science whenever observational data and evidence "speak in favor of"
or support scientific theories and everyday hypotheses. Historically,
confirmation has been closely related to the problem of induction, the
question of what to believe regarding the future in the face of
knowledge that is restricted to the past and present. One relation
between confirmation and inductive logic is that the conclusion H of
an inductively strong argument with premise E is confirmed by E. If
inductive strength comes in degrees and the inductive strength of the
argument with premise E and conclusion H is equal to r, then the
degree of confirmation of H by E is likewise said to be equal to r.
This article begins by briefly reviewing Hume's formulation of the
problem of the justification of induction. Then we jump to the middle
of the twentieth century and Hempel's pioneering work on confirmation.
After looking at Popper's falsificationism and the
hypothetico-deductive method of hypotheses testing, the notion of
probability, as it was defined by Kolmogorov, is introduced.
Probability theory is the main mathematical tool for Carnap's
inductive logic as well as for Bayesian confirmation theory. Carnap's
inductive logic is based on a logical interpretation of probability,
which will be discussed at some length. However, his heroic efforts to
construct a logical probability measure in purely syntactical terms
can be considered to have failed. Goodman's new riddle of induction
will serve to illustrate the shortcomings of such a purely syntactical
approach to confirmation. Carnap's work is nevertheless important
because today's most popular theory of confirmation – Bayesian
confirmation theory – is to a great extent the result of replacing
Carnap's logical interpretation of probability with a subjective
interpretation as degree of belief qua fair betting ratio. The rest of
the article will be concerned mainly with Bayesian confirmation
theory, although the final section will mention some alternative views
on confirmation and induction.
1. Introduction: Confirmation and Induction
Whenever observational data and evidence speak in favor of, or
support, scientific theories or everyday hypotheses, the latter are
said to be confirmed by the former. The positive result of a pregnancy
test speaks in favor of or confirms the hypothesis that the tested
woman is pregnant. The dark clouds on the sky support or confirm the
hypothesis that it will be raining.
Confirmation takes a qualitative and a quantitative form. Qualitative
confirmation is usually construed as a relation, among other things,
between three sentences or propositions: evidence E confirms
hypothesis H relative to background information B. Quantitative
confirmation is, among other things, a relation between evidence E,
hypothesis H, background information B, and a number r: E confirms H
relative to B to degree r. (Comparative confirmation – H1 is more
confirmed by E1 relative to B1 than H2 by E2 relative to B2 – is
usually derived from a quantitative notion of confirmation, and is not
discussed in this entry.)
Historically, confirmation has been closely related to the problem of
induction, the question of what to believe regarding the future in the
face of knowledge that is restricted to the past and present. David
Hume gives the classic formulation of the problem of the justification
of induction in A Treatise of Human Nature:
Let men be once fully persuaded of these two principles, that
there is nothing in any object, consider'd in itself, which can afford
us a reason for drawing a conclusion beyond it; and, that even after
the observation of the frequent or constant conjunction of objects, we
have no reason to draw any inference concerning any object beyond
those of which we have had experience; (Hume 1739/2000, book 1, part
3, section 12)
The reason is that any such inference beyond those objects of which we
had experience needs to be justified – and, according to Hume, this is
not possible.
In order to justify induction one has to provide a deductively valid
or an inductively strong argument to the effect that our inductively
strong arguments will continue to lead us to true conclusions (most of
the time) in the future. (An argument consists of a set of premises
P1, …, Pn and a conclusion C. Such an argument is deductively valid
just in case the truth of the premises guarantees the truth of the
conclusion. There is no standard definition of an inductively strong
argument, but the idea is that the premises speak in favor of or
support the conclusion.) But there is no deductively valid argument
whose premises are restricted to the past and present and whose
conclusion is about the future – and all our knowledge is about the
past and present. On the other hand, an inductively strong argument
presumably has to be inductively strong in the very sense of our
inductive practices – and thus begs the question. For more see the
introductory Skyrms (2000), the intermediate Hacking (2001), and the
advanced Howson (2000a).
Neglecting the background information B, as we will mostly do in the
following, we can state the link between induction and confirmation as
follows. The conclusion H of an inductively strong argument with
premise E is confirmed by E. If r quantifies the strength of the
inductive argument in question, the degree of confirmation of H by E
is equal to r. Let us then start the discussion of confirmation by the
first serious attempts to define the notion, and to develop a
corresponding logic of confirmation.
2. Hempel and the Logic of Confirmation
a. The Ravens Paradox
According to the Nicod criterion of confirmation (Hempel 1945),
universal generalizations of the form "All Fs are Gs," in symbols
∀x(Fx →Gx), are confirmed by their "instances" " This particular
object a is both F and G," Fa ∧ Ga. (It would be more appropriate to
call Fa → Ga rather than Fa ∧ Ga an instance of ∀x(Fx→ Gx).) The
universal generalization "All ravens are black" is thus said to be
confirmed by its instance "a is a black raven." As "a is a non-black
non-raven" is an instance of "All non-black things are non-ravens,"
the Nicod criterion says that "a is a non-black non-raven" confirms
"All non-black things are non-ravens." (It is sometimes said that a
black raven confirms the ravens hypothesis "All ravens are black." In
this case, confirmation is a relation between a non-linguistic entity
– namely, a black raven – and a hypothesis. I decided to construe
confirmation as a relation between, among other things, evidential
propositions and hypotheses, and so we have to state the above in a
clumsier way.)
One of Hempel's conditions of adequacy for any relation of
confirmation is the equivalence condition. It says that logically
equivalent hypotheses are confirmed by the same evidential
propositions. "All ravens are black" is logically equivalent to "All
non-black things are non-ravens." Therefore a non-black non-raven like
a white shoe or a red herring can be used to confirm the
ravens-hypothesis "All ravens are black." Surely, this is absurd – and
this is known as the ravens paradox.
Even worse, "All ravens are black,"∀x(Rx → Bx), is logically
equivalent to "All things that are green or not green are not ravens
or black,"∀x(Gx∨¬Gx → ¬Rx∨Bx). "a is green or not green, and a is not
raven or black" is an instance of this hypothesis. Furthermore, it is
logically equivalent to "a is not a raven or a is black." As
everything is green or not green, we get the similarly paradoxical
result that an object which is not a raven or which is black –
anything but a non-black raven which could be used to falsify the
ravens hypothesis is such an object – can be used to confirm the
ravens hypothesis that all ravens are black.
Hempel (1945), who discussed these cases of the ravens, concluded that
non-black non-ravens (as well as any other object that is not a raven
or black) can indeed be used to confirm the ravens hypothesis. He
attributed the paradoxical character of this alleged paradox to the
psychological fact that we assume there to be far more non-black
objects than ravens. However, the notion of confirmation he was
explicating was supposed to presuppose no background knowledge
whatsoever. An example by Good (1967) shows that such an unrelativized
notion of confirmation is not useful (see Hempel 1967, Good 1968).
Others have been led to the rejection of the Nicod criterion. Howson
(2000b, 113) considers the hypothesis "Everybody in the room leaves
with somebody else's hat," which he attributes to Rosenkrantz (1981).
If the background contains the information that there are only three
individuals a, b, c in the room, then the evidence consisting of the
two instances "a leaves with b's hat" and "b leaves with a's hat"
falsifies rather than confirms the hypothesis. Besides pointing to the
role played by the background information in this example, Hempel
would presumably have stressed that the Nicod criterion has to be
restricted to universal generalization in one variable only. Already
in his (1945, 13: fn. 1) he notes that R(a, b)∧ ¬R(a, b)
falsifies∀x∀y(¬[R(x, y)∧ R(y, x)] → R(x, y)∧ ¬R(x, y)), which is
equivalent to∀x∀xR(x, y), although it satisfies both the antecedent
and the consequent of the universal generalization (cf. also Carnap
1950/1962, 469f).
b. The Logic of Confirmation
After discussing the ravens, Hempel (1945) considers the following
conditions of adequacy for any relation of confirmation:
1. Entailment Condition: If an evidential proposition E logically
implies some hypothesis H, then E confirms H.
2. Special Consequence Condition: If an evidential proposition E
confirms some hypothesis H, and if H logically implies some hypothesis
H', then E also confirms H'.
3. Special Consistency Condition: If an evidential proposition E
confirms some hypothesis H, and if H is not compatible with some
hypothesis H', then E does not confirm H'.
4. Converse Consequence Condition: If an evidential proposition E
confirms some hypothesis H, and if H is logically implied by some
hypothesis H', then E also confirms H'.
(The equivalence condition mentioned above follows from 2 as well as
from 4). Hempel then shows that any relation of confirmation
satisfying 1, 2, and 4 is trivial in the sense that every evidential
proposition E confirms every hypothesis H. This is easily seen as
follows. As E logically implies itself, E confirms E according to the
entailment condition. The conjunction of E and H, E∧ H, logically
implies E, and so the converse consequence condition entails that E
confirms E∧ H. But E∧ H logically implies H; thus E confirms H by the
special consequence condition. In fact, it suffices that confirmation
satisfies 1 and 4 in order to be trivial: E logically implies and, by
1, confirms the disjunction of E and H, E∨H. As H logically implies
E∨H, E confirms H by 4.
Hempel (1945) rejects the converse consequence condition as the
culprit rendering trivial any relation of confirmation satisfying 1-4.
The latter condition has nevertheless gained popularity in the
philosophy of science – partly because it seems to be at the core of
the account of confirmation we will discuss next.
3. Popper's Falsificationism and Hypothetico-Deductive Confirmation
a. Popper's Falsificationism
Although Popper was an opponent of any kind of induction, his
falsificationism gave rise to a qualitative account of confirmation.
Popper started by observing that many scientific hypotheses have the
form of universal generalizations, say "All metals conduct
electricity." Now there can be no amount of observational data that
would verify a universal generalization. After all, the next piece of
metal could be such that it does not conduct electricity. In order to
verify this hypothesis we would have to investigate all pieces of
metal there are – and even if there were only finitely many such
pieces, we would never know this (unless there were only finitely many
space-time regions we would have to search). However, Popper's basic
insight is that these universal generalization can easily be
falsified. We only need to find a piece of metal that does not conduct
electricity in order to know that our hypothesis is false (supposing
we can check this). Popper then generalized this. He suggested that
all science should put forth bold hypotheses, which are then severely
tested (where bold means to have a high degree of falsifiability, in
other words, to have many observational consequences). As long as
these hypotheses survive their tests, scientists should stick to them.
However, once they are falsified, they should be put aside if there
are competing hypotheses that remain unfalsified.
This is not the place to list the numerous problems of Popper's
falsificationism. Suffice it to say that there are many scientific
hypotheses that are neither verifiable nor falsifiable (for example,
"Each planet has a moon"), and that falsifying instances are often
taken to be indicators of errors that lie elsewhere, say errors of
measurement or errors in auxiliary hypotheses. As Duhem and Quine
noted, confirmation is holistic in the sense that it is always a whole
battery of hypotheses that is put to test, and the arrow of error
usually does not point to a single hypothesis (Duhem 1906/1974, Quine
1953).
According to Popper's falsificationism (see Popper 1935/1994) the
hallmark of scientific (rather than meaningful, as in the early days
of logical positivism) hypotheses is that they are falsifiable:
scientific hypotheses must have consequences whose truth or falsity
can in principle (and with a grain of salt) be ascertained by
observation (with a grain of salt, because for Popper there is always
an element of conventionalism in stipulating the basis of science). If
there are no conditions under which a given hypothesis is false, this
hypothesis is not scientific (though it may very well be meaningful).
b. Hypothetico-Deductive Confirmation
The hypothetico-deductive notion of confirmation says that an
evidential proposition E confirms a hypothesis H relative to
background information B if and only if the conjunction of H and B, H∧
B, logically implies E in some suitable way (which depends on the
particular version of hypothetic-deductivism under consideration). The
intuition here is that scientific hypotheses are tested; and if a
hypothesis H survives a severe test, then, intuitively, this is
evidence in favor of H. Furthermore, scientific hypothesis are often
used for predictions. If a hypothesis H correctly predicts some
experimental outcome E by logically implying it, then, intuitively,
this is again evidence for the truth of H. Both of these related
aspects are covered by the above definition, if surviving a test is
tantamount to entailing the correct outcome.
Note that hypthetico-deductive confirmation – henceforth
HD-confirmation – satisfies Hempel's converse consequence condition.
Suppose an evidential proposition E HD-confirms some hypothesis H.
This means that H logically implies E is some suitable way. Now any
hypothesis H' which logically implies H also logically implies E. But
this means – at least under most conditions fixing the "suitable way"
of entailment – that E HD-confirms H.
Hypothetico-deductivism has run into serious difficulties. To mention
just two, there is the problem of irrelevant conjunctions and the
problem of irrelevant disjunctions. Suppose an evidential proposition
E HD-confirms some hypothesis H. Then, by the converse consequence
condition, E also HD-confirms H∧ H', for any hypothesis H' whatsoever.
Assuming that the anomalous perihelion of Mercury confirms the general
theory of relativity GTR (Earman 1992), it also confirms the
conjunction of GTR and, say, that there is life on Mars – which seems
to be wrong. Similarly, if E HD-confirms H, then E∨E' HD-confirms H,
for any evidential proposition E' whatsoever. For instance, the
disjunctive proposition of the anomalous perihelion of Mercury or
Luca's living on the second floor HD-confirms GTR (Grimes 1990,
Moretti 2004).
Another worry with HD-confirmation is that it is not clear how it
should be applied to statistical hypotheses that do not strictly
entail anything (see, however, Albert 1992). The treatment of
statistical hypotheses is no problem for probabilistic theories of
confirmation, which we will turn to now.
4. Inductive Logic
For overview articles see Fitelson (2005) and Hawthorne (2005).
a. Kolmogorov's Axiomatization
Before we turn to inductive logic, let us define the notion of
probability as it was axiomatized by Kolmogorov (1933; 1956).
Let W be a non-empty set (of outcomes or possibilities), and let A be
a field over W, that is, a set of subsets of W that contains the whole
set W and is closed under complementation (with respect to W) and
finite unions. That is, A is a field over W if and only if A is a set
of subsets of W such that
(i) W ∈A
(ii) if A ∈A, then (WA) = -A ∈A
(iii) if A ∈A and B ∈A, then (A∪B) ∈A
If (iii) is strengthened to
(iv) if A1∈A, … An ∈A, …, then (A1∪…∪An∪…) ∈A,
so that A is closed under countable (and not only finite) unions, A is
called a σ-field over W.
A function Pr: A → ℜ from the field A over W into the real numbers ℜ
is a (finitely additive) probability measure on A if and only if it is
a non-negative, normalized, and (finitely) additive measure; that is,
if and only if for all A, B ∈A
(K1) Pr(A) ≥ 0
(K2) Pr(W) = 1
(K3) if A∩B = ∅ , then Pr(A∪B) = Pr(A) + Pr(B)
The triple <W, A, Pr> with W a non-empty set, A a field over W, and Pr
a probability measure on A is called a (finitely additive) probability
space. If A is a σ-field over W and Pr: A → ℜ additionally satisfies
(K4) if A1⊇ A2 ⊇ … ⊇ An … is a decreasing sequence of elements of
A, i.e. A1∈A, … An ∈A, …, such that A1∩A2∩…∩An∩… = ∅ , then limn→∞
Pr(An) = 0,
Pr is a σ-additive probability measure on A and <W, A, Pr> is a
σ-additive probability space (Kolmogorov 1933; 1956, ch. 2). (K4)
asserts that
limn→∞ Pr(An) = Pr(A1∩A2∩…∩An∩…) = Pr(∅ ) = 0
for a decreasing sequence of elements of A. Given (K1-3), (K4) is equivalent to
(K5) if A1∈A, … An ∈A, …, and if Ai∩Aj= ∅ for all natural numbers
i, j with i ≠ j, then Pr(A1∪…∪An∪…) = Pr(A1) + … + Pr(An) + …
A probability measure Pr: A → ℜ on A is regular just in case Pr(A) > 0
for every non-empty A ∈A. Let <W, A, Pr> be a probability space, and
define A* to be the set of all A ∈A that have positive probability
according to Pr, that is, A* = {A ∈A: Pr(A) > 0}. The conditional
probability measure Pr(•|-): A x A* → ℜ on A (based on the
unconditional probability measure Pr) is defined for all A ∈A and B∈A*
by the fraction
(K6) Pr(A|B) = Pr(A∩B)/Pr(B)
(Kolmogorov 1933; 1956, ch. 1, §4). The domain of the second argument
place of Pr(•|-) has to be restricted to A*, since the fraction
Pr(A∩B)/Pr(B) is not defined when Pr(B) = 0. Note that Pr(•|B): A → ℜ
is a probability measure on A, for every B ∈A*.
Here are some immediate consequences of the Kolmogorov axioms and the
definition of conditional probability. For every probability space <W,
A, Pr> and all A, B ∈A,
Law of Negation: Pr(-A)= 1 – Pr(A)
Law of Conjunction: Pr(A∩B) = Pr(B)•Pr(A|B) whenever Pr(B) > 0
Law of Disjunction: Pr(A∪B) = Pr(A) + Pr(B) – Pr(A∩B)
Law of Total Probability: Pr(B) = ΣiPr(B|Ai)•Pr(Ai),
where the Ai form a countable partition of W, i.e. A1, … An, … is a
sequence of mutually exclusive (Ai∩Aj= ∅ for all i, j with i ≠ j) and
jointly exhaustive (A1∪…∪An∪… = W) elements of A. A special case of
the Law of Total Probability is
Pr(B) = Pr(B|A)•Pr(A) + Pr(B|-A)•Pr(-A).
Finally the definition of conditional probability is easily turned into
Bayes's Theorem: Pr(A|B) = Pr(B|A)•Pr(A)/Pr(B)
= Pr(B|A)•Pr(A)/[Pr(B|A)•Pr(A) + Pr(B|-A)•Pr(-A)]
= Pr(B|A)•Pr(A)/ΣiPr(B|Ai)•Pr(Ai),
where the Ai form a countable partition of W. The important role
played by Bayes's Theorem (in combination with some principle linking
objective chances and subjective probabilities) for confirmation will
be discussed below. For more on Bayes's Theorem see Joyce (2003).
The names of the first three laws above already indicate that
probability measures can also be defined on formal languages. Instead
of defining probability on a field A over some non-empty set W, we can
take its domain to be a formal language L, that is, a set of (possibly
open) well-formed formulas that contains the tautological sentence τ
(corresponding to the whole set W) and is closed under negation ¬
(corresponding to complementation) and disjunction ∨ (corresponding to
finite union). That is, L is a language if and only if L is a set of
well-formed formulas such that
(i) τ ∈L
(ii) if α ∈L, then ¬α ∈L
(iii) if α ∈L and β ∈L, then (α∨β) ∈ L
If L additionally satisfies
(iv) if α ∈L, then ∃xα ∈ L,
L is called a quantificational language.
A function Pr: L → ℜ from the language L into the reals ℜ is a
probability on L if and only if for all α, β ∈L,
(L0) Pr(α) = Pr(β) if α is logically equivalent (in the sense of
classical logic CL) to β
(L1) Pr(α) ≥ 0,
(L2) Pr(τ) = 1,
(L3) Pr(α∨β) = Pr(α) + Pr(β), if α∧ β is logically inconsistent
(in the sense of CL).
(L0) is not necessary, if (L2) is strengthened to: (L2+) Pr(α) = 1, if
α is logically valid. If L is a quantificational language with an
individual constant "ai" for each individual ai in the envisioned
countable domain, i = 1, 2, …, n, …, and Pr: L → ℜ additionally
satisfies
(L4) limn→∞Pr(α[a1/x]∧ …βα[an/x]) = Pr(∀xα),
Pr is called a Gaifman-Snir probability. Here "α[ai/x]" results from
"α[x]" by substituting the individual constant "ai" for all
occurrences of the individual variable "x" in "α." "x" in "α[x]"
indicates that "x" occurs free in "α," that is to say, "x" is not
bound in "α" by a quantifier like it is in "∀xα."
Given (L0-3) and the restriction to countable domains, (L4) is equivalent to
(L5) limn→∞Pr(α[a1/x]∨…∨α[an/x]) = sup{Pr(α[a1/x]∨…∨α[an/x]): n ∈N} =
Pr(∃xα),
where the equation on the right-hand side is the slightly more general
definition adopted by Gaifman & Snir (1982, 501). A probability Pr: L
→ ℜ on L is regular just in case Pr(α) > 0 for every consistent α ∈L.
For L* = {α ∈L: Pr(α) > 0} the conditional probability Pr(•|-): L x L*
→ ℜ on L (based on Pr) is defined for all α ∈L and all β ∈L* by the
fraction
(L6) Pr(α|β) = Pr(α∧ β)/Pr(β).
As before, Pr(•|β): L → ℜ is a probability on L, for every β ∈L.
Each probability Pr on a language L induces a probability space <W, A,
Pr*> with W the set Mod of all models for L, A the smallest σ-field
containing the field {Mod(α) ⊆Mod: α ∈L}, and Pr* the unique
σ-additive probability measure on A such that Pr*(Mod(α)) = Pr(α) for
all α ∈L. (A model for a language L with an individual constant for
each individual in the envisioned domain can be represented by a
function w: L → {0,1} from L into the set {0,1} such that for all α, β
∈L: w(¬α) = 1 – w(α), w(α∨β) = max{w(α), w(β)}, and w(∃xα) =
max{w(α[a/x]): "a" is an individual constant of L}.)
In conclusion, it is to be noted that some authors take conditional
probability Pr(• given -) as primitive and define probability as Pr(•
given W) or Pr(• given τ) (see Hájek 2003b). For more on probability
and its interpretations see Hájek (2003a), Hájek & Hall (2000),
Fitelson & Hájek & Hall (2005).
b. Logical Probability and Degree of Confirmation
There has always been a close connection between probability and
induction. Probability was thought to provide the basis for an
inductive logic. Early proponents of a logical conception of
probability include Keynes (1921/1973) and Jeffreys (1939/1967).
However, by far the biggest effort to construct an inductive logic was
undertaken by Carnap in his Logical Foundations of Probability
(1950/1962). Carnap starts from a simple formal language with
countably many individual constants (such as "Carl Gustav Hempel")
denoting individuals (namely, Carl Gustav Hempel) and finitely many
monadic predicates (such as "is a great philosopher of science")
denoting properties (namely, being a great philosopher of science),
but not relations (such as being a better philosopher of science
than). Then he defines a state-description to be a complete
description of each individual with respect to all the predicates. For
instance, if the language contains three individual constants "a,"
"b," and "c" (denoting the individuals a, b, and c, respectively), and
four monadic predicates "P," "Q," "R," and "S" (denoting the
properties P, Q, R, and S, respectively), then there are 23•4 state
descriptions of the form:
±Pa ∧ ±Qa ∧ ±Ra ∧ ±Sa∧ ±Pb∧ ±Qb∧ ±Rb ∧
±Sb ∧ ±Pc∧ ±Qc∧ ±Rc∧ ±Sc,
where "±" indicates that the predicate in question is either unnegated
as in "Pa" or negated as in "¬Pa." That is, a state description
determines for each individual constant "a" and each predicate "P"
whether or not Pa. Based on the notion of a state description, Carnap
then introduces the notion of a structure description, a maximal
disjunction of state descriptions which can be obtained from each
other by uniformly substituting individual constants for each other.
In the above example there are, among others, the following structure
descriptions:
(Pa∧ Qa∧ Ra∧ Sa)∧ (Pb∧ Qb∧ Rb∧ Sb)∧ (Pc∧ Qc∧ Rc∧ Sc)
((Pa∧ Qa ∧ Ra ∧ Sa) ∧ (Pb ∧ Qb ∧ Rb ∧ ¬Sb) ∧ (Pc ∧ Qc ∧ ¬Rc ∧ Sc))
∨((Pb ∧ Qb ∧ Rb ∧ Sb) ∧ (Pa ∧ Qa ∧ Ra ∧ ¬Sa) ∧ (Pc ∧ Qc ∧ ¬Rc ∧ Sc))
∨((Pc ∧ Qc ∧ Rc ∧ Sc) ∧ (Pb ∧ Qb ∧ Rb ∧ ¬Sb) ∧ (Pa ∧ Qa ∧ ¬Ra ∧ Sa))
∨((Pa ∧ Qa ∧ Ra ∧ Sa) ∧ (Pc ∧ Qc ∧ Rc ∧ ¬Sc) ∧ (Pb ∧ Qb ∧ ¬Rb ∧ Sb))
So a structure description is a disjunction of one or more state
descriptions. It says how many individuals satisfy the maximally
consistent predicates (Carnap calls them Q-predicates) that can be
formulated in the language. It may but need not say which individuals.
The first structure description above says that all three individuals
a, b, and c have the maximally consistent property Px∧ Qx∧ Rx∧ Sx. The
second structure description says that exactly one individual has the
maximally consistent property Px∧ Qx∧ Rx∧ Sx, exactly one individual
has the maximally consistent property Px∧ Qx∧ Rx∧ ¬Sx, and exactly one
individual has the maximally consistent property Px∧ Qx∧ ¬Rx∧ Sx. It
does not say which of a, b, and c as the property in question.
Each function that assigns non-negative weights wi to the state
descriptions zi whose sum Σiwi equals 1 induces a probability on the
language in question. Carnap then argues – by postulating various
principles of symmetry and invariance – that each of the finitely many
structure (not state) descriptions sj should be assigned the same
weight vj such that their sum Σjvj is equal to 1. This weight vj
should then be divided equally among the state descriptions whose
disjunction constitutes the structure description sj. The probability
so obtained is Carnap's favorite m*, which, like any other
probability, induces what Carnap calls a confirmation function (and we
have called a conditional probability): c*(H, E) = m*(H∧ E)/m*(E)
(In case the language contains countably infinitely many individual
constants, some structure descriptions are disjunctions of infinitely
many state descriptions. These state descriptions cannot all get the
same positive weight. Therefore Carnap considers the limit of the
measures m*nfor the languages Ln containing the first n individual
constants in some enumeration of the individual constants, provided
this limit exists.)
c* allows learning from experience in the sense that
c*(the n + 1st individual is P, k of the first n individuals are P) >
c*(the n + 1st individual is P, τ)
= m*(the n + 1st individual is P),
where τ is the tautological sentence. If we assigned equal weights to
the state descriptions instead of the structure descriptions, no such
learning would be possible. Let us check that c* allows learning from
experience for n = 2 in a language with three individual constants
"a," "b," and "c" and one predicate "P." There are eight state
descriptions and four structure descriptions:
z1 = Pa∧ Pb∧ Pc s1 = Pa∧ Pb∧ Pc: All three individuals are P.
z2 = Pa∧ Pb∧ ¬Pc s2 = (Pa∧ Pb∧ ¬Pc)∨(Pa∧ ¬Pb∧ Pc)∨(¬Pa∧ Pb∧ Pc):
z3 = Pa∧ ¬Pb∧ Pc Exactly two individuals are P.
z4 = Pa∧ ¬Pb∧ ¬Pc s3 = (Pa∧ ¬Pb∧ ¬Pc)∨(¬Pa∧ Pb∧ ¬Pc)∨(¬Pa∧ ¬Pb∧ Pc):
z5 = ¬Pa∧ Pb∧ Pc Exactly one individual is P.
z6 = ¬Pa∧ Pb∧ ¬Pc
z7 = ¬Pa∧ ¬Pb∧ Pc
z8 = ¬Pa∧ ¬Pb∧ ¬Pc s4 = ¬Pa∧ ¬Pb∧ ¬Pc: None of the three individuals is P.
Each structure description s1-s4 gets weight vj = 1/4 (j = 1, …, 4).
s1 = z1: v1 = m*(Pa∧ Pb∧ Pc) = 1/4
s2 = z2∨z3∨z5: v2 = m*((Pa∧ Pb∧ ¬Pc)∨(Pa∧ ¬Pb∧ Pc)∨(¬Pa∧ Pb∧ Pc)) = 1/4
s3 = z4∨z6∨z7: v3 = m*((Pa∧ ¬Pb∧ ¬Pc)∨(¬Pa∧ Pb∧ ¬Pc)∨(¬Pa∧ ¬Pb∧ Pc)) = 1/4
s4 = z8: v4 = m*(¬Pa∧ ¬Pb∧ ¬Pc) = 1/4
These weights are equally divided among the state descriptions z1-z8.
z1: w1 = m*(Pa∧ Pb∧ Pc) = 1/4 z5: w5 = m*(¬Pa∧ Pb∧ Pc) = 1/12
z2: w2 = m*(Pa∧ Pb∧ ¬Pc) = 1/12 z6: w6 = m*(¬Pa∧ Pb∧ ¬Pc) = 1/12
z3: w3 = m*(Pa∧ ¬Pb∧ Pc) = 1/12 z7: w7 = m*(¬Pa∧ ¬Pb∧ Pc) = 1/12
z4: w4 = m*(Pa∧ ¬Pb∧ ¬Pc) = 1/12 z8: w8 = m*(¬Pa∧ ¬Pb∧ ¬Pc) = 1/4
Let us now compute the values of the confirmation function c*.
c*(the 3rd individual is P, 2 of the first 2 individuals are P) =
= m*(the 3rd individual is P ∧ the first 2 individuals are
P)/m*(the first 2 individuals are P)
= m*(the first 3 individuals are P)/m*(the first 2 individuals are P)
= m*(Pa∧ Pb∧ Pc)/m*(Pa∧ Pb)
= (1/4)/(1/4 + 1/12)
= 3/4
> 1/2 = m*(Pc) = c* (the 3rd individual is P)
The general formula is (Carnap 1950/1962, 568)
c*(the n + 1st individual is P, k of the first n individuals are P)
= (k + ')/(n + κ)
= (k + ('/κ)∈κ)/(n + κ),
where ' is the "logical width" of the predicate "P" (Carnap 1950/1962,
127), that is, the number of maximally consistent properties or
Q-predicates whose disjunction is logically equivalent to "P" (' = 1
in our example: "P'). κ = 2π is the total number of Q-predicates (κ =
21 = 2 in our example: "P" and "¬P') with π being the number of
primitive predicates (π = 1 in our example: "P'). This formula is
dependent on the logical factor '/κ of the "relative width" of the
predicate "P," and the empirical factor k/n of the relative frequency
of Ps.
Later on, Carnap (1952) generalizes this to a whole continuum of
confirmation functions Cλ where the parameter λ is inversely
proportional to the impact of evidence. λ specifies how the
confirmation function Cλ weighs between the logical factor '/κ and the
empirical factor k/n. For λ = ∞, Cλ is independent of the empirical
factor k/n: Cλ(the n + 1st individual is P, k of the first n
individuals are P) = '/κ (Carnap 1952, §13). For λ = 0, Cλ is
independent of the logical factor '/κ: Cλ(the n + 1st individual is P,
k of the first n individuals are P) = k/n and thus coincides with what
is known as the straight rule (Carnap 1952, §14). c*is the special
case with λ = κ (Carnap 1952, §15). The general formula is (Carnap
1952, §9)
Cλ(the n + 1st individual is P, k of the first n individuals are
P) = (k + λ/κ)/(n + λ).
In his (1963) Carnap slightly modifies the set up and considers
families of monadic predicates {'P1," …, "Pp'} like the family of
color predicates {'red," "green," …, "blue'}. For a given family
{'P1," …, "Pp'} and each individual constant "a" there is exactly one
predicate "Pj" such that Pja. Families thus generalize {'P," "¬P'} and
correspond to random variables. Given his axioms (including A15)
Carnap (1963, 976) shows that for each family {'P1," …, "Pp'}, p ≥ 2,
Cλ(the n + 1st individual is Pj, k of the first n individuals are Pj)
= (k + λ/p)/(n + λ).
One of the peculiar features of Carnap's systems is that universal
generalizations get degrees of confirmation (alias conditional
probability) 0. Hintikka (1966) further elaborates Carnap's project in
this respect. For a neo-Carnapian approach see Maher (2004a).
Of more interest to us is Carnap's discussion of "the controversial
problem of the justification of induction" (1963, 978, emphasis in the
original). For Carnap, the justification of induction boils down to
justifying the axioms specifying a set of confirmation functions. The
"reasons are based upon our intuitive judgments concerning inductive
validity". Therefore "[i]t is impossible to give a purely deductive
justification of induction," and these "reasons are a priori" (Carnap
1963, 978). So according to Carnap, induction is justified by appeals
to intuition about inductive validity. We will see below that Goodman,
who is otherwise very skeptical about the prospects of Carnap's
project, shares this view of the justification of induction. In fact,
the view also seems to be widely accepted among current Bayesian
confirmation theorists and their desideratum/explicatum approach (see
Fitelson 2001 for an example). [According to Carnap (1952, ch. I), an
explication is "the transformation of an inexact, prescientific
concept, the explicandum, into a new exact concept, the explicatum."
(Carnap 1952, 3) The desideratum/explicatum approach consists in
stating various "intuitively plausible desiderata" the explicatum is
supposed to satisfy. Proposals for explicata that do not satisfy these
desiderata are rejected. This appeal to intuitions is fine as long as
we are doing conceptual analysis. However, contemporary confirmation
theorists also sell their accounts as normative theories. Normative
theories are not justified by appeal to intuitions, though. They are
justified relative to a goal by showing that the norms in question
further the goal at issue. See section 7.]
First, however, we will have a look at what Carnap has to say about
Hempel's conditions of adequacy.
c. Absolute and Incremental Confirmation
As we saw in the preceding section, one of Carnap's goals was to
define a quantitative notion of confirmation, explicated by a
confirmation function in the manner indicated above. It is important
to note that this quantitative concept of confirmation is a relation
between two propositions H and E (three, if we include the background
information B), a number r, and a confirmation function c. In chapters
VI and VII of his (1950/1962) Carnap discusses comparative and
qualitative concepts of confirmation. The explicans for qualitative
confirmation he offers is that of positive probabilistic relevance in
the sense of some logical probability m. That is, E qualitatively
confirms H in the sense of some logical measure m just in case E is
positively relevant to H in the sense of m, that is,
m(H∧ E) > m(H)•m(E).
If both m(H) and m(E) are positive – which is the case whenever both H
and E are not logically false, because Carnap assumes m to be regular
– this is equivalently expressed by the following inequality:
c(H, E) > c(H, τ) = m(H)
So provided both H and E have positive probability, E confirms H if
and only if E raises the conditional probability (degree of
confirmation in the sense of c) of H. Let us call this concept
incremental confirmation. Again, note that qualitative confirmation is
a relation between two propositions H and E, and a conditional
probability or confirmation function c. Incremental confirmation, or
positive probabilistic relevance, is a qualitative notion, which says
whether E raises the conditional probability (degree of confirmation
in the sense of c) of H. Its natural quantitative counterpart measures
how much E raises the conditional probability of H. This measure may
take several forms which will be discussed below.
Incremental confirmation is different from the concept of absolute
confirmation on which it is based. The quantitative explication of
absolute confirmation is given by one of Carnap's confirmation
functions c. The qualitative counterpart is to say that E absolutely
confirms H in the sense of c if and only if the degree of absolute
confirmation of H by E is sufficiently high, c(H, E) > r. So Carnap,
who offers degree of absolute confirmation c(H, E) as explication for
the quantitative notion of confirmation of H by E, and who offers
incremental confirmation or positive probabilistic relevance between E
and H as explication of the qualitative notion of confirmation, is, to
say the least, not fully consistent in his terminology. He switches
between absolute confirmation (for the quantitative notion) and
incremental confirmation (for the qualitative notion). This is
particularly peculiar, because Carnap (1950/1962, §87) is the locus
classicus for the discussion of Hempel's conditions of adequacy
mentioned in section 2b.
d. Carnap's Analysis of Hempel's Conditions
In analyzing the special consequence condition, Carnap argues that
Hempel has in mind as explicandum the following relation: "the degree
of confirmation of H by E is greater than r, where r is a fixed value,
perhaps 0 or 1/2 (Carnap 1962, 475; notation adapted);
that is, the qualitative concept of absolute confirmation. Similarly
when discussing the special consistency condition:
Hempel regards it as a great advantage of any explicatum satisfying [a
more general form of the special consistency condition 3] "that it
sets a limit, so to speak, to the strength of the hypotheses which can
be confirmed by given evidence" … This argument does not seem to have
any plausibility for our explicandum, (Carnap 1962, 477; emphasis in
original)
which is the qualitative concept of incremental confirmation,
[b]ut it is plausible for the second explicandum mentioned earlier:
the degree of [absolute] confirmation exceeding a fixed value r.
Therefore we may perhaps assume that Hempel's acceptance of [a more
general form of 3] is due again to an inadvertent shift to the second
explicandum. (Carnap 1962, 477-478)
Carnap's analysis can be summarized as follows. In presenting his
first three conditions of adequacy, Hempel was mixing up two distinct
concepts of confirmation, two distinct explicanda in Carnap's
terminology, namely,
(i) the qualitative concept of incremental confirmation (positive
probabilistic relevance) according to which E confirms H if and only
if E (has non-zero probability and) increases the degree of absolute
confirmation (conditional probability) of H, and
(ii) the qualitative concept of absolute confirmation according to
which E confirms H if and only if the degree of absolute confirmation
(conditional probability) of H by E is greater than some value r.
Hempel's second and third condition, 2 and 3, respectively, hold true
for the second explicandum (for r ≥ 1/2), but they do not hold true
for the first explicandum. On the other hand, Hempel's first condition
holds true for the first explicandum, but it does so only in a
qualified form (Carnap 1950/1962, 473) – namely only if E is not
assigned probability 0, and H is not already assigned probability 1.
This, however, means that, according to Carnap's analysis, Hempel
first had in mind the explicandum of incremental confirmation for the
entailment condition. Then he had in mind the explicandum of absolute
confirmation for the special consequence and the special consistency
conditions 2 and 3, respectively. And then, when Hempel presented the
converse consequence condition, he got completely confused and had in
mind still another explicandum or concept of confirmation (neither the
first nor the second explicandum satisfies the converse consequence
condition). This is not a very charitable analysis. It is not a good
one either, because the qualitative concept of absolute confirmation,
which Hempel is said to have had in mind for 2 and 3, also satisfies 1
– and it does so without the second qualification that H be assigned a
probability smaller than 1. So there is no need to accuse Hempel of
mixing up two concepts of confirmation. Indeed, the analysis is bad,
because Carnap's reading of Hempel also leaves open the question of
what the third explicandum for the converse consequence condition
might have been. For a different analysis of Hempel's conditions and a
corresponding logic of confirmation see Huber (2007a), respectively.
5. The New Riddle of Induction and the Demise of the Syntactic Approach
According to Goodman (1983, ch. III), the problem of justifying
induction boils down to defining valid inductive rules, and thus to a
definition of confirmation. The reason is that an inductive inference
is justified by conformity to an inductive rule, and inductive rules
are justified by their conformity to accepted inductive practices. One
does not have to follow Goodman in this respect, however, in order to
appreciate his insight that whether a hypothesis is confirmed by a
piece of evidence depends on features other than their syntactical
form.
In his (1946) he asks us to suppose a marble has been drawn from a
certain bowl on each of the ninety-nine days up to and including VE
day, and that each marble drawn was red. Our evidence can be described
by the conjunction "Marble 1 is red and … and marble 99 is red," in
symbols: Ra1∧ …∧ Ra99. Whatever the details of our theory of
confirmation, this evidence will confirm the hypothesis "Marble 100 is
red," R100. Now consider the predicate S = "is drawn by VE day and is
red, or is drawn after VE day and is not red." In terms of S rather
than R our evidence is described by the conjunction "Marble 1 is drawn
by VE day and is red or it is drawn after VE day and is not red, and
…, and marble 99 is drawn by VE day and is red or it is drawn after VE
day and is not red," Sa1∧ …∧ Sa99. If our theory of confirmation
relies solely on syntactical features of the evidence and the
hypothesis, our evidence will confirm the conclusion "Marble 100 is
drawn by VE and is red, or it is drawn after VE day and is not red,"
S100. But we know that the next marble will be drawn after VE day.
Given this, S100 is logically equivalent to the negation of R100. So
one and the same piece of evidence can be used to confirm a hypothesis
and its negation, which is certainly absurd.
One might object to this example that the two formulations do not
describe one and the same piece of evidence after all. The first
formulation in terms of R should be the conjunction "Marble 1 is drawn
by VE day and is red, and …, and marble 99 is drawn by VE day and is
red," (Da1∧ Ra1)∧ …∧ (Da99∧ Ra99). The second formulation in terms of
S should be "Marble 1 is drawn by VE day and it is drawn by VE day and
red or drawn after VE and not red, and …, and marble 99 is drawn by VE
day and it is drawn by VE day and red or drawn after VE day and not
red," (Da1∧ Sa1)∧ …∧ (Da99∧ Sa99). Now the two formulations really
describe one and the same piece of evidence in the sense of being
logically equivalent. But then the problem is whether any interesting
statement can ever be confirmed. The syntactical form of the evidence
now seems to confirm Da100∧ Ra100, equivalently Da100∧ Sa100. But we
know that the next marble is drawn after VE day; that is, we know
¬Da100. That the future resembles the past in all respects is thus
false. That it resembles the past in some respects is trivial. The new
riddle of induction is the question in which respects the future
resembles the past, and in which it does not.
It has been suggested that the puzzling character of Goodman's example
is due to its mentioning a particular point of time, namely, VE day. A
related reaction has been that gerrymandered predicates, whether or
not they involve a particular point of time, cannot be used in
inductive inferences. But there are plenty of similar examples
(Stalker 1994), and it is commonly agreed that Goodman has succeeded
in showing that a purely syntactical definition of (degree of)
confirmation won't do. Goodman himself sought to solve his new riddle
of induction by distinguishing between "projectible" predicates such
as "red" and unprojectible predicates such as "is drawn by VE day and
is red, or is drawn after VE day and is not red." The projectibility
of a predicate is in turn determined by its entrenchment in natural
language. This comes very close to saying that the projectible
predicates are the ones that we do in fact project (that is, use in
inductive inferences). (Quine's 1969 "natural kinds" are special cases
of what can be described by projectible predicates.)
6. Bayesian Confirmation Theory
Bayesian confirmation theory is by far the most popular and elaborated
theory of confirmation. It has its origins in Rudolf Carnap's work on
inductive logic (Carnap 1950/1962), but relieves itself from defining
confirmation in terms of logical probability. More or less any
subjective degree of belief function satisfying the Kolmogorov axioms
is considered to be an admissible probability measure.
a. Subjective Probability and the Dutch Book Argument
In Bayesian confirmation theory, a probability measure on a field of
propositions is usually interpreted as an agent's degree of belief
function. There is disagreement as to how broad the class of
admissible probability measures is to be construed. Some objective
Bayesians such as the early Carnap insist that the class consist of a
single logical probability measure, whereas subjective Bayesians admit
any probability measure. Most Bayesians will be somewhere in the
middle of this spectrum when it comes to the question which particular
degree of belief functions it is reasonable to adopt in a particular
situation. But they will agree that from a purely logical point of
view any (regular) probability measure is acceptable. The standard
argument for this position is the Dutch Book Argument.
The Dutch Book Argument starts with the assumption that there is a
link between subjective degrees of belief and betting ratios. It is
further assumed that it is pragmatically defective to accept a series
of bets which guarantees a sure loss, that is, a Dutch Book. By
appealing to the Dutch Book Theorem that an agent's betting ratios
satisfy the probability axioms just in case they do not make the agent
vulnerable to such a Dutch Book, it is inferred that it is
epistemically defective to have degrees of belief that violate the
probability axioms. The strength of this inference is, of course,
dependent on the link between degrees of belief and betting ratios. If
this link is identity – as it is when one defines degrees of belief as
betting ratios – the distinction between pragmatic and epistemic
defectiveness disappears, and the Dutch Book Argument is a deductively
valid argument. But this comes at the cost of rendering the link
between degrees of belief and betting ratios implausible. If the link
is weaker than identity – as it is when degrees of belief are only
measured by betting ratios – the Dutch Book Argument is not
deductively valid anymore, but it has more plausible assumptions.
The pragmatic nature of the Dutch Book Argument has led to so called
depragmatized versions. A depragmatized Dutch Book Argument starts
with a link between degrees of belief and fair betting ratios, and it
assumes that it is epistemically defective to consider a series of
bets that guarantees a sure loss as fair. Using the depragmatized
Dutch Book Theorem that an agent's fair betting ratios obey the
probability calculus if and only if the agent never considers a Dutch
Book as fair, it is then inferred that it is epistemically defective
to have degrees of belief that do not obey the probability calculus.
The thesis that an agent's degree of belief function should obey the
probability calculus is called probabilism. For more on the Dutch Book
Argument see Hájek (2005). For a different justification of
probabilism in terms of the accuracy of degrees of belief see Joyce
(1998).
b. Confirmation Measures
Let A be a field of propositions over some set of possibilities W, let
H, E, B be propositions from A, and let Pr be a probability measure on
A. We already know that H is incrementally confirmed by E relative to
B in the sense of Pr if and only if Pr(H∩E|B) > Pr(H|B)•Pr(E|B), and
that this is a relation between three propositions and a probability
space whose field contains the propositions. The central notion in
Bayesian confirmation theory is that of a confirmation measure. A real
valued function c: P → ℜ from the set P of all probability spaces <W,
A, Pr> into the reals ℜ is a confirmation measure if and only if for
every probability space <W, A, Pr> and all H, E, B ∈A:
c(H, E, B) > 0 ↔Pr(H∩E|B) > Pr(H|B)•Pr(E|B)
c(H, E, B) = 0 ↔Pr(H∩E|B) = Pr(H|B)•Pr(E|B)
c(H, E, B) < 0 ↔Pr(H∩E|B) < Pr(H|B)•Pr(E|B)
The six most popular confirmation measures are (what I now call) the
Carnap measure c (Carnap 1962), the distance measure d (Earman 1992),
the log-likelihood or Good-Fitelson measure l (Fitelson 1999 and Good
1983), the log-ratio or Milne measure r (Milne 1996), the
Joyce-Christensen measure s (Christensen 1999, Joyce 1999, ch. 6), and
the relative distance measure z (Crupi & Tentori & Gonzalez 2007).
c(H, E, B) = Pr(H∩E|B) – Pr(H|B)•Pr(E|B)
d(H, E, B) = Pr(H|E∩B) – Pr(H|B)
l(H, E, B) = log [Pr(E|H∩B)/Pr(E|-H∩B)]
r(H, E, B) = log [Pr(H|E∩B)/Pr(H|B)]
s(H, E, B) = Pr(H|E∩B) – Pr(H|-E∩B)
z(H, E, B) = [Pr(H|E∩B) – Pr(H|B)]/Pr(-H|B) if Pr(H|E∩B) ≥Pr(H|B)
= [Pr(H|E∩B) – Pr(H|B)]/Pr(H|B) if Pr(H|E∩B) < Pr(H|B)
(Mathematically speaking, there are uncountably many confirmation
measures.) For an overview article, see Eells (2005). Book length
expositions are Earman (1992) and Howson & Urbach (1989/2005).
c. Some Success Stories
Bayesian confirmation theory captures the insights of Popper's
falsificationism and hypothetico-deductive confirmation. Suppose
evidence E falsifies hypothesis H relative to background information B
in the sense that B∩H∩E = ∅ . Then Pr(E∩H|B) = 0, and so Pr(E∩H|B) = 0
< Pr(H|B)•Pr(E|B), provided both Pr(H|B) and Pr(E|B) are positive. So
as long as H is not already known to be false (in the sense of having
probability 0 conditional on B) and E is a possible outcome (one with
positive probability conditional on B), falsifying E incrementally
disconfirms H relative to B in the sense of Pr.
Remember, E HD-confirms H relative to B if and only if the conjunction
of H and B logically implies E (in some suitable way). In this case
Pr(E∩H|B) = Pr(H|B), provided Pr(B) > 0. Hence as long as Pr(E|B) < 1,
we have
Pr(E∩H|B) > Pr(H|B)•Pr(E|B),
which means that E incrementally confirms H relative to B in the sense
of Pr (Kuipers 2000).
If the conjunction of H and B logically implies E, but E is already
known to be true in the sense of having probability 1 conditional on
B, E does not incrementally confirm H relative to B in the sense of
Pr. In fact, no E which receives probability 1 conditional on B can
incrementally confirm any H whatsoever. This is the so called problem
of old evidence (Glymour 1980). It is a special case of a more general
phenomenon. The following is true for many confirmation measures (d,
l, and r, but not s). If H is positively relevant to E given B, the
degree to which E incrementally confirms H relative to B is greater,
the smaller the probability of E given B. Similarly, if H is
negatively relevant for E given B, the degree to which E disconfirms H
relative to B is greater, the smaller the probability of E given B
(Huber 2005a). If Pr(E|B) = 1 we have the problem of old evidence. If
Pr(E|B) = 0 we have the above mentioned problem that E cannot
disconfirm hypotheses it falsifies.
Some people simply deny that the problem of old evidence is a problem.
Bayesian confirmation theory, it is said, does not explicate whether
and how much E confirms H relative to B. It explicates whether E is
additional evidence for H relative to B, and how much additional
confirmation E provides for H relative to B. If E already has
probability 1 conditional on B, it is part of the background
knowledge, and so does not provide any additional evidence for H. More
generally, the more we already believe in E, the less additional
(dis)confirmation this provides for positively (negatively) relevant
H. This reply does not work in case E is a falsifier of H with
probability 0 conditional on B, for in this case Pr(H|E∩B) is not
defined. It also does not agree with the fact that the problem of old
evidence is taken seriously in the literature on Bayesian confirmation
theory (Earman 1992, ch. 5). An alternative view (Joyce 1999, ch. 6)
sees several different but equally legitimate concepts of confirmation
at work. The intuition behind one concept is the reason for the
implausibility of the explication of another.
In contrast to hypothetico-deductivism, Bayesian confirmation theory
has no problem with assigning degrees of incremental confirmation to
statistical hypotheses. Such alternative statistical hypotheses H1,
…Hn, … are taken to specify the probability of an outcome E. The
probabilities Pr(E|H1), …Pr(E|Hn), … are called the likelihoods of the
hypotheses Hi. Together with their prior probabilities Pr(Hi) the
likelihoods determine the posterior probabilities of the Hi via
Bayes's Theorem:
Pr(Hi|E) = Pr(E|Hi)•Pr(Hi)/[ΣjPr(E|Hj)•Pr(Hj) + Pr(E|H)•Pr(H)]
The so called "catchall" hypothesis H is the negation of the
disjunction or union of all the alternative hypotheses Hi, and so it
is equivalent to -(H1∪…∪Hn∪…). It is important to note the implicit
use of something like the principal principle (Lewis 1980) in such an
application of Bayes's Theorem. The probability measure Pr figuring in
the above equation is an agent's degree of belief function. The
statistical hypotheses Hi specify the objective chance of the outcome
E as Chi(E). Without a principle linking objective chances to
subjective degrees of belief, nothing guarantees that the agent's
conditional degree of belief in E given Hi, Pr(E|Hi), is equal to the
chance of E as specified by Hi, Chi(E). The principal principle says
that an agent's conditional degree of belief in a proposition A given
the information that the chance of A is equal to r (and no further
inadmissible information) should be r, Pr(A|Ch(A) = r) = r. For more
on the principal principle see Hall (1994), Lewis (1994), Thau (1994),
as well as Vranas (2004a). Spohn shows that the principal principle is
a special case of the reflection principle (van Fraassen 1984; 1995).
The latter principle says that an agent's current conditional degree
of belief in A given that her future belief in A equals r should be r,
Prnow(A|Prlater(A) = r) = r provided Prnow(Prlater(A)=r) > 0.
Bayesian confirmation theory can also handle the ravens paradox. As we
have seen, Hempel thought that "a is neither black nor a raven"
confirms "All ravens are black" relative to no or tautological
background information. He attributed the unintuitive character of
this claim to a conflation of it and the claim that "a is neither
black nor a raven" confirms "All ravens are black" relative to our
actual background knowledge A – and the fact that A contains the
information that there are more non-black objects than ravens. The
latter information is reflected in our degree of belief function Pr by
the inequality
Pr(¬Ba|A) > Pr(Ra|A).
If we further assume that the probabilities of finding a non-black
object as well as finding a raven are independent of whether or not
all ravens are black,
Pr(¬Ba|∀x(Rx → Bx)∧ A) = Pr(¬Ba|A),
Pr(Ra|∀x(Rx → Bx)∧ A) = Pr(Ra|A),
we can infer (when we assume all probabilities to be defined) that
Pr(∀x(Rx → Bx)|Ra∧ Ba∧ A) > Pr(∀x(Rx → Bx)|¬Ra∧ ¬Ba∧ A) >
Pr(∀x(Rx → Bx)|A).
So Hempel's intuitions are vindicated by Bayesian confirmation theory
to the extent that the above independence assumptions are plausible
(or there are weaker assumptions entailing a similar result), and to
the extent he also took non-black non-ravens to confirm the ravens
hypothesis relative to our actual background knowledge. For more, see
Vranas (2004b).
Let us finally consider the problem of irrelevant conjunction in
Bayesian confirmation theory. HD-confirmation satisfies the converse
consequence condition, and so has the undesirable feature that E
confirms H∧ H' relative to B whenever E confirms H relative to B, for
any H' whatsoever. This is not true for incremental confirmation. Even
if Pr(E∧ H|B) > Pr(E|B)•Pr(H|B), it need not be the case that Pr(E∧ H∧
H'|B) > Pr(E|B)•Pr(H∧ H'|B). However, the following special case is
also true for incremental confirmation.
If H∧ B logically implies E, then E incrementally confirms H∧ H'
relative to B, for any H' whatsoever (whenever the relevant
probabilities are defined).
In the spirit of the last paragraph, one can, however, show that H∧ H'
is less confirmed by E relative to B than H alone (in the sense of the
distance measure d and the Good-Fitelson measure l) if H' is an
irrelevant conjunct to H given B with respect to E in the sense that
Pr(E|H∧ H'∧ B) = Pr(E|H∧ B)
(Hawthorne & Fitelson 2004). If H∧ B logically implies E, then every
H' such that Pr(H∧ H'∧ B) > 0 is irrelevant in this sense. For more
see Fitelson (2002), Hawthorne & Fitelson (2004), Maher (2004b).
7. Taking Stock
Let us grant that Bayesian confirmation theory adequately explicates
the concept of confirmation. If so, then this is the concept
scientists use when they say that the anomalous perihelion of Mercury
confirms the general theory of relativity. It is also the concept more
ordinary epistemic agents use when they say that, relative to what
they have experienced so far, the dark clouds on the sky are evidence
for rain. The question remains what happened to Hume's problem of the
justification of induction. We know – by definition – that the
conclusion of an inductively strong argument is well-confirmed by its
premises. But does that also justify our acceptance of that
conclusion? Don't we first have to justify our definition of
confirmation before we can use it to justify our inductive inferences?
It seems we would have to, but, as Hume argued, such a justification
of induction is not possible. All we could hope for is an adequate
description of our inductive practices. As we have seen, Goodman took
the task of adequately describing induction as being tantamount to its
justification (Goodman 1983, ch. III, ascribes a similar view to Hume,
which is somehow peculiar, because Hume argued that a justification of
induction is impossible). In doing so he appealed to deductive logic,
which he claimed to be justified by its conformity to accepted
practices of deductive reasoning. But that is not so. Deductive logic
is not justified because it adequately describes our practices of
deductive reasoning – it doesn't. The rules of deductive logic are
justified relative to the goal of truth preservation in all possible
worlds. The reasons are that (i) in going from the premises of a
deductively valid argument to its conclusion, truth is preserved in
all possible worlds (this is known as soundness); and that (ii) any
argument with that property is a deductively valid argument (this is
known as completeness). Similarly for the rules of nonmonotonic logic,
which are justified relative to the goal of truth preservation in all
"normal" worlds (for normality see e.g. Koons 2005). The reason is
that all and only nonmonotonically valid inferences are such that
truth is preserved in all normal worlds when one jumps from the
premises to the conclusion (Kraus & Lehmann & Magidor 1990, for a
survey see Makinson 1994). More generally, the justification of a
canon of normative principles – such as the rules of deductive logic,
the rules of nonmonotonic logic, or the rules of inductive logic – are
only justified relative to a certain goal when one can show that
adhering to these normative principles in some sense furthers the goal
in question.
Similarly to Goodman, Carnap sought to justify the principles of his
inductive logic by appeals to intuition (cf. the quote in section 4b).
Contemporary Bayesian confirmation theorists with their
desideratum/explicatum approach follow Carnap and Goodman at least
insofar as they apparently do not see the need for justifying their
accounts of confirmation by more than appeals to intuition. These are
supposed to show that their definitions of confirmation are adequate.
But the alleged impossibility of justifying induction does not entail
that its adequate description or explication in form of a particular
theory of confirmation is sufficient to justify inductive inferences
based on that theory. Moreover, as noted by Reichenbach (1938; 1940),
a justification of induction is not impossible after all. Hume was
right in claiming that there is no deductively valid argument with
knowable premises and the conclusion that inductively strong arguments
will always lead us to true conclusions. But that is not the only
conclusion that would justify induction. Reichenbach was mainly
interested in the limiting relative frequencies of particular outcomes
in various sequences of events. He could show that a particular
inductive rule – the straight rule that conjectures that the limiting
relative frequency is equal to the observed relative frequency – will
lead us to the true limiting relative frequency, if any inductive rule
does. However, the straight rule is not the only rule with this
property. Therefore its justification relative to the goal of
discovering limiting relative frequencies is at least incomplete. If
we want to keep the analogy to deductive logic, we can put things as
follows: Reichenbach was able to establish the soundness, but not the
completeness, of his inductive logic (that is, the straight rule) with
respect to the goal of eventually arriving at the true limiting
relative frequency. (Reichenbach himself provides an example that
proves the incompleteness of the straight rule with respect to this
goal.)
While soundness in this sense is not sufficient for a justification of
the straight rule, such results provide more reasons than appeals to
intuition. They are necessary conditions for the justification of a
normative rule of inference relative to a particular goal of inquiry.
A similar view about the justification of induction is held by formal
learning theory. Here one considers the objective reliability with
which a particular method (such as the straight rule or a particular
confirmation measure) finds out the correct answer to a given
question. The use of a method to answer a question is only justified
when the method reliably answers the question, if any method does. As
different questions differ in their complexity, there are different
senses of reliability. A method may correctly answer a question after
finitely many steps and with a sign that the question is answered
correctly – as when we answer the question whether the first observed
raven is black by saying "yes" if it is, and "no" otherwise. Or it may
answer the question after finitely many steps and with a sign that it
has done so when the answer is "yes," but not when the answer is "no"
– as when we answer the question whether there exists a black raven by
saying "yes" when we first observe a black raven, and by saying "no"
otherwise. Or it may stabilize to the correct answer in the sense that
the method conjectures the right answer after finitely many steps and
continues to do so forever without necessarily giving a sign that it
has arrived at the correct answer – as when we answer the question
whether the limiting relative frequency of black ravens among all
ravens is greater than .5 by saying "yes" as long as the observed
relative frequency is greater than .5, and by saying "no" otherwise
(under the assumption that this limit exists). And so on. This
provides a classification of all problems in terms of their
complexity. The use of a particular method for answering a question of
a certain complexity is only justified if the method reliably answers
the question in the sense of reliability determined by the complexity
of the question. A discussion of Bayesian confirmation theory from the
point of view of formal learning theory can be found in Kelly &
Glymour (2004). Schulte (2002) gives an introduction to the main
philosophical ideas of formal learning theory. A technically advanced
book length exposition is Kelly (1996). The general idea is the same
as before. A rule is justified relative to a certain goal to the
extent that the rule furthers that goal.
So can we justify particular inductive rules in the form of
confirmation measures along these lines? We had better, for otherwise
there might be inductive rules that would reliably lead us to the
correct answer about a question where our inductive rules won't (cf.
Putnam 1963a; see also his 1963b). Before answering this question, let
us first be clear which goal confirmation is supposed to further. In
other words, why should we accept well-confirmed hypotheses rather
than any other hypotheses? A natural answer is that science and our
more ordinary epistemic enterprises aim at true hypotheses. The
justification for confirmation would then be that we should accept
well-confirmed hypotheses, because we are in some sense guaranteed to
arrive at true hypotheses if (and only if) we stick to well-confirmed
hypotheses. Something along these lines is true for absolute
confirmation according to which degree of confirmation is equal to
probability conditional on the data. More precisely, the Gaifman and
Snir convergence theorem (Gaifman & Snir 1982) says that for almost
every world or model w for the underlying language – that is, all
worlds w except, possibly, for those in a set of measure 0 (in the
sense of the measure Pr* on the σ-field A from section 4a) – the
probability of a hypothesis conditional on the first n data sentences
from w converges to its truth value in w (1 for true, 0 for false). It
is assumed here that the set of all data sentences separates the set
of all worlds (in the sense that for any two distinct worlds there is
a data sentence which is true in the one and false in the other
world). If we accept a hypothesis as true as soon as its probability
is greater than .5 (or any other positive threshold value < 1), and
reject it as false otherwise, we are guaranteed to almost surely
arrive at true hypotheses after finitely many steps. That does not
mean that no other method can do equally well. But it is more than to
simply appeal to our intuitions, and a necessary condition for the
justification of absolute confirmation relative to the goal of truth.
See also Earman (1992, ch. 9) and Juhl (1997).
A more limited result is true for incremental confirmation. Based on
the Gaifman and Snir convergence theorem one can show for every
confirmation measure c and almost all worlds w that there is an n such
that for all later m: the conjunction of the first m data sentences
confirms hypotheses that are true in w to a non-negative degree, and
it confirms hypotheses that are false in w to a non-positive degree
(the set of all data sentences is again assumed to separate the set of
all worlds). Even if this more limited result were a satisfying
justification for the claim that incremental confirmation furthers the
goal of truth, the question remains why one has to go to incremental
confirmation in order to arrive at true theories. It also remains
unclear what degrees of incremental confirmation are supposed to
indicate, for it is completely irrelevant for the above result whether
a positive degree of confirmation is high or low – all that matters is
that it is positive. This is in contrast to absolute confirmation.
There a high number represents a high probability – that is, a high
probability of being true – which almost surely converges to the truth
value itself. To make these vague remarks more vivid, let us consider
an example.
Suppose my 35 year old friend is pregnant and I am curious as to who
is the father. I know that it is either the 35 year old Alberto or the
55 year old Ben or the 55 year old Cesar. My initial degree of belief
function Pr is such that
Pr(A) = .9, Pr(B) = Pr(C) = .05, Pr(A∧ B) = Pr(A∧ C) = Pr(B∧ C) = 0,
Pr(A∨B) = Pr(A∨C) = .95, Pr(B∨C) = .1, Pr(A∨B∨C) = 1,
Pr(A∧ G) = .4, Pr(B∧ G) = .03, Pr(C∧ G) = .03, Pr(G) = .46,
where A is the proposition that Alberto is the father, and similarly
for B and C. G is the proposition that the father has grey hair. [More
precisely, the probability space is <L, Pr> with L the propositional
language over the set of propositional variables {A, B, C, G} and Pr
such that Pr(A∧ G) = .4, Pr(B∧ G) = .03, Pr(C∧ G) = .03, Pr(A∧ ¬G) =
.5, Pr(B∧ ¬G) = .02, Pr(C∧ ¬G) = .02, Pr(A∧ B) = Pr(A∧ C) = Pr(B∧ C) =
Pr(¬A∧ ¬B∧ ¬C)= 0.] This is a fairly reasonable degree of belief
function. Most men at the age of 55 I know have grey hair. Less than
50% of the men of age 35 I know have grey hair. And I tend to use the
principal principle whenever I can (assuming a close connection
between objective chances and relative frequencies). Now suppose I
learn that the father has grey hair. My new degrees of belief are
Pr(A|G) = 40/46, Pr(B|G) = 3/46, Pr(C|G) = 3/46,
Pr(A∨B|G) = Pr(A∨C|G) = 43/46, Pr(B∨C|G) = 6/46, Pr(A∨B∨C|G) = 1.
G incrementally confirms B, C, B∨C, A∨C, B∨C, it neither incrementally
confirms nor incrementally disconfirms A∨B∨C, and it incrementally
disconfirms A.
However, my degree of belief in A is still more than thirteen times my
degree of belief in B and my degree of belief in C. And whether I have
to bet on these propositions or whether I am just curious as to who is
the father of my friend's baby, all I care about after having received
evidence G will be my new degrees of belief in the various answers –
and my utilities, including my desire to answer the question. I will
be willing to bet on A at less favorable odds than on either B or C or
even their disjunction; and should my friend tell me she is going to
marry the father of her baby – she assuming that I know who it is – I
would buy my wedding present on the assumption that she is going to
marry Alberto (unless, of course, I can ask her first). In this
situation, incremental confirmation and degrees of incremental
confirmation are at best misleading.
[What is important is a way of updating my old degree of belief
function by the incoming evidence. The above example assumes evidence
to come in the form of a proposition that I become certain of. In this
case, probabilism says I should update my degree of belief function by
Strict Conditionalization:
If Pr is your subjective probability at time t, and between t and
t' you learn E and no logically stronger proposition in the sense that
your new degree of belief in E is 1, then your new subjective
probability at time t' should be Pr(•|E).
As Jeffrey (1983) observes, we usually do not learn by becoming
certain of a proposition. Evidence often merely changes our degrees of
belief in various propositions. Jeffrey Conditionalization is a more
general update rule than Strict Conditionalization:
If Pr is your subjective probability at time t, and between t and
t' your degrees of belief in the countable partition {E1, …, En, …}
change from Pr(Ei) to pi∈ [0,1] (with Pr(Ei) = pi for Pr(Ei) ∈ {0,1}),
and your positive degrees of belief do not change on any superset
thereof, then your new subjective probability at time t' should be
Pr*, where for all A, Pr*(A) = ΣiPr(A|Ei)∈pi.
For evidential input of the above form, Jeffrey Conditionalization
turns regular probability measures into regular probability measures,
provided no contingent evidential proposition receives an extreme
value p ∈ {0,1}. Radical probabilism (Jeffrey 2004) urges you not to
assign such extreme values, and to have a regular initial degree of
belief function – that is, whenever you can (but you can't always).
Field (1978) proposes an update rule for evidence of a different
format.
This is also the place to mention different formal frameworks besides
probability theory. For an overview, see Huber (2008a).]
More generally, degrees of belief are important to us, because
together with our desires they determine which acts it is rational for
us to take. The usual recommendation according to rational choice
theory for choosing one's acts is to maximize one's expected utility
(the mathematical representation of one's desires), that is, the
quantity
EU(a) = Σs∈Su(a(s))•Pr(s).
Here S is an exclusive and exhaustive set of states, u is the agent's
utility function over the set of outcomes a(s) which are the results
of an act a in a state s (acts are identified with functions from
states s to outcomes), and Pr is the agent's probability measure on a
field over S (Savage 1972). From this decision-theoretic point of view
all we need – besides our utilities – are our degrees of belief
encoded in Pr. Degrees of confirmation encoding how much one
proposition increases the probability of another are of no use here.
In the above examples, I only consider the propositions A, B, C,
because they are sufficiently informative to answer my question. If
truth were the only thing I am interested in, I would be happy with
the tautological answer that somebody is the father of my friend's
baby, A∨B∨C. But I am not. The reason is that I want to know what is
going on out there – not only in the sense of having true beliefs, but
also in the sense of having informative beliefs. In terms of decision
theory, my decisions do not only depend on my degrees of belief – they
also depend on my utilities. This is the idea behind the
plausibility-informativeness theory (Huber 2008b), according to which
epistemic utilities reduce to informativeness values. If we take as
our epistemic utilities in the above example the informativeness
values of the various answers (with positive probability) to our
question, we get
I(A) = I(B) = I(C) = 1, I(A∨B) = I(A∨C) ≈ 40/83, I(B∨C) = 60/83, I(A∨B∨C) = 0,
where the question "Who is the father of my friend's baby?" is
represented by the partition Q = {A, B, C} and the informativeness
values of the various answers are calculated according to
I(A) = 1 – [1 – ΣiPr*(Xi|A)2]/[1 – ΣiPr*(Xi)2],
a measure proposed by Hilpinen (1970). Contrary to what Hilpinen
(1970, 112) claims, I(A) does not increase with the logical strength
of A. The probability Pr* is the posterior degree of belief function
from our example, Pr(•|G). If we insert these values into the expected
utility formula,
EU(a) = Σs∅Su(a(s))∈Pr*(s) = ΣX∅Qu(a(X))∈Pr*(X) = ΣX∅QI(X)∈Pr*(X),
we get the result that the act of accepting A as answer to our
question maximizes our expected epistemic utility.
Not all is lost, however. The distance measure d turns out to measure
the expected utility of accepting H when utility is identified with
informativeness measured according to a measure proposed by Carnap &
Bar-Hillel (1953) (one can think of this measure as measuring how much
an answer informs about the most difficult question, namely, which
world is the actual one?). Similarly, the Joyce-Christensen measure s
turns out to measure the expected utility of accepting H when utility
is identified with informativeness about the data measured according
to a proposal by Hempel & Oppenheim (1948). So far, this is only
interesting. It gets important by noting that d and s can also be
justified relative to the goal of informative truth – and not just by
appealing to our intuitions about maximizing expected utility. When
based on a regular probability, there almost surely is an n such that
for all later m: relative to the conjunction of the first m data
sentences, contingently true hypotheses get a positive value and
contingently false hypotheses get a negative value. Moreover, within
the true hypotheses, logically stronger hypotheses get a higher value
than logically weaker hypotheses. The logically strongest true
hypothesis (the complete true theory about the world w) gets the
highest value, followed by all logically weaker true hypotheses all
the way down to the logically weakest true hypothesis, the tautology,
which is sent to 0. Similarly within the false hypotheses: the
logically strongest false hypothesis, the contradiction, is sent to 0,
followed by all logically weaker false hypotheses all the way down to
the logically weakest false hypothesis (the negation of the complete
theory about w). As informativeness increases with logical strength,
we can put this as follows (assuming that the underlying probability
measure is regular): d and s do not only distinguish between true and
false theories, as do all confirmation measures (as well as all
conditional probabilities). They additionally distinguish between
informative and uninformative true theories, as well as between
informative and uninformative false theories. In this sense, they
reveal the following structure of almost every world w [w(p) = w(q) =
1 in the toy example]:
informative and contingently true in w
p ∧ q
> 0 contingently true in w
p, q, p ↔q
uninformative and contingently true in w
p ∨ q, ¬p ∨ q, p ∨ ¬q
= 0 logically determined
p ∨ ¬p, p ∧ ¬p
informative and contingently false in w
¬p ∧ ¬q, p ∧ ¬q, ¬p ∧ q
< 0 contingently false in w
¬p, ¬q, p ↔¬q
uninformative and contingently false in w
¬p ∨¬q
This result is also true for the Carnap measure c, but it does not
extend to all confirmation measures. It is false for the Milne measure
r, which does not distinguish between informative and uninformative
false theories. And it is false for the Good-Fitelson measure l, which
distinguishes neither between informative and uninformative true
theories nor between informative and uninformative false theories. For
more see Huber (2005b).
The reason c, d, and s have this property of distinguishing between
informative and uninformative truth and falsehood is that they are
probabilistic assessment functions in the sense of the
plausibility-informativeness theory (Huber 2008b) – and the above
result is true for all probabilistic assessment functions (not only
those that can be expressed as expected utilities). The
plausibility-informativeness theory agrees with traditional philosophy
that truth is an epistemic goal. Its distinguishing thesis is that
there is a second epistemic goal besides truth, namely,
informativeness, which has to be taken into account when we evaluate
hypotheses. Like confirmation theory, the plausibility-informativeness
theory assigns numbers to hypotheses in the light of evidence. But
unlike confirmation theory, it does not appeal to intuitions when it
comes to the question why one is justified in accepting hypotheses
with high assessment values. The plausibility-informativeness theory
answers this question by showing that accepting hypotheses according
to the recommendation of an assessment function almost surely leads
one to (the most) informative (among all) true hypotheses (against,
this can be seen as a soundness result). (The corresponding
completeness result that only acceptances according to the
recommendations of assessment functions almost surely lead to
informative true hypotheses does not hold. For a discussion of this,
see Huber 2008b, sec. 6.2.)
It is idle to speculate what Hume would have said to all this ado.
Suffice it to note that his problem would not have got off the ground
without our desire for informativeness.
8. References and Further Reading
* Albert, Max (1992), "Die Falsifikation Statistischer
Hypothesen." Journal for General Philosophy of Science 23, 1-32.
* Alchourrón, Carlos E. & Gärdenfors, Peter & Makinson, David
(1985), "On the Logic of Theory Change: Partial Meet Contraction and
Revision Functions." Journal of Symbolic Logic 50, 510-530.
* Carnap, Rudolf (1950/1962), Logical Foundations of Probability.
2nd ed. Chicago: University of Chicago Press.
* Carnap, Rudolf (1952), The Continuum of Inductive Methods.
Chicago: University of Chicago Press.
* Carnap, Rudolf (1963), "Replies and Systematic Expositions.
Probability and Induction. " In P.A. Schilpp (ed.), The Philosophy of
Rudolf Carnap. La Salle, IL: Open Court, 966-998.
* Carnap, Rudolf & Bar-Hillel, Yehoshua (1953), An Outline of a
Theory of Semantic Information. Technical Report 247. Research
Laboratory of Electronics, MIT. Reprinted in Y. Bar-Hillel (1964),
Language and Information. Selected Essays on Their Theory and
Application. Reading, MA: Addison-Wesley, 221-274.
* Christensen, David (1999), "Measuring Confirmation. " Journal of
Philosophy 96, 437-461.
* Crupi, Vincenzo and Tentori, Katya, and Gonzalez, Michel (2007),
On Bayesian Measures of Evidential Support: Theoretical and Empirical
Issues. Philosophy of Science 74, 229-252.
* Duhem, Pierre (1906/1974), The Aim and Structure of Physical
Theory. New York: Atheneum.
* Earman, John (1992), Bayes or Bust? A Critical Examination of
Bayesian Confirmation Theory. Cambridge, MA: MIT Press.
* Eells, Ellery (2005), "Confirmation Theory. " In J. Pfeifer & S.
Sarkar (eds.), The Philosophy of Science. An Encyclopedia. Oxford:
Routledge.
* Field, Hartry (1978), "A Note on Jeffrey Conditionalization. "
Philosophy of Science 45, 361-367.
* Fitelson, Branden (1999), "The Plurality of Bayesian Measures of
Confirmation and the Problem of Measure Sensitivity. " Philosophy of
Science 66 (Proceedings), S362-S378.
* Fitelson, Branden (2001), Studies in Bayesian Confirmation
Theory. PhD Dissertation. Madison, WI: University of
Wisconsin-Madison.
* Fitelson, Branden (2002), "Putting the Irrelevance Back Into the
Problem of Irrelevant Conjunction. " Philosophy of Science 69,
611-622.
* Fitelson, Branden (2005), "Inductive Logic. " In J. Pfeifer & S.
Sarkar (eds.), The Philosophy of Science. An Encyclopedia. Oxford:
Routledge.
* Fitelson, Branden & Hájek, Alan & Hall, Ned (2005),
"Probability. " In J. Pfeifer & S. Sarkar (eds.), The Philosophy of
Science. An Encyclopedia. Oxford: Routledge.
* Gaifman, Haim & Snir, Marc (1982), "Probabilities over Rich
Languages, Testing, and Randomness." Journal of Symbolic Logic 47,
495-548.
* Gärdenfors, Peter (1988), Knowledge in Flux. Modeling the
Dynamics of Epistemic States. Cambridge, MA: MIT Press.
* Gärdenfors, Peter & Rott, Hans (1995), "Belief Revision. " In
D.M. Gabbay & C.J. Hogger & J.A. Robinson (eds.), Handbook of Logic in
Artificial Intelligence and Logic Programming. Vol. 4. Epistemic and
Temporal Reasoning. Oxford: Clarendon Press, 35-132.
* Glymour, Clark (1980), Theory and Evidence. Princeton: Princeton
University Press.
* Good, Irving John (1967), "The White Shoe is a Red Herring."
British Journal for the Philosophy of Science 17, 322.
* Good, Irving John (1968), "The White Shoe qua Herring is Pink."
British Journal for the Philosophy of Science 19, 156-157.
* Good, Irving John (1983), Good Thinking: The Foundations of
Probability and Its Applications. Minneapolis: University of Minnesota
Press.
* Goodman, Nelson (1946), "A Query on Confirmation." Journal of
Philosophy 43, 383-385.
* Goodman, Nelson (1983), Fact, Fiction, and Forecast. 4th ed.
Cambridge, MA: Harvard University Press.
* Grimes, Thomas R. (1990), "Truth, Content, and the
Hypothetico-Deductive Method." Philosophy of Science 57, 514-522.
* Hacking, Ian (2001), An Introduction to Probability and
Inductive Logic. Cambridge: Cambridge University Press.
* Hájek, Alan (2003a), "Interpretations of Probability." In E.N.
Zalta (ed.), Stanford Encyclopedia of Philosophy.
* Hájek, Alan (2003b), "What Conditional Probability Could Not
Be." Synthese 137, 273-323.
* Hájek, Alan (2005), "Scotching Dutch Books?" Philosopical
Perspectives 19 (Epistemology), 139-151.
* Hájek, Alan & Hall, Ned (2000), "Induction and Probability." In
P. Machamer & M. Silberstein (eds.), The Blackwell Guide to the
Philosophy of Science. Oxford: Blackwell, 149-172.
* Hall, Ned (1994), "Correcting the Guide to Objective Chance."
Mind 103, 505-518.
* Hawthorne, James (2005), "Inductive Logic." In E.N. Zalta (ed.),
Stanford Encyclopedia of Philosophy.
* Hawthorne, James & Fitelson, Branden (2004), "Re-solving
Irrelevant Conjunction with Probabilistic Independence." Philosophy of
Science 71, 505-514.
* Hempel, Carl Gustav (1945), "Studies in the Logic of
Confirmation." Mind 54, 1-26, 97-121.
* Hempel, Carl Gustav (1962), "Deductive-Nomological vs.
Statistical Explanation." In H. Feigl & G. Maxwell (eds.), Scientific
Explanation, Space and Time. Minnesota Studies in the Philosophy of
Science 3. Minneapolis: University of Minnesota Press, 98-169.
* Hempel, Carl Gustav (1967), "The White Shoe: No Red Herring."
British Journal for the Philosophy of Science 18, 239-240.
* Hempel, Carl Gustav & Oppenheim, Paul (1948), "Studies in the
Logic of Explanation." Philosophy of Science 15, 135-175.
* Hilpinen, Risto (1970), "On the Information Provided by
Observations." In J. Hintikka & P. Suppes (eds.), Information and
Inference. Dordrecht: D. Reidel, 97-122.
* Hintikka, Jaakko (1966), "A Tw-Dimensional Continuum of
Inductive Methods." In J. Hintikka & P. Suppes (eds.), Aspects of
Inductive Logic. Amsterdam: North-Holland, 113-132.
* Hitchcock, Christopher R. (2001), "The Intransitivity of
Causation Revealed in Graphs and Equations." Journal of Philosophy 98,
273-299.
* Howson, Colin (2000a), Hume's Problem: Induction and the
Justification of Belief. Oxford: Oxford University Press.
* Howson, Colin (2000b), "Evidence and Confirmation." In W.H.
Newton-Smith (ed.), A Companion to the Philosophy of Science. Oxford:
Blackwell, 108-116.
* Howson, Colin & Urbach, Peter (1989/2005), Scientific Reasoning:
The Bayesian Approach. 3rd ed. La Salle, IL: Open Court.
* Huber, Franz (2005a), "Subjective Probabilities as Basis for
Scientific Reasoning?" British Journal for the Philosophy of Science
56, 101-116.
* Huber, Franz (2005b), "What Is the Point of Confirmation?"
Philosophy of Science 75, 1146-1159.
* Huber, Franz (2008a) "Formal Epistemology." In E. N. Zalta
(ed.), Stanford Encyclopedia of Philosophy.
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