In mathematics, any form of probabilistic proof obtained through the application of a probabilistic method is not considered as a legitimate way of gaining mathematical knowledge. In a series of papers, Don Fallis has defended the thesis that there are no epistemic reasons justifying mathematicians’ rejection of probabilistic proofs. This paper identifies such an epistemic reason. More specifically, it is argued here that if one adopts a conception of mathematical knowledge in which an epistemic subject can (...) know a mathematical proposition based solely on a probabilistic proof, one is then forced to admit that such an epistemic subject can know several lottery propositions based solely on probabilistic evidence. Insofar as knowledge of lottery propositions on the basis of probabilistic evidence alone is denied by the vast majority of epistemologists, it is concluded that this constitutes an epistemic reason for rejecting probabilistic proofs as a means of acquiring mathematical knowledge. (shrink)

In a series of papers, Don Fallis points out that although mathematicians are generally unwilling to accept merely probabilistic proofs, they do accept proofs that are incomplete, long and complicated, or partly carried out by computers. He argues that there are no epistemic grounds on which probabilistic proofs can be rejected while these other proofs are accepted. I defend the practice by presenting a property I call ‘transferability’, which probabilistic proofs lack and acceptable proofs have. I also (...) consider what this says about the similarities between mathematics and, on the one hand natural sciences, and on the other hand philosophy. (shrink)

The author revisits the Blue Bus Problem, a famous thought-experiment in law involving probabilistic proof, and presents simple Bayesian solutions to different versions of the blue bus problem. In addition, the author expresses his solutions in standard and visual formats, i.e. in terms of probabilities and natural frequencies.

Is it right to convict a person of a crime on the basis of purely statistical evidence? Many who have considered this question agree that it is not, posing a direct challenge to legal probabilism – the claim that the criminal standard of proof should be understood in terms of a high probability threshold. Some defenders of legal probabilism have, however, held their ground: Schoeman (1987) argues that there are no clear epistemic or moral problems with convictions based on (...) purely statistical evidence, and speculates that our aversion to such convictions may be nothing more than an irrational bias. More recently, Hedden and Colyvan (2019, section VI) describe our reluctance to convict on the basis of purely statistical evidence as an ‘intuition’, but suggest that there may be no ‘in principle’ problem with such convictions (see also Papineau, forthcoming, section 6). In this paper, I argue that there is, in some cases, an in principle problem with a conviction based upon statistical evidence alone – namely, it commits us to a precedent which, if consistently followed through, could lead to the deliberate conviction of an innocent person. I conclude with some reflections on the idea that the criminal justice system should strive to maximise the accuracy of its verdicts – and the related idea that we should each strive to maximise the accuracy of our beliefs. (shrink)

Discusses whether legal proof is merely probabilistic, focusing on the famous proof paradox.

In a recent paper, Pettigrew reports a generalization of the celebrated accuracy-dominance theorem due to Predd et al., but Pettigrew’s proof is incorrect. I will explain the mistakes and provide a correct proof.

This note discusses three issues that Allen and Pardo believe to be especially problematic for a probabilistic interpretation of standards of proof: (1) the subjectivity of probability assignments; (2) the conjunction paradox; and (3) the non-comparative nature of probabilistic standards. I offer a reading of probabilistic standards that avoids these criticisms.

A fundamental problem in understanding the nature of time is explaining its directionality. This 1990 PhD thesis re-examines the concepts of time flow, the physical directionality of time, and the semantics of tensed language. Several novel results are argued for that contradict the orthodox anti-realist views still dominant in the subject. Specifically, the concept of "metaphysical time flow" is supported as a valid scientific concept, and argued to be intrinsic to the directionality of objective probabilities in quantum mechanics; the common (...) claims that quantum probability theory is time reversible is shown to be based on an analytic error, stemming from a false choice for the criterion for reversibility of probabilistic theories (recognized by Satosi Watanabe in the 1950s but ignored in all philosophical discussions); and a consistent semantics for tensed language (adapted from the tree model of Storrs McCall) is constructed, showing that the common rejection of "time flow" as having no meaningful semantics is false. These debates are still ongoing in almost exactly the same state they were pre-1990, and there is appears to be no visible progress in the subject. Critical points made against errors in the orthodox account (which has been sustained for 70 years by the anti-realist philosophy of time, typified by the "Pittsburg School" of Grunbaum-Earman-Norton-Roberts), are still not recognized in the philosophy of time or physics. Some key technical proofs in this thesis have been published in physics proper. See p.ii-iii for full original abstract. (This pdf is uploaded from the Massey University archive.). (shrink)

The paper is a continuation of another paper published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The (...) relevant mathematical structure is Hilbert arithmetic in a wide sense, in the framework of which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual to each other. A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction. If the former is the case, the application of set theory and arithmetic can remain disjunctively divided: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. If the latter is the case, the proof is thoroughly within set theory. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided in both cases. The idea of “arithmetic mechanics” is sketched: it might deduce the basic physical dimensions of mechanics (mass, time, distance) from the axioms of arithmetic after a relevant generalization, Furthermore, a future Part III of the paper is suggested: FLT by mediation of Hilbert arithmetic in a wide sense can be considered as another expression of Gleason’s theorem in quantum mechanics: the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information. The availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent to the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem. (shrink)

Abstract: Four main forms of Doomsday Argument (DA) exist—Gott’s DA, Carter’s DA, Grace’s DA and Universal DA. All four forms use different probabilistic logic to predict that the end of the human civilization will happen unexpectedly soon based on our early location in human history. There are hundreds of publications about the validity of the Doomsday argument. Most of the attempts to disprove the Doomsday Argument have some weak points. As a result, we are uncertain about the validity of (...) DA proofs and rebuttals. In this article, a meta-DA is introduced, which uses the idea of logical uncertainty over the DA’s validity estimated based on a virtual prediction market of the opinions of different scientists. The result is around 0.4 for the validity of some form of DA, and even smaller for “Strong DA”, which predicts the end of the world in the near term. We discuss many examples of the validity of the DA in real life as an instrument to prove it “experimentally”. We also show that DA becomes strongest if it is based on the idea of the “natural reference class” of observers, that is, the observers who know about the DA (i.e. a Self-Referenced DA). Such a DA predicts that there is a high probability of a global catastrophe with human extinction in the 21st century, which aligns with what we already know based on analysis of different technological risks. (shrink)

It is well known that the probabilistic relation of confirmation is not transitive in that even if E confirms H1 and H1 confirms H2, E may not confirm H2. In this paper we distinguish four senses of confirmation and examine additional conditions under which confirmation in different senses becomes transitive. We conduct this examination both in the general case where H1 confirms H2 and in the special case where H1 also logically entails H2. Based on these analyses, we argue (...) that the Screening-Off Condition is the most important condition for transitivity in confirmation because of its generality and ease of application. We illustrate our point with the example of Moore’s “proof” of the existence of a material world, where H1 logically entails H2, the Screening-Off Condition holds, and confirmation in all four senses turns out to be transitive. (shrink)

In legal proceedings, a fact-finder needs to decide whether a defendant is guilty or not based on probabilistic evidence. We defend the thesis that the defendant should be found guilty just in case it is rational for the fact-finder to believe that the defendant is guilty. We draw on Leitgeb’s stability theory for an appropriate notion of rational belief and show how our thesis solves the problem of statistical evidence. Finally, we defend our account of legal proof against (...) challenges from Staffel and compare it to a recent competitor put forth by Moss. (shrink)

Medicine, like law, is a pragmatic, probabilistic activity. Both require that decisions be made on the basis of available evidence, within a limited time. In contrast to law, medicine, particularly evidence-based medicine as it is currently practiced, aspires to a scientific standard of proof, one that is more certain than the standards of proof courts apply in civil and criminal proceedings. But medicine, as Dr. William Osler put it, is an "art of probabilities," or at best, a (...) "science of uncertainty." One can better practice medicine by using other evidentiary standards in addition to the "scientific." To employ only the scientific standard of proof is inappropriate, if not impossible; furthermore, as this review will show, its application in medicine is fraught with bias. Evidence is information. It supports or undermines a proposition, whether a hypothesis in science, a diagnosis in medicine, or a fact or point in question in a legal investigation. In medicine, physicians marshal evidence to make decisions on how to best prevent, diagnose, and treat disease, and improve health. In law, courts decide the facts and render justice. Judges and juries assess evidence to establish liability, to settle custody and medical issues, and to determine a defendant's guilt or innocence. (shrink)

A probability aggregation rule assigns to each profile of probability functions across a group of individuals (representing their individual probability assignments to some propositions) a collective probability function (representing the group's probability assignment). The rule is “non-manipulable” if no group member can manipulate the collective probability for any proposition in the direction of his or her own probability by misrepresenting his or her probability function (“strategic voting”). We show that, except in trivial cases, no probability aggregation rule satisfying two mild (...) conditions (non-dictatorship and consensus preservation) is non-manipulable. (shrink)

There is a long tradition in formal epistemology and in the psychology of reasoning to investigate indicative conditionals. In psychology, the propositional calculus was taken for granted to be the normative standard of reference. Experimental tasks, evaluation of the participants’ responses and psychological model building, were inspired by the semantics of the material conditional. Recent empirical work on indicative conditionals focuses on uncertainty. Consequently, the normative standard of reference has changed. I argue why neither logic nor standard probability theory provide (...) appropriate rationality norms for uncertain conditionals. I advocate coherence based probability logic as an appropriate framework for investigating uncertain conditionals. Detailed proofs of the probabilistic non-informativeness of a paradox of the material conditional illustrate the approach from a formal point of view. I survey selected data on human reasoning about uncertain conditionals which additionally support the plausibility of the approach from an empirical point of view. (shrink)

This book tells the story of modern ethics, namely the story of a discourse that, after the Renaissance, went through a methodological revolution giving birth to Grotius’s and Pufendorf’s new science of natural law, leaving room for two centuries of explorations of the possible developments and implications of this new paradigm, up to the crisis of the Eighties of the eighteenth century, a crisis that carried a kind of mitosis, the act of birth of both basic paradigms of the two (...) following centuries: Kantian ethics and utilitarianism. The new science of natural law carried a fresh start for ethics, resulting from a mixture of the Old and the New. It was, as suggested by Schneewind, an attempt at rescuing the content of Scholastic and Stoic doctrines on a new methodological basis. The former was the claim of existence of objective and universal moral laws; the latter was the self-aware attempt at justifying a minimal kernel of such laws facing skeptical doubt. What Bentham and Kant did was precisely carrying this strategy further on, even if restructuring it each of them around one out of two alternative basic claims. The nineteenth- and twentieth-century critics of the Enlightenment attacked both not on their alleged failure in carrying out their own projects, but precisely on having adopted Grotius’s and Pufendorf’s project. What counter-enlightenment has been unable to spell out is which alternative project could be carried out facing the modern condition of pluralism, while on the contrary, if we takes a closer look at developments in twentieth-century ethics or at on-going discussions on practical issues, we might feel inclined to believe that Grotius’s and Pufendorf’s project is as up-to-date as ever. -/- Table of Contents -/- Preface I. Fathers of the Reformation and Schoolmen 1.1. Luther: passive justice and the good deeds; 1.2. Calvin: voluntarism and predestination; 1.3. Baroque Scholasticism; 1.4. Casuistry and Institutiones morales -/- II Neo-Platonists, neo-Stoics, neo-Sceptics 2.1. Aristotelian, neo-Platonic, neo-Epicurean and neo-Cynic Humanists; 2.2. Oeconomica and the art of living; 2.3. Neo-Stoics; 2.4. Neo-Sceptics; 2.5. Moralistic literature -/- III Neo-Augustinians 3.l. The Jansenists on natura lapsa, sufficient grace, pure love; 3.2. Nicole on the impossibility of self-knowledge; 3.3. Nicole on self-love and charity; 3.4. Nicole against civic virtue, for Christian civility; 3.5. Malebranche on general laws and necessary evil; 3.6. Malebranche on Neo-Augustinianism and Platonism. -/- IV Grotius, Pufendorf and the new moral science 4.1. Grotius against Aristotle and the sceptics; 4.2. Mersenne and Gassendi; 4.3. Descartes on ethics as the last branch of philosophy’s tree; 4.4. Hobbes on scepticism and the new moral science; 4.5. Spinoza on the new moral science as a descriptive science;4.6. Locke on voluntarism and probabilism; 4.7. Pufendorf on natural law as an exact science; 4.8. Pufendorf on physical and moral entities; 10. Pufendorf on self-preservation -/- V The empiricist version of the new moral science: from Cumberland to Paley 5.1. Cumberland against Hobbesian voluntarism; 5.2. Cumberland and theological consequentialism; 5.3. Cumberland on universal benevolence and self-love; 5.4. Shaftesbury on the moral sense; 5.5. Hutcheson on natural law and moral faculties; 5.6. Gay, Brown, Paley and theological consequentialism. -/- VI The rationalist version of the new moral science: from Cudworth to Price 6.1. The Cambridge Platonists; 6.2. Shaftesbury on the moral sense; 6.3. Butler and a third way between voluntarism and scepticism; 6.4. Price and the rational character of moral truths; -/- VII Leibniz’s compromise between the new moral science and Aristotelianism 1.Leibniz against voluntarism; 2.Leibniz against the division between the physical and the moral good; 3.Leibniz on la place d’autrui and theological consequentialism; 4.Thomasius, Wolff, Crusius -/- VIII French eighteenth-century philosophers without the new moral science 8.1. The genealogy of our ideas of virtue and vice; 8.2. Maupertuis and moral arithmetic 8.3. The philosophes and the harmony of interests; 8.4. Rousseau on corruption, self-love, and virtue; 8.5. Sade on the merits of vice -/- IX Experimental moral science: Hume and Adam Smith 9.1. Mandeville’s paradox; 9.2. Hutcheson on the law of nature and moral faculties; 9.3. Hume on experimental moral philosophy and the intermediate principles; 9.4. Hume’s Law; 9.5. Hume on the fellow-feeling; 9.6. Hume on natural and artificial virtues and disinterested pleasure for utility; 9.7. Adam Smith’s anti-realist metaethics; 9.8. Adam Smith on self-deception and the paradox of happiness; 9.9. Adam Smith on sympathy and the impartial spectator; 9.10. Adam Smith on the twofold criterion for moral judgement and its paradox; 9.11. Reid on the refutation of scepticism and the self-evidence of duty -/- X Kantian ethics 10.1. Kantian metaethics: moral epistemology; 10.2. Kantian metaethics: moral ontology; 10.3. Kantian metaethics: moral psychology; 10.4. Kantian normative ethics; 10.5. Kant on the impracticability of applied ethics; 10.6. Kantian moral anthropology; 10.7. Civilisation and moralisation; 10.8. Theology on a moral basis and the origins of evil; 10.9. Fichte and the transformation of theoretical philosophy into practical philosophy XI Bentham and utilitarianism 11.1. Bentham’s linguistic theory; 11.2. Bentham’s moral ontology, psychology, and theory of action; 11.3. The principle of greatest happiness; 11.4. The critique of religious ethics; 11.5. The new morality; 11.6. Interest and duty; 11.7. Virtues; 11.8. Private ethics and legislation -/- XII Followers of the Enlightenment: liberal Judaism and Liberal Theology 12.1. Mendelssohn; 12.2. Salomon Maimon; 12.3. Haskalā and liberal Judaism; 12.4. Liberal Theology. -/- XIII Counter-Enlighteners 13.1.Romanticism and the fulfilment of individuality as the Summum Bonum; 13.2. Hegel on history as the making of liberty; 13.3. Hegel on the unhappy consciousness and the beautiful soul; 13.4. Hegel on Morality and Sittlichkeit; 13.5. Marx on ideology, alienation, and praxis; 13.6. Schopenhauer on compassion; 13.7. Kierkegaard on faith beyond ethics. -/- XIV Followers of the Enlightenment: intuitionists and utilitarian 14.1 Whewell‘s criticism of utilitarianism; 14.2 Whewell on morality and the philosophy of morality; 14.3 Whewell on the Supreme Norm; 14.4 Whewell on the conflict between duties; 14.5 Mill and the proof of the principle of utility; 14.6 Mill’s eudemonistic utilitarianism; 14.7 Mill on rules -/- XV Followers of the Enlightenment: neo-Kantians and positivists 15.1. French spiritualism; 15.2. Neo-Kantians: the Marburg school; 15.3. Neo-Kantians: the Marburg school; 15.4. Comte’s positivism and the invention of altruism; 15.5. Social Darwinism; 15.6. Wundt and an ethic of humankind -/- XVI Post-enlighteners: Sidgwick 16.1. Criticism of intuitionism; 16.2. On ethical egoism; 16.3. Criticism of utilitarianism -/- XVII Post-enlighteners: Durkheim 17.1. Sociology as physics of customs; 17.2. Morality as physics of customs and as practical science; 17.3. On Kantian ethics and utilitarianism; 17.4. The variability of moralities;17.5. Social solidarity as end and justification of morality; 17.6. Secular morality as “sociodicy”; XVIII Post-enlighteners: Nietzsche 18.1. On the Dionysian; 18.2. On the deconstruction of the world of values 18.3 On the twofold genealogy of moralities; 18.4. On ascetics and nihilism; 18.5. Normative ethics of self-fulfilment -/- Bibliography / Index of names / Index of concepts -/- . (shrink)

I set up two axiomatic theories of inductive support within the framework of Kolmogorovian probability theory. I call these theories ‘Popperian theories of inductive support’ because I think that their specific axioms express the core meaning of the word ‘inductive support’ as used by Popper (and, presumably, by many others, including some inductivists). As is to be expected from Popperian theories of inductive support, the main theorem of each of them is an anti-induction theorem, the stronger one of them saying, (...) in fact, that the relation of inductive support is identical with the empty relation. It seems to me that an axiomatic treatment of the idea(s) of inductive support within orthodox probability theory could be worthwhile for at least three reasons. Firstly, an axiomatic treatment demands from the builder of a theory of inductive support to state clearly in the form of specific axioms what he means by ‘inductive support’. Perhaps the discussion of the new anti-induction proofs of Karl Popper and David Miller would have been more fruitful if they had given an explicit definition of what inductive support is or should be. Secondly, an axiomatic treatment of the idea(s) of inductive support within Kolmogorovian probability theory might be accommodating to those philosophers who do not completely trust Popperian probability theory for having theorems which orthodox Kolmogorovian probability theory lacks; a transparent derivation of anti-induction theorems within a Kolmogorovian frame might bring additional persuasive power to the original anti-induction proofs of Popper and Miller, developed within the framework of Popperian probability theory. Thirdly, one of the main advantages of the axiomatic method is that it facilitates criticism of its products: the axiomatic theories. On the one hand, it is much easier than usual to check whether those statements which have been distinguished as theorems really are theorems of the theory under examination. On the other hand, after we have convinced ourselves that these statements are indeed theorems, we can take a critical look at the axioms—especially if we have a negative attitude towards one of the theorems. Since anti-induction theorems are not popular at all, the adequacy of some of the axioms they are derived from will certainly be doubted. If doubt should lead to a search for alternative axioms, sheer negative attitudes might develop into constructive criticism and even lead to new discoveries. -/- I proceed as follows. In section 1, I start with a small but sufficiently strong axiomatic theory of deductive dependence, closely following Popper and Miller (1987). In section 2, I extend that starting theory to an elementary Kolmogorovian theory of unconditional probability, which I extend, in section 3, to an elementary Kolmogorovian theory of conditional probability, which in its turn gets extended, in section 4, to a standard theory of probabilistic dependence, which also gets extended, in section 5, to a standard theory of probabilistic support, the main theorem of which will be a theorem about the incompatibility of probabilistic support and deductive independence. In section 6, I extend the theory of probabilistic support to a weak Popperian theory of inductive support, which I extend, in section 7, to a strong Popperian theory of inductive support. In section 8, I reconsider Popper's anti-inductivist theses in the light of the anti-induction theorems. I conclude the paper with a short discussion of possible objections to our anti-induction theorems, paying special attention to the topic of deductive relevance, which has so far been neglected in the discussion of the anti-induction proofs of Popper and Miller. (shrink)

The previous two parts of the paper demonstrate that the interpretation of Fermat’s last theorem (FLT) in Hilbert arithmetic meant both in a narrow sense and in a wide sense can suggest a proof by induction in Part I and by means of the Kochen - Specker theorem in Part II. The same interpretation can serve also for a proof FLT based on Gleason’s theorem and partly similar to that in Part II. The concept of (probabilistic) measure (...) of a subspace of Hilbert space and especially its uniqueness can be unambiguously linked to that of partial algebra or incommensurability, or interpreted as a relation of the two dual branches of Hilbert arithmetic in a wide sense. The investigation of the last relation allows for FLT and Gleason’s theorem to be equated in a sense, as two dual counterparts, and the former to be inferred from the latter, as well as vice versa under an additional condition relevant to the Gödel incompleteness of arithmetic to set theory. The qubit Hilbert space itself in turn can be interpreted by the unity of FLT and Gleason’s theorem. The proof of such a fundamental result in number theory as FLT by means of Hilbert arithmetic in a wide sense can be generalized to an idea about “quantum number theory”. It is able to research mathematically the origin of Peano arithmetic from Hilbert arithmetic by mediation of the “nonstandard bijection” and its two dual branches inherently linking it to information theory. Then, infinitesimal analysis and its revolutionary application to physics can be also re-realized in that wider context, for example, as an exploration of the way for physical quantity of time (respectively, for time derivative in any temporal process considered in physics) to appear at all. Finally, the result admits a philosophical reflection of how any hierarchy arises or changes itself only thanks to its dual and idempotent counterpart. (shrink)

This paper outlines a formal recursive wager resolution calculus (WRC) that provides a novel conceptual framework for sentential logic via bridge rules that link wager resolution with truth values. When paired with a traditional truth-centric criterion of logical soundness WRC generates a sentential logic that is broadly truth-conditional but not truth-functional, supports the rules of proof employed in standard mathematics, and is immune to the most vexing features of their traditional implementation. WRC also supports a novel probabilistic criterion (...) of logical soundness, the fair betting probability criterion (FBP). It guarantees that the conclusion of an FBP-valid argument is at least as credible as a conjunction of premises, and also that the conclusion is true if the premises are. In addition, WRC provides a platform for a novel non-probabilistic, computationally simpler criterion of logical soundness – the criterion of Super-validity - that issues the same logical appraisals as FBP, and hence the same guarantees. (shrink)

The article is devoted to the analysis of the evident scientific method of Galen, which establishes the necessity of correct diagnosis of diseases, determination of true symptoms and causes of diseases, which results in the choice of the exact method of treatment. The article focuses on how Galen seeks to achieve reliable knowledge based on an undeniable logical necessity. Logical reliability is contrasted with “dialectical”, that is, probabilistic judgments, often leading to the opposite of what was originally asserted in (...) them. Probabilistic judgments were characteristic of the Stoic philosophical school, with which Galen hotly argues, asserting his understanding of truth. Truth, in his opinion, is achieved through facts based on true premises. The criterion of truth for Galen is the study of the device of a living organism, and the logical conclusions depend on the accuracy of knowledge of human anatomy. Thus, the nature of the proof judgments of Galen depends on their practical content and not on the formal and logical structure offered by the Stoics when studying the functions of the human body, which ultimately leads to incorrect methods of treatment. (shrink)

J. Schauder introduced the notion of basis in a Banach space in 1927. If a Banach space has a basis then it is also separable. The problem whether every separable Banach space has a Schauder basis appeared for the first time in 1931 in Banach's book "Theory of Linear Operations". If a Banach space has a Schauder basis it also has the approximation property. A Banach space X has the approximation property if for every Banach space Y the finite rank (...) operators are dense in the closed subspace of all compact operators from Y to X. Both problems were solved in the negative in 1972 by Per Enflo. His proof was almost immediately simplified by A. M. Davie, who using a probabilistic lemma constructed a separable closed subspace of l_inifnity without the approximation property. In this thesis we present some of the equivalent properties to the approximation property due to A. Grothendieck, and we make a detailed exposition of the proof by A. M. Davie. (shrink)

Aristotle divided arguments that persuade into the rhetorical (which happen to persuade), the dialectical (which are strong so ought to persuade to some degree) and the demonstrative (which must persuade if rightly understood). Dialectical arguments were long neglected, partly because Aristotle did not write a book about them. But in the sixteenth and seventeenth century late scholastic authors such as Medina, Cano and Soto developed a sound theory of probable arguments, those that have logical and not merely psychological force but (...) fall short of demonstration. Informed by late medieval treatments of the law of evidence and problems in moral theology and aleatory contracts, they considered the reasons that could render legal, moral, theological, commercial and historical arguments strong though not demonstrative. At the same time, demonstrative arguments became better understood as Galileo and other figures of the Scientific Revolution used mathematical proof in arguments in physics. Galileo moved both dialectical and demonstrative arguments into mathematical territory. (shrink)

An annotated bibliography of legal epistemology.

Eyal Shahar’s essay review [1] of James Penston’s remarkable book [2] seems more inspired playful academic provocation than review or essay, expressing dramatic views of impossible validity. The account given of modern biostatistical causation reveals the slide from science into the intellectual confusion and non-science RCTs have created: “…. the purpose of medical research is to estimate the magnitude of the effect of a causal contrast, for example the probability ratio of a binary outcome …” But Shahar’s world is simultaneously (...) not probabilistic, but of absolute uncertainty: “We should have no confidence in any type of evidence ….. We should have no confidence at all”. Shahar’s "Causal contrast" is attractive. It seems to make sense, but bypasses in two words the means of establishing causation by the scientific method. This phrase assumes a numeric statistically significant “contrast” is causal rather than a potential correlation requiring further investigation. The concept of “causal contrast” is a slippery slope from sense into biostatistical non-science. This can be illustrated with an hypothetical RCT where 100% of interventions exhibit a posited treatment effect and 0% of placebo controls. Internal validity is seemingly quite reasonably assumed satisfied (common-sense dictating the likelihood of an awesome magnificent fraud, bias or plain error of the magnitude required is infinitesimal). Scientific method appears satisfied. The RCT demonstrates: (1) strict regularity of outcome in the presence of posited cause; (2) the absence of outcome in its absence and (3) an intervention (experiment) showing the direction of causation is from posited cause to posited effect. Now travel further down the slope from science. Assume 50% of interventions and 0% of controls are positive. We compromise scientific method, but justify this by assuming a large subgroup which we say surely must on these figures be exhibiting the posited treatment effect. But what of 10% of interventions and 9% of placebo controls exhibiting the posited treatment effect? Our biostatistician says the 1% “causal contrast” is statistically significant. But we have: (1) minimal evidence of regularity; (2) the posited outcome irrespective of presence of posited cause and (3) our intervention is at the highest equivocal in demonstrating any form of causation. This is not science. It is, however, where biostatistics has unthinkingly taken us, as Penston has shown comprehensively [2]. We, the audience of published medical research, are now for the 10% / 9% example well down the slope from science. An unattractive hypothesis results requiring numerous assumptions similar to these:- "There is a 'contrast' which is ‘causal’, albeit the method employed is not scientific. An effect of the intervention has been observed in a very small subgroup. This subgroup is susceptible to treatment. The similar number of placebo controls exhibiting the outcome sought is irrelevant, because the 1% difference between intervention and controls is statistically significant. The statistical analysis is valid and reliable. The RCT’s internal validity is sufficiently satisfied. No funding or bias or fraud has affected the results or their analysis.” As Penston notes: “Confirming and refuting the results of research is crucial to science …. But … there’s no way of testing the results of any particular large-scale RCT or epidemiological study. Each study … is left hanging in the air, unsupported.” It gets worse. To identify a rare serious adverse reaction of a frequency of 1:10,000 can require a trial of 200,000 or larger split between controls and interventions. This is not done. But for every 100 who prospectively benefit from the intervention, 9,900 also receive it. And for every 100 benefiting one person (who likely gains no benefit) will suffer a serious unidentified adverse reaction. This is also without taking account of more common adverse reactions whether serious or otherwise. References [1] Shahar, E. (2011) Research and medicine: human conjectures at every turn. International Journal of Person Centered Medicine 1 (2), 250-253. [2] Penston, J. (2010). stats.con: How we’ve been fooled by statistics-based research in medicine. London: The London Press, UK. (shrink)

There is a middle ground of imperfect knowledge in fields like medicine and the social sciences. It stands between our day-to-day relatively certain knowledge obtained from ordinary basic observation of regularities in our world and our knowledge from well-validated theories in the physical sciences. -/- The latter enable reliable prediction a great deal of the time of the happening of events never before experienced. The former enable prediction only of what has happened before and beyond that of educated guesses which (...) may sometimes prove right and others not when we test them. -/- The imperfection of our knowledge between those limits is a consequence of complexity. -/- Reductionist empiricism fails when faced with complexity but we all still have to live in a complex world where reductionist science cannot help us devise reliable theories from which we can predict the behaviour of the world around us. We cannot predict reliably. We can only monitor actively and manage actively the world around us if we want to try to control our environment. We have a limited prediction horizon. -/- Science and empiricism work well if we want to send Voyager I and II over 3 billion miles away but not for most other things. -/- In medicine how and when to apply probabilistic, conjectural, incomplete medical theories and explanations requires professional expertise, intuition and judgement (non scientific knowledge) which is essential. Medical diagnosis is a skill of predicting from knowledge of what has happened before that the past will recur in the current patient applying expertise and intuition from knowledge and experience of prior cases and probabilistic medical research and theory. The physician is left to an educated guess as to how the future will develop for the particular individual patient the focus of his or her current attention. (shrink)

Here, the author tries to build the structure of the Theory of computation based on considering time as a fuzzy concept. In fact, there are reasons to consider time as a fuzzy concept. In this article, the author doesn’t go to this side but note that Brower and Husserl views on the concept of time were similar [8]. Some reasons have been given for it in [3]. Throughout this article, the author presents the Theory of Computation with Fuzzy Time. Given (...) the classic definition of Turing Machine, the concept of Time is modified to Fuzzy time. This new term calls as Theory TC* [2] and this type of computation “Fuzzy time Computation”. We have relatively large number of fundamental unsolved problems in Complexity Theory. In the new theory, some of the major obstacles and unsolved problems have been solved [2]. It should be noted that in this article, the writer considers fuzzy number associated to instants of time as a symmetric one. The point about the symmetry is in the proof of Lemma 3, although it is generalizable. In particular, the new classes of complexity Theory, P*, NP*, BPP* in the TC* analogues to the definitions of P, NP, BPP defines as their natural alternative definition. Here, we will see P*≠ NP*, P*= BPP*. Finally, we have Theorem 4. (shrink)

Here, we try to build the structure of a Theory of computation based on considering time as a fuzzy concept. Actually, there are some reasons to consider time as a fuzzy concept. In this article, we don’t go to this side but we remind that Brower and Husserl ideas about the concept of time were similar [14]. Throughout this article, we present the Theory of Computation with Fuzzy Time. Considering the classical definition of Turing Machine we change and modify the (...) concept of Time to Fuzzy time. We call this new Theory TC* and this type of computation “Fuzzy time Computation”. We have relatively large number of fundamental unsolved problems in Complexity Theory. In the new Theory some of the major obstacles and unsolved problems are solved. It should be mentioned that in this article, we consider fuzzy number a symmetric one. The point about the symmetry is in the proof of Lemma 3, although we are able to generalize it. More specifically, we define the new classes of complexity Theory, P*, NP*, BPP* in TC* analogues to the definitions P, NP, BPP as their natural substituted definition. We show P*≠ NP*, P*= BPP*. Finally, we have Theorem 4, P≠NP . (shrink)

The aim of the paper is to develop general criteria of argumentative validity and adequacy for probabilistic arguments on the basis of the epistemological approach to argumentation. In this approach, as in most other approaches to argumentation, proabilistic arguments have been neglected somewhat. Nonetheless, criteria for several special types of probabilistic arguments have been developed, in particular by Richard Feldman and Christoph Lumer. In the first part (sects. 2-5) the epistemological basis of probabilistic arguments is discussed. With (...) regard to the philosophical interpretation of probabilities a new subjectivist, epistemic interpretation is proposed, which identifies probabilities with tendencies of evidence (sect. 2). After drawing the conclusions of this interpretation with respect to the syntactic features of the probability concept, e.g. one variable referring to the data base (sect. 3), the justification of basic probabilities (priors) by judgements of relative frequency (sect. 4) and the justification of derivative probabilities by means of the probability calculus are explained (sect. 5). The core of the paper is the definition of '(argumentatively) valid derivative probabilistic arguments', which provides exact conditions for epistemically good probabilistic arguments, together with conditions for the adequate use of such arguments for the aim of rationally convincing an addressee (sect. 6). Finally, some measures for improving the applicability of probabilistic reasoning are proposed (sect. 7). (shrink)

Are probabilism and special relativity compatible? Dieks argues that they are. But the possible universe he specifies, designed to exemplify both probabilism and special relativity, either incorporates a universal “now”, or amounts to a many world universe, or fails to have any one definite overall Minkowskian-type space-time structure. Probabilism and special relativity appear to be incompatible after all. What is at issue is not whether “the flow of time” can be reconciled with special relativity, but rather whether explicitly probabilistic (...) versions of quantum theory should be rejected because of incompatibility with special relativity. (shrink)

In this paper I expound an argument which seems to establish that probabilism and special relativity are incompatible. I examine the argument critically, and consider its implications for interpretative problems of quantum theory, and for theoretical physics as a whole.

Should we use the same standard of proof to adjudicate guilt for murder and petty theft? Why not tailor the standard of proof to the crime? These relatively neglected questions cut to the heart of central issues in the philosophy of law. This paper scrutinises whether we ought to use the same standard for all criminal cases, in contrast with a flexible approach that uses different standards for different crimes. I reject consequentialist arguments for a radically flexible standard (...) of proof, instead defending a modestly flexible approach on non-consequentialist grounds. The system I defend is one on which we should impose a higher standard of proof for crimes that attract more severe punishments. This proposal, although apparently revisionary, accords with a plausible theory concerning the epistemology of legal judgments and the role they play in society. (shrink)

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

The explanatory role of natural selection is one of the long-term debates in evolutionary biology. Nevertheless, the consensus has been slippery because conceptual confusions and the absence of a unified, formal causal model that integrates different explanatory scopes of natural selection. In this study we attempt to examine two questions: (i) What can the theory of natural selection explain? and (ii) Is there a causal or explanatory model that integrates all natural selection explananda? For the first question, we argue that (...) five explananda have been assigned to the theory of natural selection and that four of them may be actually considered explananda of natural selection. For the second question, we claim that a probabilistic conception of causality and the statistical relevance concept of explanation are both good models for understanding the explanatory role of natural selection. We review the biological and philosophical disputes about the explanatory role of natural selection and formalize some explananda in probabilistic terms using classical results from population genetics. Most of these explananda have been discussed in philosophical terms but some of them have been mixed up and confused. We analyze and set the limits of these problems. (shrink)

The best and most popular argument for probabilism is the accuracy-dominance argument, which purports to show that alethic considerations alone support the view that an agent’s degrees of belief should always obey the axioms of probability. I argue that extant versions of the accuracy-dominance argument face a problem. In order for the mathematics of the argument to function as advertised, we must assume that every omniscient credence function is classically consistent; there can be no worlds in the set of dominance-relevant (...) worlds that obey some other logic. This restriction cannot be motivated on alethic grounds unless we’re also willing to accept that rationality requires belief in every metaphysical necessity, as the distinction between a priori logical necessities and a posteriori metaphysical ones is not an alethic distinction. To justify the restriction to classically consistent worlds, non-alethic motivation is required. And thus, if there is a version of the accuracy-dominance argument in support of probabilism, it isn’t one that is grounded in alethic considerations alone. (shrink)

This paper offers a probabilistic treatment of the conditions for argument cogency as endorsed in informal logic: acceptability, relevance, and sufficiency. Treating a natural language argument as a reason-claim-complex, our analysis identifies content features of defeasible argument on which the RSA conditions depend, namely: change in the commitment to the reason, the reason’s sensitivity and selectivity to the claim, one’s prior commitment to the claim, and the contextually determined thresholds of acceptability for reasons and for claims. Results contrast with, (...) and may indeed serve to correct, the informal understanding and applications of the RSA criteria concerning their conceptual dependence, their function as update-thresholds, and their status as obligatory rather than permissive norms, but also show how these formal and informal normative approachs can in fact align. (shrink)

Nick Shea’s Representation in Cognitive Science commits him to representations in perceptual processing that are about probabilities. This commentary concerns how to adjudicate between this view and an alternative that locates the probabilities rather in the representational states’ associated “attitudes”. As background and motivation, evidence for probabilistic representations in perceptual processing is adduced, and it is shown how, on either conception, one can address a specific challenge Ned Block has raised to this evidence.

How can different individuals' probability functions on a given sigma-algebra of events be aggregated into a collective probability function? Classic approaches to this problem often require 'event-wise independence': the collective probability for each event should depend only on the individuals' probabilities for that event. In practice, however, some events may be 'basic' and others 'derivative', so that it makes sense first to aggregate the probabilities for the former and then to let these constrain the probabilities for the latter. We formalize (...) this idea by introducing a 'premise-based' approach to probabilistic opinion pooling, and show that, under a variety of assumptions, it leads to linear or neutral opinion pooling on the 'premises'. This paper is the second of two self-contained, but technically related companion papers inspired by binary judgment-aggregation theory. (shrink)

Suppose several individuals (e.g., experts on a panel) each assign probabilities to some events. How can these individual probability assignments be aggregated into a single collective probability assignment? This article reviews several proposed solutions to this problem. We focus on three salient proposals: linear pooling (the weighted or unweighted linear averaging of probabilities), geometric pooling (the weighted or unweighted geometric averaging of probabilities), and multiplicative pooling (where probabilities are multiplied rather than averaged). We present axiomatic characterisations of each class of (...) pooling functions (most of them classic, but one new) and argue that linear pooling can be justified procedurally, but not epistemically, while the other two pooling methods can be justified epistemically. The choice between them, in turn, depends on whether the individuals' probability assignments are based on shared information or on private information. We conclude by mentioning a number of other pooling methods. (shrink)

This article introduces the notion of the directedness of a process, which underlies event causation as well as the persistence of things. Using this notion it investigates what happens in typical cases of active event causation. Causes never necessitate their effects because even non- probabilistic causes can be counteracted.

The epistemic modal auxiliaries must and might are vehicles for expressing the force with which a proposition follows from some body of evidence or information. Standard approaches model these operators using quantificational modal logic, but probabilistic approaches are becoming increasingly influential. According to a traditional view, must is a maximally strong epistemic operator and might is a bare possibility one. A competing account—popular amongst proponents of a probabilisitic turn—says that, given a body of evidence, must \ entails that \\) (...) is high but non-maximal and might \ that \\) is significantly greater than 0. Drawing on several observations concerning the behavior of must, might and similar epistemic operators in evidential contexts, deductive inferences, downplaying and retractions scenarios, and expressions of epistemic tension, I argue that those two influential accounts have systematic descriptive shortcomings. To better make sense of their complex behavior, I propose instead a broadly Kratzerian account according to which must \ entails that \ = 1\) and might \ that \ > 0\), given a body of evidence and a set of normality assumptions about the world. From this perspective, must and might are vehicles for expressing a common mode of reasoning whereby we draw inferences from specific bits of evidence against a rich set of background assumptions—some of which we represent as defeasible—which capture our general expectations about the world. I will show that the predictions of this Kratzerian account can be substantially refined once it is combined with a specific yet independently motivated ‘grammatical’ approach to the computation of scalar implicatures. Finally, I discuss some implications of these results for more general discussions concerning the empirical and theoretical motivation to adopt a probabilisitic semantic framework. (shrink)

There is wide support in logic, philosophy, and psychology for the hypothesis that the probability of the indicative conditional of natural language, P(if A then B), is the conditional probability of B given A, P(B|A). We identify a conditional which is such that P(if A then B)=P(B|A) with de Finetti's conditional event, B|A. An objection to making this identification in the past was that it appeared unclear how to form compounds and iterations of conditional events. In this paper, we illustrate (...) how to overcome this objection with a probabilistic analysis, based on coherence, of these compounds and iterations. We interpret the compounds and iterations as conditional random quantities which, given some logical dependencies, may reduce to conditional events. We show how the inference to B|A from A and B can be extended to compounds and iterations of both conditional events and biconditional events. Moreover, we determine the respective uncertainty propagation rules. Finally, we make some comments on extending our analysis to counterfactuals. (shrink)

We investigate a basic probabilistic dynamic semantics for a fragment containing conditionals, probability operators, modals, and attitude verbs, with the aim of shedding light on the prospects for adding probabilistic structure to models of the conversational common ground.

This paper examines a promising probabilistic theory of singular causation developed by David Lewis. I argue that Lewis' theory must be made more sophisticated to deal with certain counterexamples involving pre-emption. These counterexamples appear to show that in the usual case singular causation requires an unbroken causal process to link cause with effect. I propose a new probabilistic account of singular causation, within the framework developed by Lewis, which captures this intuition.

Comparativism is the view that comparative confidences (e.g., being more confident that P than that Q) are more fundamental than degrees of belief (e.g., believing that P with some strength x). In this paper, I outline the basis for a new, non-probabilistic version of comparativism inspired by a suggestion made by Frank Ramsey in `Probability and Partial Belief'. I show how, and to what extent, `Ramseyan comparativism' might be used to weaken the (unrealistically strong) probabilistic coherence conditions that (...) comparativism traditionally relies on. (shrink)

All extant purely probabilistic measures of explanatory power satisfy the following technical condition: if Pr(E | H1) > Pr(E | H2) and Pr(E | ∼H1) < Pr(E | ∼H2), then H1’s explanatory power with respect to E is greater than H2’s explanatory power with respect to E. We argue that any measure satisfying this condition faces three serious problems—the Problem of Temporal Shallowness, the Problem of Negative Causal Interactions, and the Problem of Nonexplanations. We further argue that many such (...) measures face a fourth problem—the Problem of Explanatory Irrelevance. (shrink)

This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them. I concentrate on Michael Dummett’s and Dag Prawitz’ philosophical motivations and give precise characterisations of the crucial notions of harmony and stability, placed in the context of proving normalisation results in systems of natural deduction. I point out a problem for defining the meaning of negation in this framework and prospects for an account of the meanings (...) of modal operators in terms of rules of inference. (shrink)

We often have some reason to do actions insofar as they promote outcomes or states of affairs, such as the satisfaction of a desire. But what is it to promote an outcome? I defend a new version of 'probabilism about promotion'. According to Minimal Probabilistic Promotion, we promote some outcome when we make that outcome more likely than it would have been if we had done something (anything) else. This makes promotion easy and reasons cheap.

Prawitz conjectured that proof-theoretic validity offers a semantics for intuitionistic logic. This conjecture has recently been proven false by Piecha and Schroeder-Heister. This article resolves one of the questions left open by this recent result by showing the extensional alignment of proof-theoretic validity and general inquisitive logic. General inquisitive logic is a generalisation of inquisitive semantics, a uniform semantics for questions and assertions. The paper further defines a notion of quasi-proof-theoretic validity by restricting proof-theoretic validity to (...) allow double negation elimination for atomic formulas and proves the extensional alignment of quasi-proof-theoretic validity and inquisitive logic. (shrink)

I put forward a micro realistic, probabilistic version of quantum theory, which specifies the precise nature of quantum entities thus solving the quantum wave/particle dilemma, and which both reproduces the empirical success of orthodox quantum theory, and yields predictions that differ from orthodox quantum theory for as yet unperformed experiments.

Real human agents, even when they are rational by everyday standards, sometimes assign different credences to objectively equivalent statements, such as ‘Orwell is a writer’ and ‘E.A. Blair is a writer’, or credences less than 1 to necessarily true statements, such as not-yet-proven theorems of arithmetic. Anna Mahtani calls this the phenomenon of ‘opacity’. Opaque credences seem probabilistically incoherent, which goes against a key modelling assumption of probability theory. I sketch a modelling strategy for capturing opaque credence assignments without abandoning (...) probabilistic coherence. I draw on ideas from judgement-aggregation theory, where we face similar challenges of defining the ‘objects of judgement’. (shrink)