Confirmation A4

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Confirmation
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Stanford Encyclopedia Human cognition and behavior heavily relies on the notion that evidence
(data, premises) can affect the credibility of hypotheses (theories,
of Philosophy conclusions). This general idea seems to underlie sound and effective
inferential practices in all sorts of domains, from everyday reasoning up to
the frontiers of science. Yet it is also clear that, even with extensive and
truthful evidence available, drawing a mistaken conclusion is more than a
mere possibility. For painfully concrete examples, one only has to
consider missed medical diagnoses (see Winters et al. 2012) or judicial
Edward N. Zalta Uri Nodelman Colin Allen R. Lanier Anderson
errors (see Liebman et al. 2000). The Scottish philosopher David Hume
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(1711–1776) is usually credited for having disclosed the theoretical roots
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arguably, Hume’s line of thought cuts deeper than this: see Howson 2000;
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also see Lange 2011 and Varzi 2008). In most cases of interest, Hume
pointed out, many alternative candidate hypotheses remain logically
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infirming) hypotheses has to be grasped by more nuanced tools than plain
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Confirmation
Copyright c 2015 by the author inductive logic, they are endorsed (see Meehl 1990, 110). In contemporary
Vincenzo Crupi philosophy, confirmation theory can be roughly described as the area
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1
Confirmation Vincenzo Crupi
plausible models of non-deductive reasoning. Its central technical term 1.2 Two paradoxes and other difficulties
—confirmation—has often been used more or less interchangeably with 2. Hypothetico-deductivism
“evidential support”, “inductive strength”, and the like. Here we will 2.1 HD vs. Hempelian confirmation
generally comply with this liberal usage, although more subtle conceptual 2.2 Back to black (ravens)
and terminological distinctions could be usefully drawn. 2.3 Underdetermination and the Duhemian challenge
2.4 The extended HD menu
Confirmation theory has proven a rather difficult endeavour. In principle, 3. Bayesian confirmation theories
it would aim at providing understanding and guidance for tasks such as 3.1 Probabilistic confirmation as firmness
diagnosis, prediction, and learning in virtually any area of inquiry. Yet 3.2 Strengths and infirmities of firmness
popular accounts of confirmation have often been taken to run into 3.3 Probabilistic relevance confirmation
troubles even when faced with toy philosophical examples. Be that as it 3.4 Differences, ratios, and partial entailment
may, there is at least one real-world kind of activity which has remained a 3.5 New evidence, old evidence, and total evidence
prevalent target and benchmark, i.e., scientific reasoning, and especially 3.6 Paradoxes probabilified and other elucidations
key episodes from the history of modern and contemporary natural Bibliography
science. The motivation for this is easily figured out. Mature sciences Academic Tools
seem to have been uniquely effective in relying on observed evidence to Other Internet Resources
establish extremely general, powerful and sophisticated theories. Indeed, Related Entries
being capable of receiving genuine support from empirical evidence is
itself a very distinctive trait of scientific hypotheses as compared to other
kinds of statements. A philosophical characterization of what science is 1. Confirmation by instances
would then seem to require an understanding of the logic of confirmation.
And so, traditionally, confirmation theory has come to be a central In a seminal essay on induction, Jean Nicod (1924) offered the following
concern of philosophers of science. important remark:
In the following, major approaches to confirmation theory are overviewed Consider the formula or the law: F entails G. How can a particular
according to a classification that is relatively standard (see Earman and proposition, or more briefly, a fact affect its probability? If this fact
Salmon 1992; Norton 2005): confirmation by instances (Section 1), consists of the presence of G in a case of F, it is favourable to the
hypothetico-deductivism and its variants (Section 2), and probabilistic law […]; on the contrary, if it consists of the absence of G in a
(Bayesian) approaches (Section 3). case of F, it is unfavourable to this law. (219, notation slightly
adapted)
1. Confirmation by instances
1.1 Hempel’s theory Nicod’s work was an influential source for Carl Gustav Hempel’s (1943,
2 Stanford Encyclopedia of Philosophy Spring 2015 Edition 3

1945) early studies in the logic of confirmation. In Hempel’s view, the key so on. Following Hempel, we will take universally quantified material
valid message of Nicod’s statement is that the observation report that an conditionals as canonical logical representations of relevant hypotheses.
object a displays properties F and G (e.g., that a is a swan and is white) So, for instance, we will count a statement of the form ∀x(Fx → Gx) as an
confirms the universal hypothesis that all F-objects are G-objects (namely, adequate rendition of, say, “all pieces of copper conduct electricity”.
that all swans are white). Apparently, it is by means of this kind of
confirmation by instances that one can obtain supporting evidence for In Hempel’s theory, evidence statement e is said to confirm hypothesis h
statements such as “sodium salts burn yellow”, “wolves live in a pack”, or just in case it entails, not h in its full extension, but suitable instantiations
“planets move in elliptical orbits” (also see Russell 1912, Ch. 6). We will of h. The technical notion of the e-development of h is devised to identify
now see the essential features of Hempel’s analysis of confirmation. precisely those relevant instantiations, that is, the consequences of h as
restricted to the individuals involved in e. More precisely, Hempelian
1.1 Hempel’s theory confirmation can be defined as follows:
Let L be the set of the closed sentences of a first-order logical language L Hempelian confirmation
(finite, for simplicity) and consider h, e ∈ L . Also let e, the evidence For any h, e ∈ L such that e is consistent and contains individual
statement, be consistent and contain individual constants only (no constants only (no quantifier):
quantifier), and let I(e) be the set of all constants occurring (non-
i. evidence e directly Hempel-confirms hypothesis h if and only if
vacuously) in e. So, for example, if e = Qa ∧ Ra, then I(e) = {a} , and if
e ⊨ deve (h) ; e Hempel-confirms h if and only if, for some s ∈ L,
e = Qa ∧ Qb , then I(e) = {a, b} . (The non-vacuity clause is meant to
e ⊨ deve (s) and s ⊨ h ;
ensure that if sentence e happens to be, say, Qa ∧ Qb ∧ (Rc ∨ ¬R), then
ii. evidence e directly Hempel-disconfirms hypothesis h if and only
I(e) still is {a, b} , for e does not really state anything non-trivial about the
if e ⊨ deve (¬h) ; e Hempel-disconfirms h if and only if, for some
individual denoted by c. See Sprenger 2011a, 241–242.) Hempel’s theory
s ∈ L, e ⊨ deve (s) and s ⊨ ¬h;
relies on the technical construct of the development of hypothesis h for
iii. evidence e is Hempel-neutral for hypothesis h otherwise.
evidence e, or the e-development of h, indicated by deve (h) . Intuitively,
deve (h) is all that (and only what) h says once restricted to the individuals In each of clauses (i) and (ii), Hempelian confirmation (disconfirmation,
mentioned (non-vacuously) in e, i.e., exactly those denoted by the respectively) is a generalization of direct Hempelian confirmation
elements of I(e). (disconfirmation). To retrieve the latter as a special case of the former, one
only has to posit s = h (¬h), respectively, for disconfirmation).
The notion of the e-development of hypothesis h can be given an entirely
general and rigorous definition, but we’ll not need this level of detail here. By direct Hempelian confirmation, evidence statement e that, say, object a
Suffice it to say that the e-development of a universally quantified is a white swan, swan(a) ∧ white(a), confirms hypothesis h that all swans
material conditional ∀x(Fx → Gx) is just as expected, that is: Fa → Ga are white, ∀x(swan(x) → white(x)), because the former entails the e-
in case I(e) = {a} ; (Fa → Ga) ∧ (Fb → Gb) in case I(e) = {a, b} , and development of the latter, that is, swan(a) → white(a). This is a desired

result, according to Hempel’s reading of Nicod. By (indirect) Hempelian (h) ∀x(raven(x) → black(x)), i.e., all ravens are black;
confirmation, moreover, swan(a) ∧ white(a) also confirms that a
(e) raven(a) ∧ black(a), i.e., a is a black raven;
particular further object b will be white, if it’s a swan, i.e.,
(e∗ ) ¬black(a∗ ) ∧ ¬raven(a∗ ) , i.e., a∗ is a non-black non-raven (say, a
swan(b) → white(b) (to see this, just set s = ∀x(swan(x → white(x))). green apple).
The second possibility considered by Nicod (“the absence of G in a case
Is hypothesis h confirmed by e and e∗ alike? One would want to say no,
of F ”) can be accounted for by Hempelian disconfirmation. For the
but Hempel’s theory is unable to draw this distinction. Let’s see why.
evidence statement e that a is a non-white swan—swan(a) ∧ ¬white(a)—
entails (in fact, is identical to) the e-development of the hypothesis that As we know, e (directly) Hempel-confirms h, according to Hempel’s
there exist non-white swans—∃x(swan(x) ∧ ¬white(x))—which in turn is reconstruction of Nicod. By the same token, e∗ (directly) Hempel-
just the negation of ∀x(swan(x) → white(x)). So the latter is disconfirmed confirms the hypothesis that all non-black objects are non-ravens, i.e.,
by the evidence in this case. And finally, e = swan(a) ∧ ¬white(a) also h∗ = ∀x(¬black(x) → ¬raven(x)) . But h∗ ⊨ h (h and h∗ are just logically
Hempel-disconfirms that a particular further object b will be white, if it’s a equivalent). So, e∗ (the observation report of a non-black non-raven), like
swan, i.e., swan(b) → white(b), because the negation of the latter, e (black raven), does (indirectly) Hempel-confirm h (all ravens are black).
swan(b) ∧ ¬white(b), is entailed by s = ∀x(swan(x) → ¬white(x)) and Indeed, as ¬raven(a) entails raven(a) → black(a), it can be shown that h
e ⊨ deve (s) . is (directly) Hempel-confirmed by the observation of any object that is not
a raven (an apple, a cat, a shoe, or whatever), apparently disclosing
So, to sum up, we have four illustrations of how Hempel’s theory
puzzling “prospects for indoor ornithology” (Goodman 1955, 71).
articulates Nicod’s basic ideas, to wit:
Blite (Goodman 1955). Consider the peculiar predicate “blite”, defined as
(the observation report of) a white swan (directly) Hempel-confirms
follows: an object is blite just in case (i) it is black if examined at some
that all swans are white;
moment t up to some future time T (say, the next expected appearance of
(the observation report of) a white swan also Hempel-confirms that a
Halley’s comet, in 2061) and (ii) it is white if examined afterwards. So we
further swan will be white;
posit blite(x) ≡ (ext≤T (x) → black(x)) ∧ (¬ext≤T (x) → white(x)) . Now
(the observation report of) a non-white swan (directly) Hempel-
consider the following statements:
disconfirms that all swans are white;
(the observation report of) a non-white swan also Hempel- (h) ∀x(raven(x) → black(x)), i.e., all ravens are black;
disconfirms that a further swan will be white. (h∗ ) ∀x(raven(x) → blite(x)) , i.e., all ravens are blite;
(e) e = raven(a) ∧ ext≤T (a) ∧ black(a) , i.e., a is a raven observed no
1.2 Two paradoxes and other difficulties later than T and it is black.
The ravens paradox (Hempel 1937, 1945). Consider the following Does e confirm hypotheses h and h∗ alike? Here again, one would want to
statements:

say no, but Hempel’s theory is unable to draw the distinction. For one can Salmon 1992, 54, and Sprenger 2011a, 243, for more on this). Yet the
check that the e-developments of h and h∗ are both entailed by e. Thus, e Hempelian notion of confirmation turns out to be very restrictive, too, on
(the report of a raven examined no later than T and found to be black) other accounts. For suppose that hypothesis h and evidence e do not share
does Hempel-confirm h∗ (all ravens are blite) just as it confirms h (all any piece of non-logical vocabulary. h might be, say, Newton’s law of
ravens are black). Moreover, e also Hempel-confirms the statement that a universal gravitation (connecting force, distances and masses), while e
raven will be white if examined after T , because this is a logical could be the description of certain spots on a telescopic image.
consequence of h∗ (which is directly Hempel-confirmed by e). And Throughout modern physics, significant relations of confirmation and
finally, suppose disconfirmation were taken to obtain between statements like these.
Indeed, telescopic sightings have been crucial evidence for Newton’s law
blurple(x) ≡ (ext≤T (x) → black(x)) ∧ (¬ext≤T (x) → purple(x)). as applied to celestial bodies. However, as their non-logical vocabularies
are disjoint, e and h must simply be logically independent, and so must be
We then have that the very same evidence statement e Hempel-confirms
e and deve (h) (with very minor caveats, this follows from Craig’s so-
the hypothesis that all ravens are blurple, and thus also its implication that
called interpolation theorem, see Craig 1957). In such circumstances, there
a raven will be purple if examined after T !
can be nothing but Hempel-neutrality between evidence and hypothesis.
A seemingly obvious idea, here, is that there must be something inherently So Hempel’s original theory seems to lack the resources to capture a key
wrong with predicates such as blite or blurple (and perhaps non-raven and feature of inductive inference in science as well as in several other
non-black, too) and thus a principled way to rule them out as “unnatural”. domains, i.e., the kind of “vertical” relationships of confirmation (and
Then one could restrict confirmation theory accordingly, i.e., to “natural disconfirmation) between the description of observed phenomena and
kinds” only (see, e.g., Quine 1970). Yet this point turns out be very hypotheses concerning underlying structures, causes and processes.
difficult to pursue coherently and it has not borne much fruit in this
To overcome the latter difficulty, Clark Glymour (1980a) embedded a
discussion. After all, for all we know, it is a perfectly “natural” feature of
refined version of Hempelian confirmation by instances in his analysis of
a token of the “natural kind” water that it is found in one physical state for
scientific reasoning. In Glymour’s revision, hypothesis h is confirmed by
temperatures below 0 degrees Celsius and in an entirely different state for
some evidence e even if appropriate auxiliary hypotheses and assumptions
temperatures above that threshold. So why should the time threshold T in
must be involved for e to entail the relevant instances of h. This important
blite or blurple be a reason to dismiss those predicates? (The water
theoretical move turns confirmation into a three-place relation concerning
example comes from Howson 2000, 31–32. See Schwartz 2011, 399 ff.,
the evidence, the target hypothesis, and (a conjunction of) auxiliaries.
for a more general assessment of this issue.)
Originally, Glymour presented his sophisticated neo-Hempelian approach
The above, widely known “paradoxes” then suggest that Hempel’s in stark contrast with the competing traditional view of so-called
analysis of confirmation is too liberal: it sanctions the existence of hypothetico-deductivism (HD). Despite his explicit intentions, however,
confirmation relations that are intuitively very unsound (see Earman and several commentators have pointed out that, partly because of the due
recognition of the role of auxiliary assumptions, Glymour’s proposal and

HD end up being plagued by similar difficulties (see, e.g., Horwich 1983, therein).
Woodward 1983, and Worrall 1982). In the next section, we will discuss
the HD framework for confirmation and also compare it with Hempelian 2. Hypothetico-deductivism
confirmation. It will thus be convenient to have a suitable extended
definition of the latter, following the remarks above. Here is one that The central idea of hypothetico-deductive (HD) confirmation can be
serves our purposes: roughly described as “deduction-in-reverse”: evidence is said to confirm a
hypothesis in case the latter, while not entailed by the former, is able to
Hempelian confirmation (extended) entail it, with the help of suitable auxiliary hypotheses and assumptions.
For any h, e, k ∈ L such that e contains individual constants only (no The basic version (sometimes labelled “naïve”) of the HD notion of
quantifier), k = deve (α) for some α ∈ L containing quantifiers only confirmation can be spelled out thus:
(no individual constant) and such that α ⊭ h, and e ∧ k is consistent:
HD-confirmation
i. e directly Hempel-confirms h relative to k if and only if For any h, e, k ∈ L such that h ∧ k is consistent:
e ∧ k ⊨ deve (h) ; e Hempel-confirms h relative to k if and only if,
for some s ∈ L, e ∧ k ⊨ deve (s) and s ∧ k ⊨ h ; i. e HD-confirms h relative to k if and only if h ∧ k ⊨ e and k ⊭ e ;
ii. e directly Hempel-disconfirms h relative to k if and only if ii. e HD-disconfirms h relative to k if and only if h ∧ k ⊨ ¬e , and
e ∧ k ⊨ deve (¬h) ; e Hempel-disconfirms h relative to k if and k ⊭ ¬e ;
only if, for some s ∈ L, e ∧ k ⊨ deve (s) a and s ∧ k ⊨ ¬h ; iii. e is HD-neutral for hypothesis h relative to k otherwise.
iii. e is Hempel-neutral for h relative to k otherwise.
Note that clause (ii) above represents HD-disconfirmation as plain logical
One can see that in the above definition the auxiliary assumptions in k are inconsistency of the target hypothesis with the data (given the auxiliaries)
the e-development of further closed constant-free hypotheses (in fact, (see Hempel 1945, 98).
equations as applied to specific measured values, in typical examples from
Glymour 1980a), where such hypotheses are meant to be conjoined in a
2.1 HD vs. Hempelian confirmation
single statement (α) for convenience. This implies that the only terms HD-confirmation and Hempelian confirmation convey different intuitions
occurring (non-vacuously) in k are individual constants already occurring (see Huber 2008a for an original analysis). They are, in fact, distinct and
(non-vacuously) in e. For an empty α (that is, tautologous: α = ⊤ ), k must strictly incompatible notions. This will be effectively illustrated by the
be empty too, and the original (restricted) definition of Hempelian consideration of the following conditions.
confirmation applies. As for the proviso that α ⊭ h, it rules out undesired
cases of circularity—akin to so-called “macho” bootstrap confirmation, as Entailment condition (EC)
discussed in Earman and Glymour 1988 (for more on Glymour’s theory For any h, e, k ∈ L , if e ∧ k is consistent, e ∧ k ⊨ h and k ⊭ h , then e
and its developments, see Douven and Meijs 2006, and references confirms h relative to k.

Confirmation complementarity (CC) Evidence e verifies h conclusively, and yet it does not HD-confirm it,
For any h, e, k ∈ L , e confirms h relative to k if and only if e simply because h ⊭ e. So the observation of a black swan turns out to be
disconfirms ¬h relative to k . HD-neutral for the hypothesis that black swans exist! The same example
shows how HD-confirmation violates (CC), too. In fact, while HD-neutral
Special consequence condition (SCC) for h, e HD-disconfirms its negation ¬h that no swan is black,
For any h, e, k ∈ L , if e confirms h relative to k and h ∧ k ⊨ h∗ , then ∀x(swan(x) → ¬black(x)), because the latter is obviously inconsistent
e confirms h∗ relative to k. with (refuted by) e.
On the implicit proviso that k is empty (that is, tautologous: k = ⊤ ), The violation of (EC) and (CC) by HD-confirmation is arguably a reason
Hempel (1943, 1945) himself had put forward (EC) and (SCC) as for concern, for those conditions seem highly plausible. The special
compelling adequacy conditions for any theory of confirmation, and consequence condition (SCC), on the other hand, deserves separate and
devised his own proposal accordingly. As for (CC), he took it as a plain careful consideration. As we will see later on, (SCC) is a strong constraint,
definitional truth (1943, 127). Moreover, Hempelian confirmation and far from sacrosanct. For now, let us point out one major philosophical
(extended) satisfies all conditions above (of course, for arguments h, e and motivation in its favor. (SCC) has often been invoked as a means to ensure
k for which it is defined). HD-confirmation, on the contrary, violates all of the fulfilment of the following condition (see, e.g., Hesse 1975, 88;
them. Let us briefly discuss each one in turn. Horwich 1983, 57):
It is rather common for a theory of ampliative (non-deductive) reasoning Predictive inference condition (PIC)
to retain classical logical entailment as a special case, that is, to be “super- For any e, k ∈ L , if e confirms ∀x(Fx → Gx) relative to k , then e
classical” (see, e.g., Antonelli 2012, for this terminology). That’s confirms F(a) → G(a) relative to k.
essentially what (EC) implies for confirmation. Now given appropriate e,
h and k, if e ∧ k entails h, we readily get that e Hempel-confirms h relative In fact, (PIC) readily follows from (SCC) and so it is satisfied by
to k in two simple steps. First, given that e and k are both quantifier-free, Hempel’s theory. It says that, if evidence e confirms “all Fs are Gs”, then
deve (e ∧ k) = e ∧ k according to Hempel’s full definition of dev (see it also confirms that a further object will be G, if it is F. Notably, this does
Hempel 1943, 131). Then it trivially follows that e ∧ k ⊨ deve (e ∧ k) , so not hold for HD-confirmation. Here is why. Given k = Fa (i.e., the
e ∧ k is (directly) Hempel-confirmed and its logical consequence h is assumption that a comes from the F population), we have that e = Ga
likewise confirmed (indirectly). Logical entailment is thus retained as an HD-confirms h = ∀x(Fx → Gx) , because the latter entails the former
instance of Hempelian confirmation in a fairly straightforward way. HD- (given k ). (That’s the HD reconstruction of Nicod’s insight, see below.)
confirmation, on the contrary, does not fulfil (EC). Here is one odd We also have, of course, that h entails h∗ = Fb → Gb . And yet, contrary
example (see Sprenger 2011a, 234). With k = ⊤, just let e be the to (PIC), since h∗ does not entail e (given k), it is not HD-confirmed by it
observation report that object a is a black swan, swan(a) ∧ black(a), and either. The troubling conclusion is that the observation that a swan is
h be the hypothesis that black swans exist, ∃x(swan(x) ∧ black(x)). white (or that a million of them are, for that matters) does not HD-confirm

the prediction that a further swan will be found to be white. Interestingly, the introduction of auxiliary hypotheses and assumptions
shows that the issues surrounding Nicod’s remarks can become
2.2 Back to black (ravens) surprisingly subtle. Consider the following statements (Maher’s 2006
example):
One attractive feature of HD-confirmation is that it fares well enough with
the ravens paradox. As the hypothesis h that all ravens are black does not (α1 ) ∀x(white(x) → ¬black(x))
entail that some generally sampled object a will be a black raven, the HD (α2 ) ∃x(swan(x)) → ∃y(swan(y) ∧ black(y))
view of confirmation is not committed to the eminently Hempelian
implication that e = raven(a) ∧ black(a) confirms h. Likewise, α1 simply specifies that no object is both white and black, while α2 says
¬black(a) ∧ ¬raven(a) does not HD-confirm that all non-black objects that, if there are swans at all, then there also is some black swan. Also
are non-raven. The derivation of the paradox, as presented above, is thus posit, again, e = swan(a) ∧ white(a) . Under α1 and α2 , the observation of
blocked. a white swan clearly disconfirms (indeed, refutes) the hypothesis h that all
swans are white. Hempel’s theory (extended) faces difficulties here,
Indeed, HD-confirmation yields a substantially different reading of because for k = deve (α1 ∧ α2 ) it turns out that e ∧ k is inconsistent. But
Nicod’s insight as compared to Hempel’s theory (Okasha 2011 has an HD-confirmation gets this case right, thus capturing appropriate boundary
important discussion of this distinction). Here is how it goes. If object a is conditions for Nicod’s generally sensible claims. For, with k = α1 ∧ α2 ,
assumed to have been taken among ravens—so that, crucially, the one has that h ∧ k is consistent and entails ¬e (for it entails that no swan
auxiliary assumption k = raven(a) is made—and a is checked for color exists), so that e HD-disconfirms (refutes) h relative to k .
and found to be black, then, yes, the latter evidence, black(a), HD-
confirms that all ravens are black (h) relative to k. By the same token, HD-confirmation, however, is also known to suffer from distinctive
¬black(a) HD-disconfirms h relative to the same assumption “paradoxical” implications. One of the most frustrating is surely the
k = raven(a) . And, again, this is as it should be, in line with Nicod’s following (see Osherson, Smith, and Shafir 1986, 206, for further specific
mention of “the absence of G [here, non-black as evidence] in a case of F problems).
[here, raven as an auxiliary assumption]”. It is also true that an object that
The irrelevant conjunction paradox. Suppose that e confirms h relative to
is found not to be a raven HD-confirms h, but only relative to
(possibly empty) k . Let statement q be logically consistent with e ∧ h ∧ k,
k = ¬black(a) , that is, if a is assumed to have been taken among non-
but otherwise entirely irrelevant for all of those conjuncts. Does e confirm
black objects to begin with; and this seems acceptable too (after all, while
h ∧ q (relative to k ) as it does with h? One would want to say no, and this
sampling from non-black objects, one might have found the
implication can be suitably reconstructed in Hempel’s theory. HD-
counterinstance of a raven, but didn’t). Unlike Hempel’s theory,
confirmation, on the contrary, can not draw this distinction: it is easy to
moreover, HD-confirmation does not yield the debatable implication that,
show that, on the conditions specified, if the HD clause for confirmation is
by itself (that is, given k = ⊤ ), the observation of a non-raven a,
satisfied for e and h (given k ), so it is for e and h ∧ q (given k ). (This is
¬raven(a), must confirm h.

simply because, if h ∧ k ⊨ e , then h ∧ q ∧ k ⊨ e , too, by the monotonicity 2.3 Underdetermination and the Duhemian challenge
of classical logical entailment.)
The issues above look contrived and artificial to some people’s taste—
Kuipers (2000, 25) suggested that one can live with the irrelevant even among philosophers. Many have suggested a closer look at real-
conjunction problem because, on the conditions specified, e would still not world inferential practices in the sciences as a more appropriate
HD-confirm q alone (given k ), so that HD-confirmation can be benchmark for assessment. For one thing, the very idea of hypothetico-
“localized”: h is the only bit of the conjunction h ∧ q that gets any deductivism has often been said to stem from the origins of Western
confirmation on its own, as it were. Other authors have been reluctant to science. As reported by Simplicius of Cilicia (sixth century A.D.) in his
bite the bullet and have engaged in technical refinements of the “naïve” commentary on Aristotle’s De Caelo, Plato had challenged his pupils to
HD view. In these proposals, the spread of HD-confirmation upon identify combinations of “ordered” motions by which one could account
frivolous conjunctions can be blocked at the expense of some additional for (namely, deduce) the planets’ wandering trajectories across the
logical machinery (see Gemes 1993, 1998; Schurz 1991, 1994). heavens as observed by the Earth. As a matter of historical fact,
mathematical astronomy has engaged in just this task for centuries:
Finally, it should be noted that HD-confirmation offers no substantial scholars have been trying to define geometrical models from which the
relief from the blite paradox. On the one hand, apparent motion of celestial bodies would derive.
e = raven(a) ∧ ext≤T (a) ∧ black(a) does not, as such, HD-confirm either
h = ∀x(raven(x) → black(x)) or h∗ = ∀x(raven(x) → blite(x)) , that is, It is fair to say that, at its roots, the kind of challenges that the HD
for empty k . On the other hand, if object a is assumed to have been framework faces with scientific reasoning is not so different from the main
sampled from ravens before T (that is, given k = raven(a) ∧ ext≤T (a)) , puzzles that arise from philosophical considerations of a more formal
then black(a) is entailed by both “all ravens are black” and “all ravens are kind. Still, the two areas turn out to be complementary in important ways.
blite” and therefore HD-confirms each of them. So HD-confirmation, too, The following statement will serve as a useful starting point to extend the
sanctions the existence of confirmation relations that seem intuitively scope of our discussion.
unsound (indeed, indefinitely many of them: as we know, other variations
of h∗ can be conceived at will). One could insist that HD does handle the The underdetermination theorem (UT) for “naïve” HD-
blite paradox after all, because black(a) (given k as above) does not HD- confirmation
confirms that a raven will be white if examined after T (Kuipers 2000, 29 For any contingent h, e ∈ L , if h and e are logically consistent, there
ff.). Unfortunately (as pointed out by Schurz 2005, 148) black(a) does not exists some k ∈ L such that e HD-confirms h relative to k.
HD-confirm that a raven will be black if examined after T either (again,
(UT) is an elementary logical fact that has been long recognized (see, e.g.,
given k as above). That’s because, as already pointed out, HD-
Glymour 1980a, 36). On a purely formal side, just positing k = h → e
confirmation fails the predictive inference condition (PIC) in general. So,
will do for a proof. To appreciate how (UT) can spark any philosophical
all in all, HD-confirmation can not tell black from blite any more than
interest, one has to combine it with some insightful remarks first put
Hempel-confirmation can.

forward by Pierre Duhem (1906) and then famously revived by Quine impressive list of important empirical facts from their main hypothesis
(1951) in a more radical style. (Indeed, (UT) essentially amounts to the along with appropriate auxiliaries, diffraction phenomena being only one
“entailment version” of “Quinean underdetermination” in Laudan 1990, major example. But many particle theorists’ reaction was to retain their
274.) hypothesis nonetheless and to reshape other parts of the “theoretical maze”
(i.e., k ; the term is Popper’s, 1963, p. 330) to recover those observed facts
Duhem (he himself a supporter of the HD view) pointed out that in mature as consequences of their own proposal. And as we’ve seen, if the bare
sciences such as physics most hypotheses or theories of real interest can logic of naïve HD was to be taken strictly, surely they could have claimed
not be contradicted by any statement describing observable states of their overall hypothesis to be confirmed too, just as much as their
affairs. Taken in isolation, they simply do not logically imply, nor rule out, opponents.
any observable fact, essentially because (unlike “all ravens are black”)
they involve the mention of unobservable entities and processes. So, in Importantly, they didn’t. In fact, it was quite clear that particle theorists,
effect, Duhem emphasized that, typically, scientific hypotheses or theories unlike their wave-theory opponents, were striving to remedy weaknesses
are logically consistent with any piece of checkable evidence. Unless, of rather than scoring successes (see Worrall 1990). But why, then? Because,
course, the logical connection is underpinned by auxiliary hypotheses and as Duhem himself clearly realized, the logic of naïve HD “is not the only
assumptions suitably bridging the gap between the observational and non- rule for our judgments” (1906, 217). The lesson of (UT) and the
observational vocabulary, as it were. But then, once auxiliaries are in play, Duhemian insight is not quite, it seems, that naïve HD is the last word and
logic alone guarantees that some k exists such that h ∧ k is consistent, scientific inference is unconstrained by stringent rational principles, but
h ∧ k ⊨ e , and k ⊭ e , so that confirmation holds in naïve HD terms (that’s rather that the HD view has to be strengthened in order to capture the real
just the UT result above). Apparently, when Duhem’s point applies, the nature of evidential support in rational scientific inference. At least, that’s
uncritical supporter of whatever hypothesis h can legitimately claim (naïve the position of a good deal of philosophers of science working within the
HD) confirmation from any e by simply shaping k conveniently. In this HD framework broadly construed. It has even been maintained that “no
sense, hypothesis assessment would be radically “underdetermined” by serious twentieth-century methodologist” has ever subscribed to the naïve
any amount of evidence practically available. HD view above “without crucial qualifications” (Laudan 1990, 278; also
see Laudan and Leplin 1991, 466).
Influential authors such as Thomas Kuhn (1962/1970) (but see Laudan
1990, 268, for a more extensive survey) relied on Duhemian insights to So the HD approach to confirmation has yielded a number of more
suggest that confirmation by empirical evidence is too weak a force to articulated variants to meet the challenge of underdetermination.
drive the evaluation of theories in science, often inviting conclusions of a Following (loosely) Norton (2005), we will now survey an instructive
relativistic flavor (see Worrall 1996 for an illuminating reconstruction sample of them.
along these lines). Let us briefly consider a classical case, which Duhem
himself thoroughly analyzed: the wave vs. particle theories of light in 2.4 The extended HD menu
modern optics. Across the decades, wave theorists were able to deduce an

Naïve HD can be enriched by a resolute form of predictivism. According Douglas and Magnus 2013 offer more recent views and rich lists of further
to this approach, the naïve HD clause for confirmation is too weak references.)
because e must have been predicted in advance from h ∧ k. Karl Popper’s
(1934/1959) account of the “corroboration” of hypotheses famously As a possible response to the difficulties above, naïve HD can be enriched
embedded this view, but squarely predictivist stances can be traced back to by the use-novelty criterion (UN) instead. The UN reaction to the
early modern thinkers like Christiaan Huygens (1629–1695) and Gottfried underdetermination problem is more conservative than the temporal
Wilhelm Leibniz (1646–1716), and in Duhem’s work itself. The predictivist strategy. According to this view, to improve on the weak naïve
predictivist sets a high bar for confirmation. Her favorite examples HD clause for confirmation one only has to rule out one particular class of
typically include stunning episodes in which the existence of previously cases, i.e., those in which the description of a known fact, e, served as a
unknown objects, phenomena, or whole classes of them is anticipated: the constraint in the construction of h ∧ k. The UN view thus comes equipped
phases of Venus for Copernican astronomy or the discovery of Neptune with a rationale. If h ∧ k was shaped on the basis of e, UN advocates point
for Newtonian physics, all the way up to the Higgs boson for so-called out, then it was bound to get that state of affairs right; the theory never ran
standard model of subatomic particles. any risk of failure, thus did not achieve any particularly significant success
either. Precisely in these cases, and just for this reason, the evidence e
The predictivist solution to the underdetermination problem is fairly must not be double-counted: by using it for the construction of the theory,
radical: many of the relevant factual consequences of h ∧ k will be already its confirmational power becomes “dried out”, so to speak.
known when this theory is articulated, and so unfit for confirmation.
Critics have objected that predictivism is in fact far too restrictive. There The UN completion of naïve HD originated from Lakatos and some of his
seem to be many cases in which already known phenomena clearly do collaborators (see Lakatos and Zahar 1975 and Worrall 1978; also see
provide support to a new hypothesis or theory. Zahar (1973) first raised Giere 1979, 161–162, and Gillies 1989 for similar views), although
this problem of “old evidence”, then made famous by Glymour (1980a, 85 important hints in the same direction can be found at least in the work of
ff.) as a difficulty for Bayesianism (see Section 3 below). Examples of this William Whewell (1840/1847). Consider the touchstone example of
kind abound in the history of science as elsewhere, but the textbook Mercury again. According to Zahar (1973), Einstein did not need to rely
illustration has become the precession of Mercury’s perihelion, a lasting on the Mercury data to define theory and auxiliaries as to match
anomaly for Newtonian physics: Einstein’s general relativity calculations observationally correct values for the perihelion precession (also see
got this long-known fact right, thereby gaining a remarkable piece of Norton 2011a; and Earman and Janssen 1993 for a very detailed, and more
initial support for the new theory. In addition to this problem with old nuanced, account). Being already known, the fact was not of course
evidence, HD predictivism also seems to lack a principled rationale. After predicted in a strictly temporal sense, and yet, on Zahar’s reading, it could
all, the temporal order of the discovery of e and of the articulation of h and have been: it was “use-novel” and thus fresh for use to confirm the theory.
k may well be an entirely accidental historical contingency. Why should it For a more mundane illustration, so-called cross-validation techniques
bear on the confirmation relationship among them? (See Giere 1983 and represent a routine application of the UN idea in statistical settings (as
Musgrave 1974 for classical discussions of these issues. Harker 2008 and pointed out by Schurz 2014, 92; also see Forster 2007, 592 ff.). According

to some commentators, however, the UN criterion needs further retain when inference takes place at higher levels of generality and
elaboration (see Hitchcock and Sober 2004 and Lipton 2005), while others theoretical commitment, where the hypothesis space is typically much too
have criticized it as essentially wrong-headed (see Howson 1990 and poorly ordered to fit routine error-statistical analyses. Indeed, Laudan
Mayo 1991, 2014; also see Votsis 2014). (1997, 315; also see Musgrave 2010) spotted in this approach the risk of a
“balkanization” of scientific reasoning, namely, a restricted focus on
Yet another way to enrich naïve HD is to combine it with eliminativism. scattered pieces of experimental inference (but see Mayo 2010 for a
According to this view, the naïve HD clause for confirmation is too weak defense).
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 Naïve HD can also be enriched by the notion of simplicity. According to
that e may have occurred for some reason other than the truth of h. Briefly this view, the naïve HD clause for confirmation is too weak because h ∧ k
put, the occurrence of e must be such that most alternatives to h can be must be a simple (enough), unified way to account for evidence e. A
safely ruled out. The founding figure of eliminativism is Francis Bacon classic reference for the simplicity view is Newton’s first law of
(1561–1626). John Stuart Mill (1843/1872) is a major representative in philosophizing in the Principia (“admit no more causes of natural things
later times and Deborah Mayo’s “error-statistical” approach to hypothesis than such as are both true and sufficient to explain their appearances”),
testing arguably develops this tradition (Mayo 1996 and Mayo and Spanos echoing very closely Ockham’s razor. This basic idea has never lost its
2010; see Bird 2010, Kitcher 1993, 219 ff., and Meehl 1990 for other appeal—even up to recent times (see, e.g., Quine and Ullian 1970, 69 ff.;
contemporary variations). Sober 1975; Zellner, Keuzenkamp, and McAleer 2002; Scorzato 2013).
Eliminativism is most credible when experimentation is at issue (see, e.g., Despite Thomas Kuhn’s (1957, 181) suggestions to the contrary, the
Guala 2012). Indeed, the appeal to Bacon’s idea of crucial experiment success of Copernican astronomy over Ptolemy’s system has remained an
(instantia crucis) and related notions (e.g., “severe testing”) is a fairly influential case study fostering the simplicity view (Martens 2009).
reliable mark of eliminativist inclinations. Experimentation is, to a large Moreover, in ordinary scientific problems such as curve fitting, formal
extent, precisely an array of techniques to keep undesired interfering criteria of model selection are applied where the paucity of parameters can
factors at a minimum by active manipulation and deliberate control (think be interpreted naturally as a key dimension of simplicity (Forster and
of the blinding procedure in medical trials, with h the hypothesized Sober 1994). Traditionally, two main problems have proven pressing, and
effectiveness of a novel treatment and e a relative improvement in clinical frustrating, for the simplicity approach. First, how to provide a sufficiently
endpoints for a target subsample of patients thus treated). When this kind coherent and illuminating explication of this multifaceted and elusive
of control obtains, popular statistical tools are supposed to allow for the notion (see Riesch 2010); and second, how to justify the role of simplicity
calculation of the probability of e in case h is false meant as a “relative as a properly epistemic (rather than merely pragmatic) virtue (see Kelly
frequency in a (real or hypothetical) series of test applications” (Mayo 2007, 2008).
1991, 529), and to secure a sufficiently low value to validate the positive
outcome of the test. It is much less clear how firm a grip this approach can Finally, naïve HD can be enriched by the appeal to explanation. Here, the

naïve HD clause for confirmation is meant to be too weak because h ∧ k of hypothetico-deductivism (see Earman 1992, 64, and Glymour 1980b)
must be able (not only to entail, but) to explain e. By this move, the HD might have been exaggerated. For all its difficulties, HD has proven fairly
approach embeds the slogan of the so-called inference to the best resilient at least as a basic framework to elucidate some key aspects of
explanation view: “observations support the hypothesis precisely because how hypotheses can be confirmed by the evidence (see Betz 2013, Gemes
it would explain them” (Lipton 2000, 185; also see Lipton 2004). 2005, and Sprenger 2011b for consonant points of view).
Historically, the main source for this connection between explanation and
support is found in the work of Charles Sanders Peirce (1839–1914). 3. Bayesian confirmation theories
Janssen (2003) offers a particularly neat contemporary exhibit, explicitly
aimed at “curing cases of the Duhem-Quine disease” (484; also see Bayes’s theorem is a very central element of the probability calculus (see
Thagard 1978, and Douven 2011 for a relevant survey). Quite unlike Joyce 2008). For historical reasons, Bayesian has become a standard label
eliminativist approaches, explanationist analyses tend to focus on large- to allude to a range of approaches and positions sharing the common idea
scale theories and relatively high-level kinds of evidence. Dealing with that probability (in its modern, mathematical sense) plays a crucial role in
Einstein’s general relativity, for instance, Janssen (2003) greatly rational belief, inference, and behavior. According to Bayesian
emphasizes its explanation of the equivalence of inertial and gravitational epistemologists and philosophers of science, (i) rational agents have
mass (essentially a brute fact in Newtonian physics) over the resolution of credences differing in strength, which moreover (ii) satisfy the probability
the puzzle of Mercury’s perihelion. Explanationist accounts are also axioms, and can thus be represented in probabilistic form. (In non-
distinctively well-equipped to address inference patterns from non- Bayesian models (ii) is rejected, but (i) may well be retained: see Huber
experimental sciences (Cleland 2011). and Schmidt-Petri 2009, Levi 2008, and Spohn 2012.) Well-known
arguments exist in favor of this position (see, e.g., Easwaran 2011a;
The problems faced by these approaches are similar to those affecting the Leitgeb and Pettigrew 2010a,b; Pettigrew 2011; Predd et al. 2009; Skyrms
simplicity view. Agreement is still lacking on the nature of scientific 1987; Vineberg 2011), although there is no lack of difficulties and
explanation (see Woodward 2011) and it is not clear how far an criticism (see, e.g., Easwaran 2011b; Hájek 2008; Kelly and Glymour
explanationist variant of HD can go without a sound analysis of that 2004; Norton 2011b).
notion. Moreover, some critics have wondered why the relationship of
confirmation should be affected by an explanatory connection with the Beyond the core ideas above, however, the theoretical landscape of
evidence per se (see Salmon 2001). Bayesianism is quite as hopelessly diverse as it is fertile. Surveys and state
of art presentations are already numerous, and ostensibly growing (see,
The above discussion does not display an exhaustive list (nor are the listed e.g., Good 1971; Hartmann and Sprenger 2010; Jeffrey 2004; Joyce 2011;
options mutually exclusive, for that matter: see, e.g., Baker 2003; also see Oaksford and Chater 2007; Talbott 2011; Weisberg 2011). For the present
Worrall 2010 for some overlapping implications in an applied setting of purposes, attention can be restricted to a classification that is still fairly
real practical value). And our sketched presentation hardly allows for any coarse-grained, and based on just two dimensions or criteria.
conclusive assessment. It does suggest, however, that reports of the death

First, there is a distinction between permissivism and impermissivism (see Let us posit a set P of probability functions representing possible states of
White 2005, and Meacham 2014, for this terminology). For permissive belief about a domain that is described in a finite language L with L the
Bayesians (often otherwise labelled “subjectivists”), accordance with the set of its closed sentences. From now on, unless otherwise specified,
probability axioms is the only clear-cut constraint on the credences of a whenever considering some h, e, k ∈ L and P ∈ P , we will invariably
rational agent. In impermissive forms of Bayesianism (often otherwise rely on the following provisos:
called “objective”), further constraints are put forward that significantly
restrict the range of rational credences, possibly up to one single “right” i. both e ∧ k and h ∧ k are consistent;
probability function in any given setting. Second, there are different ii. P(e ∧ k), P(h ∧ k) > 0;
attitudes towards so-called principle of total evidence (TE) for the iii. P(k) > P(h ∧ k) (unless k ⊨ h);
credences on which a reasoner relies. TE Bayesians maintain that the iv. P(e ∧ k) > P(e ∧ h ∧ k) (unless e ∧ k ⊨ h ); and
relevant credences should be represented by a probability function P v. P(e ∧ h ∧ k) > 0 , as long as e ∧ h ∧ k is consistent.
which conveys the totality of what is known to the agent. For non-TE
(These assumptions are convenient and critical for technical reasons, but
approaches, depending on the circumstances, P may (or should) be set up
not entirely innocent. Festa 1999 and Kuipers 2000, 44 ff., discuss some
so that portions of the evidence available are in fact bracketed.
limiting cases that are left aside here owing to these constraints.)
(Unsurprisingly, further subtleties arise as soon as one delves a bit further
into the precise meaning and scope of TE; see Fitelson 2008 and A probabilistic theory of confirmation can be spelled out through the
Williamson 2002, Chs. 9–10, for important discussions.) definition of a function CP (h, e ∣ k) : {L3 × P} → ℜ representing the
degree of confirmation that hypothesis h receives from evidence e relative
Of course, many intermediate positions exist between extreme forms of
to k and probability function P. CP (h, e ∣ k) will then have relevant
permissivism and impermissivism so outlined, and more or less the same
probabilities as its building blocks, according to the following basic
applies for the TE issue. The above distinctions are surely rough enough,
“postulate” of probabilistic confirmation:
but useful nonetheless. Impermissive TE Bayesianism has served as a
received view in Bayesian philosophy of science since Carnap (P0) Formality
(1950/1962). But impermissivism is easily found in combination with non- There exists a function g such that, for any h, e, k ∈ L and any P ∈ P ,
TE positions, too (see, e.g., Maher 1996). TE permissivism seems a good CP (h, e ∣ k) = g[P(h ∧ e ∣ k), P(h ∣ k), P(e ∣ k)] .
approximation of De Finetti’s (2008) stance, while non-TE permissivism
is arguably close to a standard view nowadays (see, e.g., Howson and Note that the probability distribution over the algebra generated by h and
Urbach 2006). No more than this will be needed to begin our exploration e, conditional on k , is entirely determined by P(h ∧ e ∣ k) , P(h ∣ k) and
of Bayesian confirmation theories. P(e ∣ k) . Hence, (P0) simply states that CP (h, e ∣ k) depends on that
distribution, and nothing else. (The label for this assumption is taken from
3.1 Probabilistic confirmation as firmness Tentori, Crupi, and Osherson 2007, 2010.)

Hempelian and HD confirmation, as discussed above, are qualitative Theorem 1 provides a simple axiomatic characterization of the class of
theories of confirmation. They only tell us whether evidence e confirms confirmation functions that are strictly increasing with the final probability
(disconfirms) hypothesis h given k. However, assessments of the amount of the hypothesis given the evidence (and k ). All the functions in this class
of support that some evidence brings to a hypothesis are commonly are ordinally equivalent, meaning that they imply the same rank order of
involved in scientific reasoning, as well as in other domains, if only in the CP (h, e ∣ k) and CP∗ (h∗ , e∗ ∣ k ∗ ) for any h, h∗ , e, e∗ , k, k ∗ ∈ L and any
form of comparative judgments such as “hypothesis h is more strongly P, P∗ ∈ P.
confirmed by e1 than by e2 ” or “e confirms h1 to a greater extent than h2 ”.
Consider, for instance, the following principle, a veritable cornerstone of By (P0), (P1) and (P2), we thus have CP (h, e ∣ k) = f [P(h ∣ e ∧ k)] ,
probabilistic confirmation in all of its variations (see Crupi, Chater, and implying that the more likely h is given the evidence the more it is
Tentori 2013 for a list of references): confirmed. This approach explicates confirmation precisely as the overall
credibility of a hypothesis (firmness is Carnap’s 1950/1962 telling term,
(P1) Final probability xvi). In this view, “Bayesian confirmation theory is little more than the
For any h, e1 , e2 , k ∈ L and any P ∈ P , CP (h, e1 ∣ k) ⋛ CP (h, e2 ∣ k) examination of [the] properties” of the posterior probability function
if and only if P(h ∣ e1 ∧ k) ⋛ P(h ∣ e2 ∧ k). (Howson 2000, 179).
(P1) is itself a comparative, or ordinal, principle, stating that, for any fixed As we will see, the ordinal level of analysis is a solid and convenient
hypothesis h, the final (or posterior) probability and confirmation always middleground between a purely qualitative and a thoroughly quantitative
move in the same direction in the light of data, e (given k ). Interestingly, (metric) notion of confirmation. To begin with, ordinal notions are in
(P0) and (P1) are already sufficient to single out one traditional class of general sufficient to move “upwards” to the qualitative level as follows:
measures of probabilistic confirmation, if conjoined with the following
(see Crupi and Tentori forthcoming, and also Törnebohm 1966, 81): Qualitative confirmation from ordinal relations (QC)
For any h, e, k ∈ L and any P ∈ P :
(P2) Local equivalence
For any h1 , h2 , e, k ∈ L and any P ∈ P , if h1 and h2 are logically e CP -confirms h relative to k if and only if
equivalent given e and k, then CP (h1 , e ∣ k) = CP (h2 , e ∣ k). CP (h, e ∣ k) > CP (¬h, e ∣ k);
e CP -disconfirms h relative to k if and only if
The following can then be shown: CP (h, e ∣ k) < CP (¬h, e ∣ k);
e is CP -neutral for h relative to k if and only if
Theorem 1 CP (h, e ∣ k) = CP (¬h, e ∣ k).
(P0), (P1) and (P2) hold if and only if there exists a strictly increasing
function f such that, for any h, e, k ∈ L and any P ∈ P , Given Theorem 1, (P0), (P1) and (P2) can be combined with the
CP (h, e ∣ k) = f [P(h ∣ e ∧ k)] . definitions in (QC) to derive the following qualitative notion of
probabilistic confirmation as firmness:

Confirmation as firmness (F -confirmation, qualitative) CP (h, e1 ∧ e2 ∣ k) = CP (h, e1 ∣ k + CP (h, e2 ∣ e1 ∧ k).

For any h, e, k ∈ L and any P ∈ P :
Although extraneous to F-confirmation, Strict Additivity will prove of use
e F-confirms h relative to k if and only if P(h ∣ e ∧ k) > 1/2 ; later on for the discussion of further variants of Bayesian confirmation
e F-disconfirms h relative to k if and only if P(h ∣ e ∧ k) < 1/2 ; theory.
e is F-neutral for h relative to k if and only if P(h ∣ e ∧ k) = 1/2 .
3.2 Strengths and infirmities of firmness
The point of qualitative F-confirmation is thus straightforward: h is said to
be (dis)confirmed by e (given k) if it is more likely than not to be true Confirmation as firmness shares a number of structural properties with
(false). (Sometimes a threshold higher than a probability 1/2 is identified, Hempelian confirmation. It satisfies the Special Consequence Condition,
but this complication would add little for our present purposes.) thus the Predictive Inference Condition too. It satisfies the Entailment
Condition and, in virtue of (P1), extends it smoothly to the following
Also, the ordinal notion of confirmation is arguably of greater theoretical ordinal counterpart:
significance than its quantitative counterpart, because ordinal divergences,
unlike purely quantitative differences, imply opposite comparative Entailment condition (ordinal extension) (EC-Ord)
judgments for some evidence-hypothesis pairs. A refinement from the For any h, e1 , e2 , k ∈ L and any P ∈ P such that k ⊭ h :
ordinal to a properly metrical level can still be of theoretical interest,
i. if, e1 ∧ k ⊨ h and e2 ∧ k ⊭ h , then h is more confirmed by e1
however, and much useful for tractability and applications. For example,
than by e2 relative to k , that is, CP (h, e1 ∣ k) > CP (h, e2 ∣ k);
one can have 0 as a convenient neutrality threshold for confirmation as
ii. if, e1 ∧ k ⊨ h and e2 ∧ k ⊨ h, then h is equally confirmed by e1
firmness, provided that the following functional representation is adopted
and by e2 relative to k , that is, CP (h, e1 ∣ k) = CP (h, e2 ∣ k).
(see Peirce 1878 for an early occurrence):
F(h, e ∣ k) = log [
P(¬h ∣ e ∧ k) ]
P(h ∣ e ∧ k) According to (EC-Ord) not only is classical entailment retained as a case
of confirmation, it also represents a limiting case: it is the strongest
= log Odds(h ∣ e ∧ k) possible form of confirmation that a fixed hypothesis h can receive.
(The base of the logarithm can be chosen at convenience, as long as it is F-confirmation also satisfies Confirmation Complementarity and,
strictly greater than 1.) moreover, extends it to its appealing ordinal counterpart (see Crupi, Festa,
and Buttasi 2010, 85–86), that is:
A quantitative requirement that is often put forward is the following
stringent form of additivity (see Milne forthcoming): Confirmation complementarity (ordinal extension) (CC-Ord)
CP (¬h, e ∣ k) is a strictly decreasing function of CP (h, e ∣ k) , that is,
Strict additivity (SA) for any h, h∗ , e, e∗ , k ∈ L and any P ∈ P,
For any h, e1 , e2 , k ∈ L and any P ∈ P :
∗ ∗
P (h, e ∣ k) ⋛ P( , ∣ k)
CP (h, e ∣ k) ⋛ CP (h∗ , e∗ ∣ k) Bayesianism (see Hawthorne 2011 and Williamson 2011 for contemporary
variations). In fact, F-confirmation fits most neatly a classical form of TE
if and only if impermissivism à la Carnap, where one assumes that k = ⊤, that P is an
“objective” initial probability based on essentially logical considerations,
CP (¬h, e ∣ k) ⋚ CP (¬h∗ , e∗ ∣ k).
and that all the non-logical information available is collected in e. Now, it
(CC-Ord) neatly reflects Keynes’ (1921, 80) remark that “an argument is is true that the spirit of the Carnapian project never lost its appeal entirely
always as near to proving or disproving a proposition, as it is to disproving (see, e.g., Festa 2003, Franklin 2001, Maher 2010, Paris 2011). However,
or proving its contradictory”. Indeed, quantitatively, the measure that project seems to have missed its avowed goals. Overall, the idea of a
F(h, e ∣ k) instantiates Confirmation Complementarity in a simple and “logical” interpretation of P got stuck into difficulties that are now largely
elegant way, that is, it satisfies CP (h, e ∣ k) = −CP (¬h, e ∣ k). seen as insurmountable (e.g., Earman and Salmon 1992, 85–89; Gillies
2000, Ch. 3; Hájek 2012; Howson and Urbach 2006, 59–72; van Fraassen
F-confirmation also implies another attractive quantitative result, 1989, Ch. 12; Zabell 2011). And arguably, lacking some robust and
alleviating the ailments of the irrelevant conjunction paradox. In the effective impermissivist policy, the account of confirmation as firmness
statement below, indicating this result, the irrelevance of q for hypothesis ends up loosing much of its philosophical momentum. The issues
h and evidence e (relative to k ) is meant to amount to the probabilistic surrounding the ravens and blite paradoxes provide a useful illustration.
independence of q from h, e and their conjunction (given k), that is, to
P(h ∧ q ∣ k) = P(h ∣ k)P(q ∣ k), P(e ∧ q ∣ k) = P(e ∣ k)P(q ∣ k) , and Consider again h = ∀x(raven(x) → black(x)), and the main analyses of
P(h ∧ e ∧ q ∣ k) = P(h ∧ e ∣ k)P(q ∣ k) , respectively. “the observation that a is a black raven” encountered so far, that is:
Confirmation upon irrelevant conjunction (ordinal solution) (CIC) i. k = ⊤ and e = raven(a) ∧ black(a) , and
For any h, e, q, k ∈ L and any P ∈ P, if e confirms h relative to k and ii. k = raven(a) and e = black(a).
q is irrelevant for h and e relative to k, then
In both cases, whether e F-confirms h or not (relative to k ) critically
CP (h, e ∣ k) > CP (h ∧ q, e ∣ k). depends on P: if the prior P(h ∣ k) is low enough, e won’t do no matter
what under either (i) or (ii); and if it is high enough, h will be F -confirmed
So, even in case it is qualitatively preserved across the tacking of q onto h, either way. As a consequence, the F-confirmation view, by itself, does not
the positive confirmation afforded by e is at least bound to quantitatively offer any definite hint as to when, how, and why Nicod’s remarks apply or
decrease thereby. not.
Partly because of appealing formal features such as those mentioned so For the purposes of our discussion, the following condition reveals another
far, there is a long list of distinguished scholars advocating the firmness debatable aspect of the firmness explication of confirmation.
view of confirmation, from Keynes (1921) and Hosiasson-Lindenbaum
(1940) onwards, most often coupled with some form of impermissive

Consistency condition (Cons) (P3) implies that any hypothesis is equally “confirmed” by empty
For any h, h∗ , e, k ∈ L and any P ∈ P , if k ⊨ ¬(h ∧ h∗ ) then e evidence. We will say that CP (h, e ∣ k) represents the probabilistic
confirms h given k if and only if e disconfirms h∗ given k . relevance notion of confirmation, or relevance-confirmation, if and only if
it satisfies (P0), (P1) and (P3). These conditions are sufficient to derive the
(Cons) says that evidence e can never confirm incompatible hypotheses. following, purely qualitative principle, according to the definitional
But consider, by way of illustration, a clinical case of an infectious disease method in (QC) above (see Crupi and Tentori 2014, 82).
of unknown origin, and suppose that e is the failure of antibiotic treatment.
Arguably, there is nothing wrong in saying that, by discrediting bacteria as Probabilistic relevance confirmation (qualitative)
possible causes, the evidence confirms (viz. provides support for) any of a For any h, e, k ∈ L and any P ∈ P :
number of alternative viral diagnoses. This judgment clashes with (Cons),
though, which then seems an overly strong constraint. e relevance-confirms h relative to k if and only if
P(h ∣ e ∧ k) > P(h ∣ k);
Notably, (Cons) was defended by Hempel (1945) and, in fact, one can e relevance-disconfirms h relative to k if and only if
show that it follows from the conjunction of (qualitative) Confirmation P(h ∣ e ∧ k) < P(h ∣ k);
Complementary and the Special Consequence Condition, and so from both e is relevance-neutral for h relative to k if and only if
Hempelian and F-confirmation. This is but one sign of how stringent the P(h ∣ e ∧ k) = P(h ∣ k).
Special Consequence Condition is. Mainly because of the latter, both the
Hempelian and the firmness views of confirmation must depart from the The point of relevance confirmation is that the credibility of a hypothesis
plausible HD idea that hypotheses are generally confirmed by their can be changed in either a positive (confirmation in a strict sense) or
verified consequences (see Hempel 1945, 103–104). We will come back to negative way (disconfirmation) by the evidence concerned (given k).
this while discussing our next topic: a very different Bayesian explication Confirmation (in the strict sense) thus reflects an increase from initial to
of confirmation, based on the notion of probabilistic relevance. final probability, whereas disconfirmation reflects a decrease (see
Achinstein 2005 for some diverging views on this very idea).
3.3 Probabilistic relevance confirmation
The qualitative notions of confirmation as firmness and as relevance are
We’ve seen that the firmness notion of probabilistic confirmation can be demonstrably incompatible. Unlike firmness, relevance confirmation can
singled out through one ordinal constraint, (P2), in addition to the not be formalized by the final probability alone, or any increasing function
fundamental principles (P0)–(P1). The counterpart condition for the so- thereof. To illustrate, the probability of an otherwise very rare disease (h)
called relevance notion of probabilistic confirmation is the following: can be quite low even after a relevant positive test result (e); yet h is
relevance-confirmed by e to the extent that its probability rises thereby. By
(P3) Tautological evidence the same token, the probability of the absence of the disease (¬h) can be
For any h1 , h2 , k ∈ L and any P ∈ P , CP (h1 , ⊤ ∣ k) = CP (h2 , ⊤ ∣ k). quite high despite the positive test result (e), yet ¬h is relevance-

disconfirmed by e to the extent that its probability decreases thereby. (relative to k ). This is because, mathematically, it is perfectly possible for
Perhaps surprisingly, the distinction between firmness and relevance both P(h ∣ e ∧ k) and P(h ∣ ¬e ∧ k) to be arbitrarily high above 1/2 .
confirmation—“extremely fundamental” and yet “sometimes unnoticed”, Condition (CompE), on the contrary, ensures that only one between the
as Salmon (1969, 48–49) put it—had to be stressed time and again to complementary statements e and ¬e can confirm hypothesis h (relative to
achieve theoretical clarity in philosophy (e.g., Popper 1954; Peijnenburg k ). (To be precise, HD-confirmation also satisfies condition CompE, yet it
2012) as well as in other domains concerned, such as artificial intelligence would fail the above example all the same, although for a different reason,
and the psychology of reasoning (see Horvitz and Heckerman 1986; that is, because the connection between h and e is plausibly one of
Crupi, Fitelson, and Tentori 2008; Shogenji 2012). probabilistic dependence but not of logical entailment.)
The qualitative notion of relevance confirmation already has some Remarks such as the foregoing have induced some contemporary Bayesian
interesting consequences. It implies, for instance, the following theorists to dismiss the notion of confirmation as firmness altogether,
remarkable fact: concluding with I.J. Good (1968, 134) that “if you had P(h ∣ e ∧ k) close
to unity, but less than P(h ∣ k), you ought not to say that h was confirmed
Complementary Evidence (CompE) by e” (also see Salmon 1975, 13). Let us comply with this suggestion and
For any h, e, k ∈ L and any P ∈ P, e confirms h relative to k if and proceed to consider the ordinal (and quantitative) notions of relevance
only if ¬e disconfirms h relative to k. confirmation.
The importance of (CompE) can be illustrated as follows. Consider the
3.4 Differences, ratios, and partial entailment
case of a father suspected of abusing his son. Suppose that the child does
claim that s/he has been abused (label this evidence e). A forensic Just as with firmness, the ordinal analysis of relevance confirmation can
psychiatrist, when consulted, declares that this confirms guilt (h). be characterized axiomatically. With the relevance notion, however, a
Alternatively, suppose that the child is asked and does not report having larger set of options arises. Consider the following principles.
been abused (¬e). As pointed out by Dawes (2001), it may well happen
that a forensic psychiatrist will nonetheless interpret this as evidence (P4) Disjunction of alternative hypotheses
confirming guilt (suggesting that violence has prompted the child’s For any e, h1 , h2 , k ∈ L and any P ∈ P, if k ⊨ ¬(h1 ∧ h2 ) , then
denial). One might want to argue that, other things being equal, this kind
CP (h1 , e ∣ k) ⋛ CP (h1 ∨ h2 , e ∣ k)
of “heads I win, tails you lose” judgment would be inconsistent, and thus
in principle untenable. Whoever concurs with this line of argument (as if and only if
Dawes 2001 himself did) is likely to be relying on the relevance notion of
confirmation. In fact, no other notion of confirmation considered so far P(h2 ∣ e ∧ k) ⋛ P(h2 ∣ k).
provides a general foundation for this judgment. F-confirmation, in
particular, would not do, for it does allow that both e and ¬e confirm h

(P5) Law of likelihood ii. (P5) holds if and only if CP (h, e ∣ k) is a probability ratio
For any e, h1 , h2 , k ∈ L and any P ∈ P, measure, that is, if there exists a strictly increasing function f
such that, for any h, e, k ∈ L and any P ∈ P,
CP (h1 , e ∣ k) ⋛ CP (h2 , e ∣ k) P(h∣e∧k)
CP (h, e ∣ k) = f [ P(h∣k) ];
if and only if iii. (P6) holds if and only if CP (h, e ∣ k) is a likelihood ratio
measure, that is, if there exists a strictly increasing function f
P(e ∣ h1 ∧ k) ⋛ P(e ∣ h2 ∧ k). such that, for any h, e, k ∈ L and any P ∈ P,
P(e∣h∧k)
CP (h, e ∣ k) = f [ P(e∣¬h∧k) ].
(P6) Modularity (for conditionally independent data)
For any e1 , e2 , h, k ∈ L and any P ∈ P, if If a strictly additive behavior (SA above) is imposed, one functional form
is singled out for the quantitative representation of confirmation
P(e1 ∣ ±h ∧ e2 ∧ k) = P(e1 ∣ ±h ∧ k),
corresponding to each of the clauses above, that is (see Milne
then forthcoming):
CP (h, e1 ∣ e2 ∧ k) = CP (h, e1 ∣ k). i. DP (h, e ∣ k) = P(h ∣ e ∧ k) − P(h ∣ k);

P(h∣e∧k)
ii. RP (h, e ∣ k) = log[ P(h∣k)
];
All the above conditions occur more or less widely in the literature (see P(e∣h∧k)
iii. LP (h, e ∣ k) = log[ P(e∣¬h∧k) ].
Crupi, Chater, and Tentori 2013 and Crupi and Tentori forthcoming for
references and discussion). Interestingly, they’re all pairwise incompatible (The bases of the logarithms are assumed to be strictly greater than 1.)
on the background of the Formality and the Final Probability principles
(P0 and P1 above). Indeed, they sort out the relevance notion of Before discussing briefly this set of alternative quantitative measures of
confirmation into three distinct, classical families of measures, as follows relevance confirmation, we will address one further related issue.
(Crupi, Chater, and Tentori 2013; Crupi and Tentori forthcoming;
Heckerman 1988): It is a long-standing idea, going back to Carnap at least, that confirmation
theory should yield an inductive logic that is analogous to classical
Theorem 2 deductive logic in some suitable sense, thus providing a theory of partial
Given (P0) and (P1): entailment, and partial refutation. Now, the deductive-logical notions of
entailment and refutation (contradiction) exhibit the following well-known
i. (P4) holds if and only if CP (h, e ∣ k) is a probability difference properties:
measure, that is, if there exists a strictly increasing function f
such that, for any h, e, k ∈ L and any P ∈ P, Contraposition of entailment
CP (h, e ∣ k) = f [P(h ∣ e ∧ k) − P(h ∣ k)]; Entailment is contrapositive, but not commutative. That is, it holds

that e entails h (e ⊨ h) if and only if ¬h entails ¬e (¬h ⊨ ¬e), while it the discussion in Crupi and Tentori 2013), a neat confirmation-theoretic
does not hold that e entails h if and only if h entails e (h ⊨ e). generalization of logical entailment (and refutation) is possible after all.
Interestingly, relative distance measures can be additive, but only for
Commutativity of refutation uniform pairs of arguments (both confirmatory, or both disconfirmatory),
Refutation, on the contrary, is commutative, but not contrapositive. as shown by Milne (forthcoming). (See Crupi, Tentori, and Gonzalez
That is, it holds that e refutes h (e ⊨ ¬h) if and only if h refutes e 2007; Crupi, Festa, and Buttasi 2010; and Crupi and Tentori 2013, 2014,
(h ⊨ ¬e), while it does not hold that e refutes h if and only if ¬h for further discussions of the properties of relative distance measures and
refutes ¬e (¬h ⊨ ¬¬e). their intuitive motivations. Also see Mura 2008 for a related analysis.)
The confirmation-theoretic counterparts are fairly straightforward: The plurality of alternative probabilistic measures of relevance
confirmation has prompted some scholars to be skeptical or dismissive of
(P7) Contraposition of confirmation
the prospects for a quantitative theory of confirmation (see, e.g., Howson
For any e, h, k ∈ L and any P ∈ P, if e relevance-confirms h relative
2000, 184–185, and Kyburg and Teng 2001, 98 ff.). However, as we will
to k, then CP (h, e ∣ k) = CP (¬e, ¬h ∣ k).
see shortly, quantitative analyses of relevance confirmation have proved
(P8) Commutativity of disconfirmation important for handling a number of puzzles and issues that plagued
For any e, h, k ∈ L and any P ∈ P, if e relevance-disconfirms h competing approaches. Moreover, various arguments in the philosophy of
relative to k, then CP (h, e ∣ k) = CP (e, h ∣ k). science and beyond have been shown to depend critically (and sometimes
unwittingly) on the choice of one confirmation measure (or some of them)
The following can then be proven (Crupi and Tentori 2013): rather than others (Festa 1999, 2012, Fitelson 1999, Brössel 2013, Glass
2013, Roche and Shogenji 2014, and Rusconi et al. 2014).
Theorem 3
Given (P0) and (P1), (P7) and (P8) hold if and only if CP (h, e ∣ k) is a Recently, arguments have been offered by Huber (2008b) and Milne
relative distance measure, that is, if there exists a strictly increasing (forthcoming) in favor of D, by Fitelson (2001) and Zalabardo (2009) in
function f such that, for any h, e, k ∈ L and any P ∈ P, favor of L, and by Crupi and Tentori (2010) in favor of Z , while Hájek and
CP (h, e ∣ k) = f [Z(h, e ∣ k)], where: Joyce (2008, 123) have seen different measures as possibly capturing
⎧ P(h ∣ e ∧ k) − P(h ∣ k)
“distinct, complementary notions of evidential support” (also see
⎪ if P(h ∣ e ∧ k) ≥ P(h ∣ k) Schlosshauer and Wheeler 2011 and Steel 2007 for tempered forms of
⎪ 1 − P(h ∣ k)
Z(h, e ∣ k) = ⎨ pluralism). The case of measure R deserves some more specific
⎪ P(h ∣ e ∧ k) − P(h ∣ k) comments. Following Fitelson (2007), one could see R as conveying key
⎪ if P(h ∣ e ∧ k) < P(h ∣ k)
⎩ P(h ∣ k) tenets of so-called “likelihoodist” position about evidential reasoning (see
Royall 1997 for a classical statement, and Chandler 2013 and Sober 1990
So, despite some pessimistic suggestions (see, e.g., Hawthorne 2012, and for consonant arguments and inclinations). There seems to be some

consensus, however, that compelling objections can be raised against the again by (i), the confirmatory impact will be stronger the more surprising
adequacy of R as a proper measure of relevance confirmation (see, in (unlikely) the evidence was unless h was conjoined to k . So, under TE,
particular, Crupi, Festa, and Buttasi 2010, 85–86; Eells and Fitelson 2002; relevance confirmation turns out to embed a squarely predictivist version
Gillies 1986, 112; and compare Milne 1996 with Milne forthcoming). In of hypothetico-deductivism! As we know, this neutralizes the charge of
what follows, too, it will be convenient to restrict our discussion to D, L underdetermination, yet it comes at the usual cost: the old evidence
and Z as candidate measures. All the results to be presented below are problem. In fact, if TE is in force, then clause (ii) of (SP) implies that no
invariant for whatever choice among these three options, and across statement that is known to be true (thus assigned probability 1) can ever
ordinal equivalence with each of them (but those results do not always have confirmatory import.
extend to measures ordinally equivalent to R).
Interestingly, the Bayesian predictivist has an escape (neatly anticipated,
3.5 New evidence, old evidence, and total evidence and criticized, by Glymour 1980a, 91–92). Consider Einstein and Mercury
once again. As effectively pointed out by Norton (2011a, 7), Einstein was
Let us go back to a classical HD case, where the (consistent) conjunction extremely careful to emphasize that the precession phenomenon had been
h ∧ k (but not k alone) entails e. The following can be proven: derived “without having to posit any special [auxiliary] hypotheses at all”.
Why? Well, presumably because if one had allowed herself to arbitrarily
Surprising prediction theorem (SP)
devise ad hoc auxiliaries (within k , in our notation) then one could have
For any e, h, k ∈ L and any P ∈ P such that h ∧ k ⊨ e and k ⊭ e :
been pretty much certain in advance to find a way to get Mercury’s data
i. if P(e ∣ k) < 1, then e relevance-confirms h relative to k and right (recall: that’s the lesson of the underdetermination theorem). But
CP (h, e ∣ k) is a decreasing function of P(e ∣ k); getting those data right with auxiliaries k that were not thus adjusted—that
ii. if P(e ∣ k) = 1, then e is relevance-neutral for h relative to k. would have been a natural consequence had the theory of general
relativity been true and it would have been surprising otherwise.
Formally, it is fairly simple to show that (SP) characterizes relevance Arguably, this line of argument exploits much of the use-novelty idea
confirmation (see, e.g., Crupi, Festa, and Buttasi 2010, 80; Hájek and within a predictivist framework. The crucial points are (i) that the
Joyce 2008, 123), but the philosophical import of this result is nonetheless evidence implied is not a verified empirical statement e but the logical fact
remarkable. For illustrative purposes, it is useful to assume the that h ∧ k entails e, and (ii) that the existence of this connection of
endorsement of the principle of total evidence (TE) as a default position entailment was not to be obviously anticipated at all, precisely because
for the Bayesian. This means that P is assumed to represent actual degrees h ∧ k and e are such that the latter did not serve as a constraint to specify
of belief of a rational agent, that is, given all the background information the former. On these conditions, it seems that h can be confirmed by this
available. Then, by clause (i) of (SP), we have that the occurrence of e, a kind of “second-order” (logical) evidence in line with (SP) while TE is
consequence of h ∧ k (but not of k alone), confirms h relative to k concurrently preserved.
provided that e was initially uncertain to some degree (even given k). In
other words: e must have been predicted on the basis of h ∧ k . Moreover, At least two main problems arise, however. The first one is more technical

in nature. Modelling rational uncertainty concerning logical facts (such as for different hypotheses (thus mathematically implying P(e ∣ k) < 1 ) and
h ∧ k ⊨ e ) by probabilistic means is no trivial task. Garber (1983) put serve as a basis to capture formally the confirmatory impact of e (see
forward the most influential proposal, but doubts have been raised that it Hawthorne 2005 for an argument along these lines). Permissivists, on the
might not be well-behaved (van Fraassen 1988; also see Niiniluoto 1983). other hand, can not coherently rely on these considerations to articulate a
Second, and more substantially, this solution of the old evidence problem non-TE position. They must invoke counterfactual degrees of belief
can be charged of being an elusive change of the subject: for it was instead, suggesting that P should be reconstructed as representing the
Mercury’s data, not anything else, that had to be recovered as having beliefs that the agent would have, had she not known that e was true (see
confirmed (and still confirming, some would add) Einstein’s theory. Howson 1991 for a statement and discussion; also see Jeffrey 1995 and
That’s the kind of judgment that confirmation theory must capture, and Wagner 2001 for important and relevant technical results, and Steele and
which remains unattainable for the predictivist Bayesian. (Earman 1992, Werndl 2013 for an intriguing case-study from climate science).
131 voiced this complaint forcefully. Hints for a possible rejoinder appear
in Eells’s 1990 thorough discussion; see also Skyrms 1983.) 3.6 Paradoxes probabilified and other elucidations
Bayesians that are unconvinced by the predictivist position are naturally The theory of Bayesian confirmation as relevance indicates when and why
lead to dismiss TE and allow for the assignment of initial probabilities the HD idea works: if h ∧ k (but not k ) entails e, then h is relevance-
lower than 1 even to statements that were known all along. Of course, this confirmed by e (relative to k ) because the latter increases the probability
brings the underdetermination problem back, for now k can still be of the former—provided that P(e ∣ k) < 1 . Admittedly, the meaning of the
concocted ad hoc to have known evidence e following from h ∧ k and latter proviso partly depends on how one handles the problem of old
moreover P(e ∣ k) < 1 is not prevented by TE anymore, thus potentially evidence. Yet it seems legitimate to say that Bayesian relevance
licencing arbitrary confirmation relations. Two moves can be combined to confirmation (unlike the firmness view) retains a key point of ordinary
handle this problem. First, unlike HD, the Bayesian framework has the scientific practice which is embedded in HD and yields further elements of
formal resources to characterize the auxiliaries themselves as more or less clarification. Consider the following illustration.
likely and thus their adoption as relatively safe or suspicious (the standard
(e1 ) tigers carry the ND1 gene
Bayesian treatment of auxiliary hypotheses is developed along these lines
(e2 ) elephants carry the ND1 gene
in Dorling 1979 and Howson and Urbach 2006, 92–102, and it is critically
(e∗2 ) lions carry the ND1 gene
discussed in Rowbottom 2010, Strevens 2001, and Worrall 1993; also see
(h) all mammals carry the ND1 gene
Christensen 1997 for an important analysis of related issues). Second, one
has to provide indications as to how TE should be relaxed. Non-TE Qualitative confirmation theories comply with the idea that h is confirmed
Bayesians of the impermissivist strand often suggest that objective both by e1 ∧ e2 and by e1 ∧ e∗2 . In the HD case, it is clear that h entails
likelihood values concerning the outcome e—P(e ∣ h ∧ k) —can be both conjunctions, given of course k stating that tigers, lions, and
specified for the competing hypotheses at issue quite apart from the fact elephants are all mammals (an Hempelian account could also be given
that e may have already occurred. Such values would typically be diverse

easily). Bayesian relevance confirmation unequivocally yields the same now amounts to the finding of a black raven, e = raven(a) ∧ black(a) ,
qualitative verdict. There is more, however. Presumably, one might also versus a non-black non-raven, e∗ = ¬black(a) ∧ ¬raven(a) ? We’ve
want to say that h is more strongly confirmed by e1 ∧ e2 than by e1 ∧ e∗2 , already seen that, for either Hempelian or HD-confirmation, e and e∗ are
because the former offers a more varied and diverse body of positive on a par: both Hempel-confirm h, none HD-confirms it. In the former
evidence (interestingly, on experimental investigation, this pattern prevails case, the original Hempelian version of the ravens paradox immediately
in most people’s judgment, including children, see Lo et al. 2002). Indeed, arises; in the latter, it is avoided, but at a cost: e is declared flatly
the variety of evidence is a fairly central issue in the analysis of irrelevant for h—a bit of a radical move. Can the Bayesian do any better?
confirmation (see, e.g., Bovens and Hartmann 2002, Schlosshauer and Quite so. Consider the following conditions:
Wheeler 2011, and Viale and Osherson 2000). In the illustrative case
above, higher variety is readily captured by lower probability: it just seems i. P[raven(a) ∣ h] = P[raven(a)] > 0
a priori less likely that species as diverse as tigers and elephants share ii. P[¬raven(a) ∧ black(a) ∣ h] = P[¬raven(a) ∧ black(a)]
some unspecified genetic trait as compared to tigers and lions, that is,
Roughly, (i) says that the size of the ravens population does not depend on
P(e1 ∧ e2 ∣ k) < P(e1 ∧ e∗2 ∣ k). By (SP) above, then, one immediately
their color (in fact, on h), and (ii) that the size of the population of black
gets from the relevance confirmation view the sound implication that
non-raven objects also does not depend on the color of ravens. Note that
CP (h, e1 ∧ e2 ∣ k) > CP (h, e1 ∧ e∗2 ∣ k).
both (i) and (ii) seem fairly sound as far as our best understanding of our
Principle (SP) is also of much use in the ravens problem. Posit actual world is concerned. It is easy to show that, in relevance-
h = ∀x(raven(x) → black(x)) once again. Just as HD, Bayesian relevance confirmation terms, (i) and (ii) are sufficient to imply that
confirmation directly implies that e = black(a) confirms h given e = raven(a) ∧ black(a) , but not e∗ = ¬raven(a) ∧ ¬black(a) , confirms
k = raven(a) and e∗ = ¬raven(a) confirms h given k ∗ = ¬black(a) h, that is CP (h, e) > CP (h, e∗ ) = 0 (this observation is due to Mat
(provided, as we know, that P(e ∣ k) < 1 and P(e∗ ∣ k ∗ ) < 1). That’s Coakley). So the Bayesian relevance approach to confirmation can make a
because h ∧ k ⊨ e and h ∧ k ∗ ⊨ e∗ . But of course, to have h confirmed, principled difference between e and e∗ in both ordinal and qualitative
sampling ravens and finding a black one is intuitively more significant terms. (A much broader analysis is provided by Fitelson and Hawthorne
than failing to find a raven while sampling the enormous set of the non- 2010, Hawthorne and Fitelson 2010 [Other Internet Resources]. Notably,
black objects. That is, it seems, because the latter is very likely to obtain their results include the full specification of the sufficient and necessary
anyway, whether or not h is true, so that P(e∗ ∣ k ∗ ) is actually quite close conditions for the main inequality CP (h, e) > CP (h, e∗ ) .)
to unity. Accordingly, (SP) implies that h is indeed more strongly
In general, Bayesian (relevance) confirmation theory implies that the
confirmed by black(a) given raven(a) than it is by ¬raven(a) given
evidential import of an instance of some generalization will often depend
¬black(a)—that is, CP (h, e ∣ k) > CP (h, e∗ ∣ k ∗ ) —as long as the
on the credence structure, and relies on its formal representation, P, as a
assumption P(e ∣ k) < P(e∗ ∣ k ∗ ) applies.
tool for more systematic analyses. Consider another instructive example.
What then if the sampling in not constrained (k = ⊤) and the evidence Assume that a denotes some company from some (otherwise unspecified)

sector of the economy, and label the latter predicate S. So, k = Sa . You relevance strand can avail herself of the same quantitative strategy of
are informed that a increased revenues in 2014, represented as e = Ra . “damage control” for the main specific paradox of HD-confirmation, i.e.,
Does this confirm h = ∀x(Sx → Rx) ? It does, at least to some degree, one the irrelevant conjunction problem. (See statement (CIC) above, and Crupi
would say. For an expansion of the whole sector (recall that you have no and Tentori 2010, Fitelson 2002. Also see Chandler 2007 for criticism,
clue what this is) surely would account for the data. That’s a and Moretti 2006 for a related debate.)
straightforward HD kind of reasoning (and a suitable Hempelian
counterpart reconstruction would concur). But does e also confirm We’re left with one last issue to conclude our discussion, to wit, the blite
h∗ = Sb → Rb for some further company b? Well, another obvious paradox. Recall that blite is so defined:
account of the data e would be that company a has gained market shares at
blite(x) ≡ (ext≤T (x) → black(x)) ∧ (¬ext≤T (x) → white(x)).
the expenses of some competitor, so that e might well seem to support
¬h∗ , if anything (the revenues example is inspired by a remark in Blok, As always heretofore, we posit h = ∀x(raven(x) → black(x)),
Medin, and Osherson 2007, 1362). h∗ = ∀x(raven(x) → blite(x)). We then consider the set up where
k = raven(a) ∧ ext≤T (a), e = black(a), and P(e ∣ k) < 1. Various
It can be shown that the Bayesian notion of relevance confirmation allows
authors have noted that, with Bayesian relevance confirmation, one has
for this pattern of judgments, because (given k ) evidence e above increases
that P(h ∣ k) > P(h∗ ∣ k) is sufficient to imply that
the probability of h but may well have the opposite effect on h∗ (see Sober ∗
CP (h, e ∣ k) > CP (h , e ∣ k) (see Gaifman 1979, 127–128; Sober 1994,
1994 for important remarks along similar lines). Notably, h entails h∗ by
229–230; and Fitelson 2008, 131). So, as long as the black hypothesis is
plain instantiation, and so contradicts ¬h∗ . As a consequence, the
perceived as initially more credible than its blite counterpart, the former
implication that CP (h, e ∣ k) is positive while CP (h∗ , e ∣ k) is not clashes
will be more strongly confirmed than the latter. Of course,
with each of the following, and proves them unduly restrictive: the Special
P(h ∣ k) > P(h∗ ∣ k) is an entirely commonsensical assumption, yet these
Consequence Condition (SCC), the Predictive Inference Condition (PIC),
same authors have generally, and quite understandably, failed to see this
and the Consistency Condition (Cons). Note that these principles were all
result as philosophically illuminating. Lacking some interesting, non-
evaded by HD-confirmation, but all implied by confirmation as firmness
question-begging story as to why that inequality should obtain, no solution
(see above).
of the paradox seems to emerge. More modestly, one could point out that a
At the same time, the most compelling features of F-confirmation, which measure of relevance confirmation CP (h, e ∣ k) implies (i) and (ii) below.
the HD model was unable to capture, are retained by confirmation as
i. Necessarily (that is, for any P ∈ P ), e confirms h relative to k.
relevance. In fact, all our measures of relevance confirmation (D, L, and
ii. Possibly (that is, for some P ∈ P ), each one of the following obtains:
Z ) entail the ordinal extension of the Entailment Condition (EC) as well as
e confirms that a raven will be black if examined after T , that is,
CP (h, e ∣ k) = −CP (¬h, e ∣ k) and thereby Confirmation
(raven(b) ∧ ¬ext≤T (b)) → black(b), relative to k; and
Complementarity in all of its forms (qualitative, ordinal, and quantitative).
e does not confirm that a raven will be white if examined after
Moreover, the Bayesian confirmation theorist of either the firmness or the

T , that is, (raven(b) ∧ ¬ext≤T (b)) → white(b), relative to k. –––, 2013, “Contrastive Confirmation: Some Competing Accounts”,
Synthese, 190: 129–138.
Without a doubt, (i) and (ii) fall far short of a satisfactory solution of the Christensen, D., 1997, “What Is Relative Confirmation?”, Noûs, 3: 370–
blite paradox. Yet it seems at least a legitimate minimal requirement for a 384.
compelling solution (if any exists) that it implies both. It is then of interest Cleland, C.E., 2011, “Prediction and Explanation in Historical Natural
to note that confirmation as firmness is inconsistent with (i), while Science”, British Journal for the Philosophy of Science, 62: 551–582.
Hempelian and HD-confirmation are inconsistent with (ii). Craig, W., 1957, “The Uses of the Herbrand-Gentzen Theorem in Relating
Model Theory and Proof Theory”, Journal of Symbolic Logic, 3:
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Confirmation as firmness (F -confirmation, qualitative) CP (h, e1 ∧ e2 ∣ k) = CP (h, e1 ∣ k + CP (h, e2 ∣ e1 ∧ k).

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CP (h, e1 ∣ e2 ∧ k) = CP (h, e1 ∣ k). i. DP (h, e ∣ k) = P(h ∣ e ∧ k) − P(h ∣ k);

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Routledge, 2002. Reidel, pp. 47–81.

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