This article investigates the relationship among the weak Pareto principle, the strong Pareto principle, and positive responsiveness in the context of voting. First, it is shown that under a mild domain condition, if an anonymous and neutral collective choice rule (CCR) is complete and transitive, then the weak Pareto principle and the strong Pareto principle are equivalent. Next, it is shown that under another mild domain condition, if a neutral CCR (...) is transitive, then the strong Pareto principle and positive responsiveness are equivalent. By applying these results, we obtain a new characterization of the method of majority decision. (shrink)

In this paper, the independence of irrelevant alternatives and the Pareto principle are simultaneously weakened in the Arrovian framework of social choice. Moreover, we also relax transitivity of social preferences. We show that impossibility remains under weaker versions of Arrow’s original conditions. Our results complement the recent work by Coban and Sanver (Soc Choice Welf 43(4):953–961, 2014).

A principle claiming equal entitlement to continued life has been strongly defended in the literature as a fundamental social value. We refer to this principle as ‘equal value of life'. In this paper we argue that there is a general incompatibility between the equal value of life principle and the weak Pareto principle and provide proof of this under mild structural assumptions. Moreover we demonstrate that a weaker, age-dependent version of the equal value of (...) life principle is also incompatible with the weak Pareto principle. However, both principles can be satisfied if transitivity of social preference is relaxed to quasi-transitivity. Footnotes1 The authors are grateful to Luc Bovens, Kristian Schultz Hansen, Søren Holm, Franz Huber, Wiebke Kuklys, Gabriella Pigozzi, and two anonymous referees for detailed and very helpful comments on earlier versions of this paper. Any errors and shortcomings are the responsibility of the authors alone. (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.

The paper argues that the notion of Pareto efficiency builds on two normative assumptions: the more general consequentialist norm of any efficiency criterion, and the strong no-harm principle of the prohibition of any redistribution during the economic process that hurts at least one person. These normative concerns lead to a constrained and static notion of efficiency in mainstream economics, ignoring dynamic efficiency gains from more equal allocations of resources. The paper argues that a weak no-harm principle (...) instead provides an endogenous efficiency criterion, which shifts attention away from equilibrium analysis in hypothetically perfect markets towards an evolutionary analysis of efficiency in real-world, non-equilibrium markets. Moreover, such an evolutionary notion of efficiency would be less normative than the Paretian concept. (shrink)

We analyze the decisiveness structures associated with acyclical collective choice rules. In particular, we examine the consequences of adding anonymity to weak Pareto, thereby complementing earlier results on acyclical social choice. Both finite and countably infinite populations are considered. As established in contributions by Donald Brown and by Jeffrey Banks, acyclical social choice is closely linked to prefilters in the presence of the weak Pareto principle. We introduce the notion of a conditional prefilter and use (...) it to generalize their results. In the finite-population case, adding anonymity implies that the collection of decisive sets is a special case of a conditional prefilter. We then identify the decisiveness structure that results from adding anonymity to the weak Pareto principle. Moving to infinite populations, we obtain the decisive set that consists of the entire population as a possibility, along with a new class of prefilters that we refer to as symmetric free Fréchet prefilters. The choice of the term Fréchet prefilter is motivated by the observation that they share a defining property with the well-known Fréchet filter—namely, that all sets in the requisite collection are such that their complement is finite. (shrink)

Philip Pettit (2001) has suggested that there are parallels between his republican account of freedom and Amartya Sen's (1970) account of freedom as decisive preference. In this paper, I discuss these parallels from a social-choice-theoretic perspective. I sketch a formalization of republican freedom and argue that republican freedom is formally very similar to freedom as defined in Sen's “minimal liberalism” condition. In consequence, the republican account of freedom is vulnerable to a version of Sen's liberal paradox, an inconsistency between universal (...) domain, freedom, and the weak Pareto principle. I argue that some standard escape routes from the liberal paradox – those via domain restriction – are not easily available to the republican. I suggest that republicans need to take seriously the challenge of the impossibility of a Paretian republican. Footnotes1 I am grateful to Luc Bovens, Geoffrey Brennan, Keith Dowding, Philip Pettit, and Wlodek Rabinowicz for very helpful comments and discussion. (shrink)

Peter Singer has famously argued that people living in affluent western countries are morally obligated to donate money to famine relief. The central premise in his argument is that, If it is in our power to prevent something bad from happening, without thereby sacrificing anything of comparable moral importance, we ought, morally, to do so. The present paper offers an argument to the effect that affluent people ought to support foreign aid projects based on a much weaker ethical premise. The (...) new premise states that, If it is in our power to prevent something bad from happening, without thereby sacrificing anything of moral importance, we ought, morally, to do so. This premise, supplemented with a notion of final value drawing on Amartya Sen''s concept of freedom as capabilities and functionings, is conceived as a special version of a weak, egalitarian Pareto principle. (shrink)

A double ambiguity has been charged against Rawls’s difference principle (DP). Is it Maximin, Leximin, or something else? Usually, following A. Sen, scholars identify DP with the so-called Leximin. One argues here that one has to distinguish 1° the Leximin, 2° the Maximin (as rule of justice formally analogous to the maximin rule of decision), represented by the figure in L of the perfectly substitutable goods, and 3° the genuine DP. When the augmentation of inequality benefits the worse off, (...) only Pareto-strong improvements are permitted. Leximin would also permit Pareto-weak improvements too (after the first maximum D), where only the richest improves: from (2, 3) to (2, 5), say. This is forbidden by DP. With two classes, unlike Maximin, DP has no curve of indifference and is always decisive, as Leximin is. For undecisive Rules of Justice, which admit indifferent curves, I propose to add a lexically secondary rule, to break ties. That move is able to clarify the links and the differences between on the one hand Maximin alone, with its typical indifference curves in L, and on the other hand, the DP properly understood and the Leximin, which both have no indifferent curves. With two classes of persons (best off/worse off), DP appears more egalitarian than Leximin, because it's secondary rule is MinIn (Minimization of Inequality). But the intuition behind the distinction is that it cannot possible “fair” that only the best off improves in a productive social cooperation. (shrink)

It is possible that the world contains infinitely many agents that have positive and negative levels of well-being. Theories have been developed to ethically rank such worlds based on the well-being levels of the agents in those worlds or other qualitative properties of the worlds in question, such as the distribution of agents across spacetime. In this thesis I argue that such ethical rankings ought to be consistent with the Pareto principle, which says that if two worlds contain (...) the same agents and some agents are better off in the first world than they are in the second and no agents are worse off than they are in the second, then the first world is better than the second. I show that if we accept four axioms – the Pareto principle, transitivity, an axiom stating that populations of worlds can be permuted, and the claim that if the ‘at least as good as’ relation holds between two worlds then it holds between qualitative duplicates of this world pair – then we must conclude that there is ubiquitous incomparability between infinite worlds. I show that this is true even if the populations of infinite worlds are disjoint or overlapping, and that we cannot use any qualitative properties of world pairs to rank these worlds. Finally, I argue that this incomparability result generates puzzles for both consequentialist and non-consequentialist theories of objective and subjective permissibility. (shrink)

We study a principle of ‘Non-Interference’ in social welfare judgements. Non-Interference captures aspects of liberal approaches (particularly a Millian approach) to social decision making. In its full generality, Non-Interference produces an impossibility result: together with Weak Pareto Optimality, it implies that a social welfare ordering must be dictatorial. However, interesting restricted versions of Non-Interference are compatible with standard social welfare orderings.

Frege puzzles exploit cognitive differences between co-referential terms. Traditionally, they were handled by some version of Frege’s distinction between sense and reference, which avoided disruptive consequences for epistemology. However, the Fregean programme did not live up to its original promise, and was undermined by the development of theories of direct reference; for semantic purposes, its prospects now look dim. In particular, well-known analogues of Frege puzzles concern pairs of uncontentious synonyms; attempts to deal with them by distinguishing idiolects or postulating (...) ‘narrow contents’ or elaborate forms of context-sensitivity are inadequate or semantically implausible. Although ascriptions of knowledge, belief, and other attitudes are ubiquitous in epistemology, epistemologists have not properly come to terms with the surprising consequences of anti-Fregean semantic accounts of attitude ascriptions. In ‘A Puzzle about Belief ’, Saul Kripke shows that natural-seeming disquotational principles for ascribing belief lead to apparently unacceptable consequences, including outright contradictions, in problem cases. Such disquotation principles, I argue, are best regarded not as conceptual connections but just as heuristics in the psychological sense, quick and easy ways of assessing belief ascriptions, usually accurate under normal conditions but far from 100% reliable. I discuss similar heuristics for ascribing knowledge and other attitudes. That the principles have a merely heuristic status need not be pre-theoretically manifest to their users. This view vindicates Kripke’s conclusion that it would be wrong-headed to draw semantic conclusions from Frege puzzles. I discuss the epistemological consequences of an anti-Fregean approach to Frege puzzles, including for Kripke’s cases of the contingent a priori and the necessary a posteriori, but also for evidence, for epistemic modalities, and for epistemic and subjective conceptions of probability. Anna Mahtani’s recent identification of Frege puzzles for the ex ante Pareto Principle as used in welfare economics provides an interesting example. I suggest a model-building methodology as the most promising way of handling at least some of the difficulties. A final issue is the choice between different anti-Fregean approaches to semantics, from very coarse-grained intensional approaches on which sentences express functions from possible worlds to truth-values to more fine-grained hyperintensional approaches on which sentences express functions from possible or impossible worlds to truth-values or else Russellian structured propositions. Some of the hyperintensional theories violate semantic compositionality. More generally, since our attitude ascriptions rely on heuristics, they should be expected to exhibit some level of error; although hyperintensional approaches may be able slightly to reduce the level of postulated error, they do so at the cost of vastly increased theoretical complexity, and so have weak explanatory power. Methodologically, a simple intensional approach does better. (shrink)

The Pareto principle states that if the members of society express the same preference judgment between two options, this judgment is compelling for society. A building block of normative economics and social choice theory, and often borrowed by contemporary political philosophy, the principle has rarely been subjected to philosophical criticism. The paper objects to it on the ground that it indifferently applies to those cases in which the individuals agree on both their expressed preferences and their reasons (...) for entertaining them, and those cases in which they agree on their expressed preferences, while differing on their reasons. The latter are cases of "spurious unanimity", and it is normatively inappropriate, or so the paper argues, to defend unanimity preservation at the social level for them, so the Pareto principle is formulated much too broadly. The objection seems especially powerful when the principle is applied in an ex ante context of uncertainty, in which individuals can disagree on both their probabilities and utilities, and nonetheless agree on their preferences over prospects. (shrink)

We address the question of how finitely additive moral value theories (such as utilitarianism) should rank worlds when there are an infinite number of locations of value (people, times, etc.). In the finite case, finitely additive theories satisfy both Weak Pareto and a strong anonymity condition. In the infinite case, however, these two conditions are incompatible, and thus a question arises as to which of these two conditions should be rejected. In a recent contribution, Hamkins and Montero (2000) (...) have argued in favor of an anonymity-like isomorphism principle and against Weak Pareto. After casting doubt on their criticism of Weak Pareto, we show how it, in combination with certain other plausible principles, generates a plausible and fairly strong principle for the infinite case. We further show that where locations are the same in all worlds, but have no natural order, this principle turns out to be equivalent to a strengthening of a principle defended by Vallentyne and Kagan (1997), and also to a weakened version of the catching-up criterion developed by Atsumi (1965) and by von Weizsäcker (1965). Footnotes1 For valuable comments, we would like to thank Marc Fleurbaey, Bart Capéau, Joel Hamkins, Barbara Montero, Tim Mulgan, and two anonymous referees for this journal. (shrink)

In this paper I investigate the possibility of a dictatorship in the context of Harsanyi's Social Aggregation Theorem. Preliminarily, some propositions about Harsanyi's Theorem are presented using an alternative principle that I name Quasi-strong Pareto, which is the latter part of Strong Pareto. Then I define dictatorship as a requirement that social preference agrees with a dictator's preference or those of members of dictatorial group even if their preferences strictly contradict those of all other people in the (...) society. Conclusively, although in each version of Harsanyi's Theorem with Pareto Indifference, Weak Preference Pareto or Weak Pareto the social utility function may have a form of dictatorship, however if individuals' vNM utility functions are all 'individualistic' and Quasi-strong Pareto is satisfied, then the dictatorship is excluded. (shrink)

This essay develops an ideal of public legal reason--a normative theory of legal reasons that is appropriate for a society characterized by religious and moral pluralism. One of the implications of this theory is that normative theorizing about public and private law should eschew reliance on the deep premises of deontology or consequentialism and should instead rely on what the author calls public values--values that can be affirmed without relying on the deep and controversial premises of particular comprehensive moral doctrines. (...) The ideal of public legal reason is then applied to a particular question--whether welfarism (a particular form of normative law and economics) provides the sort of reasons that appropriate for legal practice. The answer to that question is no--to the extent that welfarism contends that the normative assessment of legal policies should rely exclusively on information about individual preferences, welfarism relies on deep and controversial premises of consequentialist moral theory that fail the test of public reason. The essay also investigates the thesis--advanced by Louis Kaplow and Steven Shavell--that any fairness principle (a nonwelfarist method of policy assessment) can violate weak Pareto (making everyone worse off). Whatever the implications of Kaplow and Shavell's argument, it does not show that welfarism can provide public legal reasons. The essay concludes that law's justifications should rely on normative principles that are accessible to reasonable citizens, whether they are theists or atheists, deontologists or consequentialists, moral philosophers or economists. Law's deliberations should be shallow and not deep. Law's reason should be public. (shrink)

This paper shows that weak Pareto and strong Pareto are equivalent under continuity and strong quasi-concavity. We use a framework that incorporates welfare as well as non-welfare information.

The purpose of the paper is a discussion of the meaning and relevance of the Pareto principle in economics. To begin with, the principle is briefly retraced in Pareto’s own writings. Its contemporary meaning was, however, developed in the context of the “New Welfare Economics”. While Pareto technically employed the principle in order to describe an equilibrium situation, Kaldor and Hicks developed it somewhat differently as a yardstick for economic policy formulation. Sometimes, the (...) class='Hi'>principle is also discussed as a decision rule, and in this context some critics - though not the present author - believe it to have a conservative bias. Finally, recent discussions center around the incompatibility of the Pareto principle and “liberal” values. This conflict might be of limited relevance, only, due to a misconstrued formalism. (shrink)

The coherence of general good turns out to conflict with the widely accepted Pareto principle. This chapter explains the conflict and resolves it in favour of coherence. It also presents an example of a head‐on collision between coherence and the Pareto principle. The example relies on an auxiliary assumption, but one that is very plausible. The principle of personal good is immune to the difficulty raised by the probability agreement theorem. The theorem presents welfare economics (...) with a dilemma: it must abandon either the coherence of social preferences or else the traditional Pareto principle. Either way, it will have to repudiate Harsanyi's theorem. The ex‐ante school of welfare economists abandons coherence. This is the best established school. The ex‐ante school cleaves to the traditional Pareto principle. Consequently, it has to accept incoherent social preferences. (shrink)

The concept of ‘pareto superiority’ plays a central role in ethics, economics, and law. Pareto superiority is sometimes taken as a relation between outcomes, and sometimes as a relation between actions—even where the outcomes of the actions are uncertain. Whether one action is classed as (ex ante) pareto superior to another depends on the prospects under the actions for each person concerned. I argue that a person’s prospects (in this context) can depend on how that person is (...) designated. Without any constraints on acceptable designators, then, the concept of pareto superiority is ill defined and gives inconsistent results. I consider various ways of completing the definition and draw out some surprising implications. (shrink)

The Pareto principle is in fact the fundamental concept of welfare economics. However, it has serious analytical and heuristic limits, is selective and conservative in nature and use, and is heavily normative notwithstanding the pretensions by advocates of its positive character.

This article provides a systematic analysis of social choice theory without the Pareto principle, by revisiting the method of Murakami Yasusuke. This article consists of two parts. The first part investigates the relationship between rationality of social preference and the axioms that make a collective choice rule either Paretian or anti-Paretian. In the second part, the results in the first part are applied to obtain impossibility results under various rationality requirements of social preference, such as S-consistency, quasi-transitivity, semi-transitivity, (...) the interval-order property, and acyclicity. (shrink)

This paper examines social choice theory with the strong Pareto principle. The notion of conditional decisiveness is introduced to clarify the underlying power structure behind strongly Paretian aggregation rules satisfying binary independence. We discuss the various degrees of social rationality: transitivity, semi-transitivity, the interval-order property, quasi-transitivity, and acyclicity.

We provide a general argument against value incomparability, based on a new style of impossibility result. In particular, we show that, against plausible background assumptions, value incomparability creates an incompatibility between two very plausible principles for ranking lotteries: a weak "negative dominance" principle (to the effect that Lottery 1 can be better than Lottery 2 only if some possible outcome of Lottery 1 is better than some possible outcome of Lottery 2) and a weak form of ex (...) ante Pareto (to the effect that, if Lottery 1 gives an unambiguously better prospect to some individuals than Lottery 2, and equally good prospects to everyone else, then Lottery 1 is better than Lottery 2). After spelling out our results, and the arguments based on them, we consider which principle the proponent of incomparability ought to reject. (shrink)

The ex ante Pareto principle has an intuitive pull, and it has been a principle of central importance since Harsanyi’s defence of utilitarianism. The principle has been used to criticize and refine a range of positions in welfare economics, including egalitarianism and prioritarianism. But this principle faces a serious problem. I have argued elsewhere :303-323 2017) that the concept of ex ante Pareto superiority is not well defined, because its application in a choice situation (...) concerning a fixed population can depend on how the members of that population are designated. I show in this paper that in almost all cases of policy choice, there will be numerous sets of rival designators for the same fixed population. I explore two ways that we might complete the definition of ex ante Pareto superiority. I call these the ‘supervaluationist’ reading and the ‘subvaluationist’ reading. I reject the subvaluationist reading as uncharitable, and argue that the supervaluationist reading is the most promising interpretation of the ex ante Pareto principle. I end by exploring some of the implications of this principle for prioritarianism and egalitarianism. (shrink)

Warren Samuels has suggested that the Pareto Principle, when being used in policy analysis, is (1) limited, (2) selective, and (3) displays a conservative bias. In contrast to this view, in this note it is argued that the Pareto Principle is much less limited than was initially perceived (e.g. by Pareto himself) or is generally believed to be the case, that it tends to emphasize inclusiveness instead of selectivity, and that it is more likely to (...) have an innovative instead of a conservative bias. (shrink)

First, Mahtani argues that both in the game The Mug and in the Sleeping Beauty we should not defer to a trusted person under a particular designation if they do not self-identify under this designation. This invites a more complex Reflection Principle. I respond that there are more parsimonious ways to avoid the challenges posed to the Reflection Principle. Second, Mahtani argues that preferences create a hyperintensional context, which poses a challenge to the Ex-Ante Pareto Principle (...) that can be averted by supervaluation. I respond that such an appeal to supervaluation would block randomization as a fair allocation device. (shrink)

We give a level-by-level analysis of the Weak Vopěnka Principle for definable classes of relational structures ( $\mathrm {WVP}$ ), in accordance with the complexity of their definition, and we determine the large-cardinal strength of each level. Thus, in particular, we show that $\mathrm {WVP}$ for $\Sigma _2$ -definable classes is equivalent to the existence of a strong cardinal. The main theorem (Theorem 5.11) shows, more generally, that $\mathrm {WVP}$ for $\Sigma _n$ -definable classes is equivalent to the (...) existence of a $\Sigma _n$ -strong cardinal (Definition 5.1). Hence, $\mathrm {WVP}$ is equivalent to the existence of a $\Sigma _n$ -strong cardinal for all $n<\omega $. (shrink)

The Pareto principle is a normative principle about preferences that advocates concordance with unanimous preference. However, people have perspectives not just preferences. Evaluating preferences...

I discuss the trilemma that consists of the following three principles being inconsistent: 1. The Common Principle: if one distribution, A, necessarily brings a higher total sum of personal value that is distributed in a more egalitarian way than another distribution, B, A is more valuable than B. 2. (Weak) ex-ante Pareto: if one uncertain distribution, A, is more valuable than another uncertain distribution, B, for each patient, A is more valuable than B. 3. Pluralism about attitudes (...) to risk (Pluralism): the personal value of a prospect is a weighted sum of the values of the prospect’s outcomes, but the weight each outcome gets might be different from the probability the prospect assigns to the outcome. (shrink)

It is well known that S12 cannot prove the injective weak pigeonhole principle for polynomial time functions unless RSA is insecure. In this note we investigate the provability of the surjective weak pigeonhole principle in S12 for provably weaker function classes.

We study the extension 123) of the theory S21 by instances of the dual weak pigeonhole principle for p-time functions, dWPHPx2x. We propose a natural framework for formalization of randomized algorithms in bounded arithmetic, and use it to provide a strengthening of Wilkie's witnessing theorem for S21+dWPHP. We construct a propositional proof system WF , which captures the Π1b-consequences of S21+dWPHP. We also show that WF p-simulates the Unstructured Extended Nullstellensatz proof system of Buss et al. 256). We (...) prove that dWPHP is equivalent to a statement asserting the existence of a family of Boolean functions with exponential circuit complexity. Building on this result, we formalize the Nisan–Wigderson construction in a conservative extension of S21+dWPHP. (shrink)

A modern assessment of the classical Boltzmann-Schuetz argument for large-scale entropy fluctuations as the origin of our observable cosmological domain is given.The emphasis is put on the central implication of this picture which flatly contradicts the weak anthropic principle as an epistemological statement about the universe. Therefore, to associate this picture with the anthropic principle as it is usually done is unwarranted. In particular, Feynman's criticism of theanthropic principle based on the entropy-fluctuation picture is a product (...) of this semantic confusion. (shrink)

In Arrovian social choice theory assuming the independence of irrelevant alternatives, Murakami (1968) proved two theorems about complete and transitive collective choice rules that satisfy strict non-imposition (citizens’ sovereignty), one being a dichotomy theorem about Paretian or anti-Paretian rules and the other a dictator-or-inverse-dictator impossibility theorem without the Pareto principle. It has been claimed in the later literature that a theorem of Malawski and Zhou (1994) is a generalization of Murakami’s dichotomy theorem and that Wilson’s (1972) impossibility theorem (...) is stronger than Murakami’s impossibility theorem, both by virtue of replacing Murakami’s assumption of strict non-imposition with the assumptions of non-imposition and non-nullness. In this note, we first point out that these claims are incorrect: non-imposition and non-nullness are together equivalent to strict non-imposition for all transitive collective choice rules. We then generalize Murakami’s dichotomy and impossibility theorems to the setting of incomplete social preference. We prove that if one drops completeness from Murakami’s assumptions, his remaining assumptions imply (i) that a collective choice rule is either Paretian, anti-Paretian, or dis-Paretian (unanimous individual preference implies noncomparability) and (ii) that adding proposed constraints on noncomparability, such as the regularity axiom of Eliaz and Ok (2006), restores Murakami’s dictator-or-inverse-dictator result. (shrink)

The design argument for God’s existence was critically assessed when in the growth of modern science the cognitive value of teleological categories was called into question. In recent discussions dealing with anthropic principles there has appeared a new version of the design argument, in which cosmic design is described without the use of teleological terms. The weak anthropic principle (WAP), a most critical version of all these principles, describes the fine-tuning of physical parameters necessary to the genesis of (...) carbon-based life. Consequently, a new version of the philosophical design argument can be developed on the basis of the weak anthropic principle.(edited). (shrink)

The rigid relation principle, introduced in this article, asserts that every set admits a rigid binary relation. This follows from the axiom of choice, because well-orders are rigid, but we prove that it is neither equivalent to the axiom of choice nor provable in Zermelo-Fraenkel set theory without the axiom of choice. Thus, it is a new weak choice principle. Nevertheless, the restriction of the principle to sets of reals is provable without the axiom of choice.

This article is a continuation of our search for tautologies that are hard even for strong propositional proof systems like EF, cf. [Kra-wphp,Kra-tau]. The particular tautologies we study, the τ-formulas, are obtained from any.

Goncharov and Peretyat'kin independently gave necessary and sufficient conditions for when a set of types of a complete theory is the type spectrum of some homogeneous model of. Their result can be stated as a principle of second order arithmetic, which is called the Homogeneous Model Theorem (HMT), and analyzed from the points of view of computability theory and reverse mathematics. Previous computability theoretic results by Lange suggested a close connection between HMT and the Atomic Model Theorem (AMT), which (...) states that every complete atomic theory has an atomic model. The authors show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense and do the same for an analogous result of Peretyat'kin giving necessary and sufficient conditions for when a set of types is the type spectrum of some model. (shrink)

This paper analyzes the notion of a minimal belief change that incorporates new information. I apply the fundamental decision-theoretic principle of Pareto-optimality to derive a notion of minimal belief change, for two different representations of belief: First, for beliefs represented by a theory – a deductively closed set of sentences or propositions – and second for beliefs represented by an axiomatic base for a theory. Three postulates exactly characterize Pareto-minimal revisions of theories, yielding a weaker set of (...) constraints than the standard AGM postulates. The Levi identity characterizes Pareto-minimal revisions of belief bases: a change of belief base is Pareto-minimal if and only if the change satisfies the Levi identity (for “maxichoice” contraction operators). Thus for belief bases, Pareto-minimality imposes constraints that the AGM postulates do not. The Ramsey test is a well-known way of establishing connections between belief revision postulates and axioms for conditionals (“if p, then q”). Pareto-minimal theory change corresponds exactly to three characteristic axioms of counterfactual systems: a theory revision operator that satisfies the Ramsey test validates these axioms if and only if the revision operator is Pareto-minimal. (shrink)

We bring together some facts about the weak pigeonhole principle from bounded arithmetic, complexity theory, cryptography and abstract model theory. We characterize the models of arithmetic in which WPHP fails as those which are determined by an initial segment and prove a conditional separation result in bounded arithmetic, that PV + lies strictly between PV and S21 in strength, assuming that the cryptosystem RSA is secure.

The set-theoretic axiom WISC states that for every set there is a set of surjections to it cofinal in all such surjections. By constructing an unbounded topos over the category of sets and using an extension of the internal logic of a topos due to Shulman, we show that WISC is independent of the rest of the axioms of the set theory given by a well-pointed topos. This also gives an example of a topos that is not a predicative topos (...) as defined by van den Berg. (shrink)