philog.arthurpaulpedersen.org

A link for each seminar is to be posted here shortly before it begins.

Yuri Gurevich

University of Michigan

Negative probabilities: What are they for?

Abstract. The topic may sound nonsensical. The standard frequential interpretation of probabilities makes no sense for negative probabilities. Yet negative probabilities are profitably used in quantum physics and elsewhere. So what are they? What is their intrinsic meaning? We don't know. There are attempts in the literature to provide meaning for negative probabilities but, in our judgement, the problem is wide open.

Instead, we address a more pragmatic question: What are negative probabilities good for? It is not rare in science to use a concept without understanding its intrinsic meaning. Consider early uses of complex numbers. The standard quantitative interpretation of numbers makes no sense for imaginary numbers. And the intrinsic meaning of imaginary numbers wasn't clear (and is debatable even today). Yet complex numbers were profitably used to solve algebraic equations. It turned out, for example, that many real algebraic numbers cannot be expressed in radicals unless we allow non-real complex coefficients.

It turns out that the disparate quantum applications of negative probabilities can be seen as examples of a certain application template. To make this template explicit, we introduce observation spaces. An observation space S is a family of (nonnegative) probability distributions P1, P2, ... on a common sample space. A question arises whether there is a single probability distribution P (a grounding for S) which yields all P1, P2, ... as marginal distributions. That P may be necessarily signed. We solve the grounding problem for a number of observation spaces of note.

Branden Fitelson

Northeastern University

Probabilities of Conditionals & Conditional Probabilities — Revisited

Abstract. Lewis's (1976) triviality argument against The Equation (a.k.a, Adams's Thesis) rests on an implausibly strong presupposition about the nature of (epistemic) rational requirements. Interestingly, Lewis (1980) later rejected this presupposition. In his discussion of the Principal Principle, Lewis assumes something weaker, and more reasonable, about the nature of rational requirements. In this paper, I explain how to apply the insights of Lewis (1980) to repair Lewis's earlier (1976) discussion. This leads to a more reasonable rendition of The Equation — one which is (a) immune from triviality, and (b) a better candidate for a (bona fide) rational requirement.

Rohit Parikh

Computer Science, Mathematics, Philosophy
City University of New York

Cooperation, Psychological Game Theory, and Limitations of Rationality in Social Interaction

Discussion. Andrew M. Coleman's "Cooperation, Psychological Game Theory, and Limitations of Rationality in Social Interaction," Behaviorial and Brain Sciences, 26(2): 139-153, April 2003.

Rohit Parikh

Computer Science, Mathematics, Philosophy
City University of New York

Cooperation, Psychological Game Theory, and Limitations of Rationality in Social Interaction

Discussion. Andrew M. Coleman's "Cooperation, Psychological Game Theory, and Limitations of Rationality in Social Interaction," Behaviorial and Brain Sciences, 26(2): 139-153, April 2003.

Lara Buchak

Princeton University

Risk and Rationality

Abstract. When making decisions under risk, individuals are forced to consider how their choices will turn out under various circumstances, and decide how to trade off the possibility that a choice will turn out well against the possibility that it will turn out poorly. The orthodox view is that there is only one acceptable way to do this: rational individuals must maximize expected utility. I argue, however, that the orthodox theory dictates an overly narrow way in which considerations about risk can play a role in an individual’s choices; and I propose an alternative theory of rational decision-making. This new theory allows us to isolate the distinct roles that beliefs, desires, and risk-attitudes play in decision-making. It also vindicates the ordinary decision-maker, from the point of view of even ideal rationality.

Amy Greenwald

Computer Science
Brown University

Regret Minimization in Extensive-form Games

Abstract. It is well known that regret minimization in repeated normal-form games generates mediated equilibrium behavior. More specifically, external regret minimization converges to coarse correlated equilibrium, while internal regret minimization converges to correlated equilibrium. In this work, we explore the much richer space of possible deviations in extensive-form games (external, internal, counterfactual, causal, etc.), and establish the relationships among the mediated equilibria that arise when regret is minimized with respect to these deviation sets. Additionally, we define a generic algorithm, called extensive-form regret minimization (EFR), which minimizes the regret of a given deviation set chosen from a natural class -- the behavioral deviations -- that subsumes all of the aforementioned classes. EFR's computational requirements and regret bound scale closely with the complexity of the given deviation set, so we focus on a subset of the class of behavioral deviations, which we call partial sequence deviations, that is both efficient to work with and subsumes previously studied sets. Experimentally, EFR with partial sequence deviations outperforms existing regret minimization algorithms (e.g., counterfactual regret minimization) when playing games in the OpenSpiel environment.

Joint work with Dustin Morrill, Ryan D'Orazio, Marc Lanctot, James R. Wright, Reca Sarfati, and Michael Bowling

Rohit Parikh

Computer Science, Mathematics, Philosophy
City University of New York

Counterfactuals and Causality

Abstract. It is true that if Oswald did not shoot Kennedy in Dallas then someone else did. But it is doubtful that if Oswald had not shot Kennedy in Dallas someone else would have. Why are they different?

An amusing one is, "if Beethoven had died in a plane crash at the age of 25, the history of music would have been different, as also the history of aviation."

Conditionals thus have very intricate properties. Possible world semantics for conditionals has been proposed by Stalnaker and David Lewis. And Lewis has also made connections with causality.

For clearly if Oswald had not shot Kennedy in Dallas then Kennedy would have lived, is connected with the fact that we consider Oswald's shooting as the cause of Kennedy's death.

But subtle issues arise with causality and probability as Jenn McDonald pointed out last Thursday and we will go into similar issues connected with counterfactuals and causality.

We will also briefly touch on issues raised by Eric Pacuit, Eva Cogan and ourselves on the issue of knowledge, causality and responsibility.

Jenn McDonald

CUNY Graduate Center

Probabilistic Causation: de facto Dependence and Modal Relativism

Abstract. On some interpretations of quantum mechanics - on the orthodox interpretation and the GRW interpretation - the world is fundamentally probabilistic. Furthermore, at least some of these fundamental probabilities can propagate up to the macroscopic scale. If either of these is the right view, then some causes will not be nomologically sufficient for their effects. In order to make sense of the causal relation in these circumstances, we need a probabilistic theory of causation. In his 2021 book, Causation, Luke Fenton-Glynn argues that the most promising such theory is a de facto dependence view that relies on probabilistic causal models. After presenting the view, I will first observe that it leaves certain questions open. I will then argue that the best answer to these questions results in a view of causation whereby one particular thing causes a second only relative to a set of background possibilities. The argument is analogous to one I make in the deterministic case. I call this view "Modal Relativism."

Juliet Floyd

Boston University

Russell, Gödel and Early Wittgenstein

Summary. Floyd's latest scholarly triumph, Wittgenstein's Philosophy of Mathematics, is to appear this month.

Rajesh Kasturirangan

NIAS-MIT

Cognitive Epistemology

Abstract. The single scientist world is over. Data is being collected in parallel by several labs and their sensor networks. Even more importantly, we are keenly aware that biological organisms are knowledge producing beings. Our archetype is no longer the scientist collecting data in slow motion, but the animal constantly harvesting knowledge from the world and responding appropriately.

I believe that the new archetype needs a new epistemology. The main shift, as I see it, is from scientific epistemology to cognitive epistemology. In this new view, cognition is both the empirical phenomenon which grounds our capacity to acquire knowledge and the theoretical concept through which epistemology is formalized and articulated. In other words, a reversal of the usual scheme in which cognition (in which I include perception, emotion and the varied mental capacities of nonhuman species) is modeled as an instrument of acquiring knowledge, often formalized in the language of logic or probability.

In the new scheme cognition comes first: the idea is that every classical philosophical problem has a cognitive counterpart and that the cognitive version is both interesting and insightful. This is very much preliminary work: the image I want to give is that of a slow stroll through a theoretical landscape covered in mist; there's a nip in the air that invites a bracing walk. While wakling, I will introduce cognitive concerns from the very beginning, of paying attention to issues such as categorization and generalizability that don't find themselves in introductions to epistemology.

At the same time, I am conscious that I don't want to psychologize cognition, i.e., treat cognition as a mental feature of a specific species or a feature of biological organisms in general. Just as logic can both be the study of the way humans reason and the abstract foundation of reasoning, we want cognition to be connected to the way actual humans and other animals think and and the abstract foundation of what it is to live in a world.

Noson Yanofsky

Brooklyn College and CUNY Graduate Center

Diagonalization, Fixed Points, and Self-reference.

Abstract. Some of the most profound and famous theorems in mathematics and computer science of the past 150 years can simultaneously be seen as a consequence of diagonalization, as a fixed-point theorem, and as an instance of a self-referential paradox. These results include Cantor's theorems about different levels of infinity; Russell's paradox; Gödel's incompleteness theorem; Turing's halting problem; and much more. Amazingly, all these diverse theorems and all viewpoints can be seen as instances of a single simple theorem of basic category theory. We describe this theorem and show some of the instances. A large part of the talk will be a discussion of diagonalization proofs, fixed point theorems, and self- referential paradoxes that fail to be an instance of this categorical theorem. We will meet another categorical idea that unifies some of these ideas. No category theory is needed for this talk.

R. Ramanujam

Institute of Mathematical Sciences, Chennai

A Formal Model for the Emergence of Collective Memory‖Slides

Abstract. According to Susan Sontag, "What is called collective memory is not a remembering but a stipulating." Then one can ask, what is the rationale followed by a group in ascribing / stipulating collective importance to events and their remembering? We present a formal model for collective memory based on automata. Agents keep signalling within neighbourhoods, and depending on the support each signal gets, some signals "win." By agents interacting between different neighbourhoods,'inluence' spreads and sometimes, a collective signal emerges. Interestingly enough, the natural model for such memory is the Parikh automaton that computes exactly the semi-linear predicates. In the talk we discuss the philosophical and logical aspects of the problem and raise questions, some of which can be addressed in the model.

Hassan Akbar

Brooklyn College

Judgement Aggregation

Discussion. Deciding courses of action as a group is common as the old saying goes "No man is an island." Since turning the votes of many into one outcome is such an integral part of our experiences one must ask if our method of decision making is without flaw. In the talk I will be examining various different rules for judgement aggregation to see if a perfect system exists. This will include going over different characteristics we would expect the rule to have as well as examining what characteristics we will have to relax to avoid certain outcomes.

This will be an expository talk based on some work of Christian List, Gabriela Pigozzi, and Philip Pettit.

Ali Khan

Johns Hopkins

Continuity Postulates and Solvability Axioms in Decision Theory‖Slides

Ali Khan is Abram Hutzler Professor of Political Economy with the Department of Economics at Johns Hopkins University.

Sorin Istrail

Brown University

Davidson Causality: The Unreasonable Effectiveness of Mathematics in the Regulatory Genome

Abstract. In his book The Regulatory Genome: Gene Regulatory Networks (GRN) in Development and Evolution (Academic Press 2006), Eric Davidson, the foremost experimentalist of regulatory genomics, forcefully reminds us that in the scientific method, causality is everything; all other approaches are just distractions. In contrast, Davidson — a notoriously elegant writer — offers devastating criticism of the “posterior Biology” approaches all too impatiently employed today — the “measure first” expression of thousands of genes and then “computationally infer Biology.” The last century’s luminaries of mathematical statistics taught us in no uncertain terms that causality cannot be inferred from statistical tables. Davidson aligns with them, adding to their argument a practical dose of reality. The exquisite regulatory mechanisms, locked down by evolution, can only be revealed through systematic experimental perturbations. In the absence of the ocean deep “prior Biology” knowledge, no amount of clustering statistics, or other skinny deep dives, would be able infer “Biology.” Like his mentor Max Delbruck, and with the sea urchin genome in hand, Eric Davidson become the leading liberator of quantitative principles of cell regulation, trapped in the qualitative, descriptive world of biology without genomic sequence.

In this talk we will discuss several computer science problems, inspired by our 15-years-long collaboration with Professor Davidson, who died in 2015, and rooted in his seminal research on causality, molecular biology system completeness, genomic Boolean logic.

Professor Davidson’s legacy consisted of 400+ papers and six books; he mentored about 300 Ph.D.s, postdocs and faculty in his laboratory in the Division of Biology at California Institute of Technology. He was also a beacon of critical discourse. As our beloved teacher and mentor, Davidson united us —biologists, physicists, biochemists, engineers, mathematicians and computer scientists, like in his CalTech Laboratory — in a research renaissance movement towards the quest for the functional meaning of DNA. From such research will ultimately come, by experimental demonstration, the revelation of the much-sought laws of regulatory Biology.

Eric Pacuit

University of Maryland

Epistemic Networks for Imprecise Agents‖Slides

Abstract. What is the best form for social influence to take? Are all policies which aim to increase the amount of interaction over a particular issue likely to be successful in their aims? In this talk, I will survey some models that have been proposed by economists and social epistemologists to address these questions. These models typically assume that the agents have precise beliefs about the proposition that they are trying to learn. However, in many learning situations, at least some of the agents may have imprecise beliefs about the proposition that they are trying to learn. The second part of the talk will report on some work in progress with Paul Pedersen about how best to design communication networks when some agents have imprecise beliefs.

Ada Corondo

The Graduate Center, CUNY

Nietzsche on Logic, Philosophy, and Moral Values‖Slides

Abstract. Studies in logic rarely ever mention Fredrich Nietzsche. There is very little literature on Nietzsche’s critique of classical logic and there is no indication that he followed the developments that were occurring in the field in the 19th century by contemporaneous thinkers such as George Boole, Frege, or Augustus De Morgan. Yet, logic is central to Nietzsche’s seminal work, Beyond Good and Evil: Prelude to a Philosophy of the Future, henceforth referred to as BGE. Believing that classical logic falsely reinforces the religious promise of absolutism and certainty, Nietzsche rejects the possibility of a priori truths qua truth, but embraces logic to the extent that he considers it the vehicle that systematically discharges a philosopher’s energy and morality onto the world.

In this talk I consider Nietzsche’s critique of moral values as they relate to his rejection of both a priori truths and the semantic principle of bivalence, or what he calls the “faith of opposite values”. I argue that Nietzsche’s approach to philosophy, logic, and moral values heralds the future philosophical significance of multivalent systems and paraconsistent logic.

Maryam Bibi

Brooklyn College

Limitations of Social Choice Procedures‖Slides

Discussion. We will first define a social welfare function that takes in the input of individual preference lists and outputs a single social preference list. Then we will define a "weakly reasonable" social welfare function in the sense that it describes the will of the people. This function must satisfy PAR, IIA, and monotonicity. However, the unsettling conclusion we will reach is that the only social welfare function that satisfies these conditions is a dictatorship.

This will be an expository talk based on some material from Taylor and Pacelli's Mathematics and Politics.

Todd Stambaugh

John Jay

Knowledge, Behavior, and Rationality: Rationalizability in Epistemic Games

Abstract. In strategic situations, agents base actions on knowledge and beliefs. This includes knowledge about others’ strategies and preferences over strategy profiles, but also about other external factors. Bernheim and Pearce in 1984 independently defined the game theoretic solution concept of rationalizability, which is built on the premise that rational agents will only take actions that are the best response to some situation that they consider possible. This accounts for other agents’ rationality as well, limiting the strategies to which a particular agent must respond, enabling further elimination until the strategies stabilize. We seek to generalize rationalizability to account not only for actions, but knowledge of the world as well. This will enable us to examine the interplay between action based and knowledge based rationality. We give an account of what it means for an action to be rational relative to a particular state of affairs, and in turn relative to a state of knowledge. We present a class of games, Epistemic Messaging Games (EMG), wit ha communication stage that clarifies the epistemic state among the players prior to the players’ actions. We use a history based model, which frames individual knowledge in terms of local projections of a global history. With this framework, we give an account of rationalizability for subclasses of EMG

Joe Halpern

Department of Computer Science
Cornell University

Actual Causality: A Survey

Abstract. What does it mean that an event C "actually caused" event E? The problem of defining actual causation goes beyond mere philosophical speculation. For example, in many legal arguments, it is precisely what needs to be established in order to determine responsibility. (What exactly was the actual cause of the car accident or the medical problem?) The philosophy literature has been struggling with the problem of defining causality since the days of Hume, in the 1700s. Many of the definitions have been couched in terms of counterfactuals. (C is a cause of E if, had C not happened, then E would not have happened.) In 2001, Judea Pearl and I introduced a new definition of actual cause, using Pearl's notion of structural equations to model counterfactuals. The definition has been revised twice since then, extended to deal with notions like "responsibility" and "blame", and applied in databases and program verification. I survey the last 15 years of work here, including joint work with Judea Pearl, Hana Chockler, and Chris Hitchcock. The talk will be completely self-contained.

Jongjin Kim

Korea University

Buddha versus Popper

Abstract. We discuss two approaches to life: presentism and futurism. We locate presentism within various elements of Buddhism, in the form of advice to live in the present and not to allow the future to hinder us from living in the ever present now. By contrast, futurism, which we identify with Karl Popper, advises us to think of future consequences before we act, and to act now for a better future. Of course, with its emphasis on a well-defined path to an ideal future ideally culminating in enlightenment, Buddhism undoubtedly has elements of futurism as well. We do not intend to determine which of these two approaches to time is more dominant in Buddhism, nor how the two approaches are best understood within Buddhism; but simply we intend to compare and contrast these two approaches, using those presentist elements of Buddhism as representative of presentism while contrasting them with those elements of futurism to be found in Popper and others. We will discuss various aspects of presentism and futurism, such as Ruth Millikan’s Popperian animal, the psychologist Howard Rachlin’s social and temporal discounting, and even the popular but controversial idea, YOLO (you only live once). The primary purpose of this paper is to contrast one with the other. The central question of ethics is: How should one live? Our variation on that question is: When should one live? We conjecture that the notion of flow, developed by Csikszentmihalyi, may be a better optimal choice between these two positions.

Rohit Parikh

Computer Science, Mathematics, Philosophy
City University of New York

The Logic of Knowledge Based Obligation

Abstract. Our obligations depend on what we know. If we do not know that we need to do X then there is no obligation to actually do X. However, sometimes there is also an obligation to know and hence also an obligation to inform. We look into the temporal logic of such issues, relying on work by John Horty and by Parikh and Ramanujam.

Jayant Shah

Department of Mathematics
Northeastern University

Discussion. Christian Trudeau's "From the Bankruptcy Problem and its Concede-and-Divide Solution to the Assignment Problem and its Fair Division Solution."

Nur Dean

Farmingdale State College

Polarized Population Under Imitation Dynamics in Complex Networks

Abstract. Evolutionary game theory is applied in a variety of settings, ranging from economics to socio-technical networks. The core concept in evolutionary game theory is evolutionary dynamics, which determines the composition of strategies in the population at steady state. Most evolutionary dynamics are modelled to descriptively showcase the utility derived from interactions between random pairs of players in well-mixed populations, or random pairs of neighbours in structured populations. In real-life social and socio-technical networks, it is more appropriate to evaluate a player's utility as a collection of interactions with its neighbours. To understand this; in practice, people form opinions by means of observation and imitation, by not just one friend, but a collection of friends. This paper displays a variation of the pairwise imitation dynamics where players imitate the most well-off neighbour. This process is memory-less i.e., players only use the outcome of the current game to determine their strategies in subsequent games. Empirical results demonstrate that in real-life social networks, this imitation dynamic leads to a polarized population with games that have multiple pure strategy Nash equilibria such as the Stag-Hunt game and anti-coordination games like Hawk-Dove, where an "undecided" population indefinitely swings between two strategies.

Jenn McDonald

Graduate Center, CUNY

Causal Models as Relative to Modal Profile

Abstract. A recent development in the philosophy of causation uses the framework of causal models, such as structural equation models, to define actual causation. There are two components to such a definition. The first is to identify how to define causation in terms of a given model or given class of models. The second is to provide an account of what qualifies models as given – or apt – such that they can be plugged into the first stage. A naïve hypothesis is that a model is apt just in case it is accurate. In this talk I will argue, however, that the accuracy of a model is not a determinate function of a model, an interpretation, and a situation. A given model on a given interpretation can still be deemed accurate or inaccurate of the same situation. As I demonstrate, this is because accuracy is relative to a set of background possibilities – what I call a modal profile. I argue that this reveals a heretofore hidden element in how causal models represent – that models represent situations only relative to some modal profile or other. I propose that this calls for an additional component of an interpretation: an interpretation is an assignment of content to the variables and a specification of modal profile.

Rohit Parikh

Computer Science, Mathematics, Philosophy
City University of New York

Covid-19 and Knowledge based Computation

Abstract. The purpose of this project is to combine insights from the logic of knowledge (act according to what you know), and graph theory (spread of infection follows the edges of a graph). We show how knowledge based algorithms can be used to combine safety with economic and social activity.

Arthur Paul Pedersen

Department of Computer Science
City University of New York

Discussion.Chapter 6, "Two-Person Cooperative Games" (spec. pp. 114-124), from R. Duncan Luce and Howard Raiffa's Games and Decisions.

Jayant Shah

Department of Mathematics
Northeastern University

Discussion. Robert J. Aumann and Michael Maschler's "Game Theoretic Analysis of a Bankruptcy Problem from the Talmud," Journal of Economic Theory, 36(2): 195-213, August 1985.

Rohit Parikh

Computer Science, Mathematics, Philosophy
City University of New York

Discussion. Jan van Eijck and Rineke Verbrugge's"Formal Approaches to Social Procedures," from Stanford Encyclopedia of Philosophy

Rohit Parikh

Computer Science, Mathematics, Philosophy
City University of New York

Applications of Epistemic Logic to Society

Rohit Parikh

Computer Science, Mathematics, Philosophy
City University of New York

Discussion.Chapter 4, "Conflict," from A.D. Taylor and A.M. Pacelli's Mathematics and Politics: Strategy, Voting, Power and Proof

Maryam Bibi

Brookyln College
City University of New York

Discussion.Chapter 3, "Political Power," from A.D. Taylor and A.M. Pacelli's Mathematics and Politics: Strategy, Voting, Power and Proof

Rohit Parikh

Computer Science, Mathematics, Philosophy
City University of New York

Discussion.Chapter 2, "Yes-No Voting," from A.D. Taylor and A.M. Pacelli's Mathematics and Politics: Strategy, Voting, Power and Proof

Rohit Parikh

Computer Science, Mathematics, Philosophy
City University of New York

Discussion.Chapter 1, "Social Choice," from A.D. Taylor and A.M. Pacelli's Mathematics and Politics: Strategy, Voting, Power and Proof

Barbara H. Partee

Department of Linguistics
University of Massachusetts Amherst

Language and Logic: Ideas and Controversies in the History of Formal Semantics ‖Slides

Simon Huttegger

Department of Logic and Philosophy of Science
University of California, Irvine

Discussion. On de Finetti's representation theorem for exchangeability, Chapter 7, "Unification," from Persi Diaconis and Brian Skyrms' Ten Great Ideas about Chance

Rohit Parikh

Computer Science, Mathematics, Philosophy
City University of New York

Discussion. Chapter 8, "Algorithmic Randomness," from Persi Diaconis and Brian Skyrms' Ten Great Ideas about Chance

Arthur Paul Pedersen

Department of Computer Science
City University of New York

Discussion. Chapter 6, "Inverse Inference: From Bayes and Laplace to Modern Statistics," from Persi Diaconis and Brian Skyrms' Ten Great Ideas about Chance

Cailin O'Connor

Department of Logic & Philosophy of Science
University of California, Irvine

Abstract. Standard accounts of convention include notions of arbitrariness. But many have conceived of conventionality as an all or nothing affair. In this paper, I develop a framework for thinking of conventions as coming in degrees of arbitrariness. In doing so, I introduce an information theoretic measure intended to capture the degree to which a solution to a certain social problem could have been otherwise. As the paper argues, this framework can help improve explanation aimed at the cultural evolution of social traits. Good evolutionary explanations recognize that most functional traits are also conventional, at least to some degree, and vice versa.

Paul Krasucki

Security
Accenture (Fortune Global 500 Company)

Discussion. Chapter 5, "Mathematics," from Persi Diaconis and Brian Skyrms' Ten Great Ideas about Chance

Jayant Shah

Department of Mathematics
Northeastern University

Discussion. Chapter 4, "Frequency," from Persi Diaconis and Brian Skyrms' Ten Great Ideas about Chance

Ada Coronado

Department of Philosophy
The Graduate Center, CUNY

Discussion. Chapter 3, "Psychology," from Persi Diaconis and Brian Skyrms' Ten Great Ideas about Chance

Todd Stambaugh

Department of Mathematics and Computer Science
John Jay College, CUNY

Coincidence of Bargaining Solution ‖Slides

Abstract. In 1950, a month before his dissertation on non-coorperative games was accepted at Princeton and 3 months after his famous solution concept was announced to the world in the Proceedings of the National Academy of Sciences, John Nash had published "The Bargaining Problem," in which he proposed the titular problem and gave the first solution. In the years after, several other solutions were developed, notably those by Kalai and Smorodinsky, Kalai, and Harsanyi. In this talk I will outline the problem itself, present four different solutions, and describe the precise conditions under which various sets of these solutions coincide.

Larry Moss

Department of Mathematics
Indiana University

Discussion. Chapter 2, "Judgment," from Persi Diaconis and Brian Skyrms' Ten Great Ideas about Chance

José Luis Bermúdez

Department of Philosophy
Texas A&M University

Rational Frames?