A graded Bayesian coherence notion
Frederik Herzberg
Center for Mathematical Economics (IMW), Bielefeld University
Munich Center for Mathematical Philosophy, Ludwig Maximilian University of Munich
W ORKSHOP ‘F ULL AND PARTIAL B ELIEF ’
( CO - LOCATED WITH THE 4 TH R ENÉ D ESCARTES L ECTURES )
Tilburg University
21 October 2014
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Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
1 / 56
Introduction
1
Introduction
2
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
Formalising BonJour’s first desideratum
Formalising BonJour’s second desideratum
Formalising BonJour’s third and fourth desiderata
3
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
Formalisation of the belief systems
Calculation of the third and fourth component of the coherence measure
Calculation of graded probabilistic consistency. The second component
of the coherence measure
Calculation of the first component of the coherence measure
Summary
4
Conclusion and discussion
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Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
2 / 56
Introduction
Introduction (1)
Epistemic (doxastic) justification is a central and ancient topic
of theoretical philosophy:
I
I
Can one ever be justified in believing any proposition?
If so, what are necessary and/or sufficient conditions?
Based on Aristotle’s “regress argument”, it has been argued that
there are exactly three non-skeptical views about the structure
of epistemic justification (structure of reasons).
According to this position, non-skeptical positions on epistemic
justification either assert
I
I
I
that reasons form a finite chain with some foundational proposition
at the base (foundationalism) — illustration: foundation of a house
(D ESCARTES); or
that reasons mutually support each other (coherentism) —
illustration: planks of a boat (N EURATH); or
that reasons form an infinite regress (infinitism) — illustration:
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open-ended loop.
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
3 / 56
Introduction
Introduction (2)
The philosophical discussion on epistemic justification has seen
interesting turns during the past two decades:
I
I
I
infinitist accounts of epistemic justification were revived (e.g.
K LEIN 1998ff; formally: P EIJNENBURG 2007);
a major proponent of coherentism abandoned this position
(B ON J OUR 1999, 2010) for Cartesianism;
impossibility theorems for coherence measures suggest that
coherentism defies formalisation (pioneers: K LEIN –WARFIELD
1994), formally reiterating an earlier criticism by E WING (1934).
We give a formal defense of coherentism that takes traditional
epistemology seriously:
I
I
We propose a class of coherence measures that meet the thrust of
BonJour’s desiderata.
The domain of these coherence measures will be systems of
degrees of belief.
(Our own position is a graded version of sufficiency coherentism; it
accommodates foundationalist and infinitist intuitions, but rejects
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the Principle of Inferential Justification. H. 2014b)
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
4 / 56
Introduction
The (im)possibility of formal coherence concepts (1)
There is a rich body of literature, originally closely related to the
discussion on epistemological coherentism, on the
(im)possibility of a formal graded coherence notion
(K LEIN –WARFIELD 1994, 1996; S HOGENJI 1999; A KIBA 2000;
F ITELSON 2003; B OVENS –H ARTMANN 2003a,b, 2005, 2006;
O LSSON 2002, 2005; D IETRICH –M ORETTI 2005; M EIJS –D OUVEN
2007; S CHUPBACH 2008; S IEBEL –W OLFF 2008).
The overall finding of this literature is that it appears to be very
difficult to come up with a convincing (especially
one-dimensional) graded coherence notion.
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
5 / 56
Introduction
The (im)possibility of formal coherence concepts (2)
However, the recent formal literature on coherence measures
I
I
does not interact closely with the traditional literature on
coherentism (e.g. B ON J OUR 1985, L EHRER 2000) and
in general models belief systems as sets of propositions endowed
with a unique probability measure — a very strong assumption
(psychologically and decision-theoretically less than compelling).
We suggest a new formal framework which models belief systems
as sets of conditional probability assignments, compatible
with several (even infinitely many) probability measures; they
induce a Bayesian network on the propositions.
Within that framework, we propose a formalisation of the thrust
of BonJour’s (1985) (multi-dimensional) coherence concept:
I
I
inferential connections and fragmentation are measured through
graph-theoretic concepts on the induced Bayesian network;
probabilistic consistency is measured via the size of the set of
probability measures compatible with the belief system.
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(H. 2014c)
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
6 / 56
Introduction
Criteria for a formal graded coherence notion (1)
B ON J OUR’s coherence concept provides desiderata for a formal
coherence notion:
[(I)] A system of beliefs is coherent only if it is logically
consistent.
[(II)] A system of beliefs is coherent in proportion to its degree
of probabilistic consistency.
[. . . ]
[(III)] The coherence of a system of beliefs is increased by the
presence of inferential connections between its
component beliefs and increased in proportion to the
number and strength of such connections.
[(IV)] The coherence of a system of beliefs is diminished to the
extent to which it is divided into subsystems of beliefs
which are relatively unconnected to each other by
inferential connections.
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(B ON J OUR 1985, Section 5.3, pp. 95, 98, 99)
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
7 / 56
Introduction
Criteria for a formal graded coherence notion (2)
There is, in addition, also a fifth desideratum, which however does
not admit a natural formalisation.
[(V)] The coherence of a system of beliefs is decreased in
proportion to the presence of unexplained anomalies in
the believed content of the system.
(B ON J OUR 1985, Section 5.3, pp. 95, 98, 99)
We propose a class of formal coherence concepts which satisfy the
first four of B ON J OUR’s desiderata — and ultimately might be
restricted to satisfy a formalisation of the fifth desideratum, too.
There are several classes of epistemic anomalies. A local anomaly might be
a non-foundational belief with high degree of centrality.
A global anomaly might be a belief whose omission from the belief system
would result in a substantial reduction in complexity.
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
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8 / 56
Introduction
Multiplicity of subjective probability measures (priors)
Our formal framework allows for belief systems that are
compatible with multiple probability measures.
This reflects the consensus of contemporary decision theory (and
also psychology, cf. e.g. M INSKY 1986 or O RNSTEIN 1986):
I
I
I
Building on work by E LLSBERG (1961) and G ILBOA –S CHMEIDLER
(1989), decision making under multiple priors or probabilistic
ambiguity (uncertainty in the sense of K NIGHT 1921) is studied.
There is a body of literature on probabilistic opinion pooling (e.g.
M C C ONWAY 1981 and C OOKE 1991).
The problem of aggregating probability measures can also be
studied within a comprehensive, theory of aggregating
propositional attitudes (D IETRICH –L IST 2011).
Most recently, these results have been extended:
I
I
A unified methodology for the theory of propositional-attitude
aggregation has been proposed, via universal algebra (H. 2014d).
The theory of probabilistic opinion pooling has been extended to
infinite profiles of priors — a set-theoretically delicate problem imw
which can be solved using ultrafilters (H. 2014a).
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
9 / 56
Formalisation of (the core of) BonJour’s coherence notion
1
Introduction
2
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
Formalising BonJour’s first desideratum
Formalising BonJour’s second desideratum
Formalising BonJour’s third and fourth desiderata
3
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
Formalisation of the belief systems
Calculation of the third and fourth component of the coherence measure
Calculation of graded probabilistic consistency. The second component
of the coherence measure
Calculation of the first component of the coherence measure
Summary
4
Conclusion and discussion
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
10 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
1
Introduction
2
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
Formalising BonJour’s first desideratum
Formalising BonJour’s second desideratum
Formalising BonJour’s third and fourth desiderata
3
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
Formalisation of the belief systems
Calculation of the third and fourth component of the coherence measure
Calculation of graded probabilistic consistency. The second component
of the coherence measure
Calculation of the first component of the coherence measure
Summary
4
Conclusion and discussion
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
11 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
Formal framework for belief systems
Our formal framework assumes probabilism — the thesis that
assignments of (conditional) degrees of belief have the formal
properties of probability measures.
(Cf. J OYCE 2009; L EITGEB –P ETTIGREW 2010; E ASWARAN –F ITELSON
2012; F ITELSON –M C C ARTHY 2013; W EDGWOOD 2013.)
Fix some algebra A of propositions. A belief system is a set S of
triples hA|Bkαi, where A, B ∈ A and α ∈ [0, 1].
Read hA|Bkαi ∈ S as “the belief system S assigns to A, given B, a
conditional degree of belief α”.
(A is foundational for S if hA|>k1i ∈ S.)
A belief system S is probabilistically consistent if and only if
there exists a probability measure P : A → [0, 1] such that
P(A|B) = α whenever hA|Bkαi ∈ S for any A, B ∈ A and α ∈ [0, 1].
Such a probability measure P is then said to be compatible with
S, denoted P ∈ PS .
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Such belief systems can be viewed as Bayesian networks.
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
12 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
Infinite regresses as formal belief systems
A (recipe for a) probabilistic regress is a pair α, β ∈ [0, 1]N such
that αk > βk for all k ∈ N.
A recipe for a probabilistic regress is consistent if and only if
there exist both a sequence S = hSk ik ∈N ∈ AN and a probability
measure P : A → [0, 1] such that for all k ∈ N,
0 < P(Sk +1 ) < 1,
P(Sk |Sk +1 ) = αk > βk = P(Sk |{Sk +1 )
(i.e. {hSk |Sk +1 kαk i : k ∈ N} ∪
Sk |{Sk +1 kβk : k ∈ N is consistent).
Such a pair hP, Si will be called a model for hα, βi.
Put in terms of Bayesian confirmation theory: In a regress, Sk +1
confirms Sk for all k ∈ N — so that S0 is confirmed by S1 , which
is confirmed by S2 , which is confirmed by S3 etc. ad infinitum.)
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Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
13 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
A graded formal coherence notion (1)
We propose a vector-valued coherence measure (H. 2014c):
The first component is a binary measure of logical consistency.
Herein, a belief system is logically consistent if and only if the
intersection of those propositions/events which get assigned a
high degree of belief is non-empty.
The probabilistic consistency of a belief system is measured via
the size of the set of probability measures supporting a belief
system. (This set has a distinctive geometrical structure, viz. the
intersection of several hyperplanes with a simplex, hence its size
can easily be measured as the pair consisting of the H AUSDORFF
(1918) dimension and H AUSDORFF measure.)
The number of inferential connections can be measured in
terms of graph-theoretic notions of connectivity; their strength
can be measured using a confirmation function.
The fragmentation can be measured in terms of the number of imw
maximal connected (proper) subgraphs (components).
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
14 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
A graded formal coherence notion (2)
This coherence notion is only well-defined for finite belief
systems, i.e. finite sets of conditional probability assignments.
Using A. R OBINSON’s (1961, 1966) nonstandard analysis, one
can extend this real-vector-valued coherence notion for finite belief
systems to a hyperreal-vector-valued coherence notion for
hyperfinite (“formally finite”) belief systems.
Thus, one arrives at a coherence notion which is applicable to
certain infinite belief systems defined, in particular hyperfinite
probability spaces. Such spaces are extremely rich in a
rigorous sense, viz. saturated and universal in the sense of the
model theory of stochastic processes (H OOVER –K EISLER 1984;
FAJARDO –K EISLER 2002).
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Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
15 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formalising BonJour’s first desideratum
1
Introduction
2
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
Formalising BonJour’s first desideratum
Formalising BonJour’s second desideratum
Formalising BonJour’s third and fourth desiderata
3
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
Formalisation of the belief systems
Calculation of the third and fourth component of the coherence measure
Calculation of graded probabilistic consistency. The second component
of the coherence measure
Calculation of the first component of the coherence measure
Summary
4
Conclusion and discussion
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
16 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formalising BonJour’s first desideratum
BonJour’s requirement (I) for systems of degrees of belief presupposes
a bridge principle between belief simpliciter and degrees of belief.
There is still an ongoing debate in formal epistemology on this — cf.,
e.g., A RLÓ -C OSTA –PARIKH (2005), F OLEY (2009), L EITGEB (2013,
2014) , A RLÓ -C OSTA –P EDERSEN (2012).
As a working hypothesis we choose the most well-known bridge
principle, viz. the Lockean thesis: belief simpliciter is partial belief to a
sufficiently high degree (c, say).
This seems problematic because it makes coherence dependent of
the threshold c.
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
17 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formalising BonJour’s first desideratum
A very strong parameter-independent version of desideratum (I) would
require for logical consistency:
η (S, (1/2, 1]) 6= ∅,
(1)
wherein η(S, I), for all I ⊆ [0, 1], denotes the intersection of all
propositions/events to which a probability within I is assigned by all
probability measures compatible with S:
o
\\n
η(S, I) :=
P −1 (I) : P ∈ PS
\
=
A.
A∈A
∀P∈PS P(A)∈I
A very weak parameter-independent reading of requirement (I) would
only demand:
η (S, {1}) 6= ∅.
(2)
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Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
18 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formalising BonJour’s first desideratum
Both requirements expressed in formulae (1) and (2) take an
all-or-nothing approach to logical consistency. This is perfectly in
line with B ON J OUR’s (1985) position: Logical consistency is a binary
component of the multi-faceted, non-binary, graded concept of
coherence.
We propose to choose the weak requirement, viz. (2):
1, η (S, {1}) 6= ∅
β1 (S) =
0, η (S, {1}) = ∅.
This allows for some degree of coherence even in the belief systems of
the preface or lottery paradoxes. It avoids the conclusion that most
humans hold utterly incoherent belief systems.
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Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
19 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formalising BonJour’s second desideratum
1
Introduction
2
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
Formalising BonJour’s first desideratum
Formalising BonJour’s second desideratum
Formalising BonJour’s third and fourth desiderata
3
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
Formalisation of the belief systems
Calculation of the third and fourth component of the coherence measure
Calculation of graded probabilistic consistency. The second component
of the coherence measure
Calculation of the first component of the coherence measure
Summary
4
Conclusion and discussion
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
20 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formalising BonJour’s second desideratum
A naïve reading of BonJour’s requirement (II) would look for a measure
for the degree of probabilistic consistency of a belief system in our
above formal framework. However, if one reads this requirement in
context, one finds the following paragraph:
Probabilistic consistency differs from straightforward logical
consistency in two important respects. First, it is extremely
doubtful that probabilistic inconsistency can be entirely
avoided. Improbable things do, after all, sometimes happen,
and sometimes one can avoid admitting them only by creating
an even greater probabilistic inconsistency at another point.
Second, probabilistic consistency, unlike logical consistency,
is plainly a matter of degree, depending on (a) just how many
conflicts the system contains and (b) the degree of
improbability involved in each case. (B ON J OUR 1985, p. 95)
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Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
21 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formalising BonJour’s second desideratum
However, that events which were a priori unlikely do sometimes
happen and therefore can enter a belief system a posteriori does not at
all constitute probabilistic inconsistency.
In our Bayesian framework, probabilistic consistency is already
defined in a very natural way — even though as a binary concept:
1, PS 6= ∅
β2 (S) =
0, PS = ∅.
That said, it is still interesting whether probabilistic consistency
could be a matter of degree. Here, the geometric structure of PS is
helpful. One can measure the probabilistic consistency essentially as
the H AUSDORFF (1918) dimension and H AUSDORFF measure of PS .
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Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
22 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formalising BonJour’s second desideratum
Since A ⊆ 2Ω , one can canonically — up to permutations of the
coordinates — embed the set ∆ of all probability measures defined on
A into [0, 1]card(Ω) by some map ι. The geometric representation of
subjective probability measures by ι is the key to our graded notion
of probabilistic consistency.
We shall measure the size of PS — and hence the probabilistic
consistency of the belief system S — by the pair consisting of the
H AUSDORFF dimension of the canonical image of PS under ι and
its H AUSDORFF measure:
D
E
β̃2 (S) := D (ι[PS ]) , HD(ι[PS ]) (ι[PS ]) ,
where PS is greater than PS0 if and only if either (i) D (ι[PS ]) > D (ι[PS0 ])
or (ii) D (ι[PS ]) = D (ι[PS0 ]), but HD(ι[PS ]) (ι[PS ]) > HD(ι[PS0 ]) (ι[PS0 ])
(lexicographic ordering).
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
23 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formalising BonJour’s third and fourth desiderata
1
Introduction
2
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
Formalising BonJour’s first desideratum
Formalising BonJour’s second desideratum
Formalising BonJour’s third and fourth desiderata
3
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
Formalisation of the belief systems
Calculation of the third and fourth component of the coherence measure
Calculation of graded probabilistic consistency. The second component
of the coherence measure
Calculation of the first component of the coherence measure
Summary
4
Conclusion and discussion
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
24 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formalising BonJour’s third and fourth desiderata
Belief systems as directed graphs
For BonJour’s third and fourth requirements, we suggest viewing a
belief system S as a directed graph, such that the vertices (nodes)
are propositions to which a rational agent with belief system S will
assent and such that an arrow from B to A means that B confirms A
(in the sense of Bayesian confirmation theory) with respect to the
belief system S.
This would formalise coherentist intuitions such as Quine’s and Ullian’s
“web of belief” (Q UINE –U LLIAN 1970).
However, it invokes the concept of belief simpliciter within a
framework that is built around (conditional) degrees of belief. More
careful definitions are necessary.
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
25 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formalising BonJour’s third and fourth desiderata
The extended web of belief
Let us first consider the extended web of belief HS :
The vertices are all those propositions A ∈ A that are at least
candidates for objects of full belief in the sense that P(A) > 1/2
for all P ∈ PS .
There will be an arrow between vertex B and vertex A if and only if
B confirms A in the sense of Bayesian confirmation theory (with
the belief system S in the background), i.e. if and only if
P(A|B) − P(A) > 0 for all P ∈ PS .
Note: The extended web of belief contains propositions as vertices to
which not a precise probability, but merely a lower bound is assigned
by the belief system — e.g. all events/propositions that extensionally
dominate an event/proposition to which S unconditionally assigns
some precise probability > 1/2.
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
26 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formalising BonJour’s third and fourth desiderata
The inner web of belief
Our coherence notion will be based on the inner web of belief GS :
The vertices are all those propositions A ∈ A which can be proved
to be objects of full belief for suitable thresholds c in the
Lockean thesis.
More precisely, the vertices of GS are all those propositions A ∈ A
to which precise unconditional probabilities > 1/2 are assigned by
the belief system, in the sense that there is a real number α > 1/2
such that P(A) = α for all P ∈ PS .
There will be an arrow between vertex B and vertex A if and only if
B confirms A in the sense of Bayesian confirmation theory (with
the belief system S in the background) with a precise degree of
confirmation, i.e. if and only if there exists some real number γ > 0
such that P(A|B) − P(A) = γ for all P ∈ PS . For any such A, B, we
shall refer to this positive real γ as γ(B, A). If there is, for any A, B,
no γ > 0 that would satisfy P(A|B) − P(A) = γ for all P ∈ PS , we
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simply put γ(B, A) = 0.
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
27 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formalising BonJour’s third and fourth desiderata
More restrictive notions of the inner web of belief
One could define, for any c ≥ 1/2, the c-core of beliefs as the
subgraph GSc of GS which consists of only those propositions A to
which precise unconditional probabilities > c are assigned by the
belief system, in the sense that there is a real number α > c such that
P(A) = α for all P ∈ PS . For all c ≥ 1/2, GSc as well as GS and HS are
Bayesian networks.
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
28 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formalising BonJour’s third and fourth desiderata
Two aspects of BonJour’s desideratum (III)
1
2
“The coherence of a system of beliefs is increased by the
presence of inferential connections between its component
beliefs and increased in proportion to the number [. . . ] of such
connections” (B ON J OUR 1985, p. 98; emphasis mine).
We suggest to capture this by the graph-theoretic notion of
(vertex) connectivity κ.
“The coherence of a system of beliefs is increased [. . . ] in
proportion to the [. . . ] strength of [inferential] connections
[between its component beliefs]” (B ON J OUR 1985, p. 99;
emphasis mine).
In a Bayesian setting, a natural interpretation of the “strength of an
inferential connection” from B to A is the degree by which B
confirms A (in the sense of Bayesian confirmation theory). The
most widely used measure of confirmation is the relevance
measure, (difference measure) P(A|B) − P(A).
imw
We therefore propose: β3 (S) := hκ(GS ), hγ(B, A) : A, B ∈ Aii .
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
29 / 56
Formalisation of (the core of) BonJour’s coherence notion
Formalising BonJour’s third and fourth desiderata
BonJour’s desideratum (IV)
B ON J OUR’s penultimate requirement is (IV): the relative
fragmentation of a belief system (given the overall level of
connectivity within the belief system) diminishes coherence.
A natural proposal might be to consider
FS := {F ( G : F
maximal (κ(GS ) + 1)-connected}
and then define the inverse degree of relative fragmentation as follows:
#FS , FS 6= ∅,
β4 (S) =
0,
FS = ∅.
For example, in the special case when κ(GS ) = 0 (i.e. GS is totally
disconnected), we have β4 (S) = 0.
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
30 / 56
Example: BonJour’s “ravens” challenge
1
Introduction
2
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
Formalising BonJour’s first desideratum
Formalising BonJour’s second desideratum
Formalising BonJour’s third and fourth desiderata
3
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
Formalisation of the belief systems
Calculation of the third and fourth component of the coherence measure
Calculation of graded probabilistic consistency. The second component
of the coherence measure
Calculation of the first component of the coherence measure
Summary
4
Conclusion and discussion
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
31 / 56
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
1
Introduction
2
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
Formalising BonJour’s first desideratum
Formalising BonJour’s second desideratum
Formalising BonJour’s third and fourth desiderata
3
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
Formalisation of the belief systems
Calculation of the third and fourth component of the coherence measure
Calculation of graded probabilistic consistency. The second component
of the coherence measure
Calculation of the first component of the coherence measure
Summary
4
Conclusion and discussion
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
32 / 56
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
B ON J OUR’s (1985, p. 96) challenge, as summarised by
B OVENS –H ARTMANN (2003, p. 718), is to consider the following
propositions:
R̃1 : ‘All ravens are black.’
R2 : ‘This bird is a raven.’
R3 : ‘This bird is black.’
R10 : ‘This chair is brown.’
R20 : ‘Electrons are negatively charged.’
R30 : ‘Today is Thursday.’
n
o
Let R̃ = R̃1 , R2 , R3 and R0 = R10 , R20 , R30 . One has to account for
the fact that R̃ is more coherent than R0 .”
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
33 / 56
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
Reformulation
No inferential connections among R10 , R20 , R30 are known.
In R̃ and R0 , there are no inferential connections except for the obvious
one, viz. modus ponens inference from R1 , R2 to R3 .
R̃1 is a scheme of inferential connections rather than as a proposition.
In the context of the belief system R̃, it can be replaced by a mere
single inferential connection such as:
R3 |R2 : ‘If this bird is a raven, then it is black.’
We shall henceforth study R := {R3 |R2 , R2 , R3 } en lieu of R̃.
BonJour’s challenge is to give an account of the greater coherence of
R compared to that of R0 .
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
34 / 56
Example: BonJour’s “ravens” challenge
Formalisation of the belief systems
1
Introduction
2
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
Formalising BonJour’s first desideratum
Formalising BonJour’s second desideratum
Formalising BonJour’s third and fourth desiderata
3
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
Formalisation of the belief systems
Calculation of the third and fourth component of the coherence measure
Calculation of graded probabilistic consistency. The second component
of the coherence measure
Calculation of the first component of the coherence measure
Summary
4
Conclusion and discussion
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
35 / 56
Example: BonJour’s “ravens” challenge
Formalisation of the belief systems
If the information available to a rational individual is represented by R
(or R0 , respectively), this could in a Bayesian framework rendered thus:
1
she assigns to each of the propositions/events and conditional
events in R (or R0 , respectively) a sufficiently high degree of belief;
2
all degrees of belief that she assigns to other conditional events
assume pairwise independence of the propositions constituting
the conditional event in question.
Next choose two algebras of propositions/events to which conditional
degrees of belief will be assigned, A and A0 .
By Stone’s theorem, we can identify A and A0 with power-set algebras
0
2 Ω , 2Ω .
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
36 / 56
Example: BonJour’s “ravens” challenge
Formalisation of the belief systems
An individual with information set R has a belief system S consisting of:
hRi |Ωkαi i for each i ∈ {2, 3} for some αi ∈ [0, 1] that is sufficiently
close to 1;
hR3 |R2 kα1 i for some α1 ∈ [0, 1] that is sufficiently close to 1 and
strictly greater than α3 .
Hence for the belief system S, the set Ω of states of the world only
needs four elements (ω (1) , . . . , ω (4) ) — given by the state descriptions
‘R2 and R3 ’, ‘R2 but not R3 ’, ‘R3 but not R3 ’, ‘neither R2 not R3 ’.
More formally:
D
E
ω (1) := Ṙ2 , Ṙ3
D
E
ω (3) := ¬˙ Ṙ2 , Ṙ3
D
E
ω (2) := Ṙ2 , ¬˙ Ṙ3
D
E
ω (4) := ¬˙ Ṙ2 , ¬˙ Ṙ3 .
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
37 / 56
Example: BonJour’s “ravens” challenge
Formalisation of the belief systems
An individual with information set R0 has a belief system S0 consisting
of:
0 0 0 0 0 0 0 0 0
R |Ω kα , R |R kα , R |R kα for some α10 ∈ [0, 1];
10 0 10 10 20 10 10 30 10 R |Ω kα , R |R kα , R |R kα for some α20 ∈ [0, 1];
20 0 20 20 10 02 20 30 02 R3 |Ω kα3 , R3 |R1 kα3 , R3 |R2 kα3 for some α30 ∈ [0, 1], wherein
for every i ∈ {1, 2, 3}, αi0 is sufficiently close to 1.
Hence for the belief system S0 , the set Ω0 of states of the world only
needs eight elements:
D
E
D
E
ω 0(1) := Ṙ10 , Ṙ20 , Ṙ30
ω 0(2) := Ṙ10 , Ṙ20 , ¬˙ Ṙ30
D
E
D
E
ω 0(3) := Ṙ10 , ¬˙ Ṙ20 , Ṙ30
ω 0(4) := Ṙ10 , ¬˙ Ṙ20 , ¬˙ Ṙ30
E
D
E
D
ω 0(6) := ¬˙ Ṙ10 , Ṙ20 , ¬˙ Ṙ30
ω 0(5) := ¬˙ Ṙ10 , Ṙ20 , Ṙ30
D
E
D
E
ω 0(8) := ¬˙ Ṙ10 , ¬˙ Ṙ20 , ¬˙ Ṙ30
ω 0(7) := ¬˙ Ṙ10 , ¬˙ Ṙ20 , Ṙ30
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
38 / 56
Example: BonJour’s “ravens” challenge
Formalisation of the belief systems
Now we shall
compute — or at least estimate — the several components
β1 , β2 , β3 , β4 of our coherence measure applied to S and S0 , and
show that according to our multi-dimensional coherence measure,
the coherence of S dominates that of S0 .
For the application of our coherence measure (in particular for β3 , β4 ),
we need to represent S and S0 as Bayesian networks.
The belief system S0 is represented by a completely disconnected
graph GS0 with three vertices, so that
β3 (S0 ) = h0, h0, . . . , 0ii ,
β4 (S0 ) = 0.
The belief system S consists of two vertices with an arrow from R2
(‘This bird is a raven’) to R3 (‘This bird is black’).
We shall now calculate β1 , β2 , β3 , β4 for S and S0 ; we shall also estimate
imw
β̃2 for both belief systems.
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
39 / 56
Example: BonJour’s “ravens” challenge
Calculation of the third and fourth component of the coherence
measure
1
Introduction
2
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
Formalising BonJour’s first desideratum
Formalising BonJour’s second desideratum
Formalising BonJour’s third and fourth desiderata
3
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
Formalisation of the belief systems
Calculation of the third and fourth component of the coherence measure
Calculation of graded probabilistic consistency. The second component
of the coherence measure
Calculation of the first component of the coherence measure
Summary
4
Conclusion and discussion
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
40 / 56
Example: BonJour’s “ravens” challenge
Calculation of the third and fourth component of the coherence
measure
For the calculation of the third coherence component, observe:
The (vertex) connectivity of GS is 1, the single inferential non-zero
connection in GS being the one from R2 to R3 , whose strength is
γ(R2 , R3 ) = α1 − α3 > 0.
In contrast, the (vertex) connectivity of GS0 is 0, and thus the
vector of strengths of inferential connections among the vertices in
GS0 is trivial (consists of zeroes only). Thus,
* *
++
β3 (S) =
1, α1 − α3 , 0, . . . , 0
| {z }
.
>0
0
β3 (S ) = h0, h0, . . . , 0ii .
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
41 / 56
Example: BonJour’s “ravens” challenge
Calculation of the third and fourth component of the coherence
measure
For the calculation of the fourth coherence component, observe that
the relative fragmentation of both S and S0 is zero:
GS has connectivity 1, but no 2-connected components;
GS0 has connectivity 0 (is totally disconnected) and therefore
cannot have any 1-connected components.
Thus,
β4 (S) = 0 = β4 (S0 ).
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
42 / 56
Example: BonJour’s “ravens” challenge
Calculation of graded probabilistic consistency. The second
component of the coherence measure
1
Introduction
2
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
Formalising BonJour’s first desideratum
Formalising BonJour’s second desideratum
Formalising BonJour’s third and fourth desiderata
3
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
Formalisation of the belief systems
Calculation of the third and fourth component of the coherence measure
Calculation of graded probabilistic consistency. The second component
of the coherence measure
Calculation of the first component of the coherence measure
Summary
4
Conclusion and discussion
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
43 / 56
Example: BonJour’s “ravens” challenge
Calculation of graded probabilistic consistency. The second
component of the coherence measure
We shall calculate β̃2 for S, S0 ; as a by-product, this will yield
probabilistic consistency proofs for both S and S0 .
Now, in the case of S, the canonical geometrical representation ι[PS ] is
the intersection of the four-dimensional unit cube with four
hyperplanes, one for each of the three (conditional) probability
assignment) plus the hyperplane generated by requiring the total mass
to add up to one:


x (1) + x (2) + x (3) + x (4) = 1, 





x (1) = x (1) + x (2) α1 ,
4
ι[PS ] = x ∈ [0, 1] :
.


x (1) + x (2) = α2 ,




x (1) + x (3) = α3 .
Herein, any x = x (1) , x (2) , x (3) , x (4) ∈ ι[PS ] represents a probability
(1) (2) (3) (4) measure on the power-set of Ω = ω , ω , ω , ω
that is
compatible with S and assigns probability x (i) to {ω (i) } for each
i ∈ {1, 2, 3, 4}; the three (conditional) probability assignments that imw
make up S have been encoded accordingly.
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
44 / 56
Example: BonJour’s “ravens” challenge
Calculation of graded probabilistic consistency. The second
component of the coherence measure
By elementary linear algebra, this reduces to

α1 α2



α
(1
− α1 )
2
ι[PS ] = 

α
−
α1 α2

3


1 − α2 (1 − α1 ) − α3




 .



(3)
This is a singleton and thus its H AUSDORFF dimension is zero and so
is its H AUSDORFF measure: Therefore,
β̃2 (S) = h0, 0i.
This, however, does not mean that S is probabilistically inconsistent,
just that is in a sense ‘minimally consistent’ probabilistically. The belief
system S is probabilistically consistent, yet S induces a unique
probability system that is compatible with S.
Thus, S cannot be dominated in terms of probabilistic consistency
imw
simpliciter by S0 .
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
45 / 56
Example: BonJour’s “ravens” challenge
Calculation of graded probabilistic consistency. The second
component of the coherence measure
Nevertheless, for the sake of illustration, we shall also compute β̃2 (S0 ).
A similar analysis for S0 will show that ι[PS0 ] is actually given by the
following linear equation system:


(1) + · · · + x (8) = 1,
x






(1) + x (2) + x (3) + x (4) = α0 ,


x


1




(1)
(2)
(1)
(2)
(5)
(6)
0




x
+
x
=
x
+
x
+
x
+
x
α
,


1




(1)
(3)
(1)
(3)
(5)
(7)
0


x
+
x
=
x
+
x
+
x
+
x
α
,

1 




(1)
(2)
(5)
(6)
0
x
+
x
+
x
+
x
=
α
,
8
2
,
x ∈ [0, 1] :
ι[PS0 ] =

x (1) + x (2) = x (1) + x (2) + x (3) + x (4) α20 , 







x (1) + x (5) = x (1) + x (3) + x (5) + x (7) α20 , 






(1)
(3)
(5)
(7)
0


x
+
x
+
x
+
x
=
α
,


3




(1)
(3)
(1)
(2)
(3)
(4)
0


x
+
x
=
x
+
x
+
x
+
x
α
,


3




(1)
(5)
(1)
(2)
(5)
(6)
0
x +x = x +x +x +x
α3 .
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
46 / 56
Example: BonJour’s “ravens” challenge
Calculation of graded probabilistic consistency. The second
component of the coherence measure
This can be simplified to become
ι[PS0 ]

λ



0


α1 α20 − λ






α10 α30 − λ



0

α1 − α10 α20 − α10 α30 + λ

=

α20 α30 − λ




0



α2 − α10 α20 − α20 α30 + λ






α30 − α10 α30 − α20 α30 + λ


0
(1 − α1 )(1 − α20 ) + α30 (α20 − 1) + α10 α30 − λ


















 : λ∈R
















∩[0, 1]8
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
47 / 56
Example: BonJour’s “ravens” challenge
Calculation of graded probabilistic consistency. The second
component of the coherence measure
In other words, ι[PS0 ] consists of all vectors of the eight-dimensional
unit cube that can be written in the form

 

1
0
  −1 

α10 α20

 

0 α0
  −1 

α
1
3

 

  1 

α10 (1 − α20 − α30 )

+

  −1  λ

α20 α30

 

  1 

α20 (1 − α10 − α30 )

 

  1 

α30 (1 − α10 − α20 )
−1
(1 − α10 )(1 − α20 ) + α30 (α20 − 1) + α10 α30
for some real number λ. Clearly, the H AUSDORFF dimension of ι[PS0 ] is
one, and its H AUSDORFF measure is positive.
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
48 / 56
Example: BonJour’s “ravens” challenge
Calculation of the first component of the coherence measure
1
Introduction
2
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
Formalising BonJour’s first desideratum
Formalising BonJour’s second desideratum
Formalising BonJour’s third and fourth desiderata
3
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
Formalisation of the belief systems
Calculation of the third and fourth component of the coherence measure
Calculation of graded probabilistic consistency. The second component
of the coherence measure
Calculation of the first component of the coherence measure
Summary
4
Conclusion and discussion
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
49 / 56
Example: BonJour’s “ravens” challenge
Calculation of the first component of the coherence measure
Now we calculate the first coherence component, β1 , i.e. logical
consistency.
From our above calculations of representations of PS and PS0 (under ι)
it is obvious that only the top element of the Boolean algebra is
assigned probability 1 by every probability measure that is compatible
with the respective belief system (be it S or S0 ).
Therefore, η(S, {1}) = Ω and η(S0 , {1}) = Ω0 , whence
β1 (S) = 1 = β1 (S0 ).
Just as claimed by B ON J OUR when introducing the challenge, S and S0
cannot be told apart by considering logical and probabilistic
consistency (i.e. β1 and β2 ) alone.
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
50 / 56
Example: BonJour’s “ravens” challenge
Summary
1
Introduction
2
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
Formalising BonJour’s first desideratum
Formalising BonJour’s second desideratum
Formalising BonJour’s third and fourth desiderata
3
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
Formalisation of the belief systems
Calculation of the third and fourth component of the coherence measure
Calculation of graded probabilistic consistency. The second component
of the coherence measure
Calculation of the first component of the coherence measure
Summary
4
Conclusion and discussion
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
51 / 56
Example: BonJour’s “ravens” challenge
Summary
As we have seen,
β1 (S) = β1 (S0 ),
β2 (S) = β2 (S0 ),
β3 (S) > β3 (S0 ),
β4 (S) = β4 (S0 ).
(4)
This is fully in line with B ON J OUR’s (1985, p. 95f) expectations:
the two belief systems are indistinguishable in terms of logical and
probabilistic consistency, but
S is more coherent than S0 — on account of inferential
connections.
Thus, our formal coherence notion has passed the test of B ON J OUR’s
challenge.
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
52 / 56
Conclusion and discussion
1
Introduction
2
Formalisation of (the core of) BonJour’s coherence notion
Formal framework
Formalising BonJour’s first desideratum
Formalising BonJour’s second desideratum
Formalising BonJour’s third and fourth desiderata
3
Example: BonJour’s “ravens” challenge
Preparations for the formalisation of BonJour’s challenge
Formalisation of the belief systems
Calculation of the third and fourth component of the coherence measure
Calculation of graded probabilistic consistency. The second component
of the coherence measure
Calculation of the first component of the coherence measure
Summary
4
Conclusion and discussion
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
53 / 56
Conclusion and discussion
Conclusion and discussion
We have formalised doxastic systems as families of conditional
degree-of-belief assignments, rather than sets of propositions.
Doxastic systems can thereby be studied as Bayesian networks,
permitting the use of new mathematical concepts.
The core of B ON J OUR’s coherence concept can be formalised in
this formal framework.
There is potential for relatively straightforward generalisation:
I
I
One can replace the Lockean thesis with another bridge principle,
such as those discussed by L EITGEB (2013, 2014).
One can replace the relevance measure of confirmation
The weights of the components have not been specified and may
be dependent by context. Some norms would be desirable for this.
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
54 / 56
Appendix
Selected References
Selected References I
H. Arló-Costa and R. Parikh.
Conditional probability and defeasible inference.
Journal of Philosophical Logic, 34(1):97–119, 2005.
H. Arló-Costa and A.P. Pedersen.
Belief and probability: A general theory of probability cores.
International Journal of Approximate Reasoning, 53(3):293–315, 2012.
L. BonJour.
The structure of empirical knowledge.
Harvard University Press, Cambridge, MA, 1985.
L. Bovens and S. Hartmann.
Solving the riddle of coherence.
Mind, 112(448):601–633, 2003.
R. Foley.
Beliefs, degrees of belief, and the Lockean Thesis.
In F. Huber and C. Schmidt-Petri, editors, Degrees of Belief, volume 342 of Synthese Library, pages 37–47. Springer,
Dordrecht, 2009.
F.S. Herzberg.
Aggregating infinitely many probability measures.
Theory and Decision, (forthcoming), 2014.
F.S. Herzberg.
The dialectics of infinitism and coherentism: Inferential justification versus holism and coherence.
Synthese, 191(4):701–723, 2014.
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
imw
55 / 56
Appendix
Selected References
Selected References II
F.S. Herzberg.
A graded Bayesian coherence notion.
Erkenntnis, (forthcoming), 2014.
F.S. Herzberg.
Universal algebra and general aggregation: Many-valued propositional-attitude aggregators as MV-homomorphisms.
Journal of Logic and Computation, (forthcoming), 2014.
K. Lehrer.
Theory of knowledge.
Westview Press, Boulder, CO, 2000.
H. Leitgeb.
Reducing belief simpliciter to degrees of belief.
Annals of Pure and Applied Logic, 2013.
H. Leitgeb.
The stability theory of belief.
Philosophical Review, 2014.
W.V.O. Quine and J. Ullian.
The web of belief.
Random House, New York, 1970.
imw
Frederik Herzberg (IMW / MCMP)
A graded Bayesian coherence notion
Full and Partial Belief, 2014
56 / 56