This essay develops a joint theory of rational (all-or-nothing) belief and degrees of belief. The theory is based on three assumptions: the logical closure of rational belief; the axioms of probability for rational degrees of belief; and the so-called Lockean thesis, in which the concepts of rational belief and rational degree of belief figure simultaneously. In spite of what is commonly believed, this essay will show that this combination of principles is satisfiable (and indeed nontrivially so) and that (...) the principles are jointly satisfied if and only if rational belief is equivalent to the assignment of a stably high rational degree of belief. Although the logical closure of belief and the Lockean thesis are attractive postulates in themselves, initially this may seem like a formal “curiosity”; however, as will be argued in the rest of the essay, a very reasonable theory of rational belief can be built around these principles that is not ad hoc and that has various philosophical features that are plausible independently. In particular, this essay shows that the theory allows for a solution to the Lottery Paradox, and it has nice applications to formal epistemology. The price that is to be paid for this theory is a strong dependency of belief on the context, where a context involves both the agent's degree of belief function and the partitioning or individuation of the underlying possibilities. But as this essay argues, that price seems to be affordable. (shrink)

In this commentary on Rotts paper Stability, Strength and Sensitivity: Converting Belief into Knowledge, I discuss two problems of the stability theory of knowledge which are pointed out by Rott. I conclude that these problems offer no reason for rejecting the stability theory, but might be grounds for deviating from the standard AGM account of belief revision which Rott presupposes.

We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness, and are stable in some cardinal. Assuming the singular cardinal hypothesis, we prove a full characterization of the stability cardinals, and connect the stability spectrum with the behavior of saturated models.We deduce that if a class is stable on a tail of cardinals, then it has no long splitting chains. This indicates that there is a clear notion of superstability in this framework.We also (...) present an application to homogeneous model theory: for [Formula: see text] a homogeneous diagram in a first-order theory [Formula: see text], if [Formula: see text] is both stable in [Formula: see text] and categorical in [Formula: see text] then [Formula: see text] is stable in all [Formula: see text]. (shrink)

In this paper we compare Leitgeb’s stability theory of belief and Spohn’s ranking-theoretic account of belief. We discuss the two theories as solutions to the lottery paradox. To compare the two theories, we introduce a novel translation between ranking functions and probability functions. We draw some crucial consequences from this translation, in particular a new probabilistic belief notion. Based on this, we explore the logical relation between the two belief theories, showing that models of Leitgeb’s theory correspond (...) to certain models of Spohn’s theory. The reverse is not true. Finally, we discuss how these results raise new questions in belief theory. In particular, we raise the question whether stability is rightly thought of as a property pertaining to belief. (shrink)

This book presents a coherent and systematic exposition of the mathematical theory of the problems of optimization and stability. Both of these are topics central to economic analysis since the latter is so much concerned with the optimizing behaviour of economic agents and the stability of the interaction processes to which this gives rise. The topics covered include convexity, mathematical programming, fixed point theorems, comparative static analysis and duality, the stability of dynamic systems, the calculus of (...) variations and optimal control theory. The authors present a more detailed and wide-ranging discussion of these topics than is to be found in the few books which attempt a similar coverage. Although the text deals with fairly advanced material, the mathematical prerequisites are minimised by the inclusion of an integrated mathematical review designed to make the text self-contained and accessible to the reader with only an elementary knowledge of calculus and linear algebra. A novel feature of the book is that it provides the reader with an understanding and feel for the kinds of mathematical techniques most useful for dealing with particular economic problems. This is achieved through an extensive use of a broad range of economic examples.This is suitable for use in advanced undergraduate and postgraduate courses in economic analysis and should in addition prove a useful reference work for practising economists. (shrink)

InStabilizing Dynamics Roy Weintraub provides a history of stability theory from the work of Hicks and Samuelson in the late 1930s to the Gale and Scarf counterexamples in the 1960s. Unlike his earlier work in the history of general equilibrium theory this recent contribution is not an attempt to fit the Walrasian program into the narrow framework of some particular philosophy of natural science. Rather, the theme inStabilizing Dynamicsis broadly social constructivist. Simply put, the constructivist view of (...) science is “that scientific knowledge itself is constructed socially, in communities of scientists: Knowledge is constructed, not found”. (shrink)

Harry Eckstein’s long-standing (but ever-changing) hypothesis that a nation’s political stability is a function of “congruence” between the “authority patterns” exhibited by the government and those displayed by nearly every sort of institution under that government’s aegis involved a highly complex politico-psychological theory. As a result, it was quite difficult either to confirm or disconfirm. While there have been a number of suggested revisions that apparently simplify his thesis, they suffer either from vagueness or a failure to take (...) democracy to be a system that requires an actual commitment to majoritarianism. In this paper a revision is offered that utilizes a small number of completely dichotomous variables that are easily subject to empirical tests. This revised theory (of “consonance” rather than congruence) is also explicitly majoritarian, enabling it to shed light on the essential nature of democratic regimes. (shrink)

The volume is a collection of essays devoted to the analysis of scientific change and stability. It explores the balance and tension that exist between commensurability and continuity on the one hand, and incommensurability and discontinuity on the other. Moreover, it discusses some central epistemological consequences regarding the nature of scientific progress, rationality and realism. In relation to these topics, it investigates a number of new avenues, and revisits some familiar issues, with a focus on the history and philosophy (...) of physics, and an emphasis on developments in cognitive sciences as well as on the claims of “new experimentalists”. The book constitutes fully revised versions of papers which were originally presented at the international colloquium held at the University of Nancy, France, in June 2004. (shrink)

Spekkens has introduced a toy theory (Spekkens in Phys. Rev. A 75(3):032110, 2007) in order to argue for an epistemic view of quantum states. I describe a notation for the theory (excluding certain joint measurements) which makes its similarities and differences with the quantum mechanics of stabilizer states clear. Given an application of the qubit stabilizer formalism, it is often entirely straightforward to construct an analogous application of the notation to the toy theory. This assists calculations within (...) the toy theory, for example of the number of possible states and transformations, and enables superpositions to be defined for composite systems. (shrink)

The topic of moral diversity is not only prevalent in contemporary moral and political philosophy, it is also practically relevant. Moral diversity, however, poses a significant challenge for moral theory building. John Thrasher, in his discussion of public reason theory, which includes social contract theory, argues that if one seriously considers the goal of moral constructivism and considerations of representation and stability, then moral diversity poses an insurmountable problem for most public reason theories. I agree with (...) Thrasher that moral diversity poses a significant challenge for orthodox multistage social contract theories. In fact, I even add a further problem for such theories under the assumption of deep moral diversity. Nevertheless, I argue that my recently developed multilevel social contract theory overcomes these problems. I focus on some of the underexplored features of this theory to show that multilevel social contract theory offers one conceptually coherent and plausible way to render social contract theory viable and relevant for modern diverse societies. (shrink)

We identify two aspects of the problem of maintaining perceptual stability despite an observer's eye movements. The first, visual direction constancy, is the (egocentric) stability of apparent positions of objects in the visual world relative to the perceiver. The second, visual position constancy, is the (exocentric) stability of positions of objects relative to each other. We analyze the constancy of visual direction despite saccadic eye movements.Three information sources have been proposed to enable the visual system to achieve (...) stability: the structure of the visual field, proprioceptive inflow, and a copy of neural efference or outflow to the extraocular muscles. None of these sources by itself provides adequate information to achieve visual direction constancy; present evidence indicates that all three are used.Our final question concerns how information processing operations result in a stable world. The three traditionally suggested means have been elimination, translation, or evaluation. All are rejected. From a review of the physiological and psychological evidence we conclude that no subtraction, compensation, or evaluation need take place. The problem for which these solutions were developed turns out to be a false one. We propose a “calibration” solution: correct spatiotopic positions are calculated anew for each fixation. Inflow, outflow, and retinal sources are used in this calculation: saccadic suppression of displacement bridges the errors between these sources and the actual extent of movement. (shrink)

A formalism is introduced to represent the connective organization of an evolving neuronal network and the effects of environment on this organization by stabilization or degeneration of labile synapses associated with functioning. Learning, or the acquisition of an associative property, is related to a characteristic variability of the connective organization: the interaction of the environment with the genetic program is printed as a particular pattern of such organization through neuronal functioning. An application of the theory to the development of (...) the neuromuscular junction is proposed and the basic selective aspect of learning emphasized. (shrink)

The class of geometric surgical theories is examined. The main theorem is that every stable theory that is interpretable in a geometric surgical theory is superstable of finite U-rank.

Prospect Theory (PT) is widely regarded as the most promising descriptive model for decision making under uncertainty. Various tests have corroborated the validity of the characteristic fourfold pattern of risk attitudes implied by the combination of probability weighting and value transformation. But is it also safe to assume stable PT preferences at the individual level? This is not only an empirical but also a conceptual question. Measuring the stability of preferences in a multi-parameter decision model such as PT (...) is far more complex than evaluating single-parameter models such as Expected Utility Theory under the assumption of constant relative risk aversion. There exist considerable interdependencies among parameters such that allegedly diverging parameter combinations could in fact produce very similar preference structures. In this paper, we provide a theoretic framework for measuring the (temporal) stability of PT parameters. To illustrate our methodology, we further apply our approach to 86 subjects for whom we elicit PT parameters twice, with a time lag of 1 month. While documenting remarkable stability of parameter estimates at the aggregate level, we find that a third of the subjects show significant instability across sessions. (shrink)

This paper investigates the stochastically exponential stability of reaction-diffusion impulsive stochastic cellular neural networks. The reaction-diffusion pulse stochastic system model characterizes the complexity of practical engineering and brings about mathematical difficulties, too. However, the difficulties have been overcome by constructing a new contraction mapping and an appropriate distance on a product space which is guaranteed to be a complete space. This is the first time to employ the fixed point theorem to derive the stability criterion of reaction-diffusion impulsive (...) stochastic CNN with distributed time delays. Finally, an example is provided to illustrate the effectiveness of the proposed methods. (shrink)

In everyday life we either express our beliefs in all-or-nothing terms or we resort to numerical probabilities: I believe it's going to rain or my chance of winning is one in a million. The Stability of Belief develops a theory of rational belief that allows us to reason with all-or-nothing belief and numerical belief simultaneously.

Let T be a complete theory with infinite models in a countable language. The stability function g T (κ) is defined as the supremum of the number of types over models of T of power κ. It is proved that there are only six possible stability functions, namely $\kappa, \kappa + 2^\omega, \kappa^\omega, \operatorname{ded} \kappa, (\operatorname{ded} \kappa)^\omega, 2^\kappa$.

Ecologists are challenged by the need to bridge and synthesize different approaches and theories to obtain a coherent understanding of ecosystems in a changing world. Both food web theory and regime shift theory shine light on mechanisms that confer stability to ecosystems, but from different angles. Empirical food web models are developed to analyze how equilibria in real multi-trophic ecosystems are shaped by species interactions, and often include linear functional response terms for simple estimation of interaction strengths (...) from observations. Models of regime shifts focus on qualitative changes of equilibrium points in a slowly changing environment, and typically include non-linear functional response terms. Currently, it is unclear how the stability of an empirical food web model, expressed as the rate of system recovery after a small perturbation, relates to the vulnerability of the ecosystem to collapse. Here, we conduct structural sensitivity analyses of classical consumer-resource models in equilibrium along an environmental gradient. Specifically, we change non-proportional interaction terms into proportional ones, while maintaining the equilibrium biomass densities and material flux rates, to analyze how alternative model formulations shape the stability properties of the equilibria. The results reveal no consistent relationship between the stability of the original models and the proportionalized versions, even though they describe the same biomass values and material flows. We use these findings to critically discuss whether stability analysis of observed equilibria by empirical food web models can provide insight into regime shift dynamics, and highlight the challenge of bridging alternative modelling approaches in ecology and beyond. (shrink)

There are decision problems in which rational deliberation fails to result in choosing a pure act. This phenomenon is known as decision instability and has been discussed in the literature on causal decision theory. In this paper we investigate another type of instability, called structural instability in dynamical systems theory. Structural instability indicates that certain qualitative features of the process of rational deliberation are under-determined in a decision situation. We illustrate some of the issues arising from structural instability (...) with a recent argument against causal decision theory proposed by Caspar Hare and Brian Hedden. We show that their argument is undermined by considerations arising from decision instability and structural instability. (shrink)

Grossberg and VanDieren have started a program to develop a stability theory for tame classes. We name some variants of tameness and prove the following. Let K be an AEC with Löwenheim-Skolem number ≤κ. Assume that K satisfies the amalgamation property and is κ-weakly tame and Galois-stable in κ. Then K is Galois-stable in κ⁺ⁿ for all n<ω. With one further hypothesis we get a very strong conclusion in the countable case. Let K be an AEC satisfying the (...) amalgamation property and with Löwenheim-Skolem number ℵ₀ that is ω-local and ℵ₀-tame. If K is ℵ₀-Galois-stable then K is Galois-stable in all cardinalities. (shrink)

Are risk preferences stable over time? To address this question we elicit risk preferences from the same pool of subjects at two different moments in time. To interpret the results, we use a Fechner stochastic choice model in which the revealed preference of individuals is governed by some underlying preference, together with a random error. We take cumulative prospect theory as the underlying preference model (Kahneman and Tversky, Econometrica 47:263–292, 1979; Tversky and Kahneman, Journal of Risk and Uncertainty 5:297–323, (...) 1992). We observe that the aggregate pattern of preferences is very similar in both sessions, and it matches the results reported in the literature. Most subjects are risk averse for gains, and risk seeking for losses. However, the subjects that jointly agree with the reflection effect of prospect theory are around 50%. The percentage of individuals that change their responses across sessions is quite high, 63%. Estimating the stochastic choice model we find that 72% of the subjects have an underlying preference which agrees with the reflection effect of prospect theory. The remaining 28% are mainly classified as risk averse for both gains and losses. The results reinforce the empirical validity of the reflection effect. Deviations from the reflection effect can be attributed to noise, as well as to the existence of a fraction of risk averse subjects. (shrink)

There are decision problems in which rational deliberation fails to result in choosing a pure act. This phenomenon is known as decision instability and has been discussed in the literature on causal decision theory. In this paper we investigate another type of instability, called structural instability in dynamical systems theory. Structural instability indicates that certain qualitative features of the process of rational deliberation are under-determined in a decision situation. We illustrate some of the issues arising from structural instability (...) with a recent argument against causal decision theory proposed by Caspar Hare and Brian Hedden. We show that their argument is undermined by considerations arising from decision instability and structural instability. (shrink)

The main goal of this paper is to introduce the notion of stabilizers in \(L\)-algebras and develop stabilizer theory in \(L\)-algebras. In this paper, we introduced the notions of left and right stabilizers and investigated some related properties of them. Then, we discussed the relations among stabilizers, ideal and co-annihilators. Also, we obtained that the set of all ideals of a \(CKL\)-algebra forms a relative pseudo-complemented lattice. In addition, we proved that right stabilizers in \(CKL\)-algebra are ideals. Then by (...) using the right stabilizers we produced a basis for a topology on \(L\)-algebra. We showed that the generated topology by this basis is Baire, connected, locally connected and separable and we investigated the other properties of this topology. (shrink)

Public reason theories are characterized by three conditions: constructivism, representation, and stability. Constructivism holds that justification does not rely on any antecedent moral or political values outside of the procedure of agreement. Representation holds that the reasons for the choice in the model must be rationally explicable to real agents outside the model. Stability holds that the principles chosen in the procedure should be stable upon reflection, especially in the face of diversity in a pluralistic society. Choice procedures (...) that involve at least two-stages with different information, as Rawls’s theory does, will be path-dependent and not meet the condition of representation since it will not be globally coherent. Attempts to solve this problem without eliminating the segmentation of choice in the procedure will run afoul of constructivism or stability. This problem is instructive because it highlights how public reason theories must evolve in the face of increased concerns about diversity. (shrink)

Several authors have investigated the question of whether canonical logic-based accounts of belief revision, and especially the theory of AGM revision operators, are compatible with the dynamics of Bayesian conditioning. Here we show that Leitgeb's stability rule for acceptance, which has been offered as a possible solution to the Lottery paradox, allows to bridge AGM revision and Bayesian update: using the stability rule, we prove that AGM revision operators emerge from Bayesian conditioning by an application of the (...) principle of maximum entropy. In situations of information loss, or whenever the agent relies on a qualitative description of her information state - such as a plausibility ranking over hypotheses, or a belief set - the dynamics of AGM belief revision are compatible with Bayesian conditioning; indeed, through the maximum entropy principle, conditioning naturally generated AGM revision operators. This mitigates an impossibility theorem of Lin and Kelly for tracking Bayesian conditioning with AGM revision, and suggests an approach to the compatibility problem that highlights the information loss incurred by acceptance rules in passing from probabilistic to qualitative representations of beliefs. (shrink)

In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of “generic stability” in arbitrary theories. Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure. We also give combinatorial examples of types in NSOP theories that are finitely approximated but not generically stable, as well as ϕ-types in simple theories that are definable and finitely satisfiable in (...) a small model, but not finitely approximated. Our proofs demonstrate interesting connections to classical results from Ramsey theory for finite graphs and hypergraphs. (shrink)