A graded Bayesian coherence notion Frederik Herzberg Center for - - PowerPoint PPT Presentation

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imw

A graded Bayesian coherence notion

Frederik Herzberg

Center for Mathematical Economics (IMW), Bielefeld University Munich Center for Mathematical Philosophy, Ludwig Maximilian University of Munich

WORKSHOP ‘FULL AND PARTIAL BELIEF’ (CO-LOCATED WITH THE 4TH RENÉ DESCARTES LECTURES) Tilburg University 21 October 2014

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 1 / 56

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imw 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

• f the coherence measure

Calculation of the first component of the coherence measure Summary

4

Conclusion and discussion

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 2 / 56

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imw Introduction

Introduction (1)

Epistemic (doxastic) justification is a central and ancient topic

• f theoretical philosophy:

◮ 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

• f epistemic justification (structure of reasons).

According to this position, non-skeptical positions on epistemic justification either assert

◮ that reasons form a finite chain with some foundational proposition

at the base (foundationalism) — illustration: foundation of a house (DESCARTES); or

◮ that reasons mutually support each other (coherentism) —

illustration: planks of a boat (NEURATH); or

◮ that reasons form an infinite regress (infinitism) — illustration:

• pen-ended loop.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 3 / 56

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imw Introduction

Introduction (2)

The philosophical discussion on epistemic justification has seen interesting turns during the past two decades:

◮ infinitist accounts of epistemic justification were revived (e.g.

KLEIN 1998ff; formally: PEIJNENBURG 2007);

◮ a major proponent of coherentism abandoned this position

(BONJOUR 1999, 2010) for Cartesianism;

◮ impossibility theorems for coherence measures suggest that

coherentism defies formalisation (pioneers: KLEIN–WARFIELD 1994), formally reiterating an earlier criticism by EWING (1934).

We give a formal defense of coherentism that takes traditional epistemology seriously:

◮ 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 the Principle of Inferential Justification. H. 2014b)

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 4 / 56

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imw 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 (KLEIN–WARFIELD 1994, 1996; SHOGENJI 1999; AKIBA 2000; FITELSON 2003; BOVENS–HARTMANN 2003a,b, 2005, 2006; OLSSON 2002, 2005; DIETRICH–MORETTI 2005; MEIJS–DOUVEN 2007; SCHUPBACH 2008; SIEBEL–WOLFF 2008). The overall finding of this literature is that it appears to be very difficult to come up with a convincing (especially

• ne-dimensional) graded coherence notion.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 5 / 56

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imw Introduction

The (im)possibility of formal coherence concepts (2)

However, the recent formal literature on coherence measures

◮ does not interact closely with the traditional literature on

coherentism (e.g. BONJOUR 1985, LEHRER 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

• f BonJour’s (1985) (multi-dimensional) coherence concept:

◮ 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.

(H. 2014c)

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 6 / 56

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imw Introduction

Criteria for a formal graded coherence notion (1)

BONJOUR’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

• f 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. (BONJOUR 1985, Section 5.3, pp. 95, 98, 99)

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 7 / 56

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imw 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. (BONJOUR 1985, Section 5.3, pp. 95, 98, 99) We propose a class of formal coherence concepts which satisfy the first four of BONJOUR’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 8 / 56

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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. MINSKY 1986 or ORNSTEIN 1986):

◮ Building on work by ELLSBERG (1961) and GILBOA–SCHMEIDLER

(1989), decision making under multiple priors or probabilistic ambiguity (uncertainty in the sense of KNIGHT 1921) is studied.

◮ There is a body of literature on probabilistic opinion pooling (e.g.

MCCONWAY 1981 and COOKE 1991).

◮ The problem of aggregating probability measures can also be

studied within a comprehensive, theory of aggregating propositional attitudes (DIETRICH–LIST 2011).

Most recently, these results have been extended:

◮ 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 which can be solved using ultrafilters (H. 2014a).

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 9 / 56

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imw 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

• f the coherence measure

Calculation of the first component of the coherence measure Summary

4

Conclusion and discussion

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 10 / 56

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imw 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

• f the coherence measure

Calculation of the first component of the coherence measure Summary

4

Conclusion and discussion

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 11 / 56

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imw 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. JOYCE 2009; LEITGEB–PETTIGREW 2010; EASWARAN–FITELSON 2012; FITELSON–MCCARTHY 2013; WEDGWOOD 2013.)

Fix some algebra A of propositions. A belief system is a set S of triples A|Bα, where A, B ∈ A and α ∈ [0, 1]. Read A|Bα ∈ S as “the belief system S assigns to A, given B, a conditional degree of belief α”. (A is foundational for S if A|⊤1 ∈ 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 A|Bα ∈ 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. 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

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imw 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 = Skk∈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. {Sk|Sk+1αk : k ∈ N} ∪

• Sk|∁Sk+1βk
• : k ∈ N
• is consistent).

Such a pair P, S will be called a model for α, β. 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.)

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 13 / 56

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imw 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 HAUSDORFF (1918) dimension and HAUSDORFF 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 maximal connected (proper) subgraphs (components).

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 14 / 56

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imw 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. ROBINSON’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 (HOOVER–KEISLER 1984; FAJARDO–KEISLER 2002).

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 15 / 56

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imw 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

• f the coherence measure

Calculation of the first component of the coherence measure Summary

4

Conclusion and discussion

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 16 / 56

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imw 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., ARLÓ-COSTA–PARIKH (2005), FOLEY (2009), LEITGEB (2013, 2014) , ARLÓ-COSTA–PEDERSEN (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.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 17 / 56

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imw 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]) = ∅, (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: η(S, I) := P−1 (I) : P ∈ PS

• =
• A∈A

∀P∈PS P(A)∈I

A. A very weak parameter-independent reading of requirement (I) would

• nly demand:

η (S, {1}) = ∅. (2)

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 18 / 56

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imw 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 BONJOUR’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): β1(S) = 1, η (S, {1}) = ∅ 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.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 19 / 56

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imw 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

• f the coherence measure

Calculation of the first component of the coherence measure Summary

4

Conclusion and discussion

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 20 / 56

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imw 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. (BONJOUR 1985, p. 95)

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 21 / 56

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imw 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: β2(S) = 1, PS = ∅ 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 HAUSDORFF (1918) dimension and HAUSDORFF measure of PS.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 22 / 56

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imw 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

• f 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 HAUSDORFF dimension of the canonical image of PS under ι and its HAUSDORFF measure: ˜ β2(S) :=

• D (ι[PS]) , HD(ι[PS]) (ι[PS])
• ,

where PS is greater than PS′ if and only if either (i) D (ι[PS]) > D (ι[PS′])

• r (ii) D (ι[PS]) = D (ι[PS′]), but HD(ι[PS]) (ι[PS]) > HD(ι[PS′]) (ι[PS′])

(lexicographic ordering).

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 23 / 56

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imw 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

• f the coherence measure

Calculation of the first component of the coherence measure Summary

4

Conclusion and discussion

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 24 / 56

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imw 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” (QUINE–ULLIAN 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.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 25 / 56

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imw 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.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 26 / 56

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imw 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 simply put γ(B, A) = 0.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 27 / 56

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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 Gc

S 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, Gc

S as well as GS and HS are

Bayesian networks.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 28 / 56

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imw Formalisation of (the core of) BonJour’s coherence notion Formalising BonJour’s third and fourth desiderata

Two aspects of BonJour’s desideratum (III)

1

“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” (BONJOUR 1985, p. 98; emphasis mine). We suggest to capture this by the graph-theoretic notion of (vertex) connectivity κ.

2

“The coherence of a system of beliefs is increased [. . . ] in proportion to the [. . . ] strength of [inferential] connections [between its component beliefs]” (BONJOUR 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). We therefore propose: β3(S) := κ(GS), γ(B, A) : A, B ∈ A .

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 29 / 56

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BonJour’s desideratum (IV)

BONJOUR’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: β4(S) = #FS, FS = ∅, 0, FS = ∅. For example, in the special case when κ(GS) = 0 (i.e. GS is totally disconnected), we have β4(S) = 0.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 30 / 56

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imw 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

• f the coherence measure

Calculation of the first component of the coherence measure Summary

4

Conclusion and discussion

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 31 / 56

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imw 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

• f the coherence measure

Calculation of the first component of the coherence measure Summary

4

Conclusion and discussion

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 32 / 56

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imw Example: BonJour’s “ravens” challenge Preparations for the formalisation of BonJour’s challenge

BONJOUR’s (1985, p. 96) challenge, as summarised by BOVENS–HARTMANN (2003, p. 718), is to consider the following propositions: ˜ R1: ‘All ravens are black.’ R2: ‘This bird is a raven.’ R3: ‘This bird is black.’ R′

1: ‘This chair is brown.’

R′

2: ‘Electrons are negatively charged.’

R′

3: ‘Today is Thursday.’

Let ˜ R =

• ˜

R1, R2, R3

• and R′ =
• R′

1, R′ 2, R′ 3

• . One has to account for

the fact that ˜ R is more coherent than R′.”

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 33 / 56

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imw Example: BonJour’s “ravens” challenge Preparations for the formalisation of BonJour’s challenge

Reformulation

No inferential connections among R′

1, R′ 2, R′ 3 are known.

In ˜ R and R′, there are no inferential connections except for the obvious

• ne, viz. modus ponens inference from R1, R2 to R3.

˜ R1 is a scheme of inferential connections rather than as a proposition. In the context of the belief system ˜ R, 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 ˜ R. BonJour’s challenge is to give an account of the greater coherence of R compared to that of R′.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 34 / 56

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imw 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

• f the coherence measure

Calculation of the first component of the coherence measure Summary

4

Conclusion and discussion

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 35 / 56

SLIDE 36

imw Example: BonJour’s “ravens” challenge Formalisation of the belief systems

If the information available to a rational individual is represented by R (or R′, respectively), this could in a Bayesian framework rendered thus:

1

she assigns to each of the propositions/events and conditional events in R (or R′, 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 A′. By Stone’s theorem, we can identify A and A′ with power-set algebras 2Ω, 2Ω′.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 36 / 56

SLIDE 37

imw Example: BonJour’s “ravens” challenge Formalisation of the belief systems

An individual with information set R has a belief system S consisting of: Ri|Ωαi for each i ∈ {2, 3} for some αi ∈ [0, 1] that is sufficiently close to 1; R3|R2α1 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: ω(1) :=

• ˙

R2, ˙ R3

• ω(2) :=
• ˙

R2, ˙ ¬ ˙ R3

• ω(3) :=
• ˙

¬ ˙ R2, ˙ R3

• ω(4) :=
• ˙

¬ ˙ R2, ˙ ¬ ˙ R3

• .

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 37 / 56

SLIDE 38

imw Example: BonJour’s “ravens” challenge Formalisation of the belief systems

An individual with information set R′ has a belief system S′ consisting

• f:
• R′

1|Ω′α′ 1

• ,
• R′

1|R′ 2α′ 1

• ,
• R′

1|R′ 3α′ 1

• for some α′

1 ∈ [0, 1];

• R′

2|Ω′α′ 2

• ,
• R′

2|R′ 1α′ 2

• ,
• R′

2|R′ 3α′ 2

• for some α′

2 ∈ [0, 1];

• R′

3|Ω′α′ 3

• ,
• R′

3|R′ 1α′ 3

• ,
• R′

3|R′ 2α′ 3

• for some α′

3 ∈ [0, 1], wherein

for every i ∈ {1, 2, 3}, α′

i is sufficiently close to 1.

Hence for the belief system S′, the set Ω′ of states of the world only needs eight elements: ω′(1) :=

• ˙

R′

1, ˙

R′

2, ˙

R′

3

• ω′(2) :=
• ˙

R′

1, ˙

R′

2, ˙

¬ ˙ R′

3

• ω′(3) :=
• ˙

R′

1, ˙

¬ ˙ R′

2, ˙

R′

3

• ω′(4) :=
• ˙

R′

1, ˙

¬ ˙ R′

2, ˙

¬ ˙ R′

3

• ω′(5) :=
• ˙

¬ ˙ R′

1, ˙

R′

2, ˙

R′

3

• ω′(6) :=
• ˙

¬ ˙ R′

1, ˙

R′

2, ˙

¬ ˙ R′

3

• ω′(7) :=
• ˙

¬ ˙ R′

1, ˙

¬ ˙ R′

2, ˙

R′

3

• ω′(8) :=
• ˙

¬ ˙ R′

1, ˙

¬ ˙ R′

2, ˙

¬ ˙ R′

3

• Frederik Herzberg (IMW / MCMP)

A graded Bayesian coherence notion Full and Partial Belief, 2014 38 / 56

SLIDE 39

imw 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 S′, and show that according to our multi-dimensional coherence measure, the coherence of S dominates that of S′. For the application of our coherence measure (in particular for β3, β4), we need to represent S and S′ as Bayesian networks. The belief system S′ is represented by a completely disconnected graph GS′ with three vertices, so that β3(S′) = 0, 0, . . . , 0 , β4(S′) = 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 S′; we shall also estimate ˜ β2 for both belief systems.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 39 / 56

SLIDE 40

imw 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

• f the coherence measure

Calculation of the first component of the coherence measure Summary

4

Conclusion and discussion

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 40 / 56

SLIDE 41

imw 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 GS′ is 0, and thus the vector of strengths of inferential connections among the vertices in GS′ is trivial (consists of zeroes only). Thus, β3(S) =

• 1,
• α1 − α3
• >0

, 0, . . . , 0

• .

β3(S′) = 0, 0, . . . , 0 .

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 41 / 56

SLIDE 42

imw 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 S′ is zero: GS has connectivity 1, but no 2-connected components; GS′ has connectivity 0 (is totally disconnected) and therefore cannot have any 1-connected components. Thus, β4(S) = 0 = β4(S′).

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 42 / 56

SLIDE 43

imw 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

• f the coherence measure

Calculation of the first component of the coherence measure Summary

4

Conclusion and discussion

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 43 / 56

SLIDE 44

imw Example: BonJour’s “ravens” challenge Calculation of graded probabilistic consistency. The second component of the coherence measure

We shall calculate ˜ β2 for S, S′; as a by-product, this will yield probabilistic consistency proofs for both S and S′. 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: ι[PS] =        x ∈ [0, 1]4 : x(1) + x(2) + x(3) + x(4) = 1, x(1) =

• x(1) + x(2)

α1, 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 measure on the power-set of Ω =

• ω(1), ω(2), ω(3), ω(4)

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 make up S have been encoded accordingly.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 44 / 56

SLIDE 45

imw Example: BonJour’s “ravens” challenge Calculation of graded probabilistic consistency. The second component of the coherence measure

By elementary linear algebra, this reduces to ι[PS] =            α1α2 α2 (1 − α1) α3 − α1α2 1 − α2 (1 − α1) − α3            . (3) This is a singleton and thus its HAUSDORFF dimension is zero and so is its HAUSDORFF measure: Therefore, ˜ β2(S) = 0, 0. 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 simpliciter by S′.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 45 / 56

SLIDE 46

imw 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 ˜ β2(S′). A similar analysis for S′ will show that ι[PS′] is actually given by the following linear equation system: ι[PS′] =                                  x ∈ [0, 1]8 : x(1) + · · · + x(8) = 1, x(1) + x(2) + x(3) + x(4) = α′

1,

x(1) + x(2) =

• x(1) + x(2) + x(5) + x(6)

α′

1,

x(1) + x(3) =

• x(1) + x(3) + x(5) + x(7)

α′

1,

x(1) + x(2) + x(5) + x(6) = α′

2,

x(1) + x(2) =

• x(1) + x(2) + x(3) + x(4)

α′

2,

x(1) + x(5) =

• x(1) + x(3) + x(5) + x(7)

α′

2,

x(1) + x(3) + x(5) + x(7) = α′

3,

x(1) + x(3) =

• x(1) + x(2) + x(3) + x(4)

α′

3,

x(1) + x(5) =

• x(1) + x(2) + x(5) + x(6)

α′

3.

                                 ,

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 46 / 56

SLIDE 47

imw Example: BonJour’s “ravens” challenge Calculation of graded probabilistic consistency. The second component of the coherence measure

This can be simplified to become ι[PS′] =                                    λ α′

1α′ 2 − λ

α′

1α′ 3 − λ

α′

1 − α′ 1α′ 2 − α′ 1α′ 3 + λ

α′

2α′ 3 − λ

α′

2 − α′ 1α′ 2 − α′ 2α′ 3 + λ

α′

3 − α′ 1α′ 3 − α′ 2α′ 3 + λ

(1 − α′

1)(1 − α′ 2) + α′ 3(α′ 2 − 1) + α′ 1α′ 3 − λ

            : λ ∈ R                        ∩[0, 1]8

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 47 / 56

SLIDE 48

imw Example: BonJour’s “ravens” challenge Calculation of graded probabilistic consistency. The second component of the coherence measure

In other words, ι[PS′] consists of all vectors of the eight-dimensional unit cube that can be written in the form             α′

1α′ 2

α′

1α′ 3

α′

1(1 − α′ 2 − α′ 3)

α′

2α′ 3

α′

2(1 − α′ 1 − α′ 3)

α′

3(1 − α′ 1 − α′ 2)

(1 − α′

1)(1 − α′ 2) + α′ 3(α′ 2 − 1) + α′ 1α′ 3

            +             1 −1 −1 1 −1 1 1 −1             λ for some real number λ. Clearly, the HAUSDORFF dimension of ι[PS′] is

• ne, and its HAUSDORFF measure is positive.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 48 / 56

SLIDE 49

imw 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

• f the coherence measure

Calculation of the first component of the coherence measure Summary

4

Conclusion and discussion

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 49 / 56

SLIDE 50

imw 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 PS′ (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 S′). Therefore, η(S, {1}) = Ω and η(S′, {1}) = Ω′, whence β1(S) = 1 = β1(S′). Just as claimed by BONJOUR when introducing the challenge, S and S′ cannot be told apart by considering logical and probabilistic consistency (i.e. β1 and β2) alone.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 50 / 56

SLIDE 51

imw 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

• f the coherence measure

Calculation of the first component of the coherence measure Summary

4

Conclusion and discussion

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 51 / 56

SLIDE 52

imw Example: BonJour’s “ravens” challenge Summary

As we have seen, β1(S) = β1(S′), β2(S) = β2(S′), β3(S) > β3(S′), β4(S) = β4(S′). (4) This is fully in line with BONJOUR’s (1985, p. 95f) expectations: the two belief systems are indistinguishable in terms of logical and probabilistic consistency, but S is more coherent than S′ — on account of inferential connections. Thus, our formal coherence notion has passed the test of BONJOUR’s challenge.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 52 / 56

SLIDE 53

imw 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

• f the coherence measure

Calculation of the first component of the coherence measure Summary

4

Conclusion and discussion

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 53 / 56

SLIDE 54

imw 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 BONJOUR’s coherence concept can be formalised in this formal framework. There is potential for relatively straightforward generalisation:

◮ One can replace the Lockean thesis with another bridge principle,

such as those discussed by LEITGEB (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.

Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 54 / 56

SLIDE 55

imw 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 55 / 56

SLIDE 56

imw 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. Frederik Herzberg (IMW / MCMP) A graded Bayesian coherence notion Full and Partial Belief, 2014 56 / 56