Robert Aumann presents his Agreement Theorem as the key conditional: “if two people have the same priors and their posteriors for an event A are common knowledge, then these posteriors are equal” (Aumann, 1976, p. 1236). This paper focuses on four assumptions which are used in Aumann’s proof but are not explicit in the key conditional: (1) that agents commonly know, of some prior μ, that it is the common prior; (2) that agents commonly know (...) that each of them updates on the prior by conditionalization; (3) that agents commonly know that if an agent knows a proposition, she knows that she knows that proposition (the “K K” principle); (4) that agents commonly know that they each update only on true propositions. It is shown that natural weakenings of any one of these strong assumptions can lead to countermodels to Aumann’s key conditional. Examples are given in which agents who have a common prior and commonly know what probability they each assign to a proposition nevertheless assign that proposition unequal probabilities. To alter Aumann’s famous slogan: people can “agree to disagree”, even if they share a common prior. The epistemological significance of these examples is presented in terms of their role in a defense of the Uniqueness Thesis: If an agent whose total evidence is E is fully rational in taking doxastic attitude D to P, then necessarily, any subject with total evidence E who takes a different attitude to P is less than fully rational. (shrink)

Aumann’s theorem states that no individual should agree to disagree under a range of assumptions. Political liberalism appears to presuppose these assumptions with the idealized conditions of public reason. We argue Aumann’s theorem demonstrates they nevertheless cannot be simultaneously held with what is arguably political liberalism’s most central tenet. That is, the tenet of reasonable pluralism, which implies we can rationally agree to disagree over conceptions of the good. We finish by elaborating a way of relaxing (...) one of the theorem’s axioms that arguably lends itself to a coherent account of political liberalism, namely the condition of indexical independence. (shrink)

Standard models in epistemic game theory make strong assumptions about agents’ knowledge of their own beliefs. Agents are typically assumed to be introspectively omniscient: if an agent believes an event with probability p, she is certain that she believes it with probability p. This paper investigates the extent to which this assumption can be relaxed while preserving some standard epistemic results. Geanakoplos (1989) claims to provide an Agreement Theorem using the “truth” axiom, together with the property of balancedness, (...) a significant relaxation of introspective omniscience. I provide an example which shows that Geanakoplos’s statement is incorrect. I then introduce a new property, local balancedness, which allows us both to correct Geanakoplos’s result, and to extend it to cases where the truth axiom may fail. I exploit this general Agreement Theorem to provide novel epistemic conditions for correlated and Nash equilibrium, both of which relax the assumption of introspective omniscience. In all three cases, the results are also extended to infinite state spaces. (This is Chapter 5 of my 2014 Oxford DPhil (PhD) thesis.). (shrink)

The force law of Maxwell’s classical electrodynamics does not agree with Newton’s third law of motion (N3LM), in case of open circuit magnetostatics. Initially, a generalized magnetostatics theory is presented that includes two additional physical fields B_Φ and B_l, defined by scalar functions. The scalar magnetic field B_l mediates a longitudinal Ampère force that balances the transverse Ampère force (aka the magnetic field force), such that the sum of the two forces agrees with N3LM for all stationary current distributions. Secondary (...) field induction laws are derived; a secondary curl free electric field E_l is induced by a time varying scalar magnetic field B_l, which isn’t described by Maxwell’s electrodynamics. The Helmholtz’ decomposition is applied to exclude E_l from the total electric field E, resulting into a more simple Maxwell theory. Decoupled inhomogeneous potential equations and its solutions follow directly from this theory, without having to apply a gauge condition. Field expressions are derived from the potential functions that are simpler and far field consistent with respect to the Jefimenko fields. However, our simple version of Maxwell’s theory does not satisfy N3LM. Therefore we combine the generalized magnetostatics with the simple version of Maxwell’s electrodynamics, via the generalization of Maxwell’s speculative displacement current. The resulting electrodynamics describes three types of vacuum waves: the Φ wave, the longitudinal electromagnetic (LEM) wave and the transverse electromagnetic (TEM) wave, with phase velocities respectively a, b and c. Power- and force theorems are derived, and the force law agrees with Newton’s third law only if the phase velocities satisfy the following condition: a >> b and b = c. The retarded potential functions can be found without gauge conditions, and four retarded field expressions are derived that have three near field terms and six far field terms. All six far field terms are explained as the mutual induction of two free fields. Our theory supports Rutherford’s solution of the 4/3 problem of electromagnetic mass, which requires an extra longitudinal electromagnetic momentum. Our generalized classical electrodynamics might spawn new physics experiments and electrical engineering, such as new photoelectric effects based on Φ- or LEM radiation, and the conversion of natural Φ- or LEM radiation into useful electricity, in the footsteps of dr. N. Tesla and dr. T.H. Moray. (shrink)

The paper surveys the currently available axiomatizations of common belief (CB) and common knowledge (CK) by means of modal propositional logics. (Throughout, knowledge- whether individual or common- is defined as true belief.) Section 1 introduces the formal method of axiomatization followed by epistemic logicians, especially the syntax-semantics distinction, and the notion of a soundness and completeness theorem. Section 2 explains the syntactical concepts, while briefly discussing their motivations. Two standard semantic constructions, Kripke structures and neighbourhood structures, are introduced in (...) Sections 3 and 4, respectively. It is recalled that Aumann's partitional model of CK is a particular case of a definition in terms of Kripke structures. The paper also restates the well-known fact that Kripke structures can be regarded as particular cases of neighbourhood structures. Section 3 reviews the soundness and completeness theorems proved w.r.t. the former structures by Fagin, Halpern, Moses and Vardi, as well as related results by Lismont. Section 4 reviews the corresponding theorems derived w.r.t. the latter structures by Lismont and Mongin. A general conclusion of the paper is that the axiomatization of CB does not require as strong systems of individual belief as was originally thought- only monotonicity has thusfar proved indispensable. Section 5 explains another consequence of general relevance: despite the "infinitary" nature of CB, the axiom systems of this paper admit of effective decision procedures, i.e., they are decidable in the logician's sense. (shrink)

Some arguments against the assumption that ordinary people may share common knowledge are sound. The apparent cost of such arguments is the rejection of scientific theories that appeal to common knowledge. My proposal is to accept the arguments without rejecting the theories. On my proposal, common knowledge is shared by ideally rational people, who are not just mathematically simple versions of ordinary people. They are qualitatively different from us, and theorizing about them does not lead to predictions about our behavior. (...) Nevertheless, models of action that assume common knowledge have a role to play in our understanding of collective rationality. (shrink)

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric (...) interpretation, due to Gauss and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation. (shrink)

Inspired by impossibility theorems of social choice theory, many democratic theorists have argued that aggregative forms of democracy cannot lend full democratic justification for the collective decisions reached. Hence, democratic theorists have turned their attention to deliberative democracy, according to which “outcomes are democratically legitimate if and only if they could be the object of a free and reasoned agreement among equals” (Cohen 1997a, 73). However, relatively little work has been done to offer a formal theory of democratic deliberation. (...) This article helps fill that gap by offering a formal theory of three different modes of democratic deliberation: myopic discussion, constructive discussion, and debate. We show that myopic discussion suffers from indeterminacy of long run outcomes, while constructive discussion and debate are conclusive. Finally, unlike the other two modes of deliberation, debate is path independent and converges to a unique compromise position, irrespective of the initial status quo. (shrink)

In this text the ancient philosophical question of determinism (“Does every event have a cause ?”) will be re-examined. In the philosophy of science and physics communities the orthodox position states that the physical world is indeterministic: quantum events would have no causes but happen by irreducible chance. Arguably the clearest theorem that leads to this conclusion is Bell’s theorem. The commonly accepted ‘solution’ to the theorem is ‘indeterminism’, in agreement with the Copenhagen interpretation. Here it (...) is recalled that indeterminism is not really a physical but rather a philosophical hypothesis, and that it has counterintuitive and far-reaching implications. At the same time another solution to Bell’s theorem exists, often termed ‘superdeterminism’ or ‘total determinism’. Superdeterminism appears to be a philosophical position that is centuries and probably millennia old: it is for instance Spinoza’s determinism. If Bell’s theorem has both indeterministic and deterministic solutions, choosing between determinism and indeterminism is a philosophical question, not a matter of physical experimentation, as is widely believed. If it is impossible to use physics for deciding between both positions, it is legitimate to ask which philosophical theories are of help. Here it is argued that probability theory – more precisely the interpretation of probability – is instrumental for advancing the debate. It appears that the hypothesis of determinism allows to answer a series of precise questions from probability theory, while indeterminism remains silent for these questions. From this point of view determinism appears to be the more reasonable assumption, after all. (shrink)

We introduce a ranking of multidimensional alternatives, including uncertain prospects as a particular case, when these objects can be given a matrix form. This ranking is separable in terms of rows and columns, and continuous and monotonic in the basic quantities. Owing to the theory of additive separability developed here, we derive very precise numerical representations over a large class of domains (i.e., typically notof the Cartesian product form). We apply these representationsto (1)streams of commodity baskets through time, (2)uncertain social (...) prospects, (3)uncertain individual prospects. Concerning(1), we propose a finite horizon variant of Koopmans’s (1960) axiomatization of infinite discounted utility sums. The main results concern(2). We push the classic comparison between the exanteand expostsocial welfare criteria one step further by avoiding any expected utility assumptions, and as a consequence obtain what appears to be the strongest existing form of Harsanyi’s (1955) Aggregation Theorem. Concerning(3), we derive a subjective probability for Anscombe and Aumann’s (1963) finite case by merely assuming that there are two epistemically independent sources of uncertainty. (shrink)

This short paper has two parts. First,we prove a generalisation of Aumann’s surprising impossibility result in the context of rational decision making. We then move, in the second part, to discuss the interpretational meaning of some formal setups of epistemic models, and we do so by means of presenting an interesting puzzle in epistemic logic. The aim is to highlight certain problematic aspects of these epistemic systems concerning first/third-person asymmetry which underlies both parts of the story. This asymmetry, we (...) argue, reveals certain limits of what epistemic models can be. (shrink)

The motivating question of this paper is: ‘How are our beliefs in the theorems of mathematics justified?’ This is distinguished from the question ‘How are our mathematical beliefs reliably true?’ We examine an influential answer, outlined by Russell, championed by Gödel, and developed by those searching for new axioms to settle undecidables, that our mathematical beliefs are justified by ‘intuitions’, as our scientific beliefs are justified by observations. On this view, axioms are analogous to laws of nature. They are postulated (...) to best systematize the data to be explained. We argue that there is a decisive difference between the cases. There is agreement on the data to be systematized in the scientific case that has no analog in the mathematical one. There is virtual consensus over observations, but conspicuous dispute over intuitions. In this respect, mathematics more closely resembles stereotypical philosophy. We conclude by distinguishing two ideas that have long been associated -- realism (the idea that there is an independent reality) and objectivity (the idea that in a disagreement, only one of us can be right). We argue that, while realism is true of mathematics and philosophy, these domains fail to be objective. One upshot of the discussion is that even questions of fundamental physics may fail to be objective in roughly the sense that the question, ‘Is the Parallel Postulate is true?’, understood as one of pure mathematics, fails to be. Another is a kind of pragmatism. Factual questions in mathematics, modality, logic, and evaluative areas go proxy for non-factual practical ones. -/- . (shrink)

While Classical Logic (CL) used to be the gold standard for evaluating the rationality of human reasoning, certain non-theorems of CL—like Aristotle’s and Boethius’ theses—appear intuitively rational and plausible. Connexive logics have been developed to capture the underlying intuition that conditionals whose antecedents contradict their consequents, should be false. We present results of two experiments (total n = 72), the first to investigate connexive principles and related formulae systematically. Our data suggest that connexive logics provide more plausible rationality frameworks for (...) human reasoning compared to CL. Moreover, we experimentally investigate two approaches for validating connexive principles within the framework of coherence-based probability logic (Pfeifer & Sanfilippo, 2021). Overall, we observed good agreement between our predictions and the data, but especially for Approach 2. (shrink)

In the 1951 Gibbs lecture, Gödel asserted his famous dichotomy, where the notion of informal proof is at work. G. Priest developed an argument, grounded on the notion of naïve proof, to the effect that Gödel’s first incompleteness theorem suggests the presence of dialetheias. In this paper, we adopt a plausible ideal notion of naïve proof, in agreement with Gödel’s conception, superseding the criticisms against the usual notion of naïve proof used by real working mathematicians. We explore the (...) connection between Gödel’s theorem and naïve proof so understood, both from a classical and a dialetheic perspective. (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)

Habermas’ theory of communicative action is explored as an orientation to the question of understanding which negotiates a pathway between two opposing (and complementary) theoretical frameworks—namely, hermeneutical-relational and empirical-analytical frameworks. His perspective grounds speech, action and understanding in the ethics of human relations. In his approach, understanding is fixed by particular events or situations about which intersubjective agreement must be achieved through the offer and acceptance of reasons that simultaneously orient actors to three worlds: the objective, the social and (...) the personal worlds. This approach raises the question as to whether the process of abstracting from the particular event to the general form, through such rational discourse, might create systems of understanding which silence individual expression and naming, particularly if such expression involves an identity that is not shared with others in the group. In other words, is the origin of an event or situation necessarily a null point for all actors? As this question is explored, metaphor comes to the fore as a complementary process of showing that which cannot be grounded in the dominant system of understanding. The pivot role of naming in the formal structure of Godel’s First Incompleteness Theorem for Number Theory demonstrates the challenge to Habermas’ theory. (shrink)

While the philosophers of science discuss the General Relativity, the mathematical physicists do not question it. Therefore, there is a conflict. From the theoretical point view “the question of precisely what Einstein discovered remains unanswered, for we have no consensus over the exact nature of the theory 's foundations. Is this the theory that extends the relativity of motion from inertial motion to accelerated motion, as Einstein contended? Or is it just a theory that treats gravitation geometrically in the spacetime (...) setting?”. “The voices of dissent proclaim that Einstein was mistaken over the fundamental ideas of his own theory and that their basic principles are simply incompatible with this theory. Many newer texts make no mention of the principles Einstein listed as fundamental to his theory; they appear as neither axiom nor theorem. At best, they are recalled as ideas of purely historical importance in the theory's formation. The very name General Relativity is now routinely condemned as a misnomer and its use often zealously avoided in favour of, say , Einstein's theory of gravitation What has complicated an easy resolution of the debate are the alterations of Einstein's own position on the foundations of his theory”, (Norton, 1993). Of other hand from the mathematical point view the “General Relativity had been formulated as a messy set of partial differential equations in a single coordinate system. People were so pleased when they found a solution that they didn't care that it probably had no physical significance” (Hawking and Penrose, 1996). So, during a time, the declaration of quantum theorists:“I take the positivist viewpoint that a physical theory is just a mathematical model and that it is meaningless to ask whether it corresponds to reality. All that one can ask is that its predictions should be in agreement with observation.” (Hawking and Penrose, 1996)seemed to solve the problem, but recently achieved with the help of the tightly and collectively synchronized clocks in orbit frontally contradicts fundamental assumptions of the theory of Relativity. These observations are in disagree from predictions of the theory of Relativity. (Hatch, 2004a, 2004b, 2007). The mathematical model was developed first by Grossmann who presented it, in 1913, as the mathematical part of the Entwurf theory, still referred to a curved Minkowski spacetime. Einstein completed the mathematical model, in 1915, formulated for Riemann ́s spacetimes. In this paper, we present as of General Relativity currently remains only the mathematical model, darkened with the results of Hatch and, course, we conclude that a Einstein ́s gravity theory does not exist. (shrink)

This paper defends an account of forgiveness that is sensitive to recent work on anger. Like others, we claim anger involves an appraisal, namely that someone has done something wrong. But, we add, anger has two further functions. First, anger communicates to the wrongdoer that her act has been appraised as wrong and demands she feel guilty. This function enables us to explain why apologies make it reasonable to forgo anger and forgive. Second, anger sanctions the wrongdoer for what she (...) has done. This function allows us to explore the moral status of forgiveness, including why forgiveness is typically elective. (shrink)

ABSTRACT One striking feature of On the Genealogy of Morals is how it is written. Nietzsche employs a literary style that provokes his readers' emotions. In Beyond Selflessness, Christopher Janaway argues that such a literary approach is integral to Nietzsche's philosophical goals. Feeling the emotions Nietzsche's style arouses is necessary for understanding the views he defends. I argue that Janaway's position is mistaken. The evidence at our disposal fails to establish that emotion is ever necessary for cognition. However, I maintain (...) that we do have good evidence for a slightly weaker claim. The emotionally sensitive person is epistemically better off than the cold and dispassionate person. There are some truths he or she will be more likely to believe and will have better reasons for believing. I conclude that Janaway is right to defend the philosophical importance of Nietzsche's literary writing style. His error is simply that he overstates the case. (shrink)

Following the pattern set by the early German Romantics, Kierkegaard conveys many of his insights through literature rather than academic prose. What makes him a valuable member of this tradition is the theory he develops to support it, his so-called “theory of indirect communication.” The most exciting aspect of this theory concerns the alleged importance of indirect communication: Kierkegaard claims that there are some projects only it can accomplish. This paper provides a critical account of two arguments Kierkegaard offers in (...) defense of this claim. The first argument is that he needs to use indirect communication in order to discourage people from losing themselves in the “crowd”. The second argument is that he needs to use it in order to help people out of a “monstrous illusion”. It is shown that while both arguments justify Kierkegaard’s decision to use indirect communication, neither one supports the original claim about its indispensability. (shrink)

This dissertation concerns Kierkegaard’s theory of indirect communication. A central aspect of this theory is what I call the “indispensability thesis”: there are some projects only indirect communication can accomplish. The purpose of the dissertation is to disclose and assess the rationale behind the indispensability thesis. -/- A pair of questions guides the project. First, to what does ‘indirect communication’ refer? Two acceptable responses exist: (1) Kierkegaard’s version of Socrates’ midwifery method and (2) Kierkegaard’s use of artful literary devices. Second, (...) for what end does Kierkegaard use indirect communication? There are two acceptable responses here as well: (1) helping others become religious and (2) making others aware of the nature of existence. -/- These responses are interrelated. First, Kierkegaard’s notion of religion places restrictions on the means he can use to get readers to become religious. These restrictions ultimately entail that the only viable form of religious pedagogy is the midwifery method. Second, Kierkegaard engages in the midwifery method in part by making readers aware of the nature of existence (especially religious existence). But given the problems plaguing his readers, he thinks a straightforward approach to this project will likely fail. An approach that used artful literary devices such as deception and humor would be more successful. Third, Kierkegaard believes that there is one aspect of religious existence (viz. subjectivity) that people can come to know only first-hand. As such, he cannot directly impart knowledge of subjectivity to his readers. He argues that this means he must use the midwifery method. And he thinks the most productive way to do so is to provide readers with the kind of fictional narratives found in his early pseudonymous writings. Thus artful rhetorical devices play a role here as well. -/- All of Kierkegaard’s arguments for the indispensability thesis turn on debatable assumptions. But the arguments concerning artful rhetorical devices have the additional defect of being merely probabilistic in nature. They lack the strength to support the indispensability thesis even if we grant the relevant background assumptions. Therefore, to the degree that the indispensability thesis has merit, it lies with the arguments concerning the midwifery method. (shrink)

In Religion of Existence, Noreen Khawaja suggests that Kierkegaard is an “ascetic” thinker. By this, she means that he regards religious striving as (1) requiring ceaseless renewal and (2) being an end in itself rather than a means to some further end. In this paper, I raise challenges to both parts of Khawaja’s proposal. I argue that the first part stands in tension with Kierkegaard’s assertion that his infinitely demanding account of religious existence is meant merely as a “corrective.” The (...) second part, I maintain, does not fit well with his claim that our eternal salvation is at stake in religious striving. (shrink)

This paper provides an account of Kierkegaard’s central criticism of the Danish Hegelians. Contrary to recent scholarship, it is argued that this criticism has a substantive theoretical basis and is not merely personal or ad hominem in nature. In particular, Kierkegaard is seen as criticizing the Hegelians for endorsing an unacceptable form of intellectual elitism, one that gives them pride of place in the realm of religion by dint of their philosophical knowledge. A problem arises, however, because this criticism threatens (...) to apply to Kierkegaard himself. It is shown that Kierkegaard manages to escape this problem by virtue of the humorous aspect of his work. (shrink)

Like many 19th c. thinkers, Kierkegaard embraces a cognitivist view of art. He thinks works of art matter because they can teach us in important ways. This chapter defends two striking features of Kierkegaard’s version of this theory. First, works of art do not teach “directly” by telling us truths and offering us evidence. Instead, they educate us “indirect-ly” by helping us make our own discoveries. Second, the fact that art does not teach in a straightforward manner is no defect. (...) On the contrary, it is precisely because art teaches indirectly that it teaches better than philosophy and science do. (shrink)

On one standard view, paraphrasing Kierkegaard requires no special literary talent. It demands no particular flair for the poetic. However, Kierkegaard himself rejects this view. He says we cannot paraphrase in a straightforward fashion some of the ideas he expresses in a literary format. To use the words of Johannes Climacus, these ideas defy direct communication. In this paper, I piece together and defend the justification Kierkegaard offers for this position. I trace its origins to concerns raised by Lessing and (...) Mendelssohn about the relationship between form and content in works of art. I maintain that Kierkegaard follows early German Romantic thinkers in applying these aesthetic concerns to philosophical writing. By way of conclusion, I discuss the implications of Kierkegaard’s position for contemporary scholarship. (shrink)

Kierkegaard faces an apparent dilemma. On the one hand, he concurs with the biblical injunction: we are to love our neighbors as ourselves. He takes this to imply that self-love and neighbor-love should be roughly symmetrical, similar in kind as well as degree. On the other hand, he recommends relating to others and to ourselves in disparate ways. We should be lenient, charitable, and forgiving when interacting with neighbors; the opposite when dealing with ourselves. The goal of my paper is (...) to solve this puzzle. I first consider addressing it by appealing to Gene Outka’s idea that equal love does not entail identical treatment. After rejecting this solution, I offer my own: Asymmetry between the two loves is not a moral ideal for Kierkegaard but a rehabilitative strategy. He recommends being more latitudinarian with others than with ourselves to correct against a common tendency toward the opposite extreme. (shrink)

This paper illuminates the central arguments in Sartre's UNESCO address, 'The Singular Universal." The address begins by asking whether objective facts tell us everything there is to know about Kierkegaard. Sartre's answer is negative. The question then arises as to whether we can lay hold of Kierkegaard's "irreducible subjectivity" by seeing him as alive for us today, i.e., as transhistorical. Sartre's answer here is affirmative. However, a close inspection of this answer exposes a deeper level to the address. The struggle (...) to find a place for Kierkegaard within the world of objective knowledge is an allegory. It mirrors Sartre's struggle to find a place for his existentialism within the Marxism that dominates his later thinking. (shrink)

Recent attention to the relationship between aesthetic value and cognitive value has focused on whether the latter can affect the former. In this article, I approach the issue from the opposite direction. I investigate whether the aesthetic value of a work can influence its cognitive value. More narrowly, I consider whether a work's aesthetic value ever contributes to or detracts from its philosophical value, which I take to include the truth of its claims, the strength of its arguments, and its (...) internal consistency. I argue that aesthetic value does have such an impact, at least sometimes and to some degree. The aesthetic merits of some works help to preserve their consistency, and the aesthetic defects of other works render them self-contradictory. (shrink)

In the literature on philosophical progress it is often assumed that agreement is a necessary condition for progress. This assumption is sensible only if agreement is a reliable sign of the truth, since agreement on false answers to philosophical questions would not constitute progress. This paper asks whether agreement among philosophers is (or would be) likely to be a reliable sign of truth. Insights from social choice theory are used to identify the conditions under which (...) class='Hi'>agreement among philosophers would be a reliable indicator of the truth, and it is argued that we lack good reason to think that philosophical inquiry meets these conditions. The upshot is that philosophical agreement is epistemically uninformative: agreement on the answer to a philosophical question does not supply even a prima facie reason to think that the agreed-upon view is true. However, the epistemic uninformativeness of philosophical agreement is not an indictment of philosophy's progress, because philosophy is valuable independent of its ability to generate agreement on the correct answers to philosophical questions. (shrink)

Typical agreements can be seen as joint decisions, inherently involving obligations of a distinctive kind. These obligations derive from the joint commitment' that underlies a joint decision. One consequence of this understanding of agreements and their obligations is that coerced agreements are possible and impose obligations. It is not that the parties to an agreement should always conform to it, all things considered. Unless one is released from the agreement, however, one has some reason to conform to it, (...) whatever else is true. In this sense, one is under an obligation to the other parties. The relevance of these points to the issue of political obligation is discussed. (shrink)

We investigate adverbial agreement in Sandəmarkesə (S. Marco in Lamis, Apulia) proposing phase-bound, local agreement relations, reducible to coordination, as in past and absolute participial constructions, suggesting a copulaless analysis where arguments are subjects in a small clause. With disjunct nominals with matching φ-features, the adverb agrees separately with each part in the set, otherwise resulting in ‘non-agreeing’ forms, which we test also with negative polarity items (niʃun-, ‘nobody’ and nentə, ‘nothing’). With fragment answers, the negation scopes over (...) adverbs agreeing with the two proposed topics: matching of the φ-features of both nouns values the negative operator with the same features. In fronted adverbs, agreement occurs when the question contains overt coordinated arguments, elements continuing a chain, and if coordinated arguments have matching φ-features. In agreement in topical contexts with fronted adverbs, agreement occurs with the aboutness-shift topic closely preceding them, rather than with the embedded direct object. (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)

Computer assisted theorem proving is an increasingly important part of mathematical methodology, as well as a long-standing topic in artificial intelligence (AI) research. However, the current generation of theorem proving software have limited functioning in terms of providing new proofs. Importantly, they are not able to discriminate interesting theorems and proofs from trivial ones. In order for computers to develop further in theorem proving, there would need to be a radical change in how the software functions. Recently, (...) machine learning results in solving mathematical tasks have shown early promise that deep artificial neural networks could learn symbolic mathematical processing. In this paper, I analyze the theoretical prospects of such neural networks in proving mathematical theorems. In particular, I focus on the question how such AI systems could be incorporated in practice to theorem proving and what consequences that could have. In the most optimistic scenario, this includes the possibility of autonomous automated theorem provers (AATP). Here I discuss whether such AI systems could, or should, become accepted as active agents in mathematical communities. (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)

Representation theorems are often taken to provide the foundations for decision theory. First, they are taken to characterize degrees of belief and utilities. Second, they are taken to justify two fundamental rules of rationality: that we should have probabilistic degrees of belief and that we should act as expected utility maximizers. We argue that representation theorems cannot serve either of these foundational purposes, and that recent attempts to defend the foundational importance of representation theorems are unsuccessful. As a result, we (...) should reject these claims, and lay the foundations of decision theory on firmer ground. (shrink)

Peer review is often taken to be the main form of quality control on academic research. Usually journals carry this out. However, parts of maths and physics appear to have a parallel, crowd-sourced model of peer review, where papers are posted on the arXiv to be publicly discussed. In this paper we argue that crowd-sourced peer review is likely to do better than journal-solicited peer review at sorting papers by quality. Our argument rests on two key claims. First, crowd-sourced peer (...) review will lead on average to more reviewers per paper than journal-solicited peer review. Second, due to the wisdom of the crowds, more reviewers will tend to make better judgments than fewer. We make the second claim precise by looking at the Condorcet Jury Theorem as well as two related jury theorems developed specifically to apply to peer review. (shrink)

Some recent work in formal epistemology shows that “witness agreement” by itself implies neither an increase in the probability of truth nor a high probability of truth—the witnesses need to have some “individual credibility.” It can seem that, from this formal epistemological result, it follows that coherentist justification (i.e., doxastic coherence) is not truth-conducive. I argue that this does not follow. Central to my argument is the thesis that, though coherentists deny that there can be noninferential justification, coherentists do (...) not deny that there can be individual credibility. (shrink)

A fundamental problem in science is how to make logical inferences from scientific data. Mere data does not suffice since additional information is necessary to select a domain of models or hypotheses and thus determine the likelihood of each model or hypothesis. Thomas Bayes’ Theorem relates the data and prior information to posterior probabilities associated with differing models or hypotheses and thus is useful in identifying the roles played by the known data and the assumed prior information when making (...) inferences. Scientists, philosophers, and theologians accumulate knowledge when analyzing different aspects of reality and search for particular hypotheses or models to fit their respective subject matters. Of course, a main goal is then to integrate all kinds of knowledge into an all-encompassing worldview that would describe the whole of reality. A generous description of the whole of reality would span, in the order of complexity, from the purely physical to the supernatural. These two extreme aspects of reality are bridged by a nonphysical realm, which would include elements of life, man, consciousness, rationality, mental and mathematical abstractions, etc. An urgent problem in the theory of knowledge is what science is and what it is not. Albert Einstein’s notion of science in terms of sense perception is refined by defining operationally the data that makes up the subject matter of science. It is shown, for instance, that theological considerations included in the prior information assumed by Isaac Newton is irrelevant in relating the data logically to the model or hypothesis. In addition, the concepts of naturalism, intelligent design, and evolutionary theory are critically analyzed. Finally, Eugene P. Wigner’s suggestions concerning the nature of human consciousness, life, and the success of mathematics in the natural sciences is considered in the context of the creative power endowed in humans by God. (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 (...) class='Hi'>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)

In this work, we consider the problem of the approximate hedging of a contingent claim in the minimum mean square deviation criterion. A theorem on martingaỉe representation in the case of discrete time and an application of obtained result for semi-continous market model are given.

Most work at the intersection of law and the philosophy of action focuses on criminal responsibility. Unfortunately, this focus has been at the expense of reflecting on how the philosophy of action might help illuminate our understanding of issues in civil law. In this essay, focusing on Anglo-American jurisprudence, we examine the conditions under which a party to a legal agreement is deemed to have the capacity required to be bound by that agreement. We refer to this condition (...) as the capacity condition. We begin by showing how recent work on the metaphysics of powers might ground an account of the role of capacities in the metaphysics of intentional agency. After discussing the capacity to contract in Anglo-American jurisprudence, we show how the general ontology of capacities and agency sketched can ground and help clarify legal thinking about capacity. We argue that when we couple our understanding of capacity with relevant recent data from developmental neuropsychology, a lacuna in Anglo-American contract law becomes evident. Specifically, it appears that persons in early adulthood do not clearly satisfy the conditions requisite for them to satisfy the capacity condition. In the light of this, we sketch some potential solutions that are compatible with existing legal standards and would contribute to a more consistent application of the capacity condition in contract law. (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 n = 3 as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat’s approach of infinite (...) descent. The infinite descent is linked to induction starting from n = 3 by modus tollens. An inductive series of modus tollens is constructed. The proof of the series by induction is equivalent to Fermat’s last theorem. As far as Fermat had been proved the theorem for n = 4, one can suggest that the proof for n ≥ 4 was accessible to him. An idea for an elementary arithmetical proof of Fermat’s last theorem (FLT) by induction is suggested. It would be accessible to Fermat unlike Wiles’s proof (1995). (shrink)

Stanley Cavell’s account of aesthetic judgment has two components. The first is a feeling: the judge has to see, hear, ‘dig’ something in the object being judged, there has to be an ‘emotion’ that the judge feels and expresses. The second is the ‘discipline of accounting for [the judgment]’, a readiness to argue for one’s aesthetic judgment in the face of disagreement. The discipline of accounting for one’s aesthetic judgments involves what Nick Riggle has called a norm of convergence: the (...) judge aims to get one’s audience to taste see or hear what the judge tastes or sees or hears in the object being judged. Because of the unmistakable difficulty in reaching agreement in aesthetic judgment, Riggle has denied that aesthetic judgment requires a convergence norm and has proposed instead that it requires ‘a kind of harmony of individuality’ (which Riggle calls ‘vibing’). We argue that Cavell offers a version of the convergence norm that is distinct from those that Riggle criticizes, namely Kant’s demand for agreement and Andy Egan’s presupposition of similarity in dispositions in ‘non-defective’ aesthetic conversations. Cavell’s version of the convergence norm is ‘the hope of agreement’. One can hope that one’s audience will agree with one’s aesthetic judgments even when one isn’t in a position to demand agreement or to presuppose similarity in the dispositions that would make agreement more likely. Cavell’s distinct convergence norm avoids Riggle’s criticisms and contributes to a richer account of what’s going on when we disagree about aesthetic matters. (shrink)

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

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.

One of the predominant debates in AI Ethics is the worry and necessity to create fair, transparent and accountable algorithms that do not perpetuate current social inequities. I offer a critical analysis of Reuben Binns’s argument in which he suggests using public reason to address the potential bias of the outcomes of machine learning algorithms. In contrast to him, I argue that ultimately what is needed is not public reason per se, but an audit of the implicit moral assumptions of (...) societies within which algorithms are built and applied. Public justification is appealing since it offers us the possibility to align the decision-making outcomes of the algorithm with the core moral values of stakeholders within a constitutional democratic society. My concern is that the common moral principles that form the foundation of public reason are not necessarily neutral, as they still express specific moral ideals and normative standards even if there is moral agreement by society as a whole, or among different stakeholders within society. Appealing to such normative standards may thus still lead to algorithmic outcomes being biased as common moral values may very well still be discriminatory even though they are formed from a consensus, and even if public reason is applied as a kind of filter for potential algorithmic outcomes. Hence, I argue that these implicit moral norms within society that we take as a given in public reasoning, need to be audited from generation to generation in order to effectively mitigate potential algorithmic bias. (shrink)

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)

This paper begins with a puzzle regarding Lewis' theory of radical interpretation. On the one hand, Lewis convincingly argued that the facts about an agent's sensory evidence and choices will always underdetermine the facts about her beliefs and desires. On the other hand, we have several representation theorems—such as those of (Ramsey 1931) and (Savage 1954)—that are widely taken to show that if an agent's choices satisfy certain constraints, then those choices can suffice to determine her beliefs and desires. In (...) this paper, I will argue that Lewis' conclusion is correct: choices radically underdetermine beliefs and desires, and representation theorems provide us with no good reasons to think otherwise. Any tension with those theorems is merely apparent, and relates ultimately to the difference between how 'choices' are understood within Lewis' theory and the problematic way that they're represented in the context of the representation theorems. For the purposes of radical interpretation, representation theorems like Ramsey's and Savage's just aren't very relevant after all. (shrink)

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)