A simple logic for reasoning about incomplete knowledge

The semantics of modal logics for reasoning about belief or knowledge is often described in terms of accessibility relations, which is too expressive to account for mere epistemic states of an agent. This paper proposes a simple logic whose atoms express epistemic attitudes about formulae expressed in another basic propositional language, and that allows for conjunctions, disjunctions and negations of belief or knowledge statements. It allows an agent to reason about what is known about the beliefs held by another agent. This simple epistemic logic borrows its syntax and axioms from the modal logic KD. It uses only a fragment of the S5 language, which makes it a two-tiered propositional logic rather than as an extension thereof. Its semantics is given in terms of epistemic states understood as subsets of mutually exclusive propositional interpretations. Our approach offers a logical grounding to uncertainty theories like possibility theory and belief functions. In fact, we define the most basic logic for possibility theory as shown by a completeness proof that does not rely on accessibility relations.     

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.     

Core and Ancillary Epistemic Virtues

We argue, primarily by appeal to phenomenological considerations related to the experiential aspects of agency, that belief fixation is broadly agentive; although it is rarely (if ever) voluntary, nonetheless, it is phenomenologically agentive because of its significant phenomenological similarities to voluntary-agency experience. An important consequence is that epistemic rationality, as a central feature of belief fixation, is an agentive notion. This enables us to introduce and develop a distinction between core and ancillary epistemic virtues. Core epistemic virtues involve several inter-related kinds of epistemic rationality in belief fixation. Other “habits of mind” pertinent to belief fixation constitute ancillary epistemic virtues. Finally, we discuss the relationship between both kinds of virtues, offering a unified account of epistemic virtuousness.     

Pairwise epistemic conditions for correlated rationalizability

We provide a foundation for correlated rationalizability by means of pairwise epistemic conditions imposed only on some pairs of players. Indeed, we show that pairwise mutual belief, for some pairs of players, of (i) the game payoffs, (ii) rationality, and (iii) deeming possible only strategy profiles that receive positive probability by the actual conjectures suffice for correlated rationalizability when there is a common prior. Moreover, we show that our epistemic conditions do not require nor imply mutual belief of rationality. Finally, we discuss the relationship between correlated rationalizability and Nash equilibrium on the basis of the respective pairwise epistemic conditions for each of the two concepts.     

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.     

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.     

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.

A Behavioral Interpretation of Belief Functions

Shafer’s belief functions were introduced in the seventies of the previous century as a mathematical tool in order to model epistemic probability. One of the reasons that they were not picked up by mainstream probability was the lack of a behavioral interpretation. In this paper, we provide such a behavioral interpretation and re-derive Shafer’s belief functions via a betting interpretation reminiscent of the classical Dutch Book Theorem for probability distributions. We relate our betting interpretation of belief functions to the existing literature.     

Probabilism,entropies and strictly proper scoring rules

Accuracy arguments are the en vogue route in epistemic justifications of probabilism and further norms governing rational belief. These arguments often depend on the fact that the employed inaccuracy measure is strictly proper. I argue controversially that it is ill-advised to assume that the employed inaccuracy measures are strictly proper and that strictly proper statistical scoring rules are a more natural class of measures of inaccuracy. Building on work in belief elicitation I show how strictly proper statistical scoring rules can be used to give an epistemic justification of probabilism.An agent's evidence does not play any role in these justifications of probabilism. Principles demanding the maximisation of a generalised entropy depend on the agent's evidence. In the second part of the paper I show how to simultaneously justify probabilism and such a principle. I also investigate scoring rules which have traditionally been linked with entropies.     

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.     

An analysis of the a priori and a posteriori

I present and defend a unified, non-reductive analysis of the a priori and a posteriori. It is a mistake to remove all epistemic conditions from the analysis of the a priori (as, for example, Alvin Goldman has recently suggested doing). We can keep epistemic conditions (like unrevisability) in the analysis as long as we insist that a priori and a posteriori justification admit of degrees. I recommend making the degree to which a belief’s justification is a priori or a posteriori solely dependent on the revisability relations that obtain among the faculties that deliver the belief and all other faculties.     

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.     

Belief revision conditionals: basic iterated systems

It is now well known that, on pain of triviality, the probability of a conditional cannot be identified with the corresponding conditional probability [25]. This surprising impossibility result has a qualitative counterpart. In fact, Peter Gärdenfors showed in [13] that believing ‘If A then B’ cannot be equated with the act of believing B on the supposition that A — as long as supposing obeys minimal Bayesian constraints.Recent work has shown that in spite of these negative results, the question ‘how to accept a conditional?’ has a clear answer. Even if conditionals are not truth-carriers, they do have precise acceptability conditions. Nevertheless most epistemic models of conditionals do not provide acceptance conditions for iterated conditionals. One of the main goals of this essay is to provide a comprehensive account of the notion of epistemic conditionality covering all forms of iteration.First we propose an account of the basic idea of epistemic conditionality, by studying the conditionals validated by epistemic models where iteration is permitted but not constrained by special axioms. Our modeling does not presuppose that epistemic states should be represented by belief sets (we only assume that to each epistemic state corresponds an associated belief state). A full encoding of the basic epistemic conditionals (encompassing all forms of iteration) is presented and a representation result is proved.In the second part of the essay we argue that the notion of change involved in the evaluation of conditionals is suppositional, and that such notion should be distinguished from the notion of updating (modelled by AGM and other methods). We conclude by considering how some of the recent modellings of iterated change fare as methods for iterated supposing.     

In Defense of Epistemic Circularity

In this paper I defend epistemic circularity by arguing that the “No Self-Support” principle (NSS) is false. This principle, ultimately due to Fumerton (1995), states that one cannot acquire a justified belief in the reliability of a source of belief by trusting that very source. I argue that NSS has the skeptical consequence that the trustworthiness of all of our sources ultimately depends upon the trustworthiness of certain fundamental sources – sources that we cannot justifiably believe to be reliable. This is a problem, I claim, because if the trustworthiness of all of our sources depends upon sources that we should not believe to be reliable, then a reflective individual should not trust any of his sources at all. The hidden cost of rejecting epistemic circularity is thus the unacceptable skeptical thesis that reflective individuals like you and I have no justified beliefs whatsoever.     

Preference-based unawareness

Morris, 1996, Morris, 1997 introduced preference-based definitions of knowledge and belief in standard state-space structures. This paper extends this preference-based approach to unawareness structures (Heifetz et al., 2006, Heifetz et al., 2008). By defining unawareness and knowledge in terms of preferences over acts in unawareness structures and showing their equivalence to the epistemic notions of unawareness and knowledge, we try to build a bridge between decision theory and epistemic logic. Unawareness of an event is characterized behaviorally as the event being null and its negation being null.     

An epistemic analysis of the Harsanyi transformation

Harsanyi (1967–68) proposed a method for transforming uncertainty over the strategy sets of players into uncertainty over their payoffs. The transformation appears to rely on an assumption that the players are rational, or, indeed, that they are rational and that there is common belief of rationality. Such an assumption would be awkward from the perspective of the epistemic program, which is often interested in the implications of irrationality or a lack of common belief of rationality. This paper shows that without common belief of rationality, such implications are not necessarily maintained under a Harsanyi transformation. The paper then shows how, with the belief-system model of Aumann and Brandenburger (1995), such implications can be maintained in the absence of common belief of rationality. Received: December 2000/Revised: February 2002     

Constructivism,Factoring, and Beliefs

The errors made by remedial intermediate algebra students in factoring polynomials were analyzed in the light of student definitions of factoring. Certain belief sets about factoring are found to logically entail many of the errors made. The vocabulary of epistemic semantics is used to describe how beliefs are changed. It is suggested that belief-based teaching can be successful in teaching factoring.     

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.     

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.     

On various ways of tackling incomplete information in statistics

This short paper discusses the contributions made to the featured section on Low Quality Data. We further refine the distinction between the ontic and epistemic views of imprecise data in statistics. We also question the extent to which likelihood functions can be viewed as belief functions. Finally we comment on the data disambiguation effect of learning methods, relating it to data reconciliation problems.