Degrees of Belief
Huber (Franz) & Schmidt-Petri (Christoph), Eds.
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  1. The idea that belief comes in degrees is based on the observation that we are more certain of some things than of others.
  2. Various theories try to give accounts of how measures of this confidence do or ought to behave, both as far as the internal mental consistency of the agent as well as his betting, or other, behaviour is concerned.
  3. This anthology is the first book to give a balanced overview of these theories. It also explicitly relates these debates to more traditional concerns of the philosophy of language and mind, and epistemic logic, namely how belief simpliciter does or ought to behave.
  4. The paradigmatic theory, probabilism (which holds that degrees of belief ought to satisfy the axioms of probability theory) is given most attention, but competing theories, such as Dempster-Shafer theory, possibility theory, and AGM belief revision theory are also considered.
  5. Each of these approaches is represented by one of its major proponents. The papers are specifically written to target advanced undergraduate students with a background in formal methods and beginning graduate students, but they will also serve as first point of reference for academics new to the area.
  6. Christoph Schmidt-Petri: has been awarded a PhD in Philosophy from the London School of Economics in 2005, where he also held a pre-doctoral Jacobsen Fellowship. He has been a member of the Philosophy, Probability and Modeling Group at the University of Konstanz, the Faculty of Economics at Witten/Herdecke University, and the Departments of Philosophy at the Universities of Glasgow, Saarbrücken and Leipzig. From 2002 till 2008, he has been Managing Editor of Economics and Philosophy. He has published articles in journals such as The Philosophical Quarterly, Philosophy of Science, Analyse & Kritik, and is the editor of several other books in political philosophy and the philosophy of the social sciences.
  7. Franz Huber: received his PhD from the University of Erfurt in 2003. From 2002 to 2005 he was postdoctoral researcher in the Philosophy, Probability, and Modeling group at the University of Konstanz. From 2005 to 2007 he was Ahmanson postdoctoral instructor at the California Institute of Technology and then visiting researcher at the Department of Logic and Philosophy of Science at UC Irvine. Since 2008 he is director of the Formal Epistemology Research Group at the University of Konstanz. Huber has published in journals such as Artificial Intelligence1, The British Journal for the Philosophy of Science, The Journal of Philosophical Logic, Philosophical Studies, Philosophy of Science, Studia Logica and Synthese.

  1. Belief and Degrees of Belief (Franz Huber) – 1
  2. Part I: Plain Belief and Degrees of Belief
    • Beliefs, Degrees of Belief, and the Lockean Thesis (Richard Foley) – 37
    • The Lockean Thesis and the Logic of Belief (James Hawthorne) – 49
    • Partial Belief and Flat-Out Belief (Keith Frankish) – 75
  3. Part II: What Laws Should Degrees of Belief Obey?
    • Epistemic Probability and Coherent Degrees of Belief (Colin Howson) – 97
    • Non-Additive Degrees of Belief (Rolf Haenni) – 121
    • Accepted Beliefs, Revision and Bipolarity in the Possibilistic Framework (Didier Dubois and Henri Prade) – 161
    • A Survey of Ranking Theory (Wolfgang Spohn) – 185
    • Arguments For - Or Against - Probabilism? (Alan Hajek) – 229
    • Diachronic Coherence and Radical Probabilism (Brian Skyrms) – 253
    • Accuracy and Coherence: Prospects for an Alethic Epistemology of Partial Belief (James M. Joyce) – 263
  4. Part III: Logical Approaches
    • All the Way Down: Beliefs, Non-Beliefs and Disbeliefs (Hans Rott) – 301
    • Levels of Belief in Non-monotonic Reasoning (David Makinson) – 341

"Marasoiu (Andrei) - Review of 'Degrees of Belief' (Huber (Franz) & Schmidt-Petri (Christoph), Eds)"

Source: Metapsychology Online Reviews, Nov 3rd 2009 (Volume 13, Issue 45)

Full Text
  1. Review - Degrees of Belief by Franz Huber and Christoph Schmidt-Petri (Editors), Springer, 2009
    Review by Andrei Marasoiu
    Nov 3rd 2009 (Volume 13, Issue 45)
  2. Degrees of Belief edited by Franz Huber and Christoph Schmidt-Petri is a collection of articles devoted to logical and probabilistic models of partial beliefs, i.e., beliefs which we hold to a certain degree, beliefs of which we are not necessarily certain, but partially believe them nonetheless. This collection is intended as a comprehensive survey of the work done on belief revision (i.e., how our partial beliefs change in time) and also contains some novelties in the literature. The intended audience is supposed to be familiar with propositional logic and with probability theory, but no previous acquaintance with the topics discussed in the book is required.
  3. Propositional logic is necessary because the objects of belief are identified with propositions, and probability theory is handy because the traditional way of modelling degrees of belief has been by means of subjective probabilities. According to this approach, holding a partial belief is a relation between an epistemic agent (typically, a human being), a proposition and a probability (the degree to which the agent believes the proposition). Franz Huber's introductory chapter is an excellent guide through the rest of the book. An overall assessment reveals a good balance between appeal to commonsensical intuitions concerning belief and formal constructions.
  4. The collection is divided into three parts. The first part, Plain Beliefs and Degrees of Belief, focuses on the relation between flat-out (or plain) beliefs and partial beliefs. One important question in this area is whether the Lockean thesis, which claims that when the degree of a partial belief is high enough, that belief should count as a plain belief, is true or not. Subsequently, supposing it true, how high should the threshold of partial belief be fixed so that it correspond to plain belief? These questions preoccupy the contributions of Richard Foley, James Hawthorne and Keith Frankish.
  5. The second part, What Laws Should Degrees of Belief Obey? , is devoted to quantitative approaches to partial belief, approaches which seek to measure the degree of belief. By way of contrast, the third part, Logical Approaches, deals with qualitative approaches to belief, according to which the strength with which a certain proposition is believed is determined by means of a relation similar to “agent x believes proposition y more than she believes proposition z”. Are qualitative approaches better than quantitative ones? This sounds plausible, because we do not daily measure our belief in the propositions we hold to be more or less true, but quantitative approaches also have qualitative counterparts.
  6. If the phenomena associated with believing had one ultimate and coherent account, there would be no need for so many alternative approaches. However, there is a very good reason why these alternatives exist: paradoxes emerge (the most serious one is the paradox of the lottery), and more than one solution can be offered. Here is a very sketchy account of the paradox of the lottery. Suppose you buy a ticket in a lottery of 1000 tickets. Although the objective chances of your winning equal 1 per 1000 (and the sum of all these equal chances is 1), you will ordinarily expect to gain nothing, i.e., your subjective probability will be 0 instead of 1 per 1000. But if this is the typical behavior of epistemic agents, then subjective probabilities (0+0+…) do not add up to 1, as the objective ones (1/1000+1/1000+…) do. This is known as the problem of additivity. Why should additivity be endorsed? Because it is an axiom of classical probability theory, according to which if two events A and B are possible, then if A and B are independent of one another, the probability of either of theim happening together is the sum of their individual probabilities. There are two solutions: either solve the paradox of the lottery within probability theory, or find an alternative theory to solve it.
  7. Replacing the additivity axiom in probability theory yields patterns of non-monotonic reasoning, for which David Makinson's contribution is a good overview. The two obvious solutions are either endorsing sub-additivity or supra-additivity. Sub-additivity is the option incorporated in possibility theory (see Dubois and Prade's contribution), according to which, given the necessity measures for two events A and B, the necessity measure of either of them happening is the minimal necessity measure, as the case is, of A or of B. Supra-additivity is achieved by means of DS (Dempster-Shafer) belief functions (see Rolf Haenni's contribution): if two events A and B are independent, then believing that either them of happen is stronger than believing that A happens or believing that B happens.
  8. Additivity (in subjective probabilites), sub-additivity (in possibility theory) and supra-additivity (in DS functions) exhaust the possibilities. The fourth quantitative approach, ranking theory (see Wolfgang Spohn's contribution, also the proponent of ranking theory in the 1980s) combines the advantages of the other three. One such advantage, due to Hans Rott's article, is a unified account of degrees of belief, disbelief (belief that something is not the case) and degrees of acceptance (considering how plausible something is while remaining ignorant as to its truth).
  9. But probability theory also has resources to cope with the lottery paradox, as shown by the contributions of Colin Howson and Brian Skyrms. Skyrms appeals to convex sets of probabilities. In order to make his idea intuitive, Skyrms draws an analogy between the expected prices on a market and the expected probability of a belief. Colin Howson sticks to first-order probabilites, but identifies two ways of viewing probability theory, one due to Keynes and Carnap, and the other due to Ramsey and de Finetti. He argues that endorsing the views of the latter is a good argument in favor of probabilism (the thesis that belief revision obeys the probability calculus).
  10. Given the weak conclusion that probabilism remains a viable option among many, is it also possible to show that probabilism is true? This is the import of Dutch book arguments, defended by J.Joyce and considered non-valid by Alan Hajek. A Dutch book is a series of bets each of which seems beneficial but the final result of which is costful for the epistemic agent. The argument sustained by Joyce and questioned by Hayek is that if an epistemic agent does not bet according to the probability calculus, she can be Dutch booked. However, even if Dutch book arguments are not valid, it is still the case that betting behavior provides a straightforward intuitive support to probabilism, at least a more cogent one than any intuitive representation of how possibility theory, DS functions or ranking theory work.
  11. Andrei Marasoiu is currently a student in a Master's Program in the History and Philosophy of Science at Bucharest. His main area of specialization is the philosophy of W.V. Quine.


"Huber (Franz) - Belief and Degrees of Belief"

Source: Huber (Franz) & Schmidt-Petri (Christoph) - Degrees of Belief, Introduction

  1. Introduction
  2. Objects of Belief
  3. Theories of Degrees of Belief
    • 3.1 Subjective Probabilities
      … Update Rule 1 (Strict Conditionalization)
      … Update Rule 2 (Jeffrey Conditionalization)
    • 3.2 Dempster-Shafer Belief Functions
    • 3.3 Possibility Theory
    • 3.4 Summary
  4. Belief, Degrees of Belief, and Ranking Functions
    … Update Rule 3 (Plain Conditionalization)
    … Update Rule 4 (Spohn Conditionalization)
  5. Belief Revision and Non-Monotonic Reasoning
    • 5.1 Belief and Belief Revision
    • 5.2 Belief and Non-Monotonic Reasoning

"Foley (Richard) - Beliefs, Degrees of Belief, and the Lockean Thesis"

Source: Huber (Franz) & Schmidt-Petri (Christoph) - Degrees of Belief, Part I

  1. Two questions
    • What propositions is it rational to believe? 3-fold classification of Belief, Disbelief and Suspended Judgement.
    • With what confidence is it rational to believe these propositions? More fine-grained.
  2. Close cousins, connected by the Lockean Thesis:
    • “It is rational for someone S to believe a proposition P just in case it is rational for S to have a degree of confidence in P that is sufficient for belief.”
    • Foley points out that this allows for beliefs to be rational even if S’s degree of belief only approximates to that warranted by the evidence. Exact correspondence would be too much of a constraint on rationality. The thesis seems to hold that rationality survives provided that the warranted degree of belief is sufficient, whether or not this is the exact degree of belief S actually has. I’m slightly suspicious of this, as S may respond entirely wrongly to the evidence, even though – were he to react rationally – this evidence would suffice. But I agree that we can’t expect beliefs to exactly coincide with the evidence.

"Hawthorne (James) - The Lockean Thesis and the Logic of Belief"

Source: Huber (Franz) & Schmidt-Petri (Christoph) - Degrees of Belief, Part I

  1. Introduction
    The Lockean Thesis: “It is epistemically rational for us to believe a proposition just in case it is epistemically rational for us to have a sufficiently high degree of confidence in it, sufficiently high to make our attitude to it one of belief.” (Richard Foley, 1992)
  2. Ideal Agents and the Qualitative Lockean Thesis
  3. The Logic of Comparative Confidence
    • 3.1 The Rudimentary Confidence Relations
      … Definition 1: Rudimentary Confidence Relations
    • 3.2 The Completed Confidence Relations
      … Definition 2: Properly Extendable Rudimentary Confidence Relations
      … Definition 3: Confidence Relations
  4. The Integration of Confidence and Belief
    • 4.1 The Preface and the n-Bounded Belief Logics
    • 4.2 The Lottery and the (n+1)*-Bounded Belief Logics
  5. The Logic of Belief
    … Definition 4: the Rudimentary n-Level Confidence-Belief Pairs
    … Definition 5: the Properly Extendable n-Level Confidence-Belief Pairs
    … Definition 5: the n-Level Confidence-Belief Pairs
  6. Concluding Remarks
    The Contextual Qualitative Lockean Thesis: “An agent is epistemically warranted in believing a statement in a context psi just in case she is epistemically warranted in having a sufficiently high grade of confidence in the statement – sufficiently high to make her attitude to it one of belief in context psi.”

"Frankish (Keith) - Partial Belief and Flat-Out Belief"

Source: Huber (Franz) & Schmidt-Petri (Christoph) - Degrees of Belief, Part I

  1. Introduction: A Duality
  2. Partial Belief as a Derivative of Flat-Out Belief
  3. Flat-Out Belief as a Derivative of Partial Belief
  4. Flat-Out Belief as an Intentional Disposition
  5. Flat-Out Belief as a Premising Policy
  6. Flat-Out Belief and Action
  7. Flat-Out Belief and Formal Epistemology

"Haenni (Rolf) - Non-Additive Degrees of Belief"

Source: Huber (Franz) & Schmidt-Petri (Christoph) - Degrees of Belief, Part II

  1. Introduction
    • 1.1 Properties of Degrees of Beliefs
      … Fig 1: Non-monotone and Non-additive Degrees of Belief
    • 1.2 Opinions
      … Fig 2: The Opinion Triangle with its Three Dimensions
      … Fig 3: Special Types of Opinions
    • 1.3 Related Work
      … Fig 4: Belief and Plausibility Induced by an Opinion omegah
      … Fig 5: Betting Probability Induced by an Opinion omegah
    • 1.4 Historical Roots
  2. Dempster-Shafer Theory
    • 2.1 Frame of Discernment
    • 2.2 Mass Functions
      … Fig 6: Different Classes of Mass Functions
      … Fig 7: The Complete Multi-valued Mapping of the Murder Case Example
      … Fig 8: The Evidence Reduced to a Mass Function
    • 2.3 Belief and Plausibility Functions
      … Table 1: The Connection between Mass, Belief and Plausibility Functions
      … Fig 9: The Complete Multi-valued Mapping
      … Table 2: Belief and Plausibility Functions Induced by a Hint
      … Fig 10: The Opinions for all Eta subsets of Theta
    • 2.4 Dempster’s Rule of Combination
    • 2.5 Aleatory versus Epistemic Uncertainty
      Epistemic Uncertainty: results from lack of knowledge, a property of the analysts.
      Aleatory Uncertainty: results from the fact that a system can perform in random ways.
  3. Probabilistic Argumentation
    • 3.1 Unifying Logic and Probability
      … Fig 11: Different Sets of Probabilistic Variables
    • 3.2 Degree of Support
      … Fig 12: Prior and Posterior Distributions over Probabilistic Variables
      … Fig 13: Evidence Conditioned on Various Scenarios
      … Fig 14: Supporting, Refuting, Neutral, and Inconsistent Scenarios
    • 3.3 Connection to Dempster-Shafer Theory
    • Logical and Probabilistic Reasoning
  4. Conclusion

"Howson (Colin) - Epistemic Probability and Coherent Degrees of Belief"

Source: Huber (Franz) & Schmidt-Petri (Christoph) - Degrees of Belief, Part II

  1. Introduction: the Principle of Indifference
  2. Ramsey
  3. Savage
  4. De Finetti
  5. Finite versus Countable Additivity
  6. A Different Approach
  7. Consistency and Coherence
  8. Conditionalisation
  9. Conclusion

"Dubois (Didier) & Prade (Henri) - Accepted Beliefs, Revision and Bipolarity in the Possibilistic Framework"

Source: Huber (Franz) & Schmidt-Petri (Christoph) - Degrees of Belief, Part II

  1. Introduction
  2. The Representation of Uncertainty in Qualitative Probability Theory
    • Possibility Distribution
    • Possibility Measure
    • Potential Surprise
    • Necessity Measure
    • Reasoning According to the Weakest Link
    • Other Related Settings
    • Representing Information with Possibility and Necessity Degrees
  3. Degrees of Belief versus Degrees of Truth
    • The Three Classical Epistemic States
    • Stratified Information
    • Graded Epistemic States
  4. Accepted Beliefs Induced by a Confidence Relation
    • Confidence Relations
    • Accepted Beliefs
    • Contextual Beliefs
    • Acceptance Relations
    • Big-Stepped Probabilities
    • The Lottery Paradox
  5. Non-Monotonic Reasoning and Possibility Relations
    • Representing Conditionals
    • Basic Postulates
    • Reasoning with Default Rules
    • Rational Monotony
      … Fig. 1: Non-Monotonic Entailment
    • Possibilistic Reasoning and Independence
  6. Revising Accepted Beliefs versus Revising an Acceptance Relation
  7. A Bipolar View of Information
  8. Concluding Remarks

"Spohn (Wolfgang) - A Survey of Ranking Theory"

Source: Huber (Franz) & Schmidt-Petri (Christoph) - Degrees of Belief, Part II

  1. Introduction
  2. The Theory
    • 2.1 Basics
      … 6 Definitions
    • 2.2 Reasons and their Balance
      … Definitions 7 & 8
    • 2.3 The Dynamics of Belief and the Measurement of Belief
      … Definitions 9 & 10
    • 2.4 Conditional Independence and Bayesian Nets
      … Definitions 11 & 12
    • 2.5 Objective Ranks?
  3. Ranks and Probabilities
    • 3.1 Formal Aspects
    • 3.2 Philosophical Aspects
  4. Further Comparisons
    • 4.1 Earlier and Philosophical Literature
    • 4.2 More Recent Computer Science Literature

"Hajek (Alan) - Arguments For - Or Against - Probabilism?"

Source: Huber (Franz) & Schmidt-Petri (Christoph) - Degrees of Belief, Part II

  1. Introduction
  2. The Dutch Book Argument
    • Dutch Book Theorem: If you violate probability theory, there exists a set of bets, each of which you consider fair, which collectively guarantee your loss.
    • Czech Book Theorem: If you violate probability theory, there exists a set of bets, each of which you consider fair, which collectively guarantee1 your gain.
    • 2.1 Saving the Dutch Book Argument
    • 2.2 ‘The Dutch Book Argument merely Dramatises an Inconsistency in the Attitudes of an Agent Whose Credences Violate Probability Theory”
  3. Representation Theorem-Based Arguments
    • Representation Theorem: If all your preferences satisfy certain ‘rationality’ conditions, then there exists a representation of you as an expected utility maximizer, relative to some probability and utility function.
    • Representation Theorem Argument
    • Conclusion: [All] Rational Persons Have Probability and Utility Functions
  4. The Calibration Argument
    • Calibration Theorem: If c violates the laws of probability then there is a probability function c+ that is better calibrated than c under every logically consistent assignment of truth-values to propositions.
  5. The Gradational Accuracy Argument
    • Gradational Accuracy Theorem: If c violates the laws of probability then there is a probability function c+ that strictly more accurate than c under every logically consistent assignment of truth-values to propositions.
  6. Conclusion

In-Page Footnotes ("Hajek (Alan) - Arguments For - Or Against - Probabilism?")

Footnote 1: Well, yes – but the bookie may not agree, and so won’t take your bets.

"Skyrms (Brian) - Diachronic Coherence and Radical Probabilism"

Source: Huber (Franz) & Schmidt-Petri (Christoph) - Degrees of Belief, Part II

  1. Introduction
  2. Arbitrage
  3. Degrees of Belief
  4. Probability
  5. Probabilities of Probabilities
  6. Diachronic Coherence Revisited
  7. Coherence and Conditioning
  8. Probability Kinematics
  9. Tomorrow and Tomorrow and Tomorrow
  10. Diachronic Coherence Generalised

"Joyce (James M.) - Accuracy and Coherence: Prospects for an Alethic Epistemology of Partial Belief"

Source: Huber (Franz) & Schmidt-Petri (Christoph) - Degrees of Belief, Part II

  1. Formal Framework
  2. Epistemic Utility and Scoring Rules
  3. The Principle of Admissibility
  4. Estimation and Accuracy
  5. Atomism or Holism?
  6. Content Independence
  7. Some Examples
  8. Strictly Proper Measures
  9. Convexity
  10. Prospects for a Non-Pragmatic Vindication of Probabilism
  11. Is Inadmissibility an Epistemic Defect?
  12. Homage to the Brier Score
  13. Appendix: Proof of Theorem(s)

"Rott (Hans) - All the Way Down: Beliefs, Non-Beliefs and Disbeliefs"

Source: Huber (Franz) & Schmidt-Petri (Christoph) - Degrees of Belief, Part III

  1. Introduction
  2. Degrees of Beliefs
    • 2.1 Entrenchment Relations
    • 2.2 Entrenchment Ranking Functions
    • 2.3 Entrenchment Functions and Relations
  3. Degrees of Disbeliefs
    • 3.1 Plausibility Relations
    • 3.2 Plausibility Ranking Functions
    • 3.3 Plausibility Functions and Plausibility Relations
  4. Combining Degrees of Belief and Degrees of Disbelief
    • 4.1 Rabinowicz’ Likelihood Relations
    • 4.2 Spohnian Beta Functions
    • 4.3 Spohnian Beta Functions and Rabinowicz’ Likelihood Relations
  5. Degrees for Non-Beliefs: Expectations, Disexpectations and Non-expectations
    • 5.1 Relations for Non-Beliefs
    • 5.2 Functions for Non-Beliefs
  6. Combining Degrees of Beliefs and Disbeliefs with Degrees of Non-Beliefs
    • 6.1 Combining Relations for Beliefs and Disbeliefs with Relations for Non-Beliefs
    • 6.2 Combining Functions for Beliefs and Disbeliefs with Functions for Non-Beliefs
  7. Levi on Degrees of Beliefs and Degrees of Incorrigibility
  8. Conclusion: Elusive Belief
  9. Appendix I: Some Proofs – a Few Little Lemmas for Likelihood Relations
  10. Appendix II: The Modal Logic1 of Plain Belief as Implicit in the Logic of Entrenchment Relations and Functions

"Makinson (David) - Levels of Belief in Non-monotonic Reasoning"

Source: Huber (Franz) & Schmidt-Petri (Christoph) - Degrees of Belief, Part III

  1. Introduction
  2. What is Non-Monotonic Reasoning?
  3. Three Sources of Non-Monotonicity
  4. Additional Background Assumptions
  5. Enter Levels of Belief
  6. Variations
  7. Preferred States
  8. Additional Background Rules
  9. Using Probability
  10. Conclusions

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