Condorcet's famous jurytheorem reaches an optimistic conclusion on the correctness of majority decisions, based on two controversial premises about voters: they are competent and vote independently, in a technical sense. I carefully analyse these premises and show that: whether a premise is justi…ed depends on the notion of probability considered; none of the notions renders both premises simultaneously justi…ed. Under the perhaps most interesting notions, the independence assumption should be weakened.
My aim in this paper is to explain what Condorcet’s jurytheorem is, and to examine its central assumptions, its significance to the epistemic theory of democracy and its connection with Rousseau’s theory of general will. In the first part of the paper I will analyze an epistemic theory of democracy and explain how its connection with Condorcet’s jurytheorem is twofold: the theorem is at the same time a contributing historical source, and the model (...) used by the authors to this day. In the second part I will specify the purposes of the theorem itself, and examine its underlying assumptions. Third part will be about an interpretation of Rousseau’s theory, which is given by Grofman and Feld relying on Condorcet’s jurytheorem, and about criticisms of such interpretation. In the fourth, and last, part I will focus on one particular assumption of Condorcet’s theorem, which proves to be especially problematic if we would like to apply the theorem under real-life conditions; namely, the assumption that voters choose between two options only. (shrink)
This paper generalises the classical Condorcet jurytheorem from majority voting over two options to plurality voting over multiple options. The paper further discusses the debate between epistemic and procedural democracy and situates its formal results in that debate. The paper finally compares a number of different social choice procedures for many-option choices in terms of their epistemic merits. An appendix explores the implications of some of the present mathematical results for the question of how probable majority cycles (...) (as in Condorcet's paradox) are in large electorates. (shrink)
Under the independence and competence assumptions of Condorcet’s classical jury model, the probability of a correct majority decision converges to certainty as the jury size increases, a seemingly unrealistic result. Using Bayesian networks, we argue that the model’s independence assumption requires that the state of the world (guilty or not guilty) is the latest common cause of all jurors’ votes. But often – arguably in all courtroom cases and in many expert panels – the latest such common cause (...) is a shared ‘body of evidence’ observed by the jurors. In the corresponding Bayesian network, the votes are direct descendants not of the state of the world, but of the body of evidence, which in turn is a direct descendant of the state of the world. We develop a model of jury decisions based on this Bayesian network. Our model permits the possibility of misleading evidence, even for a maximally competent observer, which cannot easily be accommodated in the classical model. We prove that (i) the probability of a correct majority verdict converges to the probability that the body of evidence is not misleading, a value typically below 1; (ii) depending on the required threshold of ‘no reasonable doubt’, it may be impossible, even in an arbitrarily large jury, to establish guilt of a defendant ‘beyond any reasonable doubt’. (shrink)
It has been argued that an epistemically rational agent’s evidence is subjectively mediated through some rational epistemic standards, and that there are incompatible but equally rational epistemic standards available to agents. This supports Permissiveness, the view according to which one or multiple fully rational agents are permitted to take distinct incompatible doxastic attitudes towards P (relative to a body of evidence). In this paper, I argue that the above claims entail the existence of a unique and more reliable epistemic standard. (...) My strategy relies on Condorcet’s JuryTheorem. This gives rise to an important problem for those who argue that epistemic standards are permissive, since the reliability criterion is incompatible with such a type of Permissiveness. (shrink)
We give a review and critique of jury theorems from a social-epistemology perspective, covering Condorcet’s (1785) classic theorem and several later refinements and departures. We assess the plausibility of the conclusions and premises featuring in jury theorems and evaluate the potential of such theorems to serve as formal arguments for the ‘wisdom of crowds’. In particular, we argue (i) that there is a fundamental tension between voters’ independence and voters’ competence, hence between the two premises of most (...)jury theorems; (ii) that the (asymptotic) conclusion that ‘huge groups are infallible’, reached by many jury theorems, is an artifact of unjustified premises; and (iii) that the (nonasymptotic) conclusion that ‘larger groups are more reliable’, also reached by many jury theorems, is not an artifact and should be regarded as the more adequate formal rendition of the ‘wisdom of crowds’. (shrink)
Recent political developments cast doubt on the wisdom of democratic decision-making. Brexit, the Colombian people's (initial) rejection of peace with the FARC, and the election of Donald Trump suggest that the time is right to explore alternatives to democracy. In this essay, I describe and defend the epistocratic system of government which is, given current theoretical and empirical knowledge, most likely to produce optimal political outcomes—or at least better outcomes than democracy produces. To wit, we should expand the suffrage as (...) wide as possible and weight citizens’ votes in accordance with their competence. As it turns out, the optimal system is closely related to J. S. Mill's plural voting proposal. I also explain how voters’ competences can be precisely determined, without reference to an objective standard of correctness and without generating invidious comparisons between voters. (shrink)
Can experimental philosophy help us answer central questions about the nature of moral responsibility, such as the question of whether moral responsibility is compatible with determinism? Specifically, can folk judgments in line with a particular answer to that question provide support for that answer. Based on reasoning familiar from Condorcet’s JuryTheorem, such support could be had if individual judges track the truth of the matter independently and with some modest reliability: such reliability quickly aggregates as the number (...) of judges goes up. In this chapter, however, I argue, partly based on empirical evidence, that although non-specialist judgments might on average be more likely than not to get things right, their individual likelihoods fail to aggregate because they do not track truth with sufficient independence. (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 jurytheorem and recent work on intersectionality theory; and the application of advanced mathematics in philosophy, with examples such as accuracy-first epistemology. (shrink)
The contemporary theory of epistemic democracy often draws on the Condorcet JuryTheorem to formally justify the ‘wisdom of crowds’. But this theorem is inapplicable in its current form, since one of its premises – voter independence – is notoriously violated. This premise carries responsibility for the theorem's misleading conclusion that ‘large crowds are infallible’. We prove a more useful jurytheorem: under defensible premises, ‘large crowds are fallible but better than small groups’. This (...)theorem rehabilitates the importance of deliberation and education, which appear inessential in the classical jury framework. Our theorem is related to Ladha's (1993) seminal jurytheorem for interchangeable (‘indistinguishable’) voters based on de Finetti's Theorem. We also prove a more general and simpler such jurytheorem. (shrink)
Epistemically immodest agents take their own epistemic standards to be among the most truth-conducive ones available to them. Many philosophers have argued that immodesty is epistemically required of agents, notably because being modest entails a problematic kind of incoherence or self-distrust. In this paper, I argue that modesty is epistemically permitted in some social contexts. I focus on social contexts where agents with limited cognitive capacities cooperate with each other (like juries).
There is a substantial class of collective decision problems whose successful solution requires interdependence among decision makers at the agenda-setting stage and independence at the stage of choice. We define this class of problems and describe and apply a search-and-decision mechanism theoretically modeled in the context of honeybees and identified in earlier empirical work in biology. The honeybees’ mechanism has useful implications for mechanism design in human institutions, including courts, legislatures, executive appointments, research and development in firms, and basic research (...) in the sciences. Our paper offers a fresh perspective on the idea of “biomimicry” in institutional design and raises the possibility of comparative institutional analysis across species. (shrink)
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)
In his 2010 paper “Philosophical Naturalism and Intuitional Methodology”, Alvin I. Goldman invokes the Condorcet JuryTheorem in order to defend the reliability of intuitions. The present note argues that the original conditions of the theorem are all unrealistic when analysed in connection to the case of intuitions. Alternative conditions are discussed.
How can democratic governments be relied upon to achieve adequate political knowledge when they turn over their authority to those of no epistemic distinction whatsoever? This deep and longstanding concern is one that any proponent of epistemic conceptions of democracy must take seriously. While Condorcetian responses have recently attracted substantial interest, they are largely undermined by a fundamental neglect of agenda-setting. I argue that the apparent intractability of the problem of epistemic adequacy in democracy stems in large part from a (...) failure to appreciate the social character of political knowledge. A social point of view brings into focus a number of vital factors that bear on our understanding of democratic epistemology and our assessment of its prospects: the essential role of inclusive deliberation, the public's agenda-setting function, institutional provisions for policy feedback, the independence of expert communities, and the knowledge-pooling powers of markets. (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)
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)
A proof of Fermat’s last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the three definitive properties of the relation of equality (identity, symmetry, and transitivity), modus tollens, axiom of induction, the proof of Fermat’s last theorem in the case of.
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)
Epistemic justifications for democracy have been offered in terms of two different aspects of decision-making: voting and deliberation, or ‘votes’ and ‘talk.’ The Condorcet JuryTheorem is appealed to as a justification in terms votes, and the Hong-Page “Diversity Trumps Ability” result is appealed to as a justification in terms of deliberation. Both of these, however, are most plausibly construed as models of direct democracy, with full and direct participation across the population. In this paper, we explore how (...) these results hold up if we vary the model so as to reflect the more familiar democratic structure of a representative hierarchy. We first recount extant analytic work that shows that representation inevitably weakens the voting results of the Condorcet JuryTheorem, but we question the ability of that result to shine light on real representative systems. We then show that, when we move from votes to talk, as modeled in Hong-Page, representation holds its own and even has a slight edge. (shrink)
Epistemic justifications for democracy have been offered in terms of two different aspects of decision-making: voting and deliberation, or 'votes' and 'talk.' The Condorcet JuryTheorem is appealed to as a justification in terms of votes, and the Hong-Page "Diversity Trumps Ability" result is appealed to as a justification in terms of deliberation. Both of these, however, are most plausibly construed as models of direct democracy, with full and direct participation across the population. In this paper, we explore (...) how these results hold up if we vary the model so as to reflect the more familiar democratic structure of a representative hierarchy. We first recount extant analytic work that shows that representation inevitably weakens the voting results of the Condorcet JuryTheorem, but we question the ability of the result to shine light on real representative systems. We then show that, when we move from votes to talk, as modeled in Hong-Page, representation holds its own and even has a slight edge. (shrink)
The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin's famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good (...) measure of the strength of the theory. I exhibit certain strong counterexamples and establish conclusively that the received view is false. Moreover, I show that the limiting constants provided by the theorem do not in any way reflect the power of formalized theories, but that the values of these constants are actually determined by the chosen coding of Turing machines, and are thus quite accidental. (shrink)
Una de las bifurcaciones en el debate contemporáneo sobre la legitimidad de la democracia explora si ésta ofrece ventajas distintivamente epistémicas frente a otras alternativas políticas. Quienes defienden la tesis de la democracia epistémica afirman que la democracia es instrumentalmente superior o equiparable a otras formas de organización política en lo que concierne a la obtención de varios bienes epistémicos. En este ensayo presento dos (grupos de) argumentos a favor de la democracia epistémica, que se inspiran en resultados formales: el (...) teorema del jurado de Condorcet [TJC] y el teorema ‘diversidad supera habilidad’ [DSH]. Pese a su gran atractivo, sostengo que estos argumentos son incapaces de respaldar dicha tesis: no brindan razones para considerar que la democracia es epistémicamente superior (o equiparable) a algunas alternativas políticas no democráticas. En su lugar, sugiero que, sin requerir un cambio radical en nuestras formas de organización política, la epistemología democrática –el estudio de las ‘circunstancias epistémicas de la democracia’– puede ofrecer valiosas lecciones de sobre cómo optimizar, en nuestra situación, instituciones y procedimientos de toma de decisiones. Para ello, primero distingo entre varias maneras de evaluar procedimientos de toma de decisión colectiva. Argumento que, al considerarlos como formas de organización política, un factor importante en la evaluación de tales procedimientos involucra asuntos fácticos sobre los cuales puede aspirarse a obtener o promover algunos bienes epistémicos. En este contexto, presento algunos de los argumentos más importantes a favor de la democracia epistémica. A continuación, reúno algunas de las objeciones sobre la aplicabilidad de dichos argumentos y ofrezco razones independientes para dudar de que ofrezcan apoyo a la tesis de la democracia epistémica. Finalmente, defiendo que la epistemología democrática puede desempeñar un papel significativo en la legitimación de formas de organización colectiva que podrían denominarse ‘democráticas’. (shrink)
This paper provides an introductory review of the theory of judgment aggregation. It introduces the paradoxes of majority voting that originally motivated the field, explains several key results on the impossibility of propositionwise judgment aggregation, presents a pedagogical proof of one of those results, discusses escape routes from the impossibility and relates judgment aggregation to some other salient aggregation problems, such as preference aggregation, abstract aggregation and probability aggregation. The present illustrative rather than exhaustive review is intended to give readers (...) new to the field of judgment aggregation a sense of this rapidly growing research area. (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 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)
In this paper, we argue that computer simulations can provide valuable insights into the performance of voting methods on different collective decision problems. This could improve institutional design, even when there is no general theoretical result to support the optimality of a voting method. To support our claim, we first describe a decision problem that has not received much theoretical attention in the literature. We outline different voting methods to address that collective decision problem. Under certain criteria of assessment akin (...) to extensions of the Condorcet JuryTheorem, we run simulations for the methods using MATLAB, in order to compare their performance under various conditions. We consider and respond to concerns about the use of simulations in the assessment of voting procedures for policymaking. (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.
In this study I analyse the performance of a democratic decision-making rule: the weighted majority rule. It assigns to each voter a number of votes that is proportional to her stakes in the decision. It has been shown that, for collective decisions with two options, the weighted majority rule in combination with self-interested voters maximises the common good when the latter is understood in terms of either the sum-total or prioritarian sum of the voters’ well-being. The main result of my (...) study is that this argument for the weighted majority rule — that it maximises the common good — can be improved along the following three main lines. (1) The argument can be adapted to other criteria of the common good, such as sufficientarian, maximin, leximin or non-welfarist criteria. I propose a generic argument for the collective optimality of the weighted majority rule that works for all of these criteria. (2) The assumption of self-interested voters can be relaxed. First, common-interest voters can be accommodated. Second, even if voters are less than fully competent in judging their self-interest or the common interest, the weighted majority rule is weakly collectively optimal, that is, it almost certainly maximises the common good given large numbers of voters. Third, even for smaller groups of voters, the weighted majority rule still has some attractive features. (3) The scope of the argument can be extended to decisions with more than two options. I state the conditions under which the weighted majority rule maximises the common good even in multi-option contexts. I also analyse the possibility and the detrimental effects of strategic voting. Furthermore, I argue that self-interested voters have reason to accept the weighted majority rule. (shrink)
In the context of EPR-Bohm type experiments and spin detections confined to spacelike hypersurfaces, a local, deterministic and realistic model within a Friedmann-Robertson-Walker spacetime with a constant spatial curvature (S^3 ) is presented that describes simultaneous measurements of the spins of two fermions emerging in a singlet state from the decay of a spinless boson. Exact agreement with the probabilistic predictions of quantum theory is achieved in the model without data rejection, remote contextuality, superdeterminism or backward causation. A singularity-free Clifford-algebraic (...) representation of S^3 with vanishing spatial curvature and non-vanishing torsion is then employed to transform the model in a more elegant form. Several event-by-event numerical simulations of the model are presented, which confirm our analytical results with the accuracy of 4 parts in 10^4 . Possible implications of our results for practical applications such as quantum security protocols and quantum computing are briefly discussed. (shrink)
In "Microaggressions: Strong Claims, Inadequate Evidence," Scott Lillenfeld argues that, despite a decade of scholarship, the Microaggression Research Program (MRP) continues to suffer serious analytic and evidentiary problems. After walking through these shortcomings, he provides 18 suggestions to help improve the reliability and utility of the MRP. In "Microaggressions and 'Evidence': Experimental or Experiential Reality?" Derald Wing Sue responds. This chapter provides background on the origin of the MRP, and referees the dispute between Lillenfeld and Sue about its contemporary status.
Democratic decision-making is often defended on grounds of the ‘wisdom of crowds’: decisions are more likely to be correct if they are based on many independent opinions, so a typical argument in social epistemology. But what does it mean to have independent opinions? Opinions can be probabilistically dependent even if individuals form their opinion in causal isolation from each other. We distinguish four probabilistic notions of opinion independence. Which of them holds depends on how individuals are causally affected by environmental (...) factors such as commonly perceived evidence. In a general theorem, we identify causal conditions guaranteeing each kind of opinion independence. These results have implications for whether and how ‘wisdom of crowds’ arguments are possible, and how truth-conducive institutions can be designed. (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)
In §8 of Remarks on the Foundations of Mathematics (RFM), Appendix 3 Wittgenstein imagines what conclusions would have to be drawn if the Gödel formula P or ¬P would be derivable in PM. In this case, he says, one has to conclude that the interpretation of P as “P is unprovable” must be given up. This “notorious paragraph” has heated up a debate on whether the point Wittgenstein has to make is one of “great philosophical interest” revealing “remarkable insight” in (...) Gödel’s proof, as Floyd and Putnam suggest (Floyd (2000), Floyd (2001)), or whether this remark reveals Wittgenstein’s misunderstanding of Gödel’s proof as Rodych and Steiner argued for recently (Rodych (1999, 2002, 2003), Steiner (2001)). In the following the arguments of both interpretations will be sketched and some deficiencies will be identified. Afterwards a detailed reconstruction of Wittgenstein’s argument will be offered. It will be seen that Wittgenstein’s argumentation is meant to be a rejection of Gödel’s proof but that it cannot satisfy this pretension. (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.
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) .
The standard representation theorem for expected utility theory tells us that if a subject’s preferences conform to certain axioms, then she can be represented as maximising her expected utility given a particular set of credences and utilities—and, moreover, that having those credences and utilities is the only way that she could be maximising her expected utility. However, the kinds of agents these theorems seem apt to tell us anything about are highly idealised, being always probabilistically coherent with infinitely precise (...) degrees of belief and full knowledge of all a priori truths. Ordinary subjects do not look very rational when compared to the kinds of agents usually talked about in decision theory. In this paper, I will develop an expected utility representation theorem aimed at the representation of those who are neither probabilistically coherent, logically omniscient, nor expected utility maximisers across the board—that is, agents who are frequently irrational. The agents in question may be deductively fallible, have incoherent credences, limited representational capacities, and fail to maximise expected utility for all but a limited class of gambles. (shrink)
I argue that Composition as Identity blocks the plural version of Cantor's Theorem, and that therefore the plural version of Cantor's Theorem can no longer be uncritically appealed to. As an example, I show how this result blocks a recent argument by Hawthorne and Uzquiano.
N. Cartwright’s results on invariance under intervention and causality (2003) are reconsidered. Procedural approach to causality elicited in this paper and contrasted with Cartwright’s apparently philosophical one unravels certain ramifications of her results. The procedural approach seems to license only a constrained notion of intervention and in consequence the “correctness to invariance” part of Cartwright’s first theorem fails for a class of cases. The converse “invariance to correctness” part of the theorem relies heavily on modeling assumptions which prove (...) to be difficult to validate in practice and are often buttressed by independently acquired evidence. (shrink)
We generalize and extend the class of Sahlqvist formulae in arbitrary polyadic modal languages, to the class of so called inductive formulae. To introduce them we use a representation of modal polyadic languages in a combinatorial style and thus, in particular, develop what we believe to be a better syntactic approach to elementary canonical formulae altogether. By generalizing the method of minimal valuations à la Sahlqvist–van Benthem and the topological approach of Sambin and Vaccaro we prove that all inductive formulae (...) are elementary canonical and thus extend Sahlqvist’s theorem over them. In particular, we give a simple example of an inductive formula which is not frame-equivalent to any Sahlqvist formula. Then, after a deeper analysis of the inductive formulae as set-theoretic operators in descriptive and Kripke frames, we establish a somewhat stronger model-theoretic characterization of these formulae in terms of a suitable equivalence to syntactically simpler formulae in the extension of the language with reversive modalities. Lastly, we study and characterize the elementary canonical formulae in reversive languages with nominals, where the relevant notion of persistence is with respect to discrete frames. (shrink)
The following essay reconsiders the ontological and logical issues around Frege’s Basic Law (V). If focuses less on Russell’s Paradox, as most treatments of Frege’s Grundgesetze der Arithmetik (GGA)1 do, but rather on the relation between Frege’s Basic Law (V) and Cantor’s Theorem (CT). So for the most part the inconsistency of Naïve Comprehension (in the context of standard Second Order Logic) will not concern us, but rather the ontological issues central to the conflict between (BLV) and (CT). These (...) ontological issues are interesting in their own right. And if and only if in case ontological considerations make a strong case for something like (BLV) we have to trouble us with inconsistency and paraconsistency. These ontological issues also lead to a renewed methodological reflection what to assume or recognize as an axiom. (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)
In this paper, I present an argument for a rational norm involving a kind of credal attitude called a quantificational credence – the kind of attitude we can report by saying that Lucy thinks that each record in Schroeder’s collection is 5% likely to be scratched. I prove a result called a Dutch Book Theorem, which constitutes conditional support for the norm. Though Dutch Book Theorems exist for norms on ordinary and conditional credences, there is controversy about the epistemic (...) significance of these results. So, my conclusion is that if Dutch Book Theorems do, in general, support norms on credal states, then we have support for the suggested norm on quantificational credences. Providing conditional support for this norm gives us a fuller picture of the normative landscape of credal states. (shrink)
Introduction to mathematical logic, part 2.Textbook for students in mathematical logic and foundations of mathematics. Platonism, Intuition, Formalism. Axiomatic set theory. Around the Continuum Problem. Axiom of Determinacy. Large Cardinal Axioms. Ackermann's Set Theory. First order arithmetic. Hilbert's 10th problem. Incompleteness theorems. Consequences. Connected results: double incompleteness theorem, unsolvability of reasoning, theorem on the size of proofs, diophantine incompleteness, Loeb's theorem, consistent universal statements are provable, Berry's paradox, incompleteness and Chaitin's theorem. Around Ramsey's theorem.
Bell inequalities are usually derived by assuming locality and realism, and therefore violations of the Bell-CHSH inequality are usually taken to imply violations of either locality or realism, or both. But, after reviewing an oversight by Bell, in the Corollary below we derive the Bell-CHSH inequality by assuming only that Bob can measure along vectors b and b' simultaneously while Alice measures along either a or a', and likewise Alice can measure along vectors a and a' simultaneously while Bob measures (...) along either b or b', without assuming locality. The violations of the Bell-CHSH inequality therefore only mean impossibility of measuring along b and b' simultaneously. (shrink)
This note clarifies an error in the proof of the main theorem of “The Ricean Objection: An Analogue of Rice’s Theorem for First-Order Theories”, Logic Journal of the IGPL, 16(6): 585–590(2008).
Many mathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemerédi's theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep by many mathematicians, and so makes for a good case (...) on which to focus in analyzing mathematical depth. After introducing the theorem, four accounts of mathematical depth will be considered. (shrink)
Two of the most important ideas in the philosophy of law are the “Coase Theorem” and the “Prisoner’s Dilemma.” In this paper, the authors explore the relation between these two influential models through a creative thought-experiment. Specifically, the paper presents a pure Coasean version of the Prisoner’s Dilemma, one in which property rights are well-defined and transactions costs are zero (i.e. the prisoners are allowed to openly communicate and bargain with each other), in order to test the truth value (...) of the Coase Theorem. In addition, the paper explores what effect (a) uncertainty, (b) exponential discounting, (c) and elasticity have on the behavior of the prisoners in the Coasean version of the dilemma. Lastly, the paper considers the role of the prosecutor (and third-parties generally) in the Prisoner’s Dilemma and closes with some parting thoughts about the complexity of the dilemma. The authors then conclude by identifying the conditions under which the Prisoner’s Dilemma refutes the Coase Theorem. (shrink)
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