Change rules for hierarchical beliefs

The paper builds a belief hierarchy as a framework common to all uncertainty measures expressing that an actor is ambiguous about his uncertain beliefs. The belief hierarchy is further interpreted by distinguishing physical and psychical worlds, associated to objective and subjective probabilities. Various rules of transformation of a belief hierarchy are introduced, especially changing subjective beliefs into objective ones. These principles are applied in order to relate different contexts of belief change, revising, updating and even focusing. The numerous belief change rules already proposed in the literature receive epistemic justifications by associating them to specific belief hierarchies and change contexts. As a result, it is shown that the resiliency of probability judgments may have some limits and be reconciled with the possibility of learning from factual messages.     

Stake-Invariant Belief

What can rational deliberation indicate about belief? Belief clearly influences deliberation. The principle that rational belief is stake-invariant rules out at least one way that deliberation might influence belief. The principle is widely, if implicitly, held in work on the epistemology of categorical belief, and it is built into the model of choice-guiding degrees of belief that comes to us from Ramsey and de Finetti. Criticisms of subjective probabilism include challenges to the assumption of additive values (the package principle) employed by defenses of probabilism. But the value-interaction phenomena often cited in such challenges are excluded by stake-invariance. A comparison with treatments of categorical belief suggests that the appeal to stake-invariance is not ad hoc. Whether or not to model belief as stake-invariant is a question not settled here.     

Varieties of Pragmatic Encroachment

According to a recent view, known as the 'pragmatic encroachment' thesis, an agent’s non-truth-related factors are relevant to the epistemic status of her beliefs. In particular, in addition to truth-related factors, practical factors are said to be relevant to the question whether or not true belief amounts to knowledge. Despite the intuitive appeal of the thesis, however, it is puzzling how practical factors can impact the truth-related factors that ground the epistemic status of one's beliefs. In this paper, I will distinguish between a strong and a weak sense of the way in which practical factors are said to be thus relevant. Their differences are explicated in terms of the nature and the extent to which practical factors are said to impact the epistemic status of one's beliefs. I begin by considering a strong version of the thesis that suggests principles according to which the practical rationality of one's actions is a necessary condition on knowledge and justification. Having noted an inadequacy in the formulation of such principles, the arguments in their support are subsequently stated and criticized. Finally, I identify two modest versions of the thesis of pragmatic encroachment and argue that they, too, fail to explain how practical factors can bear on the epistemic status of one's beliefs.     

Inferior Disagreement

Literature in the epistemology of disagreement has focused on peer disagreement: disagreement between those with shared evidence and equal cognitive abilities. Additional literature focuses on the perspective of amateurs who disagree with experts. However, the appropriate epistemic reaction from superiors who disagree with inferiors remains underexplored. Prima facie, this may seem an uninteresting set of affairs. If A is B’s superior, and A has good reason to believe she is B’s superior, A appears free to dismiss B’s disagreement. However, a closer look will show otherwise. I first distinguish competent from incompetent inferiors and then argue that disagreement from the former often gives superiors reason to adjust credence and reevaluate belief. In other words, epistemic inferiority alone is insufficient grounds for dismissing opinion. More nuanced difficulties arise with incompetent inferiors. When superiors disagree with incompetents, this might provide evidence to bolster belief credence; however, agreement from incompetents can defeat justification. In either instance, inferior opinion carries epistemic weight. Yet, this fails to cover all ground; at times, superiors learn nothing from inferior disagreement. I finish by exploring these uninformative disagreements, how to distinguish them from the informative cases, and the proper epistemic reactions thereof.     

Understanding, Integration, and Epistemic Value

Understanding enjoys a special kind of value, one not held by lesser epistemic states such as knowledge and true belief. I explain the value of understanding via a seemingly unrelated topic, the implausibility of veritism. Veritism holds that true belief is the sole ultimate epistemic good and all other epistemic goods derive their value from the epistemic value of true belief. Veritism entails that if you have a true belief that p, you have all the epistemic good qua p. Veritism is a plausible and widely held view; I argue that it is untenable. I argue that integration among beliefs possesses epistemic value independent from the good of true belief, and so has value veritism cannot account for. I argue further that this integration among beliefs comprises the distinctive epistemic value of understanding.     

Ownership, Authorship and External Justification

Some of the most well-known arguments against epistemic externalism come in the form of thought experiments involving subjects who acquire beliefs through anomolous means such as clairvoyance. These thought experiments purport to provide counterexamples to the reliabilist conception of justification: their subjects are intuitively epistemically unjustified, yet meet reliabilist externalist criteria for justification. In this article, I address a recent defence of externalism due to Daniel Breyer, who argues that externalists need not consider such subjects justified, since they fail to own those beliefs in a way required for epistemic evaluability. I argue that the concept of belief ownership Breyer adopts leaves his account open to related counterexamples, and suggest a modification, drawing on analogies between these cases and cases of delusions, such as thought insertion. I will argue that a concept of authorship developed in the literature on delusions better grounds the sense of attribution required for epistemic evaluability.     

Preference-based belief operators

We show how different kinds of belief operators derived from preferences can be defined in terms an accessibility relation of epistemic priority, and characterized by means of a vector of nested accessibility relations. The semantic structure enables us to compare and reconcile certain non-standard notions of belief that have recently been used in epistemic analyses of games.     

Reliabilist Justification: Basic,Easy, and Brute

Process reliabilists hold that in order for a belief to be justified, it must result from a reliable cognitive process. They also hold that a belief can be basically justified: justified in this manner without having any justification to believe that belief is reliably produced. Fumerton (1995), Vogel (2000), and Cohen (2002) have objected that such basic justification leads to implausible easy justification by means of either epistemic closure principles or so-called track record arguments. I argue that once we carefully distinguish closure principles from transmission principles, and epistemic consequences from epistemic preconditions, neither version of this objection succeeds.

The co-construction of arguments by middle-school students

This paper highlights the value of student collaboration in doing mathematics, demonstrates how urban, middle-school students, working together, co-constructed justifications for their solutions, and shows that certain conditions are associated with the promotion of a culture of reasoning. It is documented that students collaboratively built arguments that took the form of proof, challenged each others’ arguments, and justified these arguments in small groups and whole class discussions. In producing their mathematical justifications, students included the input of others. Finally, the way in which students, by expanding on the arguments of others, also used alternative forms of reasoning which in many cases led to even more refined arguments is discussed.     

Epistemic characterizations of iterated deletion of inferior strategy profiles in preference-based type spaces

Bonanno (Logics and the foundations of game and decision theory, Amsterdam University Press, Amsterdam, 2008) provides an epistemic characterization for the solution concept of iterated deletion of inferior strategy profiles (IDIP) by embedding strategic-form games with ordinal payoffs in non-probabilistic epistemic models which are built on Kripke frames. In this paper, we will follow the event-based approach to epistemic game theory and supplement strategic games with type space models, where each type is associated with a preference relation on the state space. In such a framework, IDIP can be characterized by the conditions that at least one player has correct beliefs about the state of the world and that there is common belief that every player is rational, has correct beliefs about the state of the world and has strictly monotone preferences. Moreover, we shall compare the epistemic motivations for IDIP and its mixed strategy variant known as strong rationalizability (SR). Presuppose the above conditions. Whenever there is also common belief that players’ preferences are representable by some expected utility function IDIP still applies. But if there is common belief that players’ preferences are representable by some expected payoff function, then SR results.     

Forecasting with imprecise probabilities

We review de Finetti’s two coherence criteria for determinate probabilities: coherence₁ defined in terms of previsions for a set of events that are undominated by the status quo – previsions immune to a sure-loss – and coherence₂ defined in terms of forecasts for events undominated in Brier score by a rival forecast. We propose a criterion of IP-coherence₂ based on a generalization of Brier score for IP-forecasts that uses 1-sided, lower and upper, probability forecasts. However, whereas Brier score is a strictly proper scoring rule for eliciting determinate probabilities, we show that there is no real-valued strictly proper IP-score. Nonetheless, with respect to either of two decision rules – Γ-maximin or (Levi’s) E-admissibility-+-Γ-maximin – we give a lexicographic strictly proper IP-scoring rule that is based on Brier score.     

Why fix it when it ain't broken? A reply to Henning's critique

This rejoinder discusses Henning's arguments against my model for exchange systems with restricted market access. It also presents an alternative rational choice model that combines Henning's central demands with my original modeling idea (according to which restricted access has effects before the optimization). This alternative approach yields similar conclusions as my earlier analysis. A comparison with Henning's results (which refer to a scenario in which access restrictions explicitly enter the optimization program) suggests that the respective model implications depend crucially on the decision where and how restricted access is introduced into a model. From this perspective, Henning's negative assessment of my work is questionable because, in the absence of empirical evidence, he has no better rationale for modeling restricted access than I do.     

A framework for managing uncertain inputs: An axiomization of rewarding

The success postulate in belief revision ensures that new evidence (input) is always trusted. However, admitting uncertain input has been questioned by many researchers. Darwiche and Pearl argued that strengths of evidence should be introduced to determine the outcome of belief change, and provided a preliminary definition towards this thought. In this paper, we start with Darwiche and Pearl’s idea aiming to develop a framework that can capture the influence of the strengths of inputs with some rational assumptions. To achieve this, we first define epistemic states to represent beliefs attached with strength, and then present a set of postulates to describe the change process on epistemic states that is determined by the strengths of input and establish representation theorems to characterize these postulates. As a result, we obtain a unique rewarding operator which is proved to be a merging operator that is in line with many other works. We also investigate existing postulates on belief merging and compare them with our postulates. In addition, we show that from an epistemic state, a corresponding ordinal conditional function by Spohn can be derived and the result of combining two epistemic states is thus reduced to the result of combining two corresponding ordinal conditional functions proposed by Laverny and Lang. Furthermore, when reduced to the belief revision situation, we prove that our results induce all the Darwiche and Pearl’s postulates as well as the Recalcitrance postulate and the Independence postulate.     

Belief and probability: A general theory of probability cores

This paper considers varieties of probabilism capable of distilling paradox-free qualitative doxastic notions (e.g., full belief, expectation, and plain belief) from a notion of probability taken as a primitive. We show that core systems, collections of nested propositions expressible in the underlying algebra, can play a crucial role in these derivations. We demonstrate how the notion of a probability core can be naturally generalized to high probability, giving rise to what we call a high probability core, a notion that when formulated in terms of classical monadic probability coincides with the notion of stability proposed by Hannes Leitgeb [32]. Our work continues by one of us in collaboration with Rohit Parikh [7]. In turn, the latter work was inspired by the seminal work of Bas van Fraassen [46]. We argue that the adoption of dyadic probability as a primitive (as articulated by van Fraassen [46]) admits a smoother connection with the standard theory of probability cores as well as a better model in which to situate doxastic notions like full belief. We also illustrate how the basic structure underlying a system of cores naturally leads to alternative probabilistic acceptance rules, like the so-called ratio rule initially proposed by Isaac Levi [34].Core systems in their various guises are ubiquitous in many areas of formal epistemology (e.g., belief revision, the semantics of conditionals, modal logic, etc.). We argue that core systems can also play a natural and important role in Bayesian epistemology and decision theory. In fact, the final part of the article shows that probabilistic core systems are naturally derivable from basic decision-theoretic axioms which incorporate only qualitative aspects of core systems; that the qualitative aspects of core systems alone can be naturally integrated in the articulation of coherence of primitive conditional probability; and that the guiding idea behind the primary qualitative features of a core system gives rise to the formulation of lexicographic decision rules.     

How Do Students Know What They Learn in Middle School Mathematics Is True?

This article presents ways in which students ascertain that what they have learned in mathematics is true. Students in the middle school (and a few from other grades) were interviewed by prospective and in‐service teachers. Students were asked what they had learned recently in mathematics and how they knew it was true. The answers were grouped by the author according to the justification schemes used by the students in their explanations. Students interviewed used three kinds of justification schemes: externally based, empirical, and analytic. For each kind, examples are provided of students' justifications. Additional insights are included from the reflections of the interviewers. Some suggestions are offered regarding how teachers can help increase their students' ability to give convincing arguments in mathematics.     

Moore's Paradox in Belief and Desire

Is there a Moore’s paradox in desire? I give a normative explanation of the epistemic irrationality, and hence absurdity, of Moorean belief that builds on Green and Williams’ normative account of absurdity. This explains why Moorean beliefs are normally irrational and thus absurd, while some Moorean beliefs are absurd without being irrational. Then I defend constructing a Moorean desire as the syntactic counterpart of a Moorean belief and distinguish it from a ‘Frankfurt’ conjunction of desires. Next I discuss putative examples of rational and irrational desires, suggesting that there are norms of rational desire. Then I examine David Wall’s groundbreaking argument that Moorean desires are always unreasonable. Next I show against this that there are rational as well as irrational Moorean desires. Those that are irrational are also absurd, although there seem to be absurd desires that are not irrational. I conclude that certain norms of rational desire should be rejected.     

Probability,fuzziness and borderline cases

An integrated approach to truth-gaps and epistemic uncertainty is described, based on probability distributions defined over a set of three-valued truth models. This combines the explicit representation of borderline cases with both semantic and stochastic uncertainty, in order to define measures of subjective belief in vague propositions. Within this framework we investigate bridges between probability theory and fuzziness in a propositional logic setting. In particular, when the underlying truth model is from Kleene's three-valued logic then we provide a complete characterisation of compositional min–max fuzzy truth degrees. For classical and supervaluationist truth models we find partial bridges, with min and max combination rules only recoverable on a fragment of the language. Across all of these different types of truth valuations, min–max operators are resultant in those cases in which there is only uncertainty about the relative sharpness or vagueness of the interpretation of the language.     

Space-time dynamic of normalized doxatons: automata models of pathological collective mentality

The paper aims to investigate influences of norms on a space-time dynamic of stirred (solutions) and non-stirred (lattices) collectives of very simple believing agents. A mental state of an agent is characterized by agent's belief in some proposition and a truth value of the proposition in agent's local vicinity. Every agent of the collective updates its belief depending on its current belief, belief of its closest neighbours and truth value of the proposition in neighbourhood of the neighbours. Each agent is represented by a finite automaton – a so-called doxaton [A. Adamtzky, Appl. Math. Comput., in press]. The doxaton takes five doxastic states, derived from the belief: knowledge, doubt, misbelief, delusion and ignorance. In the above mentioned reference, we defined a binary composition of doxastic states and investigated its algebraic structure. The composition itself is not deterministic; however, it can be made deterministic by applying norms. The norms are expressed in a priority order on the doxastic states. Space-time evolution of the solutions and the lattices of doxatons is studied in computer experiments to understand influences of norms and initial conditions on the behaviour of abstract collectives of simple agents. Diffusion and reaction of doxastic states are explored as well as formation of stationary patterns of the doxastic states.     

Imprecise probabilities for representing ignorance about a parameter

This paper distinguishes between objective probability—or chance—and subjective probability. Most statistical methods in machine learning are based on the hypothesis that there is a random experiment from which we get a set of observations. This random experiment could be identified with a chance or objective probability, but these probabilities depend on some unknown parameters. Our knowledge of these parameters is not objective and in order to learn about them, we must assess some epistemic probabilities about their values. In some cases, our objective knowledge about these parameters is vacuous, so the question is: What epistemic probabilities should be assumed? In this paper we argue for the assumption of non-vacuous (a proper subset of [0, 1]) interval probabilities. There are several reasons for this; some are based on the betting interpretation of epistemic probabilities while others are based on the learning capabilities under the vacuous representation. The implications of the selection of epistemic probabilities in different concepts as conditioning and learning are studied. It is shown that in order to maintain some reasonable learning capabilities we have to assume more informative prior models than those frequently used in the literature, such as the imprecise Dirichlet model.     

A unified method and its application to brake instability analysis involving different types of epistemic uncertainties

The key idea of the proposed method is the use of the equivalent variables named as evidence-based fuzzy variables, which are special evidence variables with fuzzy focal elements. On the basis of the equivalent variables, an uncertainty quantification model is established, in which the unified probabilistic information related to the uncertain responses of engineering systems can be computed with the aid of the fuzziness discretization and reconstruction, the belief and plausibility measures analysis, and the interval response analysis. Monte Carlo simulation is presented as a reference method to validate the accuracy of the proposed method. The proposed method then is extended to perform squeal instability analysis involving different types of epistemic uncertainties. To illustrate the feasibility and effectiveness of the proposed method, seven numerical examples of disc brake instability analysis involving different epistemic uncertainties are provided and analyzed. By conducting appropriate comparisons with reference results, the high accuracy and efficiency of the proposed method on quantifying the effects of different epistemic uncertainties on brake instability are demonstrated.