Bounded rationality is typically understood as a concession to human cognitive limitations, a departure from an ideal coherent in principle if unattainable in practice. I argue this gets the relationship backwards. Unbounded rationality is a physical impossibility, and its attendant normative standards—Bayesian updating, sure loss avoidance, expected utility maximization, logical closure—are techniques for favorable circumstances when resources permit, not ideals from which mortals regrettably fall short. The argument rests on the physics of computation: any information-processing system incurs irreducible costs in energy and time. Three independent lines of support establish this conclusion. The first runs through Landauer’s principle. The second draws on Wolpert’s stochastic thermodynamics framework, extended by Kolchinsky and Wolpert to Turing machines, where thermodynamic costs track algorithmic complexity. The third draws on quantum-mechanical and relativistic bounds that fix finite ceilings on the operations any physical system can perform in a given region of space and time. These constraints bind all physical systems, natural or artificial. Coherence conditions like Savage’s axioms and Bayesian probability evaluate global states, not local procedures. No finite physical process can construct, verify, or maintain global coherence over a realistic state space. They can still function diagnostically, as devices for flagging departures worth explaining; what they cannot do is serve as action-guiding norms from which bounded reasoners fall short. Normative theories that presuppose unbounded rationality demand the physically impossible. Bounded rationality is not a departure from ideal rationality. It is the only kind there is.

The desirable gambles framework provides a foundational approach to imprecise probability theory but relies heavily on linear utility assumptions. This paper introduces function-coherent gambles, a generalization that accommodates non-linear utility while preserving essential rationality properties. We establish core axioms for function-coherence and prove a representation theorem that characterizes acceptable gambles through continuous linear functionals. The framework is then applied to analyze various forms of discounting in intertemporal choice, including hyperbolic, quasi-hyperbolic, scale-dependent, and state-dependent discounting. We demonstrate how these alternatives to constant-rate exponential discounting can be integrated within the function-coherent framework. This unified treatment provides theoretical foundations for modeling sophisticated patterns of time preference within the desirability paradigm, bridging a gap between normative theory and observed behavior in intertemporal decision-making under genuine uncertainty.

The desirable gambles framework provides a rigorous foundation for imprecise probability theory but relies heavily on linear utility via its coherence axioms. In our related work, we introduced function-coherent gambles to accommodate non-linear utility. However, when repeated gambles are played over time -- especially in intertemporal choice where rewards compound multiplicatively -- the standard additive combination axiom fails to capture the appropriate long-run evaluation. In this paper we extend the framework by relaxing the additive combination axiom and introducing a nonlinear combination operator that effectively aggregates repeated gambles in the log-domain. This operator preserves the time-average (geometric) growth rate and addresses the ergodicity problem. We prove the key algebraic properties of the operator, discuss its impact on coherence, risk assessment, and representation, and provide a series of illustrative examples. Our approach bridges the gap between expectation values and time averages and unifies normative theory with empirically observed non-stationary reward dynamics.

The field of neuroscience and the development of artificial neural networks (ANNs) have mutually influenced each other, drawing from and contributing to many concepts initially developed in statistical mechanics. Notably, Hopfield networks and Boltzmann machines are versions of the Ising model, a model extensively studied in statistical mechanics for over a century. In the first part of this chapter, we provide an overview of the principles, models, and applications of ANNs, highlighting their connections to statistical mechanics and statistical learning theory. Artificial neural networks can be seen as high-dimensional mathematical functions, and understanding the geometric properties of their loss landscapes (i.e., the high-dimensional space on which one wishes to find extrema or saddles) can provide valuable insights into their optimization behavior, generalization abilities, and overall performance. Visualizing these functions can help us design better optimization methods and improve their generalization abilities. Thus, the second part of this chapter focuses on quantifying geometric properties and visualizing loss functions associated with deep ANNs.

Analyzing geometric properties of high-dimensional loss functions, such as local curvature and the existence of other optima around a certain point in loss space, can help provide a better understanding of the interplay between neural network structure, implementation attributes, and learning performance. In this work, we combine concepts from high-dimensional probability and differential geometry to study how curvature properties in lower-dimensional loss representations depend on those in the original loss space. We show that saddle points in the original space are rarely correctly identified as such in expected lower-dimensional representations if random projections are used. The principal curvature in the expected lower-dimensional representation is proportional to the mean curvature in the original loss space. Hence, the mean curvature in the original loss space determines if saddle points appear, on average, as either minima, maxima, or almost flat regions. We use the connection between expected curvature in random projections and mean curvature in the original space (i.e., the normalized Hessian trace) to compute Hutchinson-type trace estimates without calculating Hessian-vector products as in the original Hutchinson method. Because random projections are not suitable to correctly identify saddle information, we propose to study projections along dominant Hessian directions that are associated with the largest and smallest principal curvatures. We connect our findings to the ongoing debate on loss landscape flatness and generalizability. Finally, for different common image classifiers and a function approximator, we show and compare random and Hessian projections of loss landscapes with up to about $7\times 10^6$ parameters.

In this paper, we present a new semantic challenge to the moral error theory. Its first component calls upon moral error theorists to deliver a deontic semantics that is consistent with the error-theoretic denial of moral truths by returning the truth-value false to all moral deontic sentences. We call this the 'consistency challenge' to the moral error theory. Its second component demands that error theorists explain in which way moral deontic assertions can be seen to differ in meaning despite necessarily sharing the same intension. We call this the 'triviality challenge' to the moral error theory. Error theorists can either meet the consistency challenge or the triviality challenge, we argue, but are hard pressed to meet both.

I had the good fortune of meeting Teddy Seidenfeld early in my career and learning from him over many years how to approach probability in general, and imprecise probability in particular. I first learned about Teddy's work from Henry Kyburg, Teddy's undergraduate teacher and my Ph.D. supervisor, who introduced me to Teddy's and Larry Wasserman's work on dilation, among other striking contributions that Teddy made while I was a graduate student in the 1990s, many with his longtime collaborators Jay Kadane and Mark Schervish. I cannot overstate the impact Teddy's work has had on me. He will doubtlessly conclude that I have not learned from him well enough, but I am nonetheless grateful for his years of instruction. The field of imprecise probability has matured in no small part because of Teddy's decades of original scholarship and essential contributions to building and sustaining the ISIPTA community. Although the basic idea behind imprecise probability is (at least) 150 years old, a mature mathematical theory has only taken full form in the last 30 years. Interest in imprecise probability during this period has also grown, but many of the ideas that the mature theory serves can be difficult to apprehend to those new to the subject. Although these fundamental ideas are common knowledge in the ISIPTA community, they are expressed, when they are expressed at all, obliquely, over the course of years with students and colleagues. A single essay cannot convey the store of common knowledge from any research community, let alone the ISIPTA community. But, this essay nevertheless is an attempt to guide those familiar with the basic Bayesian framework to appreciate some of the elegant and powerful ideas that underpin the contemporary theory of lower previsions, which is the theory that most people associate with the term 'imprecise probability'.

Recent debate about the error theory has taken a 'formal turn'. On the one hand, there are those who argue that the error theory should be rejected because of its difficulties in providing a convincing formal account of the logic and semantics of moral claims. On the other hand, there are those who claim that such formal objections fail, maintaining that arguments against the error theory must be of a substantive rather than a formal kind. In this paper, we argue that formal objections to the error theory cannot be eschewed but must be met head-on.

The desirable gambles framework offers the most comprehensive foundations for the theory of lower previsions, which in turn affords the most general account of imprecise probabilities. Nevertheless, for all its generality, the theory of lower previsions rests on the notion of linear utility. This commitment to linearity is clearest in the coherence axioms for sets of desirable gambles. This paper considers two routes to relaxing this commitment. The first preserves the additive structure of the desirable gambles framework and the machinery for coherent inference but detaches the interpretation of desirability from the multiplicative scale invariance axiom. The second strays from the additive combination axiom to accommodate repeated gambles that return rewards by a non-stationary processes that is not necessarily additive. Unlike the first approach, which is a conservative amendment to the desirable gambles framework, the second is a radical departure. Yet, common to both is a method for describing rewards called discounted utility.

The purpose of this paper is to show that if one adopts conditional probabilities as the primitive concept of probability, one must deal with the fact that even in very ordinary circumstances at least some probability values may be imprecise, and that some probability questions may fail to have numerically precise answers.

Traditionally, logic has been the dominant formal method within philosophy. Are logical methods still dominant today, or have the types of formal methods used in philosophy changed in recent times? To address this question, we coded a sample of philosophy papers from the late 2000s and from the late 2010s for the formal methods they used. The results indicate that (a) the proportion of papers using logical methods remained more or less constant over that time period but (b) the proportion of papers using probabilistic methods was approximately three times higher in the late 2010s than it was in the late 2000s. Further analyses explored this change by looking more closely at specific methods, specific levels of technical engagement, and specific subdisciplines within philosophy. These analyses indicate that the increasing proportion of papers using probabilistic methods was pervasive, not confined to particular probabilistic methods, levels of sophistication, or subdisciplines.

The theory of lower previsions is designed around the principles of coherence and sure-loss avoidance, thus steers clear of all the updating anomalies highlighted in Gong and Meng's "Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss and Simpson's Paradox" except dilation. In fact, the traditional problem with the theory of imprecise probability is that coherent inference is too complicated rather than unsettling. Progress has been made simplifying coherent inference by demoting sets of probabilities from fundamental building blocks to secondary representations that are derived or discarded as needed.

Conditionals and conditional reasoning have been a long-standing focus of research across a number of disciplines, ranging from psychology through linguistics to philosophy. But almost no work has concerned itself with the question of how hearing or reading a conditional changes our beliefs. Given that we acquire much-perhaps most-of what we believe through the testimony of others, the simple matter of acquiring conditionals via others' assertion of a conditional seems integral to any full understanding of the conditional and conditional reasoning. In this paper we detail a number of basic intuitions about how beliefs might change in response to a conditional being uttered, and show how these are backed by behavioral data. In the remainder of the paper, we then show how these deceptively simple phenomena pose a fundamental challenge to present theoretical accounts of the conditional and conditional reasoning - a challenge which no account presently fully meets.

Starting from Simon’s bounded rationality notion, in this study we consider some of the links between concepts of bounded rationality and the approaches followed by economists in their analysis of the role played by economic agents’ expectations in driving the evolution of the economy through time. We argue that the degree of attention devoted to the formation of expectations by the macroeconomic theory has followed high and low cycles. In recent years, the increasing availability of survey data and the failings of models based on purely rational representative agents have prompted renewed interest in inquiries into the direct measurement of expectations and empirical studies of their formation. The intellectual legacy of Herbert Simon provides a useful guide for both these activities.

A characterization result of dilation in terms of positive and negative association admits an extremal counterexample, which we present together with a minor repair of the result. Dilation may be asymmetric whereas covariation itself is symmetric. Dilation is still characterized in terms of positive and negative covariation, however, once the event to be dilated has been specified.

The ISIPTA meetings are the primary forum for presenting and discussing advances in imprecise probabilities research. They are organized once every two years by SIPTA, the Society for Imprecise Probabilities: Theories and Applications. The first meeting was held in Ghent in 1999. It was followed by meetings in Ithaca, Lugano, Pittsburgh, Prague, Durham, Innsbruck, Compiegne, Pescara and Lugano. After twenty years, we return to Ghent for the 11th International Symposium on Imprecise Probabilities: Theories and Applications. This anniversary edition will be held from Wednesday 3 to Saturday 6 July 2019.

Two compelling principles, the Reasonable Range Principle and the Preservation of Irrelevant Evidence Principle, are necessary conditions that any response to peer disagreements ought to abide by. The Reasonable Range Principle maintains that a resolution to a peer disagreement should not fall outside the range of views expressed by the peers in their dispute, whereas the Preservation of Irrelevant Evidence (PIE) Principle maintains that a resolution strategy should be able to preserve unanimous judgments of evidential irrelevance among the peers. No standard Bayesian resolution strategy satisfies the PIE Principle, however, and we give a loss aversion argument in support of PIE and against Bayes. The theory of imprecise probability allows one to satisfy both principles, and we introduce the notion of a set-based credal judgment to frame and address a range of subtle issues that arise in peer disagreements.

Here is a portrait of experimental science. A question arises, a hypothesis is proposed, experiments are devised, then performed, and a judgment is made on how well some or another implication of that hypothesis, supposing it were true, accords with the outcomes. Such was Charles Sanders Peirce’s view of experimental inquiry at the end of the nineteenth century.1 By the close of the twentieth, a spectacular store of methods were on hand to quantify the uncertainty an experimentalist confronts, along with a logic, broadly speaking, to assess its consequences. Just as the differential calculus swept to the margins ancient bewilderment over change and how to reason about quantities that change, so have the triumphs of modern statistics pushed aside Cartesian paralysis over error and how to reason with corrigible, uncertain quantities. It is not so much that Leibniz and Newton answered Parmenides, or that Peirce set the course for Pearson and Fisher to refute the academic skeptics, but rather that in each case a genuine obstacle to inquiry was plucked from confusion and paradox and a clear way to reason around those obstacles was shown. For those who wonder what philosophical progress looks like, look no further.