In response to recent work on the aggregation of individual judgments on logically connected propositions into collective judgments, it is often asked whether judgment aggregation is a special case of Arrowian preference aggregation. We argue for the converse claim. After proving two impossibility theorems on judgment aggregation (using "systematicity" and "independence" conditions, respectively), we construct an embedding of preference aggregation into judgment aggregation and prove Arrow’s theorem (stated for strict preferences) as a corollary of our second result. Although we (...) thereby provide a new proof of Arrow’s theorem, our main aim is to identify the analogue of Arrow’s theorem in judgment aggregation, to clarify the relation between judgment and preference aggregation, and to illustrate the generality of the judgment aggregation model. JEL Classi…cation: D70, D71.. (shrink)

This paper initiates the reverse mathematics of social choice theory, studying Arrow's impossibility theorem and related results including Fishburn's possibility theorem and the Kirman–Sondermann theorem within the framework of reverse mathematics. We formalise fundamental notions of social choice theory in second-order arithmetic, yielding a definition of countable society which is tractable in RCA0. We then show that the Kirman–Sondermann analysis of social welfare functions can be carried out in RCA0. This approach yields a proof of (...) class='Hi'>Arrow's theorem in RCA0, and thus in PRA, since Arrow's theorem can be formalised as a Π01 sentence. Finally we show that Fishburn's possibility theorem for countable societies is equivalent to ACA0 over RCA0. (shrink)

Riker (1982) famously argued that Arrow’s impossibility theorem undermined the logical foundations of “populism”, the view that in a democracy, laws and policies ought to express “the will of the people”. In response, his critics have questioned the use of Arrow’s theorem on the grounds that not all configurations of preferences are likely to occur in practice; the critics allege, in particular, that majority preference cycles, whose possibility the theorem exploits, rarely happen. In this essay, I argue (...) that the critics’ rejoinder to Riker misses the mark even if its factual claim about preferences is correct: Arrow’s theorem and related results threaten the populist’s principle of democratic legitimacy even if majority preference cycles never occur. In this particular context, the assumption of an unrestricted domain is justified irrespective of the preferences citizens are likely to have. (shrink)

This paper critically engages Philip Mirowki's essay, "The scientific dimensions of social knowledge and their distant echoes in 20th-century American philosophy of science." It argues that although the cold war context of anti-democratic elitism best suited for making decisions about engaging in nuclear war may seem to be politically and ideologically motivated, in fact we need to carefully consider the arguments underlying the new rational choice based political philosophies of the post-WWII era typified by Arrow's impossibility theorem. A (...) distrust of democratic decision-making principles may be developed by social scientists whose leanings may be toward the left or right side of the spectrum of political practices. (shrink)

In this paper, I investigate the relationship between preference and judgment aggregation, using the notion of ranking judgment introduced in List and Pettit. Ranking judgments were introduced in order to state the logical connections between the impossibility theorem of aggregating sets of judgments and Arrow’s theorem. I present a proof of the theorem concerning ranking judgments as a corollary of Arrow’s theorem, extending the translation between preferences and judgments defined in List and Pettit to the conditions (...) on the aggregation procedure. (shrink)

Amalgamating evidence of different kinds for the same hypothesis into an overall confirmation is analogous, I argue, to amalgamating individuals’ preferences into a group preference. The latter faces well-known impossibility theorems, most famously “Arrow’s Theorem”. Once the analogy between amalgamating evidence and amalgamating preferences is tight, it is obvious that amalgamating evidence might face a theorem similar to Arrow’s. I prove that this is so, and end by discussing the plausibility of the axioms required for the theorem.

According to conciliatory views about the epistemology of disagreement, when epistemic peers have conflicting doxastic attitudes toward a proposition and fully disclose to one another the reasons for their attitudes toward that proposition (and neither has independent reason to believe the other to be mistaken), each peer should always change his attitude toward that proposition to one that is closer to the attitudes of those peers with which there is disagreement. According to pure higher-order evidence views, higher-order evidence for a (...) proposition always suffices to determine the proper rational response to disagreement about that proposition within a group of epistemic peers. Using an analogue of Arrow's Impossibility Theorem, I shall argue that no conciliatory and pure higher-order evidence view about the epistemology of disagreement can provide a true and general answer to the question of what disagreeing epistemic peers should do after fully disclosing to each other the (first-order) reasons for their conflicting doxastic attitudes. (shrink)

We argue that a semantics for counterfactual conditionals in terms of comparative overall similarity faces a formal limitation due to Arrow’s impossibility theorem from social choice theory. According to Lewis’s account, the truth-conditions for counterfactual conditionals are given in terms of the comparative overall similarity between possible worlds, which is in turn determined by various aspects of similarity between possible worlds. We argue that a function from aspects of similarity to overall similarity should satisfy certain plausible constraints while Arrow’s (...) impossibility theorem rules out that such a function satisfies all the constraints simultaneously. We argue that a way out of this impasse is to represent aspectual similarity in terms of ranking functions instead of representing it in a purely ordinal fashion. Further, we argue against the claim that the determination of overall similarity by aspects of similarity faces a difficulty in addition to the Arrovian limitation, namely the incommensurability of different aspects of similarity. The phenomena that have been cited as evidence for such incommensurability are best explained by ordinary vagueness. (shrink)

An analogue of Arrow’s theorem has been thought to limit the possibilities for multi-criterial theory choice. Here, an example drawn from Toy Science, a model of theories and choice criteria, suggests that it does not. Arrow’s assumption that domains are unrestricted is inappropriate in connection with theory choice in Toy Science. There are, however, variants of Arrow’s theorem that do not require an unrestricted domain. They require instead that domains are, in a technical sense, ‘rich’. Since there are (...) rich domains in Toy Science, such theorems do constrain theory choice to some extent—certainly in the model and perhaps also in real science. (shrink)

Majority cycling and related social choice paradoxes are often thought to threaten the meaningfulness of democracy. But deliberation can prevent majority cycles – not by inducing unanimity, which is unrealistic, but by bringing preferences closer to single-peakedness. We present the first empirical test of this hypothesis, using data from Deliberative Polls. Comparing preferences before and after deliberation, we find increases in proximity to single-peakedness. The increases are greater for lower versus higher salience issues and for individuals who seem to have (...) deliberated more versus less effectively. They are not merely a byproduct of increased substantive agreement. Our results both refine and support the idea that deliberation, by increasing proximity to single-peakedness, provides an escape from the problem of majority cycling. (shrink)

(This is for the Cambridge Handbook of Analytic Philosophy, edited by Marcus Rossberg) In this handbook entry, I survey the different ways in which formal mathematical methods have been applied to philosophical questions throughout the history of analytic philosophy. I consider: formalization in symbolic logic, with examples such as Aquinas’ third way and Anselm’s ontological argument; Bayesian confirmation theory, with examples such as the fine-tuning argument for God and the paradox of the ravens; foundations of mathematics, with examples such as (...) Hilbert’s programme and Gödel’s incompleteness theorems; social choice theory, with examples such as Condorcet’s paradox and Arrow’s theorem; ‘how possibly’ results, with examples such as Condorcet’s jury theorem and recent work on intersectionality theory; and the application of advanced mathematics in philosophy, with examples such as accuracy-first epistemology. (shrink)

Shows how, as a consequence of the Arrow Impossibility Theorem, objectivity in grading is chimerical, given a sufficiently knowledgeable teacher (of his students, not his subject) in a sufficiently small class. -/- PDF available from JStor only; permission to post full version previously granted by journal editors and publisher expired. -/- Unpublished reply posted gratis.

This paper applies ideas and tools from social choice theory (such as Arrow's theorem and related results) to linguistics. Specifically, the paper investigates the problem of constraint aggregation in optimality theory from a social-choice-theoretic perspective.

Suppose that the members of a group each hold a rational set of judgments on some interconnected questions, and imagine that the group itself has to form a collective, rational set of judgments on those questions. How should it go about dealing with this task? We argue that the question raised is subject to a difficulty that has recently been noticed in discussion of the doctrinal paradox in jurisprudence. And we show that there is a general impossibility theorem that (...) that difficulty illustrates. Our paper describes this impossibility result and provides an exploration of its significance. The result naturally invites comparison with Kenneth Arrow's famous theorem (Arrow, 1963 and 1984; Sen, 1970) and we elaborate that comparison in a companion paper (List and Pettit, 2002). The paper is in four sections. The first section documents the need for various groups to aggregate its members' judgments; the second presents the discursive paradox; the third gives an informal statement of the more general impossibility result; the formal proof is presented in an appendix. The fourth section, finally, discusses some escape routes from that impossibility. (shrink)

Juries, committees and experts panels commonly appraise things of one kind or another on the basis of grades awarded by several people. When everybody's grading thresholds are known to be the same, the results sometimes can be counted on to reflect the graders’ opinion. Otherwise, they often cannot. Under certain conditions, Arrow's ‘impossibility’ theorem entails that judgements reached by aggregating grades do not reliably track any collective sense of better and worse at all. These claims are made by (...) adapting the Arrow–Sen framework for social choice to study grading in groups. (shrink)

We introduce a new theorem in social choice theory built on a path integral approach which will show that, under some reasonable conditions, there is a unique way to aggregate individual preferences based on fuzzy sets into a social preference based on probabilities, and that this way is invariant under any permutation of alternatives. We then apply this theorem to the case of democratic decision making with data of the behaviour and voting preferences of voting agents and show (...) that there is a tradeoff between fairness and efficiency and that no voting system can achieve both simultaneously. (shrink)

This paper develops and explores a new framework for theorizing about the measurement and aggregation of well-being. It is a qualitative variation on the framework of social welfare functionals developed by Amartya Sen. In Sen’s framework, a social or overall betterness ordering is assigned to each profile of real-valued utility functions. In the qualitative framework developed here, numerical utilities are replaced by the properties they are supposed to represent. This makes it possible to characterize the measurability and interpersonal comparability of (...) well-being directly, without the use of invariance conditions, and to distinguish between real changes in well-being and merely representational changes in the unit of measurement. The qualitative framework is shown to have important implications for a range of issues in axiology and social choice theory, including the characterization of welfarism, axiomatic derivations of utilitarianism, the meaningfulness of prioritarianism, the informational requirements of variable-population ethics, the impossibility theorems of Arrow and others, and the metaphysics of value. (shrink)

I propose a relevance-based independence axiom on how to aggregate individual yes/no judgments on given propositions into collective judgments: the collective judgment on a proposition depends only on people’s judgments on propositions which are relevant to that proposition. This axiom contrasts with the classical independence axiom: the collective judgment on a proposition depends only on people’s judgments on the same proposition. I generalize the premise-based rule and the sequential-priority rule to an arbitrary priority order of the propositions, instead of a (...) dichotomous premise/conclusion order resp. a linear priority order. I prove four impossibility theorems on relevance-based aggregation. One theorem simultaneously generalizes Arrow’s Theorem (in its general and indifference-free versions) and the well-known Arrow-like theorem in judgment aggregation. (shrink)

In the theory of judgment aggregation, it is known for which agendas of propositions it is possible to aggregate individual judgments into collective ones in accordance with the Arrow-inspired requirements of universal domain, collective rationality, unanimity preservation, non-dictatorship and propositionwise independence. But it is only partially known (e.g., only in the monotonic case) for which agendas it is possible to respect additional requirements, notably non-oligarchy, anonymity, no individual veto power, or implication preservation. We fully characterize the agendas for which there (...) are such possibilities, thereby answering the most salient open questions about propositionwise judgment aggregation. Our results build on earlier results by Nehring and Puppe (2002), Nehring (2006), Dietrich and List (2007a) and Dokow and Holzman (2010a). (shrink)

A platitude that took hold with Kuhn is that there can be several equally good ways of balancing theoretical virtues for theory choice. Okasha recently modelled theory choice using technical apparatus from the domain of social choice: famously, Arrow showed that no method of social choice can jointly satisfy four desiderata, and each of the desiderata in social choice has an analogue in theory choice. Okasha suggested that one can avoid the Arrow analogue for theory choice by employing a strategy (...) used by Sen in social choice, namely, to enhance the information made available to the choice algorithms. I argue here that, despite Okasha’s claims to the contrary, the information-enhancing strategy is not compelling in the domain of theory choice. (shrink)

The ``doctrinal paradox'' or ``discursive dilemma'' shows that propositionwise majority voting over the judgments held by multiple individuals on some interconnected propositions can lead to inconsistent collective judgments on these propositions. List and Pettit (2002) have proved that this paradox illustrates a more general impossibility theorem showing that there exists no aggregation procedure that generally produces consistent collective judgments and satisfies certain minimal conditions. Although the paradox and the theorem concern the aggregation of judgments rather than preferences, they (...) invite comparison with two established results on the aggregation of preferences: the Condorcet paradox and Arrow's impossibility theorem. We may ask whether the new impossibility theorem is a special case of Arrow's theorem, or whether there are interesting disanalogies between the two results. In this paper, we compare the two theorems, and show that they are not straightforward corollaries of each other. We further suggest that, while the framework of preference aggregation can be mapped into the framework of judgment aggregation, there exists no obvious reverse mapping. Finally, we address one particular minimal condition that is used in both theorems – an independence condition – and suggest that this condition points towards a unifying property underlying both impossibility results. (shrink)

Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with (...) other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achilles and the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called “ontological”, on which basis “motion” studied by physics and “conclusion” studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll’s paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox therefore naturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)’s completeness theorems). (shrink)

This is a series of lectures on formal decision theory held at the University of Bayreuth during the summer terms 2008 and 2009. It largely follows the book from Michael D. Resnik: Choices. An Introduction to Decision Theory, 5th ed. Minneapolis London 2000 and covers the topics: -/- Decisions under ignorance and risk Probability calculus (Kolmogoroff Axioms, Bayes' Theorem) Philosophical interpretations of probability (R. v. Mises, Ramsey-De Finetti) Neuman-Morgenstern Utility Theory Introductory Game Theory Social Choice Theory (Sen's Paradox of (...) Liberalism, Arrow's Theorem) . (shrink)

By introducing elements of phenomenological philosophy to the analysis of human needs in economics; from Sartrean postulates as well as the nature and essence of individual’s needs, has been revealed a theorethical framework that serves to ponder human being’s existential behavior by means of their phenomenologic social choices and welfare. Defining a planning agent under strong assumptions of rationality and projective efficacious capabilities, the Arrow’s theorem has been proved for the economic agent aware of its finitude in this world.

In this study, the starting point is the well-known physical laws applied to human social life. On the basis of natural laws human actions are considered and through the prism of physical laws such concepts as use and possession are defined. A parallel is drawn between such a representation of these concepts and those conflicting views that are available in the literature regarding the concept of property. To complete the definitions of use and possession nature is introduced as a fictitious (...) owner. And on this basis, the positive possibility of a theoretical solution to the problem of initial assignment is shown. Again, on the basis of physical laws, the fundamental concept of [human] needs is introduced. It is shown that the collision of people's needs on the same thing allows uniform classifying property defined in literature as the relationship between a person and a thing or as the relationship between people because of a thing. Considering the relationship between two human beings through needs and costs, as a natural necessity, people inevitably renounce their claims to possess and use certain things in favor of other people. It is shown that this refusal forms the right of those people in whose favor this refusal is carried out to possess and use things. The right of one is the refusal of all others to own and use the thing. It is shown that the right was ensured and will always be ensured by force. The use or threat of the use of force is something that can reliably ward off a person from the unbridled realization of his needs. In the process of the formation of mankind, nature itself forces people to organize into communities that can oppose their individual members and their associations with significantly greater power. The whole, as a rule, is stronger than its part. And it is society that can reliably ensure the exercise of rights for its members. Natural laws also make it possible to resolve, on the basis of the concept of law, as a renunciation of possession and use, the issue of belonging to what nature gives us. These are natural resources and the human body. It is shown that the human body should belong to the person himself, and the resources to all members of society equally. The affiliation of all other things produced by man can be unambiguously determined within the framework of contractual relations between members of society, their associations and society as a whole. Since the right of everyone is ensured by the society, and, therefore, by each member of the society individually, a necessary condition for membership is understanding and recognition of the rights of certain things to other members of the society. And this may be the main criterion for joining full members of society, as opposed to the commonly used age criterion, which works on the bulk of people, but gives failures in many special cases. A typical example of such cases is the deprivation or infringement of the rights of persons of full legal age, but committed acts that are called unlawful in society. As part of the research on ownership issues, ownership is considered. It is shown that the necessary tool for using the objects of co-ownership is the voting of co-owners. A special case of co-ownership, when all co-owners have equal shares in co-ownership, is indistinguishable from what is called democracy. It is shown that voting in general and democracy in particular, as procedures for aggregating preferences, can have a positive decision, in refutation of the universality of the conclusion of Arrow's theorem. (shrink)

Tsui and Weymark (Economic Theory, 1997) have shown that the only continuous social welfare orderings on the whole Euclidean space which satisfy the weak Pareto principle and are invariant to individual-specific similarity transformations of utilities are strongly dictatorial. Their proof relies on functional equation arguments which are quite complex. This note provides a simpler proof of their theorem.

According to a theorem recently proved in the theory of logical aggregation, any nonconstant social judgment function that satisfies independence of irrelevant alternatives (IIA) is dictatorial. We show that the strong and not very plausible IIA condition can be replaced with a minimal independence assumption plus a Pareto-like condition. This new version of the impossibility theorem likens it to Arrow’s and arguably enhances its paradoxical value.

Several recent results on the aggregation of judgments over logically connected propositions show that, under certain conditions, dictatorships are the only propositionwise aggregation functions generating fully rational (i.e., complete and consistent) collective judgments. A frequently mentioned route to avoid dictatorships is to allow incomplete collective judgments. We show that this route does not lead very far: we obtain oligarchies rather than dictatorships if instead of full rationality we merely require that collective judgments be deductively closed, arguably a minimal condition of (...) rationality, compatible even with empty judgment sets. We derive several characterizations of oligarchies and provide illustrative applications to Arrowian preference aggregation and Kasher and Rubinsteinís group identification problem. (shrink)

In normative political theory, it is widely accepted that democracy cannot be reduced to voting alone, but that it requires deliberation. In formal social choice theory, by contrast, the study of democracy has focused primarily on the aggregation of individual opinions into collective decisions, typically through voting. While the literature on deliberation has an optimistic flavour, the literature on social choice is more mixed. It is centred around several paradoxes and impossibility results identifying conflicts between different intuitively plausible desiderata. In (...) recent years, there has been a growing dialogue between the two literatures. This paper discusses the connections between them. Important insights are that (i) deliberation can complement aggregation and open up an escape route from some of its negative results; and (ii) the formal models of social choice theory can shed light on some aspects of deliberation, such as the nature of deliberation-induced opinion change. (shrink)

How can different individuals' probability assignments to some events be aggregated into a collective probability assignment? Classic results on this problem assume that the set of relevant events -- the agenda -- is a sigma-algebra and is thus closed under disjunction (union) and conjunction (intersection). We drop this demanding assumption and explore probabilistic opinion pooling on general agendas. One might be interested in the probability of rain and that of an interest-rate increase, but not in the probability of rain or (...) an interest-rate increase. We characterize linear pooling and neutral pooling for general agendas, with classic results as special cases for agendas that are sigma-algebras. As an illustrative application, we also consider probabilistic preference aggregation. Finally, we compare our results with existing results on binary judgment aggregation and Arrovian preference aggregation. This paper is the first of two self-contained, but technically related companion papers inspired by binary judgment-aggregation theory. (shrink)

Comparative overall similarity lies at the basis of a lot of recent metaphysics and epistemology. It is a poor foundation. Overall similarity is supposed to be an aggregate of similarities and differences in various respects. But there is no good way of combining them all.

The aim of this article is to introduce the theory of judgment aggregation, a growing interdisciplinary research area. The theory addresses the following question: How can a group of individuals make consistent collective judgments on a given set of propositions on the basis of the group members' individual judgments on them? I begin by explaining the observation that initially sparked the interest in judgment aggregation, the so-called "doctinal" and "discursive paradoxes". I then introduce the basic formal model of judgment aggregation, (...) which allows me to present some illustrative variants of a generic impossibility result. I subsequently turn to the question of how this impossibility result can be avoided, going through several possible escape routes. Finally, I relate the theory of judgment aggregation to other branches of aggregation theory. Rather than offering a comprehensive survey of the theory of judgment aggregation, I hope to introduce the theory in a succinct and pedagogical way, providing an illustrative rather than exhaustive coverage of some of its key ideas and results. (shrink)

In solving judgment aggregation problems, groups often face constraints. Many decision problems can be modelled in terms the acceptance or rejection of certain propositions in a language, and constraints as propositions that the decisions should be consistent with. For example, court judgments in breach-of-contract cases should be consistent with the constraint that action and obligation are necessary and sufficient for liability; judgments on how to rank several options in an order of preference with the constraint of transitivity; and judgments on (...) budget items with budgetary constraints. Often more or less demanding constraints on decisions are imaginable. For instance, in preference ranking problems, the transitivity constraint is often contrasted with the weaker acyclicity constraint. In this paper, we make constraints explicit in judgment aggregation by relativizing the rationality conditions of consistency and deductive closure to a constraint set, whose variation yields more or less strong notions of rationality. We review several general results on judgment aggregation in light of such constraints. (shrink)

What is the relationship between degrees of belief and binary beliefs? Can the latter be expressed as a function of the former—a so-called “belief-binarization rule”—without running into difficulties such as the lottery paradox? We show that this problem can be usefully analyzed from the perspective of judgment-aggregation theory. Although some formal similarities between belief binarization and judgment aggregation have been noted before, the connection between the two problems has not yet been studied in full generality. In this paper, we seek (...) to fill this gap. The paper is organized around a baseline impossibility theorem, which we use to map out the space of possible solutions to the belief-binarization problem. Our theorem shows that, except in limiting cases, there exists no belief-binarization rule satisfying four initially plausible desiderata. Surprisingly, this result is a direct corollary of the judgment-aggregation variant of Arrow’s classic impossibility theorem in social choice theory. (shrink)

An important objection to preference-satisfaction theories of well-being is that they cannot make sense of interpersonal comparisons. A tradition dating back to Harsanyi :434, 1953) attempts to solve this problem by appeal to people’s so-called extended preferences. This paper presents a new problem for the extended preferences program, related to Arrow’s celebrated impossibility theorem. We consider three ways in which the extended-preference theorist might avoid this problem, and recommend that she pursue one: developing aggregation rules that violate Arrow’s Independence (...) of Irrelevant Alternatives condition. (shrink)

The paper discusses the sense in which the changes undergone by normative economics in the twentieth century can be said to be progressive. A simple criterion is proposed to decide whether a sequence of normative theories is progressive. This criterion is put to use on the historical transition from the new welfare economics to social choice theory. The paper reconstructs this classic case, and eventually concludes that the latter theory was progressive compared with the former. It also briefly comments on (...) the recent developments in normative economics and their connection with the previous two stages. (Published Online April 18 2006) Footnotes1 This paper suspersedes an earlier one entitled “Is There Progress in Normative Economics?” (Mongin 2002). I thank the organizers of the Fourth ESHET Conference (Graz 2000) for the opportunity they gave me to lecture on this topic. Thanks are also due to J. Alexander, K. Arrow, A. Bird, R. Bradley, M. Dascal, W. Gaertner, N. Gravel, D. Hausman, B. Hill, C. Howson, N. McClennen, A. Trannoy, J. Weymark, J. Worrall, two annonymous referees of this journal, and especially the editor M. Fleurbaey, for helpful comments. The editor's suggestions contributed to determine the final orientation of the paper. The author is grateful to the LSE and the Lachmann Foundation for their support at the time when he was writing the initial version. (shrink)

In this short survey article, I discuss Bell’s theorem and some strategies that attempt to avoid the conclusion of non-locality. I focus on two that intersect with the philosophy of probability: (1) quantum probabilities and (2) superdeterminism. The issues they raised not only apply to a wide class of no-go theorems about quantum mechanics but are also of general philosophical interest.

Extensive measurement is the standard measurement-theoretic approach for constructing a ratio scale. It involves the comparison of objects that can be concatenated in an additively representable way. This paper studies the implications of extensively measurable welfare for social choice theory. We do this in two frameworks: an Arrovian framework with a fixed population and no interpersonal comparisons, and a generalized framework with variable populations and full interpersonal comparability. In each framework we use extensive measurement to introduce novel domain restrictions, independence (...) conditions, and constraints on social evaluation. We prove a welfarism theorem for these domains and characterize the social welfare functions that satisfy the axioms of extensive measurement at both the individual and social levels. The main results are simple axiomatizations of strong dictatorship in the Arrovian framework and classical utilitarianism in the generalized framework. (shrink)

This thesis argues for the fundamental importance of the opposition between holistic and reductionistic world-views in economics. Both reductionism and holism may nevertheless underpin laissez-faire policy prescriptions. Scrutiny of the nature of the articulation between micro and macro levels in the writings of economists suggests that invisible hand theories play a key role in reconciling reductionist policy prescriptions with a holistic world. An examination of the prisoners' dilemma in game theory and Arrow's impossibility theorem in social choice theory (...) sets the scene. The prisoners' dilemma epitomises the collective irrationality coordination problems lead to. The source of the dilemma is identified as the combination of interdependence in content and independence in form of the decision making process. Arrovian impossibility has been perceived as challenging traditional views of the relationship between micro and macro levels in economics. Conservative arguments against the possibility in principle of a social welfare function are criticised here as depending on an illicit dualism. The thesis then reviews the standpoints of Smith, Hayek and Keynes. For Smith, the social desirability of individual self-seeking activity is ensured by the 'invisible hand' of a god who has moulded us so to behave, that the quantity of happiness in the world is always maximised. Hayek seeks to re-establish the invisible hand in a secular age, replacing the agency of a deity with an evolutionary mechanism. Hayek's evolutionary theory, criticised here as being based on the exploded notion of group selection, cannot underpin the desirability of spontaneous outcomes. I conclude by arguing that Keynes shares the holistic approach of Smith and Hayek, but without their reliance on invisible hand mechanisms. If spontaneous processes cannot be relied upon to generate desirable social outcomes then we have to take responsibility for achieving this ourselves by establishing the appropriate institutional framework to eliminate macroeconomic prisoners' dilemmas. (shrink)

Can we design a perfect democratic decision procedure? Condorcet famously observed that majority rule, our paradigmatic democratic procedure, has some desirable properties, but sometimes produces inconsistent outcomes. Revisiting Condorcet’s insights in light of recent work on the aggregation of judgments, I show that there is a conflict between three initially plausible requirements of democracy: “robustness to pluralism”, “basic majoritarianism”, and “collective rationality”. For all but the simplest collective decision problems, no decision procedure meets these three requirements at once; at most (...) two can be met together. This “democratic trilemma” raises the question of which requirement to give up. Since different answers correspond to different views about what matters most in a democracy, the trilemma suggests a map of the “logical space” in which different conceptions of democracy are located. It also sharpens our thinking about other impossibility problems of social choice and how to avoid them, by capturing a core structure many of these problems have in common. More broadly, it raises the idea of “cartography of logical space” in relation to contested political concepts. (shrink)

In this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea that purity and explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically (...) impure (ergodic) proof due to Furstenberg (Section 3). Furstenberg’s ergodic proof is striking because it utilizes intuitively foreign and infinitary resources to prove a finitary combinatorial result and does so in a perspicuous fashion. I claim that Furstenberg’s proof is explanatory in light of its clear expression of a crucial structural result, which provides the “reason why” Szemerédi’s theorem is true. This is, however, rather surprising: how can such intuitively different conceptual resources “get a grip on” the theorem to be proved? I account for this phenomenon by articulating a new construal of the content of a mathematical statement, which I call structural content (Section 4). I argue that the availability of structural content saves intuitive epistemic distinctions made in mathematical practice and simultaneously explicates the intervention of surprising and explanatorily rich conceptual resources. Structural content also disarms general arguments for thinking that impurity and explanatory power might come apart. Finally, I sketch a proposal that, once structural content is in hand, impure resources lead to explanatory proofs via suitably understood varieties of simplification and unification (Section 5). (shrink)

I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of ±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of ±2 it sets out to prove. Once this oversight is ameliorated from Bell's argument (...) by identifying the impediment that leads to it and local realism is implemented correctly, the bounds on the Bell-CHSH sum of expectation values work out to be ±2√2 instead of ±2, thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by any of the Bell-test experiments is not local realism but the linear additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with. I also identify similar oversight in the GHZ variant of Bell's theorem, invalidating its claim of having found an inconsistency in the premisses of the argument by EPR for completing quantum mechanics. Conceptually, the oversight in both Bell's theorem and its GHZ variant traces back to the oversight in von Neumann's theorem against hidden variable theories identified by Grete Hermann in the 1930s. (shrink)

The definition of nanoeconomics can relate to different levels and areas of economic life. First of all, this is the nanolevel of the economic system. As a human economy, nanoeconomics provides for the allocation of an individual factor within the framework of a socio-economic phenomenon. The nanoeconomic aspect is central to the definition of inclusion. So, the inclusion of a person, as the main subject of nanoeconomics, to the formation and stabilization of economic systems is the initial one in the (...) integration of an individual in relation to production processes and economic development. A person is involved in academic and social life by making decisions about their own business and integrating it into the sectoral and national economic space. It is proved that its indicators are the conditions for clustering the economic system. The study carried out a cluster analysis of the innovation system in a country with an economy in transition. In addition, the study outlined that inclusive phenomena in the economy are close to integration and are the opposite of segregation and isolation. It is noted that different institutions of integration can be used to form objective conditions for the development of babyeconomics. Public decisions of inclusion involve the use of Arrow's impossibility theorem. The research results can be used: – the individualistic functions of inclusion should be used in the formation of the babyeconomics, the human economy and the economy of nanotechnology; – states of inclusion must be created at all levels of the economic system; – a person and wealth are an individualistic aspect of an inclusive economy, because national wealth consists of individual wealth. Nanoeconomics is just beginning to be included in the systemic processes of inclusive economic phenomena, especially in countries with economies in transition. (shrink)

Bell’s theorem has fascinated physicists and philosophers since his 1964 paper, which was written in response to the 1935 paper of Einstein, Podolsky, and Rosen. Bell’s theorem and its many extensions have led to the claim that quantum mechanics and by inference nature herself are nonlocal in the sense that a measurement on a system by an observer at one location has an immediate effect on a distant entangled system. Einstein was repulsed by such “spooky action at a (...) distance” and was led to question whether quantum mechanics could provide a complete description of physical reality. In this paper I argue that quantum mechanics does not require spooky action at a distance of any kind and yet it is entirely reasonable to question the assumption that quantum mechanics can provide a complete description of physical reality. The magic of entangled quantum states has little to do with entanglement and everything to do with superposition, a property of all quantum systems and a foundational tenet of quantum mechanics. (shrink)

In this article, a possible generalization of the Löb’s theorem is considered. Main result is: let κ be an inaccessible cardinal, then ¬Con( ZFC +∃κ) .

Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though some versions of mathematical (...) instrumentalism are defeated by Gödel’s theorem, not all are. By considering inductive reasons in mathematics, we show that some mathematical instrumentalisms survive the theorem. (shrink)

Time in electromagnetism shares many features with time in other physical theories. But there is one aspect of electromagnetism's relationship with time that has always been controversial, yet has not always attracted the limelight it deserves: the electromagnetic arrow of time. Beginning with a re-analysis of a famous argument between Ritz and Einstein over the origins of the radiation arrow, this chapter frames the debate between modern Einsteinians and neo-Ritzians. It tries to find a clean statement of what the arrow (...) is and then explains how it relates to the cosmological and thermodynamic arrows, representing the most developed and sophisticated attack yet, in either the physics or philosophy literature, on the electromagnetic arrow of time. (shrink)

This essay deals with Gödel's Theorems in relationship to Philosophy of Science; firstly, in outlining Ludwig Wittgenstein's position on the limits of philosophical truth that we can derive from Gödel (and how this in turn impacts modern-philosophical conceptions of science), and secondly, the deeper uncertainty about consciousness that Gödel's theorems point to, most notably elucidated by Sir Roger Penrose.

Indigenous Sustainable Wisdom: First Nation Know-How for Global Flourishing’s contributors describe ways of being that reflect a worldview that has guided humanity for 99% of human history; they describe the practical traditional wisdom stemming from Nature-based relational cultures that were or are guided by this worldview. Such cultures did not cause the kinds of anti-Nature and de-humanizing or inequitable policies and practices that now pervade our world. Far from romanticizing Indigenous histories, Indigenous Sustainable Wisdom offers facts about how human beings, (...) with our potential for good and evil behaviors, can live in relative harmony again. Contributions cover views from anthropology, psychology, sociology, leadership, native science, native history, native art. (shrink)

It has been known for a few years that no more than Pi-1-1 comprehension is needed for the proof of "Frege's Theorem". One can at least imagine a view that would regard Pi-1-1 comprehension axioms as logical truths but deny that status to any that are more complex—a view that would, in particular, deny that full second-order logic deserves the name. Such a view would serve the purposes of neo-logicists. It is, in fact, no part of my view that, (...) say, Delta-3-1 comprehension axioms are not logical truths. What I am going to suggest, however, is that there is a special case to be made on behalf of Pi-1-1 comprehension. Making the case involves investigating extensions of first-order logic that do not rely upon the presence of second-order quantifiers. A formal system for so-called "ancestral logic" is developed, and it is then extended to yield what I call "Arché logic". (shrink)