There is a theorem that shows that it is impossible for an algorithm to jointly satisfy the statistical fairness criteria of Calibration and Equalised Odds non-trivially. But what about the recently advocated alternative to Calibration, Base Rate Tracking? Here, we show that Base Rate Tracking is strictly weaker than Calibration, and then take up the question of whether it is possible to jointly satisfy Base Rate Tracking and Equalised Odds in non-trivial scenarios. We show that it is not, thereby (...) establishing an even more general impossibility theorem. (shrink)

Agents are often assumed to have degrees of belief (“credences”) and also binary beliefs (“beliefs simpliciter”). How are these related to each other? A much-discussed answer asserts that it is rational to believe a proposition if and only if one has a high enough degree of belief in it. But this answer runs into the “lottery paradox”: the set of believed propositions may violate the key rationality conditions of consistency and deductive closure. In earlier work, we showed that this problem (...) generalizes: there exists no local function from degrees of belief to binary beliefs that satisfies some minimal conditions of rationality and non-triviality. “Locality” means that the binary belief in each proposition depends only on the degree of belief in that proposition, not on the degrees of belief in others. One might think that the impossibility can be avoided by dropping the assumption that binary beliefs are a function of degrees of belief. We prove that, even if we drop the “functionality” restriction, there still exists no local relation between degrees of belief and binary beliefs that satisfies some minimal conditions. Thus functionality is not the source of the impossibility; its source is the condition of locality. If there is any non-trivial relation between degrees of belief and binary beliefs at all, it must be a “holistic” one. We explore several concrete forms this “holistic” relation could take. (shrink)

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)

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)

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 (...) class='Hi'>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)

This essay discusses the difficulty to reconcile two paradigms about beliefs: the binary or categorical paradigm of yes/no beliefs and the probabilistic paradigm of degrees of belief. The possibility for someone to hold both types of belief simultaneously is challenged by the lottery paradox, and more recently by a general impossibility theorem by Dietrich and List (2018, 2021). The nature, relevance, and implications of the tension are explained and assessed.

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)

I have read many recent discussions of the limits of computation and the universe as computer, hoping to find some comments on the amazing work of polymath physicist and decision theorist David Wolpert but have not found a single citation and so I present this very brief summary. Wolpert proved some stunning impossibility or incompleteness theorems (1992 to 2008-see arxiv.org) on the limits to inference (computation) that are so general they are independent of the device doing the computation, (...) and even independent of the laws of physics, so they apply across computers, physics, and human behavior. They make use of Cantor's diagonalization, the liar paradox and worldlines to provide what may be the ultimate theorem in Turing Machine Theory, and seemingly provide insights into impossibility,incompleteness, the limits of computation,and the universe as computer, in all possible universes and all beings or mechanisms, generating, among other things,a non-quantum mechanical uncertainty principle and a proof of monotheism. (shrink)

I have read many recent discussions of the limits of computation and the universe as computer, hoping to find some comments on the amazing work of polymath physicist and decision theorist David Wolpert but have not found a single citation and so I present this very brief summary. Wolpert proved some stunning impossibility or incompleteness theorems (1992 to 2008-see arxiv.org) on the limits to inference (computation) that are so general they are independent of the device doing the computation, (...) and even independent of the laws of physics, so they apply across computers, physics, and human behavior. They make use of Cantor's diagonalization, the liar paradox and worldlines to provide what may be the ultimate theorem in Turing Machine Theory, and seemingly provide insights into impossibility, incompleteness, the limits of computation,and the universe as computer, in all possible universes and all beings or mechanisms, generating, among other things,a non- quantum mechanical uncertainty principle and a proof of monotheism. There are obvious connections to the classic work of Chaitin, Solomonoff, Komolgarov and Wittgenstein and to the notion that no program (and thus no device) can generate a sequence (or device) with greater complexity than it possesses. One might say this body of work implies atheism since there cannot be any entity more complex than the physical universe and from the Wittgensteinian viewpoint, ‘more complex’ is meaningless (has no conditions of satisfaction, i.e., truth-maker or test). Even a ‘God’ (i.e., a ‘device’ with limitless time/space and energy) cannot determine whether a given ‘number’ is ‘random’ nor can find a certain way to show that a given ‘formula’, ‘theorem’ or ‘sentence’ or ‘device’ (all these being complex language games) is part of a particular ‘system’. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my article The Logical Structure of Philosophy, Psychology, Mind and Language as Revealed in Wittgenstein and Searle 59p(2016). For all my articles on Wittgenstein and Searle see my e-book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Wittgenstein and Searle 367p (2016). Those interested in all my writings in their most recent versions may consult my e-book Philosophy, Human Nature and the Collapse of Civilization - Articles and Reviews 2006-2016’ 662p (2016). -/- All of my papers and books have now been published in revised versions both in ebooks and in printed books. -/- Talking Monkeys: Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet - Articles and Reviews 2006-2017 (2017) https://www.amazon.com/dp/B071HVC7YP. -/- The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle--Articles and Reviews 2006-2016 (2017) https://www.amazon.com/dp/B071P1RP1B. -/- Suicidal Utopian Delusions in the 21st century: Philosophy, Human Nature and the Collapse of Civilization - Articles and Reviews 2006-2017 (2017) https://www.amazon.com/dp/B0711R5LGX . (shrink)

I have read many recent discussions of the limits of computation and the universe as computer, hoping to find some comments on the amazing work of polymath physicist and decision theorist David Wolpert but have not found a single citation and so I present this very brief summary. Wolpert proved some stunning impossibility or incompleteness theorems (1992 to 2008-see arxiv dot org) on the limits to inference (computation) that are so general they are independent of the device doing (...) the computation, and even independent of the laws of physics, so they apply across computers, physics, and human behavior. They make use of Cantor's diagonalization, the liar paradox and worldlines to provide what may be the ultimate theorem in Turing Machine Theory, and seemingly provide insights into impossibility, incompleteness, the limits of computation, and the universe as computer, in all possible universes and all beings or mechanisms, generating, among other things, a non- quantum mechanical uncertainty principle and a proof of monotheism. There are obvious connections to the classic work of Chaitin, Solomonoff, Komolgarov and Wittgenstein and to the notion that no program (and thus no device) can generate a sequence (or device) with greater complexity than it possesses. One might say this body of work implies atheism since there cannot be any entity more complex than the physical universe and from the Wittgensteinian viewpoint, ‘more complex’ is meaningless (has no conditions of satisfaction, i.e., truth-maker or test). Even a ‘God’ (i.e., a ‘device’with limitless time/space and energy) cannot determine whether a given ‘number’ is ‘random’, nor find a certain way to show that a given ‘formula’, ‘theorem’ or ‘sentence’ or ‘device’ (all these being complex language games) is part of a particular ‘system’. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 2nd ed (2019) and Suicidal Utopian Delusions in the 21st Century 4th ed (2019) . (shrink)

This thesis consists of several independent papers in population ethics. I begin in Chapter 1 by critiquing some well-known 'impossibility theorems', which purport to show there can be no intuitively satisfactory population axiology. I identify axiological vagueness as a promising way to escape or at least mitigate the effects of these theorems. In particular, in Chapter 2, I argue that certain of the impossibility theorems have little more dialectical force than sorites arguments do. From these negative arguments I (...) move to positive ones. In Chapter 3, I justify the use of a 'veil of ignorance', starting from three more basic normative principles. This leads to positive arguments for various kinds of utilitarianism - the best such arguments I know. But in general the implications of the veil depend on how one answers what I call 'the risky existential question': what is the value to an individual of a chance of non-existence? I chart out the main options, and raise some puzzles for non-comparativism, the view that life is incomparable to non-existence. Finally, in Chapter 4, I consider the consequences for population ethics of the idea that what is normatively relevant is not personal identity, but a degreed relation of psychological connectedness. In particular, I pursue a strategy based in population ethics for understanding the controversial 'time-relative interests' account of the badness of death. (shrink)

Judgment aggregation theory, or rather, as we conceive of it here, logical aggregation theory generalizes social choice theory by having the aggregation rule bear on judgments of all kinds instead of merely preference judgments. It derives from Kornhauser and Sager’s doctrinal paradox and List and Pettit’s discursive dilemma, two problems that we distinguish emphatically here. The current theory has developed from the discursive dilemma, rather than the doctrinal paradox, and the final objective of the paper is to give the latter (...) its own theoretical development along the line of recent work by Dietrich and Mongin. However, the paper also aims at reviewing logical aggregation theory as such, and it covers impossibility theorems by Dietrich, Dietrich and List, Dokow and Holzman, List and Pettit, Mongin, Nehring and Puppe, Pauly and van Hees, providing a uniform logical framework in which they can be compared with each other. The review goes through three historical stages: the initial paradox and dilemma, the scattered early results on the independence axiom, and the so-called canonical theorem, a collective achievement that provided the theory with its specific method of analysis. The paper goes some way towards philosophical logic, first by briefly connecting the aggregative framework of judgment with the modern philosophy of judgment, and second by thoroughly discussing and axiomatizing the ‘general logic’ built in this framework. (shrink)

While a large social-choice-theoretic literature discusses the aggregation of individual judgments into collective ones, there is much less formal work on the transformation of judgments in group communication. I develop a model of judgment transformation and prove a baseline impossibility theorem: Any judgment transformation function satisfying some initially plausible conditions is the identity function, under which no opinion change occurs. I identify escape routes from this impossibility and argue that the kind of group communication envisaged by deliberative (...) democats must be "holistic": It must focus on webs of connected propositions, not on one proposition at a time, which echoes the Duhem-Quine "holism thesis" on scientific theory testing. My approach provides a map of the logical space in which different possible group communication processes are located. (shrink)

Jim Joyce argues for two amendments to probabilism. The first is the doctrine that credences are rational, or not, in virtue of their accuracy or “closeness to the truth” (1998). The second is a shift from a numerically precise model of belief to an imprecise model represented by a set of probability functions (2010). We argue that both amendments cannot be satisfied simultaneously. To do so, we employ a (slightly-generalized) impossibility theorem of Seidenfeld, Schervish, and Kadane (2012), who (...) show that there is no strictly proper scoring rule for imprecise probabilities. -/- The question then is what should give way. Joyce, who is well aware of this no-go result, thinks that a quantifiability constraint on epistemic accuracy should be relaxed to accommodate imprecision. We argue instead that another Joycean assumption — called strict immodesty— should be rejected, and we prove a representation theorem that characterizes all “mildly” immodest measures of inaccuracy. (shrink)

This chapter briefly reviews the present state of judgment aggregation theory and tentatively suggests a future direction for that theory. In the review, we start by emphasizing the difference between the doctrinal paradox and the discursive dilemma, two idealized examples which classically serve to motivate the theory, and then proceed to reconstruct it as a brand of logical theory, unlike in some other interpretations, using a single impossibility theorem as a key to its technical development. In the prospective (...) part, having mentioned existing applications to social choice theory and computer science, which we do not discuss here, we consider a potential application to law and economics. This would be based on a deeper exploration of the doctrinal paradox and its relevance to the functioning of collegiate courts. On this topic, legal theorists have provided empirical observations and theoretical hints that judgment aggregation theorists would be in a position to clarify and further elaborate. As a general message, the chapter means to suggest that the future of judgment aggregation theory lies with its applications rather than its internal theoretical development. (shrink)

The new field of judgment aggregation aims to merge many individual sets of judgments on logically interconnected propositions into a single collective set of judgments on these propositions. Judgment aggregation has commonly been studied using classical propositional logic, with a limited expressive power and a problematic representation of conditional statements ("if P then Q") as material conditionals. In this methodological paper, I present a simple unified model of judgment aggregation in general logics. I show how many realistic decision problems (...) can be represented in it. This includes decision problems expressed in languages of classical propositional logic, predicate logic (e.g. preference aggregation problems), modal or conditional logics, and some multi-valued or fuzzy logics. I provide a list of simple tools for working with general logics, and I prove impossibility results that generalise earlier theorems. (shrink)

Approval Voting offers advantages over other voting systems for single-winner elections. This manuscript analyzes three unique virtues of Approval Voting. First, the procedure does not violate the independence of irrelevant alternatives criterion for rational choice. Second, it prevents manipulation of outcomes through agenda setting. Third, it avoids intransitive majority preference cycles like Condorcet paradoxes and so escapes Arrow’s Impossibility Theorem constraints. As a result of these virtues, which are generally not shared by its best known competitors, Approval voting (...) emerges as a strong option for realizing majoritarian democratic will. (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 (...) class='Hi'>theorem. (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)

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)

This paper develops a largely neglected parallel between prudence and population ethics. Prudence is generally understood to be concerned with the balancing of well-being over time. How, precisely, well-being ought to be balanced over time, however, is a fervently debated question. I argue that developing a standard guiding such evaluations is exceedingly challenging. This is due to an often overlooked fact about prudence, namely that it shares a structural similarity with population ethics: In both contexts, we assess the comparative value (...) of populations of personstages/people, which may vary in number and level of well-being. Based on this analogy, I show that the development of an adequate theory of prudence runs into very similar impossibility results as obtain in population ethics. In particular, I prove that Arrhenius’ fifth impossibility theorem can be applied to prudence. I develop and compare four possible answers to this challenge. First, I discuss the possibility of accepting the very repugnant conclusion intrapersonally. Second, I present and further develop Donald Bruckner’s Minimax Regret approach, which gives up Transitivity. Third, I apply Jacob Nebel’s Lexical Threshold View to prudence and critically evaluate it. Lastly, I introduce what I call the Negative Lexicality View, which is based on Lexical Threshold View but overcomes some of its problems. (shrink)

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)

The literature on algorithmic fairness has examined exogenous sources of biases such as shortcomings in the data and structural injustices in society. It has also examined internal sources of bias as evidenced by a number of impossibility theorems showing that no algorithm can concurrently satisfy multiple criteria of fairness. This paper contributes to the literature stemming from the impossibility theorems by examining how informational richness affects the accuracy and fairness of predictive algorithms. With the aid of a computer (...) simulation, we show that informational richness is the engine that drives improvements in the performance of a predictive algorithm, in terms of both accuracy and fairness. The centrality of informational richness suggests that classification parity, a popular criterion of algorithmic fairness, should be given relatively little weight. But we caution that the centrality of informational richness should be taken with a grain of salt in light of practical limitations, in particular, the so-called bias-variance trade off. (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)

It is a widespread intuition that the coherence of independent reports provides a powerful reason to believe that the reports are true. Formal results by Huemer, M. 1997. “Probability and Coherence Justification.” Southern Journal of Philosophy 35: 463–72, Olsson, E. 2002. “What is the Problem of Coherence and Truth?” Journal of Philosophy XCIX : 246–72, Olsson, E. 2005. Against Coherence: Truth, Probability, and Justification. Oxford University Press., Bovens, L., and S. Hartmann. 2003. Bayesian Epistemology. Oxford University Press, prove that, under (...) certain conditions, coherence cannot increase the probability of the target claim. These formal results, known as ‘the impossibility theorems’ have been widely discussed in the literature. They are taken to have significant epistemic upshot. In particular, they are taken to show that reports must first individually confirm the target claim before the coherence of multiple reports offers any positive confirmation. In this paper, I dispute this epistemic interpretation. The impossibility theorems are consistent with the idea that the coherence of independent reports provides a powerful reason to believe that the reports are true even if the reports do not individually confirm prior to coherence. Once we see that the formal discoveries do not have this implication, we can recover a model of coherence justification consistent with Bayesianism and these results. This paper, thus, seeks to turn the tide of the negative findings for coherence reasoning by defending coherence as a unique source of confirmation. (shrink)

It is commonly thought that such topics as Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason are disparate scientific physical or mathematical issues having little or nothing in common. I suggest that they are largely standard philosophical problems (i.e., language games) which were resolved by Wittgenstein over 80 years ago. -/- Wittgenstein also demonstrated the fatal error in regarding mathematics or language or our behavior in general as a unitary coherent logical ‘system,’ rather (...) than as a motley of pieces assembled by the random processes of natural selection. “Gödel shows us an unclarity in the concept of ‘mathematics’, which is indicated by the fact that mathematics is taken to be a system” and we can say (contra nearly everyone) that is all that Gödel and Chaitin show. Wittgenstein commented many times that ‘truth’ in math means axioms or the theorems derived from axioms, and ‘false’ means that one made a mistake in using the definitions, and this is utterly different from empirical matters where one applies a test. Wittgenstein often noted that to be acceptable as mathematics in the usual sense, it must be useable in other proofs and it must have real world applications, but neither is the case with Godel’s Incompleteness. Since it cannot be proved in a consistent system (here Peano Arithmetic but a much wider arena for Chaitin), it cannot be used in proofs and, unlike all the ‘rest’ of PA it cannot be used in the real world either. As Rodych notes “…Wittgenstein holds that a formal calculus is only a mathematical calculus (i.e., a mathematical language-game) if it has an extra- systemic application in a system of contingent propositions (e.g., in ordinary counting and measuring or in physics) …” Another way to say this is that one needs a warrant to apply our normal use of words like ‘proof’, ‘proposition’, ‘true’, ‘incomplete’, ‘number’, and ‘mathematics’ to a result in the tangle of games created with ‘numbers’ and ‘plus’ and ‘minus’ signs etc., and with -/- ‘Incompleteness’ this warrant is lacking. Rodych sums it up admirably. “On Wittgenstein’s account, there is no such thing as an incomplete mathematical calculus because ‘in mathematics, everything is algorithm [and syntax] and nothing is meaning [semantics]…” -/- I make some brief remarks which note the similarities of these ‘mathematical’ issues to economics, physics, game theory, and decision theory. -/- Those wishing further comments on philosophy and science from a Wittgensteinian two systems of thought viewpoint may consult my other writings -- Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle 2nd ed (2019), Suicide by Democracy 4th ed (2019), The Logical Structure of Human Behavior (2019), The Logical Structure of Consciousness (2019, Understanding the Connections between Science, Philosophy, Psychology, Religion, Politics, and Economics and Suicidal Utopian Delusions in the 21st Century 5th ed (2019), Remarks on Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason in Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych, Berto, Floyd, Moyal-Sharrock and Yanofsky (2019), and The Logical Structure of Philosophy, Psychology, Sociology, Anthropology, Religion, Politics, Economics, Literature and History (2019). (shrink)

Abstract. As a general theory of reasoning—and as a general theory of what holds true under every possible circumstance—logic is supposed to be ontologically neutral. It ought to have nothing to do with questions concerning what there is, or whether there is anything at all. It is for this reason that traditional Aristotelian logic, with its tacit existential presuppositions, was eventually deemed inadequate as a canon of pure logic. And it is for this reason that modern quantification theory, (...) too, with its residue of existentially loaded theorems and patterns of inference, has been claimed to suffer from a defect of logical purity. The law of non-contradiction rules out certain circumstances as impossible—circumstances in which a statement is both true and false, or perhaps circumstances where something both is and is not the case. Is this to be regarded as a further ontological bias? (shrink)

Este artículo brinda algunas respuestas y alternativas a ciertos problemas y propuestas en el área de la teoría democrática. El ensayo tiene como enfoque la cuestión de distinguir sistemas que pueden parecer democráticos sin serlo de sistemas realmente democráticos. Develando algunos actores disfrazados del discurso democrático en América Latina, el artículo argumenta que es preferible la regla de la mayoría como base para la identificación del bien común por medio del interés general, que reglas de minorías, consentimiento total o (...) bases constitucionales opuestas al gobierno de la mayoría. Esto se hace sin desestimar a las minorías ni las constituciones, ni lo ideal de la unanimidad. Simplemente las ubica. La investigación incluye una nueva respuesta al problema de la supuesta irracionalidad de la democracia identificada originalmente por Platón y después formalizada por Arrow. (shrink)

In their Why Machines Will Never Rule the World, Landgrebe and Smith (2023) argue that it is impossible for artificial general intelligence (AGI) to succeed, on the grounds that it is impossible to perfectly model or emulate the “complex” “human neurocognitive system”. However, they do not show that it is logically impossible; they only show that it is practically impossible using current mathematical techniques. Nor do they prove that there could not be any other kinds of theories than those (...) in current use. Even if perfect theories were impossible or unlikely, perfection may not be needed and may even be unhelpful. (shrink)

It is common to assume that the problem of induction arises only because of small sample sizes or unreliable data. In this paper, I argue that the piecemeal collection of data can also lead to underdetermination of theories by evidence, even if arbitrarily large amounts of completely reliable experimental and observational data are collected. Specifically, I focus on the construction of causal theories from the results of many studies (perhaps hundreds), including randomized controlled trials and observational studies, where the studies (...) focus on overlapping, but not identical, sets of variables. Two theorems reveal that, for any collection of variables V, there exist fundamentally different causal theories over V that cannot be distinguished unless all variables are simultaneously measured. Underdetermination can result from piecemeal measurement, regardless of the quantity and quality of the data. Moreover, I generalize these results to show that, a priori, it is impossible to choose a series of small (in terms of number of variables) observational studies that will be most informative with respect to the causal theory describing the variables under investigation. This final result suggests that scientific institutions may need to play a larger role in coordinating differing research programs during inquiry. (shrink)

Arrhenius’s impossibility theorems purport to demonstrate that no population axiology can satisfy each of a small number of intuitively compelling adequacy conditions. However, it has recently been pointed out that each theorem depends on a dubious assumption: Finite Fine-Grainedness. This assumption states that there exists a finite sequence of slight welfare differences between any two welfare levels. Denying Finite Fine-Grainedness makes room for a lexical population axiology which satisfies all of the compelling adequacy conditions in each theorem. (...) Therefore, Arrhenius’s theorems fail to prove that there is no satisfactory population axiology. In this paper, I argue that Arrhenius’s theorems can be repurposed. Since all of our population-affecting actions have a non-zero probability of bringing about more than one distinct population, it is population prospect axiologies that are of practical relevance, and amended versions of Arrhenius’s theorems demonstrate that there is no satisfactory population prospect axiology. These impossibility theorems do not depend on Finite Fine-Grainedness, so lexical views do not escape them. (shrink)

According to the computational theory of mind , to think is to compute. But what is meant by the word 'compute'? The generally given answer is this: Every case of computing is a case of manipulating symbols, but not vice versa - a manipulation of symbols must be driven exclusively by the formal properties of those symbols if it is qualify as a computation. In this paper, I will present the following argument. Words like 'form' and 'formal' are ambiguous, as (...) they can refer to form in either the syntactic or the morphological sense. CTM fails on each disambiguation, and the arguments for CTM immediately cease to be compelling once we register that ambiguity. The terms 'mechanical' and 'automatic' are comparably ambiguous. Once these ambiguities are exposed, it turns out that there is no possibility of mechanizing thought, even if we confine ourselves to domains where all problems can be settled through decision-procedures. The impossibility of mechanizing thought thus has nothing to do with recherché mathematical theorems, such as those proven by Gödel and Rosser. A related point is that CTM involves, and is guilty of reinforcing, a misunderstanding of the concept of an algorithm. (shrink)

The new …eld of judgment aggregation aims to …nd collective judgments on logically interconnected propositions. Recent impossibility results establish limitations on the possibility to vote independently on the propositions. I show that, fortunately, the impossibility results do not apply to a wide class of realistic agendas once propositions like “if a then b” are adequately modelled, namely as subjunctive implications rather than material implications. For these agendas, consistent and complete collective judgments can be reached through appropriate quota rules (...) (which decide propositions using acceptance thresholds). I characterise the class of these quota rules. I also prove an abstract result that characterises consistent aggregation for arbitrary agendas in a general logic. (shrink)

The Turing machine halting problem can be explained by several factors, including arithmetic logic irreversibility and memory erasure, which contribute to computational uncertainty due to information loss during computation. Essentially, this means that an algorithm can only preserve information about an input, rather than generate new information. This uncertainty arises from characteristics such as arithmetic logical irreversibility, Landauer's principle, and memory erasure, which ultimately lead to a loss of information and an increase in entropy. To measure this uncertainty and loss (...) of information, the concept of arithmetic logical entropy can be used. The Turing machine and its equivalent, general recursive functions can be understood through the λ calculus and the Turing/Church thesis. However, there are certain recursive functions that cannot be fully understood or predicted by other algorithms due to the loss of information during logical-arithmetic operations. In other words, the behaviour of these functions cannot be completely determined at every stage of the computation due to a lack of information in their definition. While there are some cases where the behaviour of recursive functions is highly predictable, the lack of information in most cases makes it impossible for algorithms to determine if a program will halt or not. This inability to predict the outcome of the computation is the essence of the halting problem of the Turing machine. Even in cases where more information is available about the program, it is still difficult to determine with certainty if the program will halt or not. This also highlights the importance of the Turing oracle machine, which introduces information from outside the computation to compensate for the lack of information and ultimately decide the result of the computation. (shrink)

Compare two conceptions of validity: under an example of a modal conception, an argument is valid just in case it is impossible for the premises to be true and the conclusion false; under an example of a topic-neutral conception, an argument is valid just in case there are no arguments of the same logical form with true premises and a false conclusion. This taxonomy of positions suggests a project in the philosophy of logic: the reductive analysis of the modal conception (...) of logical consequence to the topic-neutral conception. Such a project would dispel the alleged obscurity of the notion of necessity employed in the modal conception in favour of the clarity of an account of logical consequence given in terms of tractable notions of logical form, universal generalization and truth simpliciter. In a series of publications, John Etchemendy has characterized the model-theoretic definition of logical consequence as truth preservation in all models as intended to provide just such an analysis. In this paper, I will argue that Aristotle intends to provide an account of a modal conception of logical consequence in topic-neutral terms and so is engaged in a project comparable to the one described above. That Aristotle would be engaged in this sort of project is controversial. Under the standard reading of the Prior Analytics, Aristotle does not and cannot provide an account of logical consequence. Rather, he must take the validity of the first figure syllogisms (such as the syllogism known by its medieval mnemonic ‘Barbara’: A belongs to all B; B belongs to all C; so A belongs to all C) as obvious and not needing justification; he then establishes the validity of the other syllogisms by showing that they stand in a suitable relation to the first figure syllogisms. I will argue that Aristotle does attempt to provide an account of logical consequence—namely, by appeal to certain mereological theorems. For example, he defends the status of Barbara as a syllogism by appeal to the transitivity of mereological containment. There are, as I will discuss, reasons to doubt the success of this account. But the attempt is not implausible given certain theses Aristotle holds in semantics, mereology and the theory of relations. (shrink)

Quantum computer is considered as a generalization of Turing machine. The bits are substituted by qubits. In turn, a "qubit" is the generalization of "bit" referring to infinite sets or series. It extends the consept of calculation from finite processes and algorithms to infinite ones, impossible as to any Turing machines (such as our computers). However, the concept of quantum computer mets all paradoxes of infinity such as Gödel's incompletness theorems (1931), etc. A philosophical reflection on how quantum computer might (...) implement the idea of "infinite calculation" is the main subject. (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) .

There is an extensive literature in social choice theory studying the consequences of weakening the assumptions of Arrow's Impossibility Theorem. Much of this literature suggests that there is no escape from Arrow-style impossibility theorems unless one drastically violates the Independence of Irrelevant Alternatives (IIA). In this paper, we present a more positive outlook. We propose a model of comparing candidates in elections, which we call the Advantage-Standard (AS) model. The requirement that a collective choice rule (CCR) be (...) rationalizable by the AS model is in the spirit of but weaker than IIA; yet it is stronger than what is known in the literature as weak IIA (two profiles alike on x, y cannot have opposite strict social preferences on x and y). In addition to motivating violations of IIA, the AS model makes intelligible violations of another Arrovian assumption: the negative transitivity of the strict social preference relation P. While previous literature shows that only weakening IIA to weak IIA or only weakening negative transitivity of P to acyclicity still leads to impossibility theorems, we show that jointly weakening IIA to AS rationalizability and weakening negative transitivity of P leads to no such impossibility theorems. Indeed, we show that several appealing CCRs are AS rationalizable, including even transitive CCRs. (shrink)

Famous results by David Lewis show that plausible-sounding constraints on the probabilities of conditionals or evaluative claims lead to unacceptable results, by standard probabilistic reasoning. Existing presentations of these results rely on stronger assumptions than they really need. When we strip these arguments down to a minimal core, we can see both how certain replies miss the mark, and also how to devise parallel arguments for other domains, including epistemic “might,” probability claims, claims about comparative value, and so on. A (...) popular reply to Lewis's results is to claim that conditional claims, or claims about subjective value, lack truth conditions. For this strategy to have a chance of success, it needs to give up basic structural principles about how epistemic states can be updated—in a way that is strikingly parallel to the commitments of the project of dynamic semantics. (shrink)

Standard impossibility theorems on judgment aggregation over logically connected propositions either use a controversial systematicity condition or apply only to agendas of propositions with rich logical connections. Are there any serious impossibilities without these restrictions? We prove an impossibility theorem without requiring systematicity that applies to most standard agendas: Every judgment aggregation function (with rational inputs and outputs) satisfying a condition called unbiasedness is dictatorial (or effectively dictatorial if we remove one of the agenda conditions). Our agenda (...) conditions are tight. When applied illustratively to (strict) preference aggregation represented in our model, the result implies that every unbiased social welfare function with universal domain is effectively dictatorial. (shrink)

This paper generalises the classical Condorcet jury theorem 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)

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 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)

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.

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)

The “four-color” theorem seems to be generalizable as follows. The four-letter alphabet is sufficient to encode unambiguously any set of well-orderings including a geographical map or the “map” of any logic and thus that of all logics or the DNA plan of any alive being. Then the corresponding maximally generalizing conjecture would state: anything in the universe or mind can be encoded unambiguously by four letters. That admits to be formulated as a “four-letter theorem”, and thus one can (...) search for a properly mathematical proof of the statement. It would imply the “four colour theorem”, the proof of which many philosophers and mathematicians believe not to be entirely satisfactory for it is not a “human proof”, but intermediated by computers unavoidably since the necessary calculations exceed the human capabilities fundamentally. It is furthermore rather unsatisfactory because it consists in enumerating and proving all cases one by one. Sometimes, a more general theorem turns out to be much easier for proving including a general “human” method, and the particular and too difficult for proving theorem to be implied as a corollary in certain simple conditions. The same approach will be followed as to the four colour theorem, i.e. to be deduced more or less trivially from the “four-letter theorem” if the latter is proved. References are only classical and thus very well-known papers: their complete bibliographic description is omitted. (shrink)

The aggregation of individual judgments over interrelated propositions is a newly arising field of social choice theory. I introduce several independence conditions on judgment aggregation rules, each of which protects against a specific type of manipulation by agenda setters or voters. I derive impossibility theorems whereby these independence conditions are incompatible with certain minimal requirements. Unlike earlier impossibility results, the main result here holds for any (non-trivial) agenda. However, independence conditions arguably undermine the logical structure of judgment aggregation. (...) I therefore suggest restricting independence to premises, which leads to a generalised premise-based procedure. This procedure is proven to be possible if the premises are logically independent. (shrink)

My aim in this paper is to explain what Condorcet’s jury theorem 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 jury theorem 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 jury theorem, 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)

Any intermediate propositional logic can be extended to a calculus with epsilon- and tau-operators and critical formulas. For classical logic, this results in Hilbert’s $\varepsilon $ -calculus. The first and second $\varepsilon $ -theorems for classical logic establish conservativity of the $\varepsilon $ -calculus over its classical base logic. It is well known that the second $\varepsilon $ -theorem fails for the intuitionistic $\varepsilon $ -calculus, as prenexation is impossible. The paper investigates the effect of adding critical $\varepsilon $ (...) - and $\tau $ -formulas and using the translation of quantifiers into $\varepsilon $ - and $\tau $ -terms to intermediate logics. It is shown that conservativity over the propositional base logic also holds for such intermediate ${\varepsilon \tau }$ -calculi. The “extended” first $\varepsilon $ -theorem holds if the base logic is finite-valued Gödel–Dummett logic, and fails otherwise, but holds for certain provable formulas in infinite-valued Gödel logic. The second $\varepsilon $ -theorem also holds for finite-valued first-order Gödel logics. The methods used to prove the extended first $\varepsilon $ -theorem for infinite-valued Gödel logic suggest applications to theories of arithmetic. (shrink)