given \(k\)), so thatHD-confirmation can be “localized”: \(h\) is the only bitof the conjunction \(h\wedge c\) that gets any confirmation on itsown, as it were. Other authors have been reluctant to bite the bulletand have engaged in technical refinements of the“naïve” HD view. In these proposals, the spread ofHD-confirmation upon frivolous conjunctions can be blocked at theexpense of some additional logical machinery (see Gemes 1993, 1998;Schurz 1991, 1994).

Finally, it should be noted that HD-confirmation offers no substantialrelief from the blite paradox. On the one hand, \(e = raven(a) \wedgeex_{t\le T}(a) \wedge black(a)\) doesnot, as such,HD-confirm either \(h = \forall x(raven(x) \rightarrow black(x))\) or\(h^* = \forall x(raven(x) \rightarrow blite(x))\), that is, for empty\(k\). On the other hand, if object \(a\) is assumed to have beensampled from ravens before \(T\) (that is, given \(k = raven(a) \wedgeex_{t\le T}(a))\), then \(black(a)\) is entailed by both “allravens are black” and “all ravens are blite” andtherefore HD-confirms each of these hypotheses (and indeed,indefinitely many others: as we know, further variations of \(h^*\)can be conceived at will, like the “blurple” hypothesis).One could insist that HD does handle the blite paradox after all,because \(black(a)\) (given \(k\) as above) does not HD-confirms thata raven will be white if examined after \(T\) (Kuipers 2000, 29 ff.).Unfortunately (as pointed out by Schurz 2005, 148) \(black(a)\) doesnot HD-confirm that a raven will be black if examined after \(T\)either (again, given \(k\) as above). That’s because, as alreadypointed out, HD-confirmation fails the predictive inference condition(PIC) in general. So, all in all, HD-confirmation can not tell blackfrom blite any more than Hempel-confirmation can.

2.3 Underdetermination and the Duhemian challenge

The issues above look contrived and artificial to some people’staste—even among philosophers. Many have suggested a closer lookat real-world inferential practices in the sciences as a moreappropriate benchmark for assessment. For one thing, the very idea ofhypothetico-deductivism has often been said to stem from the originsof Western science. As reported by Simplicius of Cilicia (sixthcentury A.D.) in his commentary on Aristotle’sDeCaelo, Plato had challenged his pupils to identify combinationsof “ordered” motions by which one could account for(namely, deduce) the planets’ wandering trajectories across theheavens as observed by the Earth. As a matter of historical fact,mathematical astronomy (the first mature empirical science) hasengaged in just this task for centuries: scholars have been trying todefine geometrical models from which the apparent motion of celestialbodies would derive.

It is fair to say that, at its roots, the kind of challenges that theHD framework faces with scientific reasoning is not so different fromthe main puzzles that arise from philosophical considerations of amore formal kind. Still, the two areas turn out to be complementary inimportant ways. The following statement will serve as a usefulstarting point to extend the scope of our discussion.

Underdetermination Theorem (UT) for“naïve” HD-confirmation
For any contingent \(h, e \in \bL\), if \(h\) and \(e\) are logicallyconsistent, there exists some \(k \in \bL\) such that \(e\)HD-confirms \(h\) relative to \(k\).

(UT) is an elementary logical fact that has been long recognized (see,e.g., Glymour 1980a, 36). In purely formal terms, just positing \(k =h \rightarrow e\) will do for a proof. To appreciate how (UT) canspark any philosophical interest, one has to combine it with someinsightful remarks first put forward by Pierre Duhem (1906) and thenfamously revived by Quine (1951) in a more radical style. (Indeed,(UT) essentially amounts to the “entailment version” of“Quinean underdetermination” in Laudan 1990, 274.)

Duhem (he himself a supporter of the HD view) pointed out that inmature sciences such as physics most hypotheses or theories of realinterest can not be contradicted by any statement describingobservable states of affairs. Taken in isolation, they simply do notlogically imply, nor rule out, any observable fact, essentiallybecause (unlike “all ravens are black”) they concernunobservable entities and processes. So, in effect, Duhem emphasizedthat, typically, scientific hypotheses or theoriesarelogically consistent with any piece of checkable evidence. Unless, ofcourse, the logical connection is underpinned by auxiliary hypothesesand assumptions suitably bridging the gap between the observationaland non-observational vocabulary, as it were. But then, onceauxiliaries are in play, logic alone guarantees thatsome\(k\) exists such that \(h\wedge k\) is consistent, \(h\wedge k \vDashe\), and \(k \not\vDash e\), so that confirmation holds in naïveHD terms (that’s just the UT result above). Apparently, whenDuhem’s point applies, the uncritical supporter of whateverhypothesis \(h\) can legitimately claim (naïve HD) confirmationfrom any \(e\) by simply shaping \(k\) conveniently. In this sense,hypothesis assessment would be radically “underdetermined”by any amount of evidence practically available.

Influential authors such as Thomas Kuhn (1962/1970) (but see Laudan1990, 268, for a more extensive survey) relied on Duhemian insights tosuggest that confirmation by empirical evidence is too weak a force todrive the evaluation of theories in science, often invitingconclusions of a relativistic flavor (see Worrall 1996 for anilluminating reconstruction along these lines). Let us brieflyconsider a classic case, which Duhem himself thoroughly analyzed: thewavevs. particle theories of light in modern optics. Acrossthe decades, wave theorists were able to deduce an impressive list ofimportant empirical facts from their main hypothesis along withappropriate auxiliaries, diffraction phenomena being only one majorexample. But many particle theorists’ reaction was to retaintheir hypothesis nonetheless and to reshape other parts ofthe “theoretical maze” (i.e., \(k\); the term isPopper’s, 1963, p. 330) to recover those observed facts asconsequences of their own proposal. And as we’ve seen,if the bare logic of naïve HD was to be taken strictly,surely theycould have claimed their overall hypothesis to beconfirmed too, just as much as their opponents.

Importantly, they didn’t. In fact, it was quite clear thatparticle theorists, unlike their wave-theory opponents, were strivingto remedy weaknesses rather than scoring successes (see Worrall 1990).But why, then? Because, as Duhem himself clearly realized, the logicof naïve HD “is not the only rule for our judgments”(1906, 217). The lesson of (UT) and the Duhemian insight is not quite,it seems, that naïve HD is the last word and scientific inferenceis unconstrained by stringent rational principles, but rather that theHD view has to be strengthened in order to capture the real nature ofevidential support in rational scientific inference. At least,that’s the position of a good deal of philosophers of scienceworking within the HD framework broadly construed. It has even beenmaintained that “no serious twentieth-centurymethodologist” has ever subscribed to the naïve HD viewabove “without crucial qualifications” (Laudan 1990, 278;also see Laudan and Leplin 1991, 466).

So the HD approach to confirmation has yielded a number of morearticulated variants to meet the challenge of underdetermination.Following (loosely) Norton (2005), we will now survey an instructivesample of them.

2.4 The extended HD menu

Naïve HD can be enriched by a resolute form ofpredictivism. According to this approach, the naïve HDclause for confirmation is too weak because \(e\) must have beenpredicted in advance from \(h\wedge k\). Karl Popper’s(1934/1959) account of the “corroboration” of hypothesesfamously embedded this view, but squarely predictivist stances can betraced back to early modern thinkers like Christiaan Huygens(1629–1695) and Gottfried Wilhelm Leibniz (1646–1716), andin Duhem’s work itself. The predictivist sets a high bar forconfirmation. Her favorite examples typically include stunningepisodes in which the existence of previously unknown objects,phenomena, or whole classes of them is anticipated: the phases ofVenus for Copernican astronomy or the discovery of Neptune forNewtonian physics, all the way up to the Higgs boson for so-calledstandard model of subatomic particles.

The predictivist solution to the underdetermination problem is fairlyradical: many of the relevant factual consequences of \(h\wedge k\)will be already known when this theory is articulated, and so unfitfor confirmation. Critics have objected that predictivism is in factfar too restrictive. There seem to be many cases in which alreadyknown phenomena clearly do provide support to a new hypothesis ortheory. Zahar (1973) first raised this problem of “oldevidence”, then made famous by Glymour (1980a, 85 ff.) as adifficulty for Bayesianism (seeSection 3 below). Examples of this kind abound in the history of science aselsewhere, but the textbook illustration has become the precession ofMercury’s perihelion, a lasting anomaly for Newtonian physics:Einstein’s general relativity calculations got this long-knownfact right, thereby gaining a remarkable piece of initial support forthe new theory. In addition to this problem with old evidence, HDpredictivism also seems to lack a principled rationale. After all, thetemporal order of the discovery of \(e\) and of the articulation of\(h\) and \(k\) may well be an entirely accidental historicalcontingency. Why should it bear on the confirmation relationship amongthem? (See Giere 1983 and Musgrave 1974 for classic discussions ofthese issues. Douglas and Magnus 2013 and Barnes 2018 offer morerecent views and rich lists of further references.)

As a possible response to the difficulties above, naïve HD can beenriched by theuse-novelty criterion (UN) instead. The UNreaction to the underdetermination problem is more conservative thanthe temporal predictivist strategy. According to this view, to improveon the weak naïve HD clause for confirmation one only has to ruleout one particular class of cases, i.e., those in which thedescription of a known fact, \(e\), served as a constraint in theconstruction of \(h\wedge k\). The UN view thus comes equipped with arationale. If \(h\wedge k\) was shaped on the basis of \(e\), UNadvocates point out, then it was bound to get that state of affairsright; the theory never ran any risk of failure, thus did not achieveany particularly significant success either. Precisely in these cases,and just for this reason, the evidence \(e\) must not bedouble-counted: by using it for the construction of the theory, itsconfirmational power becomes “dried out”, so to speak.

The UN completion of naïve HD originated from Lakatos and some ofhis collaborators (see Lakatos and Zahar 1975 and Worrall 1978; alsosee Giere 1979, 161–162, and Gillies 1989 for similar views),although important hints in the same direction can be found at leastin the work of William Whewell (1840/1847). Consider the touchstoneexample of Mercury again. According to Zahar (1973), Einstein did notneed to rely on the Mercury data to define theory and auxiliaries asto match observationally correct values for the perihelion precession(also see Norton 2011a; and Earman and Janssen 1993 for a verydetailed, and more nuanced, account). Being already known, the factwas not of course predicted in a strictly temporal sense, and yet, onZahar’s reading, itcould have been: it was“use-novel” and thus fresh for use to confirm the theory(see Crupi 2025 for a possible refinement and an application to theCopernican revolution). For a more mundane illustration, so-calledcross-validation techniques represent a routine applicationof the UN idea in statistical settings (as pointed out by Schurz 2014,92; also see Forster 2007, 592 ff.). According to some commentators,however, the UN criterion needs further elaboration (see Hitchcock andSober 2004 and Lipton 2005), while others have criticized it asessentially wrong-headed (see Howson 1990 and Mayo 1991, 2014; alsosee Votsis 2014).

Yet another way to enrich naïve HD is to combine it witheliminativism. According to this view, the naïve HDclause for confirmation is too weak because there must have been a low(enough) objective chance of getting the outcome \(e\) (favorable to\(h\)) if \(h\) was false, so that few possibilities exist that \(e\)may have occurred for some reason other than the truth of \(h\).Briefly put, the occurrence of \(e\) must be such that mostalternatives to \(h\) can be safely ruled out. The founding figure ofeliminativism is Francis Bacon (1561–1626). John Stuart Mill(1843/1872) is a major representative in later times, and DeborahMayo’s “error-statistical” approach to hypothesistesting arguably develops this tradition (Mayo 1996 and Mayo andSpanos 2010; see Bird 2010, Kitcher 1993, 219 ff., and Meehl 1990 forother contemporary variations).

Eliminativism is most credible when experimentation is at issue (see,e.g., Guala 2012). Indeed, the appeal to Bacon’s idea ofcrucial experiment (instantia crucis) and relatednotions (e.g., “severe testing”) is a fairly reliable markof eliminativist inclinations. Experimentation is, to a large extent,precisely an array of techniques to keep undesired interfering factorsat a minimum by active manipulation and deliberate control (think ofthe blinding procedure in medical trials, with \(h\) the hypothesizedeffectiveness of a novel treatment and \(e\) a relative improvement inclinical endpoints for a target subsample of patients thus treated).When this kind of control obtains, popular statistical tools aresupposed to allow for the calculation of the probability of \(e\) incase \(h\) is false meant as a “relative frequency in a (real orhypothetical) series of test applications” (Mayo 1991, 529), andto secure a sufficiently low value to validate the positive outcome ofthe test. It is much less clear how firm a grip this approach canretain when inference takes place at higher levels of generality andtheoretical commitment, where the hypothesis space is typically muchtoo poorly ordered to fit routine error-statistical analyses. Indeed,Laudan (1997, 315; also see Musgrave 2010) spotted in this approachthe risk of a “balkanization” of scientific reasoning,namely, a restricted focus on scattered pieces of experimentalinference (but see Mayo 2010 for a defense).

Naïve HD can also be enriched by the notion ofsimplicity. According to this view, the naïve HD clausefor confirmation is too weak because \(h\wedge k\) must be a simple(enough), unified way to account for evidence \(e\). A classicreference for the simplicity view is Newton’s first law ofphilosophizing in thePrincipia (“admit no more causesof natural things than such as are both true and sufficient to explaintheir appearances”), echoing very closely Ockham’s razor.This basic idea has never lost its appeal—even up to recenttimes (see, e.g., Quine and Ullian 1970, 69 ff.; Sober 1975; Zellner,Keuzenkamp, and McAleer 2002; Scorzato 2013).

Despite Thomas Kuhn’s (1957, 181) suggestions to the contrary,the success of Copernican astronomy over Ptolemy’s system hasremained an influential case study fostering the simplicity view(Martens 2009). Moreover, in ordinary scientific problems such ascurve fitting, formal criteria of model selection are appliedwhere the paucity of parameters can be interpreted naturally as a keydimension of simplicity (Forster and Sober 1994). Traditionally, twomain problems have proven pressing, and frustrating, for thesimplicity approach. First, how to provide a sufficiently coherent andilluminating explication of this multifaceted and elusive notion (seeRiesch 2010); and second, how to justify the role of simplicity as aproperlyepistemic (rather than merelypragmatic)virtue (see Kelly 2007, 2008).

Finally, naïve HD can be enriched by the appeal toexplanation. Here, the naïve HD clause for confirmationis meant to be too weak because \(h\wedge k\) must be able (not onlyto entail, but) to explain \(e\). By this move, the HD approach embedsthe slogan of the so-calledinference to the best explanationview: “observations support the hypothesis precisely because itwould explain them” (Lipton 2000, 185; also see Lipton 2004).Historically, the main source for this connection between explanationand support is found in the work of Charles Sanders Peirce(1839–1914). Janssen (2003) offers a particularly neatcontemporary exhibit, explicitly aimed at “curing cases of theDuhem-Quine disease” (484; also see Thagard 1978, and Douven2017 for a relevant survey). Quite unlike eliminativist approaches,explanationist analyses tend to focus on large-scale theories andrelatively high-level kinds of evidence. Dealing with Einstein’sgeneral relativity, for instance, Janssen (2003) greatly emphasizesits explanation of the equivalence of inertial and gravitational mass(essentially a brute fact in Newtonian physics) over the resolution ofthe puzzle of Mercury’s perihelion. Explanationist accounts arealso distinctively well-equipped to address inference patterns fromnon-experimental sciences (Cleland 2011).

The problems faced by these approaches are similar to those affectingthe simplicity view. Agreement is still lacking on the nature ofscientific explanation (see Woodward 2019) and it is not clear how faran explanationist variant of HD can go without a sound analysis ofthat notion (Prasetya 2024). Moreover, critics have wondered why therelationship of confirmation should be affected by an explanatoryconnection with the evidenceper se (see Salmon 2001).

The above discussion does not display an exhaustive list (nor are thelisted options mutually exclusive, for that matter: see, e.g., Baker2003; also see Worrall 2010 for some overlapping implications in anapplied setting of real practical value). And our sketchedpresentation hardly allows for any conclusive assessment. It doessuggest, however, that reports of the death of hypothetico-deductivism(see Earman 1992, 64, and Glymour 1980b) might have been exaggerated.For all its difficulties, HD has proven fairly resilient at least as abasic framework to elucidate some key aspects of how hypotheses can beconfirmed by the evidence (see Betz 2013, Gemes 2005, and Sprenger2011b for consonant points of view).

3. Bayesian confirmation theories

Bayes’s theorem is a very central element of theprobability calculus (see Joyce 2019). For historical reasons,Bayesian has become a standard label to allude to a range ofapproaches and positions sharing the common idea that probability (inits modern, mathematical sense) plays a crucial role in rationalbelief, inference, and behavior. According to Bayesian epistemologistsand philosophers of science, (i) rational agents have credencesdiffering in strength, which moreover (ii) satisfy the probabilityaxioms, and can thus be represented in probabilistic form. (Innon-Bayesian models (ii) is rejected, but (i) may well be retained:see Huber and Schmidt-Petri 2009, Levi 2008, and Spohn 2012.)Well-known arguments exist in favor of this position (see, e.g.,Easwaran 2011a; Pettigrew 2016; Skyrms 1987; Vineberg 2016), althoughthere is no lack of difficulties and criticism (see, e.g., Easwaran2011b; Hájek 2008; Kelly and Glymour 2004; Norton 2011b).

Beyond the core ideas above, however, the theoretical landscape ofBayesianism is quite as hopelessly diverse as it is fertile. Surveysand state of art presentations are already numerous, and ostensiblygrowing (see, e.g., Good 1971; Joyce 2011; Oaksford and Chater 2007;Sprenger and Hartmann 2020; Weisberg 2015). For the present purposes,attention can be restricted to a classification that is still fairlycoarse-grained, and based on just two dimensions or criteria.

First, there is an important distinction betweenpermissivismandimpermissivism (see Meacham 2014 and Kopec and Titelbaum2016 for this terminology). For permissive Bayesians (sometimesotherwise labelled “subjectivists”), accordance with theprobability axioms is the only clear-cut constraint on the credencesof a rational agent. In impermissive forms of Bayesianism (oftenotherwise called “objective”), further constraints are putforward that significantly restrict the range of rational credences,possibly up to one single “right” probability function inany given setting. Second, there are different attitudes towardsso-called principle oftotal evidence (TE) for theprobabilities on which a reasoner relies. TE Bayesians maintain thatthe relevant credences should be represented by a probability function\(P\) which conveys the totality of what is known to the agent. Fornon-TE approaches, depending on the circumstances, \(P\) may (orshould) be set up so that portions of the evidence available are infact bracketed. (Unsurprisingly, further subtleties arise as soon asone delves a bit further into the precise meaning and scope of TE; seeFitelson 2008 and Williamson 2002, Chs. 9–10, for importantdiscussions.)

Of course, many intermediate positions exist between extreme forms ofpermissivism and impermissivism so outlined, and more or less the sameapplies for the TE issue. The above distinctions are surely roughenough, but useful nonetheless. Impermissive TE Bayesianism has servedas a received view in early Bayesian philosophy of science (e.g.,Carnap 1950/1962). But impermissivism is easily found in combinationwith non-TE positions, too (see, e.g., Maher 1996). TE permissivismseems a good approximation of De Finetti’s (2008) stance, whilenon-TE permissivism is arguably close to a standard view nowadays(see, e.g., Howson and Urbach 2006). No more than this will be neededto begin our exploration of Bayesian confirmation theories.

3.1 Probabilistic confirmation asfirmness

Let us consider a set \(\bP\) of probability functions representingpossible states of belief about a domain that is described in a finitelanguage \(L\) with \(\bL\) the set of its closed sentences. From nowon, unless otherwise specified, whenever considering some \(h, e, k\in \bL\) and \(P \in \bP\), we will invariably rely on the followingprovisos:

1. both \(e\wedge k\) and \(h\wedge k\) are consistent;
2. \(P(e\wedge k), P(h\wedge k) \gt 0;\)
3. \(P(k) \gt P(h\wedge k)\) (unless \(k \vDash h\));
4. \(P(e\wedge k) \gt P(e\wedge h\wedge k)\) (unless \(e\wedge k\vDash h\)); and
5. \(P(e\wedge h\wedge k) \gt 0\), as long as \(e\wedge h\wedge k\)is consistent.

(These assumptions are convenient and critical for technical reasons,but not entirely innocent. Festa 1999 and Kuipers 2000, 44 ff.,discuss some limiting cases that are left aside here owing to theseconstraints.)

A probabilistic theory of confirmation can be spelled out through thedefinition of a function \(C_{P}(h, e\mid k): \{\bL^3 \times \bP\}\rightarrow \Re\) representing the degree of confirmation thathypothesis \(h\) receives from evidence \(e\) relative to \(k\) andprobability function \(P\). \(C_{P}(h,e\mid k)\) will then haverelevant probabilities as its building blocks, according to thefollowing basic postulate of probabilistic confirmation:

(P0) Formality
There exists a function \(g\) such that, for any \(h, e, k \in \bL\)and any \(P \in \bP\), \(C_{P}(h,e\mid k) = g[P(h\wedge e\midk),P(h\mid k),P(e\mid k)]\).

Note that the probability distribution over the algebra generated by\(h\) and \(e\), conditional on \(k\), is entirely determined by\(P(h\wedge e\mid k)\), \(P(h\mid k)\) and \(P(e\mid k)\). Hence, (P0)simply states that \(C_{P}(h, e\mid k)\) depends on that distribution,and nothing else. (The label for this assumption is taken fromTentori, Crupi, and Osherson 2007, 2010.)

Hempelian and HD confirmation, as discussed above, arequalitative theories of confirmation. They only tell uswhether evidence \(e\) confirms (disconfirms) hypothesis\(h\) given \(k\). However, assessments of theamount ofsupport that some evidence brings to a hypothesis are commonlyinvolved in scientific reasoning, as well as in other domains, if onlyin the form ofcomparative judgments such as“hypothesis \(h\) is more strongly confirmed by \(e_{1}\) thanby \(e_{2}\)” or “\(e\) confirms \(h_{1}\) to a greaterextent than \(h_{2}\)”. Consider, for instance, the followingprinciple, a veritable cornerstone of probabilistic confirmation inall of its variations (see Crupi, Chater, and Tentori 2013 for a listof references):

(P1) Final probability
For any \(h,e_{1},e_{2},k \in \bL\) and any \(P \in \bP\),\(C_{P}(h,e_{1}\mid k) \gtreqless C_{P}(h, e_{2}\mid k)\) if and onlyif \(P(h\mid e_{1} \wedge k) \gtreqless P(h\mid e_{2} \wedge k).\)

(P1) is itself a comparative, orordinal, principle, statingthat, for any fixed hypothesis \(h\), the final (or posterior)probability and confirmation always move in the same direction in thelight of data, \(e\) (given \(k\)). Interestingly, (P0) and (P1) arealready sufficient to single out one traditional class of measures ofprobabilistic confirmation, if conjoined with the following (see Crupiand Tentori 2016, 656, Schippers 2017, and also Törnebohm 1966,81):

(P2) Local equivalence
For any \(h_{1},h_{2},e,k \in \bL\) and any \(P\in \bP\), if \(h_{1}\)and \(h_{2}\) are logically equivalent given \(e\) and \(k\), then\(C_{P}(h_{1},e\mid k) = C_{P}(h_{2}, e\mid k).\)

The following can then be shown:

Theorem 1
(P0), (P1) and (P2) hold if and only if there exists a strictlyincreasing function \(f\) such that, for any \(h, e, k \in \bL\) andany \(P \in \bP\), \(C_{P}(h, e\mid k) = f[P(h\mid e\wedge k)]\).

Theorem 1 provides a simple axiomatic characterization of the class ofconfirmation functions that are strictly increasing with the finalprobability of the hypothesis given the evidence (and \(k\)) (provenin Schippers 2017). All the functions in this class areordinallyequivalent, meaning that they imply the same rank order of\(C_{P}(h, e\mid k)\) and \(C_{P^*}(h^*,e^*\mid k^*)\) for any \(h,h^*,e, e^*,k, k^* \in \bL\) and any \(P, P^* \in \bP.\)

By (P0), (P1) and (P2), we thus have \(C_{P}(h, e\mid k) = f[P(h\mid e\wedge k)]\), implying that the more likely \(h\) is given theevidence the more it is confirmed. This approach explicatesconfirmation precisely as theoverall credibility of ahypothesis (firmness is Carnap’s 1950/1962 tellingterm, xvi). In this view, “Bayesian confirmation theory islittle more than the examination of [the] properties” of theposterior probability function (Howson 2000, 179).

As we will see, the ordinal level of analysis is a solid andconvenient middle ground between a purely qualitative and a thoroughlyquantitative (metric) notion of confirmation. To begin with, ordinalnotions are in general sufficient to move “upwards” to thequalitative level as follows:

Given Theorem 1, (P0), (P1) and (P2) can be combined with thedefinitions in (QC) to derive the following qualitative notion ofprobabilistic confirmation as firmness:

The point of qualitative \(F\)-confirmation is thus straightforward:\(h\) is said to be (dis)confirmed by \(e\) (given \(k\)) if it ismore likely than not to be true (false). (Sometimes a threshold higherthan a probability \(\bfrac{1}{2}\) is identified, but thiscomplication would add little for our present purposes.)

The ordinal notion of confirmation is of high theoretical significancebecause ordinal divergences, unlike purely quantitative differences,imply opposite comparative judgments for some evidence-hypothesispairs. A refinement from the ordinal to a properly quantitative levelis also be of interest, however, and much useful for tractability andapplications. For example, one can have 0 as a convenient neutralitythreshold for confirmation as firmness, provided that the followingfunctional representation is adopted (see Peirce 1878 for an earlyoccurrence):

(The base of the logarithm can be chosen at convenience, as long as itis strictly greater than 1.)

A quantitative requirement that is often put forward is the followingstringent form of additivity:

Strict additivity (SA)
For any \(h, e_{1},e_{2},k \in \bL\) and any \(P \in \bP\),
\(\ \ \ C_{P}(h, e_{1} \wedge e_{2}\mid k) = C_{P}(h, e_{1}\mid k) +C_{P}(h, e_{2}\mid e_{1} \wedge k).\)

Although extraneous to \(F\)-confirmation, Strict Additivity willprove of use later on for the discussion of further variants ofBayesian confirmation theory.

3.2 Strengths and infirmities of firmness

Confirmation as firmness shares a number of structural properties withHempelian confirmation. It satisfies the Special ConsequenceCondition, thus the Predictive Inference Condition too. It satisfiesthe Entailment Condition and, in virtue of (P1), extends it smoothlyto the following ordinal counterpart:

According to (EC-Ord) not only is classical entailment retained as acase of confirmation, it also represents a limiting case: it is thestrongest possible form of confirmation that a fixed hypothesis \(h\)can receive.

\(F\)-confirmation also satisfies Confirmation Complementarity and,moreover, extends it to its appealing ordinal counterpart (see Crupi,Festa, and Buttasi 2010, 85–86), that is:

Confirmation complementarity (ordinal extension)(CC-Ord)
\(C_{P}(\neg h, e\mid k)\) is a strictly decreasing function of\(C_{P}(h, e\mid k)\), that is, for any \(h, h^*,e, e^*,k \in \bL\)and any \(P\in \bP,\) \(C_{P}(h, e\mid k)\gtreqless C_{P}(h^*,e^*\midk)\) if and only if \(C_{P}(\neg h, e\mid k) \lesseqgtr C_{P}(\negh^*,e^*\mid k).\)

(CC-Ord) neatly reflects Keynes’ (1921, 80) remark that“an argument is always as near to proving or disproving aproposition, as it is to disproving or proving itscontradictory”. Indeed, quantitatively, the measure \(F(h, e\midk)\) instantiates Confirmation Complementarity in a simple and elegantway, that is, it satisfies \(C_{P}(h, e\mid k) = -C_{P}(\neg h, e\midk).\)

\(F\)-confirmation also implies another attractive quantitativeresult, alleviating the ailments of the irrelevant conjunctionparadox. In the statement below, indicating this result, theirrelevance of \(c\) for hypothesis \(h\) and evidence \(e\)(relative to \(k\)) is meant to amount to the probabilisticindependence of \(c\) from \(h, e\) and their conjunction (given\(k\)), that is, to \(P(h \wedge c\mid k) = P(h\mid k)P(c\mid k),\)\(P(e \wedge c\mid k) = P(e\mid k)P(c\mid k)\), and \(P(h \wedge e\wedge c\mid k) = P(h \wedge e\mid k)P(c\mid k)\), respectively.

So, even in case it is qualitatively preserved across the tacking of\(c\) onto \(h\), the positive confirmation afforded by \(e\) is atleast bound to quantitativelydecrease thereby.

Partly because of appealing formal features such as those mentioned sofar, there is a long list of distinguished scholars advocating thefirmness view of confirmation, from Keynes (1921) andHosiasson-Lindenbaum (1940) onwards, most often coupled with some formof impermissive Bayesianism (see Hawthorne 2011 and Williamson 2011for contemporary variations). In fact, \(F\)-confirmation fits mostneatly a classical form of TE impermissivismà laCarnap, where one assumes that \(k = \top,\) that \(P\) is an“objective” initial probability based on essentiallylogical considerations, and that all the non-logical informationavailable is collected in \(e\). The spirit of the Carnapian projectnever lost its appeal entirely (see, e.g., Festa 2003, Franklin 2001,Maher 2010, Paris 2011). However, the idea of a “logical”interpretation of \(P\) got stuck into difficulties that are oftenseen as insurmountable (e.g., Earman and Salmon 1992, 85–89;Gillies 2000, Ch. 3; Hájek 2019; Howson and Urbach 2006,59–72; van Fraassen 1989, Ch. 12; Zabell 2011). And arguably,lacking some robust and effective impermissivist policy, the accountof confirmation as firmness ends up loosing much of its philosophicalmomentum. The issues surrounding the ravens and blite paradoxesprovide a useful illustration.

Consider again \(h = \forall x(raven(x) \rightarrow black(x))\), andthe main analyses of “the observation that \(a\) is a blackraven” encountered so far, that is:

1. \(k = \top\) and \(e = raven(a) \wedge black(a)\), and
2. \(k = raven(a)\) and \(e = black(a).\)

In both cases, whether \(e\) \(F\)-confirms \(h\) or not (relative to\(k\)) critically depends on \(P\): if the prior \(P(h\mid k)\) is lowenough, \(e\) won’t do no matter what under either (i) or (ii);and if it is high enough, \(h\) will be \(F\)-confirmed either way. Asa consequence, the \(F\)-confirmation view, by itself, does not offerany definite hint as to when, how, and why Nicod’s remarks applyor not.

For the purposes of our discussion, the following condition revealsanother debatable aspect of the firmness explication ofconfirmation.

Consistency condition (Cons)
For any \(h, h^*,e, k \in \bL\) and any \(P \in \bP\), if \(k \vDash\neg(h\wedge h^*)\) then \(e\) confirms \(h\) given \(k\) if and onlyif \(e\) disconfirms \(h^*\) given \(k\).

(Cons) says that evidence \(e\) can never confirm incompatiblehypotheses. But consider, by way of illustration, a clinical case ofan infectious disease of unknown origin, and suppose that \(e\) is thefailure of antibiotic treatment. Arguably, there is nothing wrong insaying that, by discrediting bacteria as possible causes, the evidenceconfirms (viz. provides some support for) any of a number ofalternative viral diagnoses. This judgment clashes with (Cons),though, which then seems an overly strong constraint.

Notably, (Cons) was defended by Hempel (1945) and, in fact, one canshow that it follows from the conjunction of (qualitative)Confirmation Complementary and the Special Consequence Condition, andso from both Hempelian and \(F\)-confirmation. This is but one sign ofhow stringent the Special Consequence Condition is. Mainly because ofthe latter, both the Hempelian and the firmness views of confirmationmust depart from the plausible HD idea that hypotheses are generallyconfirmed by their verified consequences (see Hempel 1945,103–104). We will come back to this while discussing our nexttopic: a very different Bayesian explication of confirmation, based onthe notion of probabilisticrelevance.

3.3 Probabilistic relevance confirmation

We’ve seen that the firmness notion of probabilisticconfirmation can be singled out through one ordinal constraint, (P2),in addition to the fundamental principles (P0)–(P1). Thecounterpart condition for the so-calledrelevance notion ofprobabilistic confirmation is the following:

(P3) Tautological evidence
For any \(h_{1},h_{2},k\in \bL\) and any \(P\in \bP\),\(C_{P}(h_{1},\top \mid k) = C_{P}(h_{2},\top \mid k).\)

(P3) implies that any hypothesis is equally “confirmed” byempty evidence. We will say that \(C_{P}(h, e\mid k)\) represents theprobabilistic relevance notion of confirmation, orrelevance-confirmation, if and only if it satisfies (P0), (P1) and(P3). These conditions are sufficient to derive the following, purelyqualitative principle, according to the definitional method in (QC)above (see Crupi and Tentori 2014, 82, and Crupi 2015).

The point of relevance confirmation is that the credibility of ahypothesis can bechanged in either a positive (confirmationin a strict sense) or negative way (disconfirmation) by the evidenceconcerned (given \(k\)). Confirmation (in the strict sense) thusreflects an increase from initial to final probability, whereasdisconfirmation reflects a decrease (see Achinstein 2005 for somediverging views on this very idea).

The qualitative notions of confirmation as firmness and as relevanceare demonstrably distinct. Unlike firmness, relevance confirmation cannot be formalized by the final probability alone, or any increasingfunction thereof. To illustrate, the probability of an otherwise veryrare disease \((h)\) can be quite low even after a relevant positivetest result \((e)\); yet \(h\) is relevance-confirmed by \(e\) to theextent that its probability rises thereby. By the same token, theprobability of the absence of the disease \((\neg h)\) can be quitehigh despite the positive test result \((e)\), yet \(\neg h\) isrelevance-disconfirmed by \(e\) to the extent that its probabilitydecreases thereby. Perhaps surprisingly, the distinction betweenfirmness and relevance confirmation—“extremelyfundamental” and yet “sometimes unnoticed”, asSalmon (1969, 48–49) put it—had to be stressed time andagain to achieve theoretical clarity in philosophy (e.g., Popper 1954;Peijnenburg 2012) as well as in other domains concerned, such asartificial intelligence and the psychology of reasoning (see Horvitzand Heckerman 1986; Crupi, Fitelson, and Tentori 2008; Shogenji2012).

The qualitative notion of relevance confirmation already has someinteresting consequences. It implies, for instance, the followingremarkable fact:

Complementary Evidence (CompE)
For any \(h, e, k\in \bL\) and any \(P\in \bP,\) \(e\) confirms \(h\)relative to \(k\) if and only if \(\neg e\) disconfirms \(h\) relativeto \(k.\)

The importance of (CompE) can be illustrated as follows. Consider thecase of a father suspected of abusing his child. Suppose that thechild does claim that s/he has been abused (label this evidence\(e\)). A forensic psychiatrist, when consulted, declares that thisconfirms guilt \((h)\). Alternatively, suppose that the child is askedand doesnot report having been abused \((\neg e).\) Aspointed out by Dawes (2001), it may well happen that a forensicpsychiatrist will nonetheless interpretthis as evidenceconfirming guilt (suggesting that violence has prompted thechild’s denial). One might want to argue that, other thingsbeing equal, this kind of “heads I win, tails you lose”judgment would be inconsistent, and thus in principle untenable.Whoever concurs with this line of argument (as Dawes 2001 himself did)is likely to be relying on the relevance notion of confirmation. Infact, no other notion of confirmation considered so far provides ageneral foundation for this judgment. \(F\)-confirmation, inparticular, would not do, for it does allow that both \(e\) and \(\nege\) confirm \(h\) (relative to \(k\)). This is because,mathematically, it is perfectly possible for both \(P(h\mid e \wedgek)\) and \(P(h\mid \neg e \wedge k)\) to be arbitrarily high above\(\bfrac{1}{2}.\) Condition (CompE), on the contrary, ensures thatonly one between the complementary statements \(e\) and\(\neg e\) can confirm hypothesis \(h\) (relative to \(k\)). (To beprecise, HD-confirmation also satisfies condition CompE, yet it wouldfail the above example all the same, although for a different reason,that is, because the connection between \(h\) and \(e\) is plausiblyone of probabilistic dependence but not of logical entailment.)

Remarks such as the foregoing have induced some contemporary Bayesiantheorists to dismiss the notion of confirmation as firmnessaltogether, concluding with I.J. Good (1968, 134) that “if youhad \(P(h\mid e \wedge k)\) close to unity, but less than \(P(h\midk)\), youought not to say that \(h\) was confirmed by\(e\)” (also see Salmon 1975, 13). Let us follow this suggestionand proceed to consider the ordinal (and quantitative) notions ofrelevance confirmation.

3.4 Differences, ratios, and partial entailment

Just as with firmness, the ordinal analysis of relevance confirmationcan be characterized axiomatically. With the relevance notion,however, a larger set of options arises. Consider the followingprinciples.

All the above conditions occur more or less widely in the literature(see Crupi, Chater, and Tentori 2013 and Crupi and Tentori 2016 forreferences and discussion). Interestingly, they’re all pairwiseincompatible on the background of the Formality and the FinalProbability principles (P0 and P1 above). Indeed, they sort out therelevance notion of confirmation into three distinct, classic familiesof measures, as follows (Crupi, Chater, and Tentori 2013; Crupi andTentori 2016; Heckerman 1988; Merin 2021; Sprenger and Hartmann 2020,Ch. 1):

If a strictly additive behavior (SA above) is imposed, one functionalform is singled out for the quantitative representation ofconfirmation corresponding to each of the clauses above:

1. \(D_{P}(h, e\mid k) = P(h\mid e \wedge k) - P(h\mid k);\)
2. \(R_{P}(h, e\mid k) = \log[\frac{P(h\mid e \wedge k)}{P(h\midk)}];\)
3. \(L_{P}(h, e\mid k) = \log[\frac{P(e\mid h \wedge k)}{P(e\mid \negh \wedge k)}].\)

(The bases of the logarithms are assumed to be strictly greater than1.)

Before discussing briefly this set of alternative quantitativemeasures of relevance confirmation, we will address one furtherrelated issue. It is a long-standing idea, going back to Carnap atleast, that confirmation theory should yield aninductivelogic that is analogous to classical deductive logic in somesuitable sense, thus providing a theory of partial entailment, andpartial refutation. Now, the deductive-logical notions of entailmentand refutation (contradiction) exhibit the following well-knownproperties:

The confirmation-theoretic counterparts are fairlystraightforward:

The following can then be proven (Crupi and Tentori 2013):

So, despite some pessimistic suggestions (see, e.g., Hawthorne 2018,and the discussion in Crupi and Tentori 2013), a neatconfirmation-theoretic generalization of logical entailment (andrefutation) is possible after all. Interestingly, relative distancemeasures can be additive, butonly foruniform pairsof arguments—both confirmatory or both disconfirmatory (seeMilne 2014, p. 259). (Note: Crupi, Tentori, and Gonzalez 2007; Crupi,Festa, and Buttasi 2010; and Crupi and Tentori 2013, 2014, Douven2021, and Fitelson 2021 provide further discussions of the propertiesof relative distance measures, their motivation and limitations. Alsosee Mura 2008 for a related analysis.)

The plurality of alternative probabilistic measures of relevanceconfirmation has prompted some scholars to be skeptical or dismissiveof the prospects for a quantitative theory of confirmation (see, e.g.,Howson 2000, 184–185, and Kyburg and Teng 2001, 98 ff.).However, as we will see shortly, quantitative analyses of relevanceconfirmation have proved important for handling a number of puzzlesand issues that plagued competing approaches. Moreover, variousarguments in the philosophy of science and beyond have been shown todepend critically (and sometimes unwittingly) on the choice of oneconfirmation measure (or some of them) rather than others (see Festaand Cevolani 2017, Fitelson 1999, Brössel 2013, Glass 2013, Rocheand Shogenji 2014, Rusconiet al. 2014, and van Enk2014).

Arguments have been offered by Huber (2008b) in favor of \(D\), byPark (2014), Pruss (2014), and Vassend (2015) in favor of \(L\) (alsosee Morey, Romeijn, and Rouder 2016 for an important connection withstatistics), and by Crupi and Tentori (2010) in favor of \(Z\).Hájek and Joyce (2008, 123), on the other hand, have seendifferent measures as possibly capturing “distinct,complementary notions of evidential support” (also seeSchlosshauer and Wheeler 2011, Sprenger and Hartmann 2020, Ch.1, andSteel 2007 for tempered forms of pluralism). The case of measure \(R\)deserves some more specific comments, however. Following Fitelson(2007), one could see \(R\) as conveying key tenets of so-called“likelihoodist” position about evidential reasoning (seeRoyall 1997 for a classic statement, and Chandler 2013 and Sober 1990for consonant arguments and inclinations). There seems to be someconsensus, however, that compelling objections can be raised againstthe adequacy of \(R\) as a proper measure of relevance confirmation(see, in particular, Crupi, Festa, and Buttasi 2010, 85–86;Eells and Fitelson 2002; Gillies 1986, 112; and compare Milne 1996with Milne 2010, Other Internet Resources). In what follows, too, itwill be convenient to restrict our discussion to \(D, L\) and \(Z\) ascandidate measures. All the results to be presented below areinvariant for whatever choice among these three options, and acrossordinal equivalence with each of them (but those results donot always extend to measures ordinally equivalent to\(R\)).

3.5 New evidence, old evidence, and total evidence

Let us go back to a classical HD case, where the (consistent)conjunction \(h \wedge k\) (but not \(k\) alone) entails \(e.\) Thefollowing can be proven:

Formally, it is fairly simple to show that (SP) characterizesrelevance confirmation (see, e.g., Crupi, Festa, and Buttasi 2010, 80;Hájek and Joyce 2008, 123), but the philosophical import ofthis result is nonetheless remarkable. For illustrative purposes, itis useful to assume the endorsement of the principle of total evidence(TE) as a default position for the Bayesian. This means that \(P\) isassumed to representactual degrees of belief of a rationalagent, that is, given all the background information available. Then,by clause (i) of (SP), we have that the occurrence of \(e\), aconsequence of \(h \wedge k\) (but not of \(k\) alone), confirms \(h\)relative to \(k\)provided that \(e\) was initially uncertainto some degree (even given \(k\)). In other words: \(e\)must havebeen predicted on the basis of \(h \wedge k\). Moreover, again by(i), the confirmatory impact will be stronger the more surprising(unlikely) the evidence was unless \(h\) was conjoined to \(k\). So,under TE, relevance confirmation turns out to embed a squarelypredictivist version of hypothetico-deductivism! As we know, thisneutralizes the charge of underdetermination, yet it comes at theusual cost, namely, the old evidence problem. In fact, if TE is inforce, then clause (ii) of (SP) implies that no statement that is knownto be true (thus assigned probability 1) can ever have confirmatoryimport.

Interestingly, the Bayesian predictivist has an escape (neatlyanticipated, and criticized, by Glymour 1980a, 91–92). ConsiderEinstein and Mercury once again. As effectively pointed out by Norton(2011a, 7), Einstein was extremely careful to emphasize that theprecession phenomenon had been derived “without having toposit any special [auxiliary]hypotheses atall”. Why? Well, presumably because if one had allowedherself to arbitrarily devisead hoc auxiliaries (within\(k\), in our notation) then one could have been pretty much certainin advance to find a way to get Mercury’s data right (remember:that’s the lesson of the underdetermination theorem). Butgetting those data right with auxiliaries \(k\) that were not thusadjusted—that would have been a natural consequencehadthe theory of general relativity been trueand it would have beensurprising otherwise. Arguably, this line of argument exploitsmuch of the use-novelty idea within a predictivist framework. Thecrucial points are (i) that the evidence implied is not a verifiedempirical statement \(e\) but the logical fact that \(h \wedge k\)entails \(e\), and (ii) that the existence of this connection ofentailment was not to be obviously anticipated at all, preciselybecause \(h \wedge k\) and \(e\) are such that the latter did notserve as a constraint to specify the former. On these conditions, itseems that \(h\) can be confirmed by this kind of“second-order” (logical) evidence in line with (SP)while TE is concurrently preserved.

At least two main problems arise, however. The first one is moretechnical in nature. Modelling rational uncertainty concerning logicalfacts (such as \(h \wedge k \vDash e\)) by probabilistic means is notrivial task. Garber (1983) put forward an influential proposal, butdoubts have been raised that it might not be well-behaved (e.g., vanFraassen 1988; a careful survey with further references can be foundin Eva and Hartmann 2020). Second, and more substantially, thissolution of the old evidence problem can be charged of being anelusive change of the subject: for it wasMercury’sdata, not anything else, that had to be recovered as havingconfirmed (and still confirming, some would add) Einstein’stheory. That’s the kind of judgment that confirmation theorymust capture, and which remains unattainable for the predictivistBayesian. (Earman 1992, 131, voiced this complaint forcefully. Hintsfor a possible rejoinder appear in Eells’s 1990 thoroughdiscussion; see also Skyrms 1983.)

Bayesians that are unconvinced by the predictivist position arenaturally led to dismiss TE and allow for the assignment of initialprobabilities lower than 1 even to statements that were known allalong. Of course, this brings the underdetermination problem back, fornow \(k\) can still be concoctedad hoc to have knownevidence \(e\) following from \(h \wedge k\)and moreover\(P(e\mid k)\lt 1\) is not prevented by TE anymore, thus potentiallylicencing arbitrary confirmation relations. Two moves can be combinedto handle this problem. First, unlike HD, the Bayesian framework hasthe formal resources to characterize the auxiliaries themselves asmore or less likely and thus their adoption as relatively safe orsuspicious (the standard Bayesian treatment of auxiliary hypotheses isdeveloped along these lines in Dorling 1979 and Howson and Urbach2006, 92–102, and it is critically discussed in Rowbottom 2010,Strevens 2001, and Worrall 1993; also see Christensen 1997 for animportant analysis of related issues). Second, one has to provideindications as to how TE should be relaxed. Non-TE Bayesians of theimpermissivist strand often suggest that objective likelihood valuesconcerning the outcome \(e\)—\(P(e\mid h \wedge k)\)—canbe specified for the competing hypotheses at issue quite apart fromthe fact that \(e\) may have already occurred. Such values wouldtypically be diverse for different hypotheses (thus mathematicallyimplying \(P(e\mid k)\lt 1\)) and serve as a basis to capture formallythe confirmatory impact of \(e\) (see Hawthorne 2005 and Climenhaga2024 for arguments along these lines). Permissivists, on the otherhand, can not coherently rely on these considerations to articulate anon-TE position. They must invokecounterfactual degrees ofbelief instead, suggesting that \(P\) should be reconstructed asrepresenting the beliefs that the agent would have, had she not knownthat \(e\) was true (see Howson 1991 for a statement and discussion,and Sprenger 2015 for an original recent variant; also see Jeffrey1995 and Wagner 2001 for relevant technical results, and Steele andWerndl 2013 for an intriguing case-study from climate science).

3.6 Paradoxes probabilified and other elucidations

The theory of Bayesian confirmation as relevance indicates when andwhy the HD idea works: if \(h \wedge k\) (but not \(k\)) entails\(e\), then \(h\) is relevance-confirmed by \(e\) (relative to \(k\))because the latter increases the probability of theformer—provided that \(P(e\mid k) \lt 1\). Admittedly,the meaning of the latter proviso partly depends on how one handlesthe problem of old evidence. Yet it seems legitimate to say thatBayesian relevance confirmation (unlike the firmness view)retains a key point of ordinary scientific practice which is embeddedin HD and yields further elements of clarification. Consider thefollowing illustration.

\((e_{1})\)

tigers carry the ND1 gene

\((e_{2})\)

elephants carry the ND1 gene

\((e_{2}^*)\)

lions carry the ND1 gene

\((h)\)

all mammals carry the ND1 gene

Qualitative confirmation theories comply with the idea that \(h\) isconfirmed both by \(e_{1} \wedge e_{2}\) and by \(e_{1} \wedgee_{2}^*.\) In the HD case, it is clear that \(h\) entails bothconjunctions, given of course \(k\) stating that tigers, lions, andelephants are all mammals (an Hempelian account could also be giveneasily). Bayesian relevance confirmation unequivocally yields the samequalitative verdict. There is more, however. Presumably, one mightalso want to say that \(h\) is more strongly confirmed by \(e_{1}\wedge e_{2}\) than by \(e_{1} \wedge e_{2}^*,\) because the formeroffers a more varied and diverse body of positive evidence(interestingly, on experimental investigation, this pattern prevailsin most people’s judgment, including children, see Lo et al.2002). Indeed, the variety of evidence is a fairly central issue inthe analysis of confirmation (see, e.g., Bovens and Hartmann 2002,Landes 2020, Schlosshauer and Wheeler 2011, Viale and Osherson 2000).In the illustrative case above, higher variety is readily captured bylower probability: it just seemsa priori less likely thatspecies as diverse as tigers and elephants share some unspecifiedgenetic trait as compared to tigers and lions, that is, \(P(e_{1}\wedge e_{2}\mid k)\lt P(e_{1} \wedge e_{2}^*\mid k).\) By (SP) above,then, one immediately gets from the relevance confirmation view thesound implication that \(C_{P}(h, e_{1} \wedge e_{2}\mid k)\gtC_{P}(h, e_{1} \wedge e_{2}^*\mid k).\)

Principle (SP) is also of much use in the ravens problem. Posit \(h =\forall x(raven(x)\rightarrow black(x))\) once again. Just as HD,Bayesian relevance confirmation directly implies that \(e = black(a)\)confirms \(h\) given \(k = raven(a)\) and \(e^* =\neg raven(b)\)confirms \(h\) given \(k^* =\neg black(b)\) (provided, as we know,that \(P(e\mid k)\lt 1\) and \(P(e^*\mid k^*)\lt 1).\) That’sbecause \(h \wedge k\vDash e\) and \(h \wedge k^*\vDash e^*.\) But ofcourse, to have \(h\) confirmed, sampling ravens and finding a blackone is intuitively more significant than failing to find a raven whilesampling the enormous set of the non-black objects. That is, it seems,because the latter is very likely to obtain anyway, whether or not\(h\) is true, so that \(P(e^*\mid k^*)\) is actually quite close tounity. Accordingly, (SP) implies that \(h\) is indeed more stronglyconfirmed by \(black(a)\) given \(raven(a)\) than it is by \(\negraven(b)\) given \(\neg black(b)\)—that is, \(C_{P}(h, e\midk)\gt C_{P}(h, e^*\mid k^*)\)—as long as the assumption\(P(e\mid k)\lt P(e^*\mid k^*)\) applies.

What then if the sampling in not constrained \((k = \top)\) and theevidence now amounts to the finding of a black raven, \(e = raven(a)\wedge black(a)\), versus a non-black non-raven, \(e^* =\neg black(a)\wedge \neg raven(a)\)? We’ve already seen that, for eitherHempelian or HD-confirmation, \(e\) and \(e^*\) are on a par: bothHempel-confirm \(h\), none HD-confirms it. In the former case, theoriginal Hempelian version of the ravens paradox immediately arises;in the latter, it is avoided, but at a cost: \(e\) is declared flatlyirrelevant for \(h\)—a bit of a radical move. Can the Bayesiando any better? Quite so. Consider the following conditions:

1. \(P[raven(a)\mid h] = P[raven(a)] \gt 0\)
2. \(P[\neg raven(a) \wedge black(a)\mid h] = P[\neg raven(a) \wedgeblack(a)]\)

Roughly, (i) says that the size of the ravens population does notdepend on their color (in fact, on \(h\)), and (ii) that the size ofthe population of blacknon-raven objects also does notdepend on the color of ravens. Note that both (i) and (ii) seem fairlysound as far as our best understanding of our actual world isconcerned. It is easy to show that, in relevance-confirmation terms,(i) and (ii) are sufficient to imply that \(e = raven(a) \wedgeblack(a)\), butnot \(e^* = \neg raven(a) \wedge \negblack(a)\), confirms \(h\), that is \(C_{P}(h,e) \gt C_{P}(h,e^*) =0\) (this observation is due to Mat Coakley). So the Bayesianrelevance approach to confirmation can make a principled differencebetween \(e\) and \(e^*\) in both ordinaland qualitativeterms. (A broader analysis is provided by Fitelson and Hawthorne 2010,Hawthorne and Fitelson 2010 [Other Internet Resources]. Notably, theirresults include the full specification of the sufficientandnecessary conditions for the main inequality \(C_{P}(h, e) \gtC_{P}(h, e^*)\).)

In general, Bayesian (relevance) confirmation theory implies that theevidential import of an instance of some generalization will oftendepend on the credence structure, and relies on its formalrepresentation, \(P\), as a tool for more systematic analyses.Consider another instructive example. Assume that \(a\) denotes somecompany from some (otherwise unspecified) sector of the economy, andlabel the latter predicate \(S\). So, \(k = Sa\). You are informedthat \(a\) increased revenues in 2019, represented as \(e = Ra\). Doesthis confirm \(h = \forall x(Sx \rightarrow Rx)\)? It does, at leastto some degree, one would say. For an expansion of the whole sector(recall that you have no clue what this is) surely would account forthe data. That’s a straightforward HD kind of reasoning (and asuitable Hempelian counterpart reconstruction would concur). But does\(e\) also confirm \(h^* = Sb \rightarrow Rb\) for some furthercompany \(b\)? Well, another obvious account of the data \(e\) wouldbe that company \(a\) has gained market shares at the expenses of somecompetitor, so that support from \(e\) to \(h^*,\) may appear quiteunwarranted (the revenues example is inspired by a remark in Blok,Medin, and Osherson 2007, 1362).

It can be shown that the Bayesian notion of relevance confirmationallows for this pattern of judgments, because (given \(k\)) evidence\(e\) above increases the probability of \(h\) but may well have theopposite effect on \(h^*\) (see Sober 1994 for important remarks alongsimilar lines). Notably, \(h\) entails \(h^*\) by plain instantiation,and so contradicts \(\neg h^*\). As a consequence, the implicationthat \(C_{P}(h,e\mid k)\) is positive while \(C_{P}(h^*,e\mid k)\) isnot clashes with each of the following, and proves them undulyrestrictive: the Special Consequence Condition (SCC), the PredictiveInference Condition (PIC), and the Consistency Condition (Cons). Notethat these principles were all evaded by HD-confirmation, but allimplied by confirmation as firmness (see above).

At the same time, the most compelling features of \(F\)-confirmation,which the HD model was unable to capture, are retained by confirmationas relevance. In fact, all our measures of relevance confirmation(\(D, L\), and \(Z\)) entail the ordinal extension of the EntailmentCondition (EC) as well as \(C_{P}(h, e\mid k) = -C_{P}(\neg h, e\midk)\) and thereby Confirmation Complementarity in all of its forms(qualitative, ordinal, and quantitative). Moreover, the Bayesianconfirmation theorist of either the firmness or the relevance strandcan avail herself of the same quantitative strategy of “damagecontrol” for the main specific paradox of HD-confirmation, i.e.,the irrelevant conjunction problem. (See statement (CIC) above, andCrupi and Tentori 2010, Fitelson 2002. Also see Chandler 2007 forcriticism, and Moretti 2006 for a related debate.)

We’re left with one last issue to conclude our discussion, towit, the blite paradox. Recall that \(blite\) is so defined:

As always heretofore, we assume \(h = \forall x(raven(x)\rightarrowblack(x)),\) \(h^* = \forall x(raven(x)\rightarrow blite(x)).\) Wethen consider the set up where \(k = raven(a) \wedge ex_{t\le T}(a),\)\(e= black(a),\) and \(P(e\mid k)\lt 1.\) Various authors have notedthat, with Bayesian relevance confirmation, one has that \(P(h\midk)\gt P(h^*\mid k)\) is sufficient to imply that \(C_{P}(h, e\midk)\gt C_{P}(h^*,e\mid k)\) (see Gaifman 1979, 127–128; Sober1994, 229–230; and Fitelson 2008, 131). So, as long as the blackhypothesis is perceived as initially more credible than its blitecounterpart, the former will be more strongly confirmed than thelatter. Of course, \(P(h\mid k)\gt P(h^*\mid k)\) is an entirelycommonsensical assumption, yet these same authors have generally, andquite understandably, failed to see this result as philosophicallyilluminating. Lacking some interesting, non-question-begging story asto why that inequality should obtain, no solution of the paradox seemsto emerge. More modestly, one could point out that a measure ofrelevance confirmation \(C_{P}(h, e\mid k)\) implies (i) and (ii)below.

1. Necessarily (that is, for any \(P\in \bP\)), \(e\) confirms \(h\)relative to \(k\).
2. Possibly (that is, for some \(P\in \bP\)), each one of thefollowing obtains:
   • \(e\) confirms that a raven will be black if examined after \(T\),that is, \((raven(b)\wedge \neg ex_{t\le T}(b)) \rightarrowblack(b),\) relative to \(k\);and
   • \(e\) doesnot confirm that a raven will be white ifexamined after \(T\), that is, \((raven(b)\wedge \neg ex_{t\le T}(b))\rightarrow white(b),\) relative to \(k\).

Without a doubt, (i) and (ii) fall far short of a full and satisfactorysolution of the blite paradox. Yet it seems at least a legitimateminimal requirement for a compelling solution (if any exists) that itimplies both. It is then of interest to note that confirmation asfirmness is inconsistent with (i), while Hempelian and HD-confirmationare inconsistent with (ii).

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• Hawthorne, J. and B. Fitelson, 2010,An Even Better Solution to the Paradox of the Ravens, manuscript available online (PDF).
• Milne, P., 2010,Measuring Confirmation (PDF), slides for a talk at the 7th Annual Formal EpistemologyWorkshop, Konstanz, 2–4 September 2010.

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Acknowledgments

I would like to thank Gustavo Cevolani, Paul Dicken, and Jan Sprengerfor useful comments on previous drafts of this entry, and Prof. WonbaeChoi for helping me correcting a mistake.