Probabilities figure centrally in much of the literature on the semantics of conditionals. I find this surprising: it accords a special status to conditionals that other parts of language apparently do not share. I critically discuss two notable ‘probabilities first’ accounts of counterfactuals, due to Edgington and Leitgeb. According to Edgington, counterfactuals lack truth values but have probabilities. I argue that this combination gives rise to a number of problems. According to Leitgeb, counterfactuals have truth conditions-roughly, a counterfactual is true (...) when the corresponding conditional chance is sufficiently high. I argue that problems arise from the disparity between truth and high chance, between approximate truth and high chance, and from counterfactuals for which the corresponding conditional chances are undefined. However, Edgington, Leitgeb and I can unite in opposition to Stalnaker and Lewis-style ‘similarity’ accounts of counterfactuals. (shrink)

We argue that indeterminate probabilities are not only rationally permissible for a Bayesian agent, but they may even be rationally required . Our first argument begins by assuming a version of interpretivism: your mental state is the set of probability and utility functions that rationalize your behavioral dispositions as well as possible. This set may consist of multiple probability functions. Then according to interpretivism, this makes it the case that your credal state is indeterminate. Our second argument begins (...) with our describing a world that plausibly has indeterminate chances. Rationality requires a certain alignment of your credences with corresponding hypotheses about the chances. Thus, if you hypothesize the chances to be indeterminate, your will inherit their indeterminacy in your corresponding credences. Our third argument is motivated by a dilemma. Epistemic rationality requires you to stay open-minded about contingent matters about which your evidence has not definitively legislated. Practical rationality requires you to be able to act decisively at least sometimes. These requirements can conflict with each other-for thanks to your open-mindedness, some of your options may have undefined expected utility, and if you are choosing among them, decision theory has no advice to give you. Such an option is playing Nover and Hájek’s Pasadena Game , and indeed any option for which there is a positive probability of playing the Pasadena Game. You can serve both masters, epistemic rationality and practical rationality, with an indeterminate credence to the prospect of playing the Pasadena game. You serve epistemic rationality by making your upper probability positive-it ensures that you are open-minded. You serve practical rationality by making your lower probability 0-it provides guidance to your decision-making. No sharp credence could do both. (shrink)

In this paper I want to discuss some basic problems of inductive logic, i.e. of the attempt to solve the problem of induction by means of a calculus of logical probability. I shall try to throw some light upon these problems by contrasting inductive logic, based on logical probability, and working with undefined samples of observations, with mathematical statistics, based on statistical probability, and working with representative random samples.

This reissue of D. A. Gillies highly influential work, first published in 1973, is a philosophical theory of probability which seeks to develop von Mises’ views on the subject. In agreement with von Mises, the author regards probability theory as a mathematical science like mechanics or electrodynamics, and probability as an objective, measurable concept like force, mass or charge. On the other hand, Dr Gillies rejects von Mises’ definition of probability in terms of limiting frequency and (...) claims that probability should be taken as a primitive or undefined term in accordance with modern axiomatic approaches. This of course raises the problem of how the abstract calculus of probability should be connected with the ‘actual world of experiments’. It is suggested that this link should be established, not by a definition of probability, but by an application of Popper’s concept of falsifiability. In addition to formulating his own interesting theory, Dr Gillies gives a detailed criticism of the generally accepted Neyman Pearson theory of testing, as well as of alternative philosophical approaches to probability theory. The reissue will be of interest both to philosophers with no previous knowledge of probability theory and to mathematicians interested in the foundations of probability theory and statistics. (shrink)

It is a curious fact that in the many writings on the theory of probability, one rarely finds an instance of a clear cut definition of the fundamental problem of the subject. On intuitive grounds, one grasps the intent of most of the authors: it is to define the concept of probability and to show its relationship to the other concepts of science; and yet since the criterion of adequacy in the answering of this question is rarely stated, (...) one must feel that the problem itself has remained undefined. (shrink)

This chapter contains sections titled: Introduction The Mathematical Theory of Probability The Philosophical Theory of Probability Conclusion References.

We are in the midst of a fundamental shift, as more enterprises start to use services enabled by cloud technologies. This will heavily impact IT services providers, who must now consider strategies for coping with profound changes in the marketplace or risk being left behind. This research will be of interest to IT services vendors, as well as consumers of IT services.

The standard treatment of conditional probability leaves conditional probability undefined when the conditioning proposition has zero probability. Nonetheless, some find the option of extending the scope of conditional probability to include zero-probability conditions attractive or even compelling. This article reviews some of the pitfalls associated with this move, and concludes that, for the most part, probabilities conditional on zero-probability propositions are more trouble than they are worth.

In our current era, a new rapidly spreading pandemic disease called coronavirus disease, caused by a virus identified as a novel coronavirus, is becoming a crucial threat for the whole world. Currently, the number of patients infected by the virus is expanding exponentially, but there is no commercially available COVID-19 medication for this pandemic. However, numerous antiviral drugs are utilized for the treatment of the COVID-19 disease. Identification of the appropriate antivirus medicine to treat the infection of COVID-19 is still (...) a complicated and uncertain decision. This study’s key objective is to develop a novel approach called _q_-rung orthopair probabilistic hesitant fuzzy rough set, which incorporates the _q_-rung orthopair fuzzy set, probabilistic hesitant fuzzy set, and rough set structures. New _q_-ROPHFR aggregation operators have been established: the _q_-ROPHFR Einstein weighted averaging operator and the _q_-ROPHFR Einstein weighted geometric operator. In this study, we explored some basic features of the developed operators. Afterward, to demonstrate the viability and feasibility of the established decision-making approach in real-world applications, a case study related to selecting drugs for COVID-19 pandemic is addressed. Furthermore, a comprehensive comparison with the _q_-rung orthopair probabilistic hesitant fuzzy rough TOPSIS technique is also presented to illustrate the benefits of the new framework. The obtained results confirm the reliability and effectiveness of the proposed approach for finding uncertainty in real-world decision-making. (shrink)

Pascal’s Wager involves expected utilities. In this chapter, we examine the Wager in light of two main features of expected utility theory: utilities and probabilities. We discuss infinite and finite utilities, and zero, infinitesimal, extremely low, imprecise, and undefined probabilities. These have all come up in recent literature regarding Pascal’s Wager. We consider the problems each creates and suggest prospects for the Wager in light of these problems.

Nature-inspired optimization techniques are useful tools in electrical engineering problems to minimize or maximize an objective function. In this paper, we use the firefly algorithm to improve the optimal solution for the problem of directional overcurrent relays (DOCRs). It is a complex and highly nonlinear constrained optimization problem. In this problem, we have two types of design variables, which are variables for plug settings (PSs) and the time dial settings (TDSs) for each relay in the circuit. The objective function is (...) to minimize the total operating time of all the basic relays to avoid unnecessary delays. We have considered four models in this paper which are IEEE (3-bus, 4-bus, 6-bus, and 8-bus) models. From the numerical results, it is obvious that the firefly algorithm with certain parameter settings performs better than the other state-of-the-art algorithms. (shrink)

Measuring and evaluating a learner’s learning ability is always the focus of every person whose aim is to develop strategies and plans for their learners to improve the learning process. For example, classroom assessments, self-assessment using computer systems such as Intelligent Tutoring Systems, and other approaches are available. Assessment of metacognition is one of these techniques. Having the ability to evaluate and monitor one’s learning is known as metacognition. An individual can then propose adjustments to their learning process based on (...) this assessment. By monitoring, improving, and planning their activities, learners who can manage their cognitive skills are better able to manage their knowledge about a particular subject. It is common knowledge that students’ metacognitive and self-assessment skills and abilities have been extensively studied, but no research has been carried out on the mistakes that novice developers make because they do not use their self-assessment abilities enough. This study aims to assess the metacognitive skills and abilities of novice software developers working in the industry and to describe the consequences of awareness of metacognition on their performance. In the proposed study, we experimented with novice software developers and collected data using Devskiller and a self-assessment log to analyze their use of self-regulation skills. The proposed study showed that when developers are asked to reflect upon their work, they become more informed about their habitual mistakes, and using a self-assessment log helps them highlight their repetitive mistakes and experiences which allows them to improve their performance on future tasks. (shrink)

With the increasing volume of data transmission through insecure communication channels, big data security has become one of the important concerns in the cybersecurity domain. To address these concerns and keep data safe, a robust privacy-preserving cryptosystem is necessary. Such a solution relies on chaos encryption algorithms over standard cryptographic methods that possess multistage encryption levels, including high speed, high security, low compute overheads, and procedural power, among other characteristics. In this work, a secure image encryption scheme is proposed using (...) linear feedback shift register and chaos-based quantum chaotic map. The focus of the scheme is mainly dependent on the secret keys from the input of the algorithm. The threat landscape, the statistical test analysis, along critical comparisons with other schemes indicate that the presented algorithm is significantly secure and is resistant to a wide range of different attacks such as differential and statistical attacks. The proposed method has sufficiently higher sensitivity and security when compared to existing encryption algorithms. Several security parameters validated the security of proposed work such as correlation coefficient analyses among the neighboring pixels, entropy, the number of pixels change rate, unified average change intensity, mean square error, brute force, key sensitivity, and peak signal to noise ratio analyses. The randomness of the ciphers produced by the proposed technique is also passed through NIST-800-22. The results of NIST indicate that the ciphers are highly random and do not produce any type of periodicity or pattern. (shrink)

The 3D Prandtl fluid flow through a bidirectional extending surface is analytically investigated. Cattaneo–Christov fluid model is employed to govern the heat and mass flux during fluid motion. The Prandtl fluid motion is mathematically modeled using the law of conservations of mass, momentum, and energy. The set of coupled nonlinear PDEs is converted to ODEs by employing appropriate similarity relations. The system of coupled ODEs is analytically solved using the well-established mathematical technique of HAM. The impacts of various physical parameters (...) over the fluid state variables are investigated by displaying their corresponding plots. The augmenting Prandtl parameter enhances the fluid velocity and reduces the temperature and concentration of the fluid. The momentum boundary layer boosts while the thermal boundary layer mitigates with the rising elastic parameter strength. Furthermore, the enhancing thermal relaxation parameter ) reduces the temperature distribution, whereas the augmenting concentration parameter drops the strength of the concentration profile. The increasing Prandtl parameter declines the fluid temperature while the augmenting Schmidt number drops the fluid concentration. The comparison of the HAM technique with the numerical solution shows an excellent agreement and hence ascertains the accuracy of the applied analytical technique. This work finds applications in numerous fields involving the flow of non-Newtonian fluids. (shrink)

Many real-world optimization problems usually require a large number of conflicting objectives to be optimized simultaneously to obtain solution. It has been observed that these kinds of many-objective optimization problems often pose several performance challenges to the traditional multi-objective optimization algorithms. To address the performance issue caused by the different types of MaOPs, recently, a variety of many-objective particle swarm optimization has been proposed. However, external archive maintenance and selection of leaders for designing the MaOPSO to real-world MaOPs are still (...) challenging issues. This work presents a MaOPSO based on entropy-driven global best selection strategy to solve the many-objective software package restructuring problem. EMPSO makes use of the entropy and quality indicator for the selection of global best particle. To evaluate the performance of the proposed approach, we applied it over the five MaOSPR problems. We compared it with eight variants of MaOPSO, which are based on eight different global best selection strategies. The results indicate that the proposed EMPSO is competitive with respect to the existing global best selection strategies based on variants of MaOPSO approaches. (shrink)

According to orthodox (Kolmogorovian) probability theory, conditional probabilities are by definition certain ratios of unconditional probabilities. As a result, orthodox conditional probabilities are undefined whenever their antecedents have zero unconditional probability. This has important ramifications for the notion of probabilistic independence. Traditionally, independence is defined in terms of unconditional probabilities (the factorization of the relevant joint unconditional probabilities). Various “equivalent” formulations of independence can be given using conditional probabilities. But these “equivalences” break down if conditional probabilities are (...) permitted to have conditions with zero unconditional probability. We reconsider probabilistic independence in this more general setting. We argue that a less orthodox but more general (Popperian) theory of conditional probability should be used, and that much of the conventional wisdom about probabilistic independence needs to be rethought. (shrink)

Naive versions of decision theory take probabilities and utilities as primitive and use expected value to give norms on rational decision. However, standard decision theory takes rational preference as primitive and uses it to construct probability and utility. This paper shows how to justify a version of the naive theory, by taking dominance as the most basic normatively required preference relation, and then extending it by various conditions under which agents should be indifferent between acts. The resulting theory can (...) make all the decisions of classical expected utility theory, plus more in cases where expected utilities are infinite or undefined. Although the theory requires similarly strong assumptions to classical expected utility theory, versions of the theory can be developed with slightly weaker assumptions, without having to prove a new representation theorem for the weaker theory. This alternate foundation is particularly useful if probability is prior to preference, as suggested by the recent program to base probabilism on accuracy and alethic considerations rather than pragmatic ones. (shrink)

Following Nancy Cartwright and others, I suggest that most (if not all) theories incorporate, or depend on, one or more idealizing assumptions. I then argue that such theories ought to be regimented as counterfactuals, the antecedents of which are simplifying assumptions. If this account of the logic form of theories is granted, then a serious problem arises for Bayesians concerning the prior probabilities of theories that have counterfactual form. If no such probabilities can be assigned, the the posterior probabilities will (...) be undefined, as the latter are defined in terms of the former. I argue here that the most plausible attempts to address the problem of probabilities of conditionals fail to help Bayesians, and, hence, that Bayesians are faced with a new problem. In so far as these proposed solutions fail, I argue that Bayesians must give up Bayesianism or accept the counterintuitive view that no theories that incorporate any idealizations have ever really been confirmed to any extent whatsoever. Moreover, as it appears that the latter horn of this dilemma is highly implausible, we are left with the conclusion that Bayesianism should be rejected, at least as it stands. (shrink)

It has often been noted that there is some tension between engaging in decision-making and believing that one’s choices might be predetermined. The possibility that our choices are predetermined forces us to consider, in our decisions, act-state pairs which are inconsistent, and hence to which we cannot assign sensible utilities. But the reasoning which justifies two-boxing in Newcomb’s problem also justifies associating a non-zero causal probability with these inconsistent act-state pairs. Put together these undefined utilities and non-zero probabilities (...) entail that expected utilities are undefined whenever it is a possibility that our choices are predetermined. There are three ways to solve the problem, but all of them suffer serious costs: always assume that, contrary to our evidence, the outcome of our present decision-making is not predetermined; give up the reasoning that justifies unconditional two-boxing in Newcomb’s problem; or allow epistemically impossible outcomes to contribute to expected utility, leading to the wrong results in a series of cases introduced by Ahmed :665–685, 2014a, Evidence, decision and causality, Cambridge University Press, Cambridge, 2014b). However they choose to respond, causal decision theorists cannot remain silent: the intuitive tension between decision-making and the possibility of predetermination can be made precise, and resolving it will require giving up something. Causal decision theorists have a predetermination problem. (shrink)

Following Nancy Cartwright and others, I suggest that most theories incorporate, or depend on, one or more idealizing assumptions. I then argue that such theories ought to be regimented as counterfactuals, the antecedents of which are simplifying assumptions. If this account of the logical form of theories is granted, then a serious problem arises for Bayesians concerning the prior probabilities of theories that have counterfactual form. If no such probabilities can be assigned, then posterior probabilities will be undefined, as (...) the latter are defined in terms of the former. I argue here that the most plausible attempts to address the problem of probabilities of conditionals fail to help Bayesians, and, hence, that Bayesians are faced with a new problem. In so far as these proposed solutions fail, I argue that Bayesians must give up Bayesianism or accept the counterintuitive view that no theories that incorporate any idealizations have ever really been confirmed to any extent whatsoever. Moreover, as it appears that the latter horn of this dilemma is highly implausible, we are left with the conclusion that Bayesianism should be rejected, at least as it stands. (shrink)

Given a physical system, one knows that there is a logical duality between its properties and its states. In this paper, we choose its states as the undefined notions of our axiomatic construction. In fact, by means of well-motivated assumptions expressed in terms of a transition probability function defined on the set of all pure states of the system, we construct a system of elementary propositions, i.e., a complete orthomodular atomic lattice satisfying the covering law. We also study (...) in this framework the important notion of compatibility of propositions, and we define the superpositions and the mixtures of the states of the physical system. (shrink)

Philosophy and statistics have studied two causal species, deterministic and probabilistic. There's a third species, however, hitherto unanalysed: underdeterministic causal phenomena, which are non-deterministic yet non-probabilistic. Here, I formulate a framework for modelling them. -/- Consider a simple case. If I go out, I may stumble into you but also may miss you. If I don’t go out, we won't meet. I go out. We meet. My going out is a cause of our encounter even if there was no determinate (...) probability of us meeting conditional on my going out. The cause is neither deterministic (it didn't necessitate the effect) nor probabilistic (the relevant conditional probabilities are undefined). Rather, it's underdeterministic: it raises the modal status of the effect from causally impossible to possible. -/- Here, I won't offer a theory of such token causes but develop the prerequisite for any such theory: the underdeterministic framework. The framework is like the deterministic structural-equations framework but with one consequential difference---an equation can return multiple values. This change allows me to define causal possibility and necessity, and corresponding notions of interventionist might- and would-counterfactuals. I also define conditional independence, which obeys the graphoid axioms, and prove that underdeterministic models satisfy the causal Markov condition. The framework can causally model situations that other frameworks cannot: decision-making under bounded uncertainty, games with multiple equilibria, infinite fair lotteries, and any other non-deterministic situation where indeterminacies are essentially non-probabilistic, or where we have a reason not to use probabilities. (shrink)

This paper presents some impossibility results for certain views about what you should do when you are uncertain about which moral theory is true. I show that under reasonable and extremely minimal ways of defining what a moral theory is, it follows that the concept of expected moral choiceworthiness is undefined, and more generally that any theory of decision-making under moral uncertainty must generate pathological results.

Various philosophers accept moral views that are impartial, additive, and risk-neutral with respect to moral betterness. But, if that risk neutrality is spelt out according to expected value theory alone, such views face a dire reductio ad absurdum. If the expected sum of value in humanity's future is undefined--if, e.g., the probability distribution over possible values of the future resembles the Pasadena game, or a Cauchy distribution--then those views say that no option is ever better than any other. (...) And, as I argue, this holds in practice: our evidence supports such a probability distribution. Indeed, it supports a probability distribution that cannot be evaluated even if we adopt one of the various extensions of expected value theory proposed in the literature. Must we therefore reject all impartial, additive, risk-neutral moral theories? It turns out that we need not. I develop an alternative solution: a new method of extending expected value theory, which allows us to deal with this distribution and to salvage those moral views. I also examine how this solution affects one of the most notable implications of those views--namely, longtermism. (shrink)

This paper uses a non-distributive system of Boolean fractions (a|b), where a and b are 2-valued propositions or events, to express uncertain conditional propositions and conditional events. These Boolean fractions, 'a if b' or 'a given b', ordered pairs of events, which did not exist for the founders of quantum logic, can better represent uncertain conditional information just as integer fractions can better represent partial distances on a number line. Since the indeterminacy of some pairs of quantum events is due (...) to the mutual inconsistency of their experimental conditions, this algebra of conditionals can express indeterminacy. In fact, this system is able to express the crucial quantum concepts of orthogonality, simultaneous verifiability, compatibility, and the superposition of quantum events, all without resorting to Hilbert space. A conditional (a|b) is said to be "inapplicable" (or "undefined") in those instances or models for which b is false. Otherwise the conditional takes the truth-value of proposition a. Thus the system is technically 3-valued, but the 3rd value has nothing to do with a state of ignorance, nor to some half-truth. People already routinely put statements into three categories: true, false, or inapplicable. As such, this system applies to macroscopic as well as microscopic events. Two conditional propositions turn out to be simultaneously verifiable just in case the truth of one implies the applicability of the other. Furthermore, two conditional propositions (a|b) and (c|d) reside in a common Boolean sub-algebra of the non-distributive system of conditional propositions just in case b = d, their conditions are equivalent. Since all aspects of quantum mechanics can be represented with this near classical logic, there is no need to adopt Hilbert space logic as ordinary logic, just a need perhaps to adopt propositional fractions to do logic, just as we long ago adopted integer fractions to do arithmetic. The algebra of Boolean fractions is a natural, near-Boolean extension of Boolean algebra adequate to express quantum logic. While this paper explains one group of quantum anomalies, it nevertheless leaves no less mysterious the 'influence-at-a-distance', quantum entanglement phenomena. A quantum realist must still embrace non-local influences to hold that "hidden variables" are the measured properties of particles. But that seems easier than imaging wave-particle duality and instant collapse, as offered by proponents of the standard interpretation of quantum mechanics. (shrink)

According to Nancy Cartwright, a causal law holds just when a certain probabilistic condition obtains in all test situations which in turn satisfy a set of background conditions. These background conditions are shown to be inconsistent and, on separate account, logically incoherent. I offer a corrective reformulation which also incorporates a strategy for problems like Hesslow's thrombosis case. I also show that Cartwright's recent argument for modifying the condition to appeal to singular causes fails.Proposed modifications of the theory's probabilistic condition (...) to handle effects with extreme probabilities (0 or 1) are found unsatisfactory. I propose a unified solution which also handles extreme causes. Undefined conditional probabilities give rise to three good, but non-equivalent, ways of formulating the theory. Various formulations appear in the literature. I give arguments to eliminate all but one candidate. Finally, I argue for a crucial new condition clause, and show how to extend the results beyond a simple probabilistic framework. (shrink)

We trace self-reference phenomena to the possibility of naming functions by names that belong to the domain over which the functions are defined. A naming system is a structure of the form ,{ }), where D is a non-empty set; for every a∈ D, which is a name of a k-ary function, {a}: Dk → D is the function named by a, and type is the type of a, which tells us if a is a name and, if it is, (...) the arity of the named function. Under quite general conditions we get a fixed point theorem, whose special cases include the fixed point theorem underlying Gödel's proof, Kleene's recursion theorem and many other theorems of this nature, including the solution to simultaneous fixed point equations. Partial functions are accommodated by including “undefined” values; we investigate different systems arising out of different ways of dealing with them. Many-sorted naming systems are suggested as a natural approach to general computatability with many data types over arbitrary structures. The first part of the paper is a historical reconstruction of the way Gödel probably derived his proof from Cantor's diagonalization, through the semantic version of Richard. The incompleteness proof–including the fixed point construction–result from a natural line of thought, thereby dispelling the appearance of a “magic trick”. The analysis goes on to show how Kleene's recursion theorem is obtained along the same lines. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:How was Haiti?Sadath Sayeed"She smelled of milk and urine. Chacko marveled at how someone so small and undefined, so vague in her resemblances, could so completely command the attention, the love, the sanity of a grown man."—Arundhati Roy from The God of Small ThingsFather and SonTwenty minutes before I was to be taxied to the airport in Port-au-Prince, the baby boy handed to me did not breathe continuously. (...) He didn't breathe at all. Born a few hours earlier at home—probably in one of the hundreds of make-shift tent cities. I did ask if the baby cried after he came out of mom. Yes. He didn't suckle well and stopped crying. Dad decided to get help. Climbed onto a moped with his baby wrapped in a blanket. Must have tucked him under one arm and rode between piles of rubble to the closest health care facility.People at this other place tried to help his baby boy breathe. They pushed air into his mouth like you are doing. After a few minutes, they said that they couldn't help anymore."Go to the University hospital."From a shirt sleeve pocket, he pulled a crumpled white piece of paper with some writing scribbled on it. I gathered it documented this last visit. They took the time to hand him a note before they sent him on his way.What did Dad think? Thanks for the note. Okay, now what? Swaddle my baby. Tuck him under one arm. Hop on my moped.Strained ResourcesL'Université d'État d'Haïti has been the main teaching hospital for all of the country for decades. Many, if not most, formally trained Haitian health care providers have spent some time in Port-au-Prince at this institution. As important as it is, it has suffered from neglect over many, many years. The earthquake made matters exponentially worse. [End Page 98]Dad must have been directed to the pediatric service when he arrived. There, a few Haitian pediatric residents and nurses were doing what they did every morning. Tending to an overcrowded patient population of babies, children, and parents. The pediatrics building was left uninhabitable. All inpatient and outpatient care took place on a street where the tents had been erected.In the ICU, tent twenty beds filled a space probably meant for ten military style cots. Metal cribs and full adult hospital beds were jammed next to one another to fill every possible nook. Parents slept on the adult beds with their sick children or underneath the cribs. Siblings too. Outside, hundreds of parents would line up starting at six a.m. and stand in line for hours so that they could get their children seen in a couple of smaller tents that were used to run the outpatient clinics. Moms would fan their kids under the shade of trees on the side of the road. Feed them some milk or juice while they waited. Smile at us passers-by. They seemed to possess a bottomless well of patience.Power cords with extension wires littered the street-hospital floor under the canopy.When it rained, we were only modestly concerned about electrocution. A single 1950's style metal desk with two worn out chairs greeted visitors to the entry way and was usually occupied by some of the medical staff. A single, large metal fan that had collected enough dust and soot in its circular cage to look bearded hummed a breeze toward the desk.A bottomless well of patience and resiliency. In the popular media, this last word has been thrown around a lot to describe the character of the Haitian people. It is insufficient. Their adversity is truly absurd. Sense cannot be made of the scene in the pediatrics tents months after the earthquake struck, nor elsewhere on that campus, let alone outside its walls.The local medical staff hadn't been paid a dime in many months even prior to the earthquake. They suffered unimaginable losses during the earthquake. They still came to work."We love our children. They love us too."What so many of us think of as charity... (shrink)

This study discusses the treatise of Jalāl al-Dīn Davvānī (d. 908/1502) named Risāla fī taʻrīf ʻilm. This treatise criticizes a definition of knowledge adopted by some theologians in the late period (mutaʾakhkhirīn). The definition of knowledge at issue consists of three components: Attribution, discernment, no possibility of contradiction. Knowledge is an attribute as a category and with this attribution, a discernment is obtained. As a result of this process knowledge is acquired and there should be no possibility of this knowledge (...) to be inconsistent. Davvānī examines eight possibilities of the definition on knowledge. In each of these eight possibilities a different problems arises. According to Davvānī, the definition mentioned is problematic in all probability. Nonetheless, Davvānī does not propose another definition of knowledge that can replace the definition of knowledge that he criticizes. This article represents the process of reaching the definition of knowledge stated in the history of theological thought. The definitions on knowledge made from the first period (al-mutaqaddimīn) are summarized and the strengths and weaknesses of these definitions are explained. There are two main approaches regarding the nature of knowledge in Islamic thought: According to the first approach pioneered by Fakhr al-Dīn Rāzī (d. 606/1202), absolute knowledge is a priory (badīhī) and undefinable. According to the second one, the nature of absolute knowledge is a posteriori (naẓarī) and definable. The first part of this article includes the definitions and criticisms of knowledge in Islamic thought. The second part discusses the claim of originality. In the second part, Davvānī’s treatise will be evaluated for the first time. According to the conclusion of the treatise, the definition made to find the essence of knowledge is unfounded in all respect. It is important to evaluate the treatise as a reflection of anti-essentialism in the history of Islamic thought. Most contemporary studies do not count this treatise among Davvānī’s works. In the biography book al-Aʿlām written by Khayr al-Dīn al-Ziriklī (d. 1976), Risāla fī taʻrīf ʻilm is counted among the works of Davvānī. In the introduction of more than one manuscript, the work is clearly attributed to Davvānī. As a result of the double-sided research, it has been ensured that the work belongs to Davvānī. However, in other two manuscripts containing the same treatise, the work is shown as belonging to an unknown person named Şāirzāda. But there is no information found about this person. This name seems more like a copyist (mustansıh) than an author. Possibly, such a mistake occurred as a result of confusing the two names. Davvānī’s small volume treatise sheds light on the intellectual atmosphere and level of its period. The definition of knowledge mentioned in the classical theology book of ‘Aḍud al-Dīn al-Ījī (d. 756/1355) called al-Mawāqif is superior to other definitions of knowledge. Instead of this definition of knowledge, which is presented as perfect, Jurjānī (d. 816/1413) preferred the definition of knowledge attributed to al-Māturīdī (d. 333/944) in the Sharḥ al-Mawāqif. This preference has been accepted as an implicit criticism for future generations, including Davvānī. Those who claimed that Jurjānī did not taken a random choice, carried on their criticism by going over the definition. The fine craftsmanship revealed in the treatise reveals that critical thinking is used intensively. The properly understanding of the treatise depends on knowing the science of logic and following it with intense attention. The definition of knowledge is examined from various angles with the help of concepts such as conception, ratification, judgment and contradiction (nakz) of the science of logic. In the science of logic, every layer of knowledge such as certainty, conjecture, imitation, doubt is investigated one by one. In the science of kalām, attention is paid that the ideal level of knowledge is precise. An uncertain knowledge is not considered knowledge according to kalām. Because the main goal of theologians is to build religious beliefs on the obtained certain knowledge. It is not accepted to base knowledge to a probable information on imprecise knowledge. For a knowledge among the knowns to reach the level of certainty, it must be correct and exclude other possibilities. Another condition is required for certainty in the definition of knowledge that Davvānī examines in his treatise is that the opposite is wrong. The fact that a piece of information is considered true depends on the fact that its reverse is false. In this treatise, it is discussed how to look for the contradictions or discrepancy that are wanted to be excluded. Knowledge is inseparable from the human mind, from outside of the world, or from the relationship between the two. (shrink)

Perceptual Confidence is the thesis that perceptual experiences can be probabilistic. This thesis has been defended and criticised based on a variety of phenomenological, epistemological, and explanatory arguments. One gap in these arguments is that they neglect the question of whether perceptual experiences satisfy the formal conditions that define the notion of probability to which Perceptual Confidence is committed. Here, we focus on this underexplored question and argue that perceptual experiences do not satisfy such conditions. But if they do (...) not, then ascriptions of perceptual confidence are undefined; and so, Perceptual Confidence cannot be true. (shrink)

This paper argues that if the world is irreducibly stochastic, then both Salmon's S-R model of explanation and Fetzer's C-R model of explanation have the following undesirable consequence: the objective probability (associated with the model's relevance condition) of any actual macro-event is either undefined or else, if defined, it equals one--so that the event is not even a candidate for a probabilistic explanation. This result follows from the temporal ambiguity of ontic probability in an irreducibly stochastic world. (...) It is argued further that an analogous difficulty faces those theories of probabilistic causality which depend upon the notions of contributing and counteracting causes. Because of the problem of temporal ambiguity, it is not possible to objectively label a particular event as a contributing (or a counteracting) cause of some subsequent event. The argument is carried through in detail for a recent theory of probabilistic causality proposed by Paul Humphreys. (shrink)

Dozens of articles have addressed the challenge that gambles having undefined expectation pose for decision theory. This paper makes two contributions. The first is incremental: we evolve Colyvan's ``Relative Expected Utility Theory'' into a more viable ``conservative extension of expected utility theory" by formulating and defending emendations to a version of this theory proposed by Colyvan and H\'ajek. The second is comparatively more surprising. We show that, so long as one assigns positive probability to the theory that there (...) is a uniform bound on the utility of possible gambles (and assuming a uniform bound on the amount of utility that can accrue in a fixed amount of time), standard principles of anthropic reasoning (as formulated by Brandon Carter) place lower and upper bounds on the expected values of gambles advertised as having no expectation--even assuming that with positive probability, all gambles advertised as having infinite expected utility are administered faithfully. Should one accept the uniform bound premises, this reasoning thus dissolves (or nearly dissolves, in some cases) several puzzles in infinite decision theory. (shrink)

_ Source: _Volume 10, Issue 1, pp 98 - 115 What has come to be known as the ‘linguistic turn’ in historical theory over the past forty years or so has finished what the two World Wars began in demolishing the confidence that the historical discipline possessed at the turn of the twentieth century. This confidence was most memorably expressed by Lord Acton that one day we would possess ‘ultimate history’. Today most historians are probably more inclined to subscribe to (...) Pieter Geyl’s view that history is ‘an argument without end’. Yet the jettisoning of a teleological goal for historical accounts does not mean that we have to also part with the idea of progress; we just need a new definition of it. In this article I argue that we should adopt an evolutionary epistemology of history which sees progress as something pushed from behind, rather than aiming at an undefined point in the future; but this is not the only advantage an evolutionary epistemology can offer us. I go on to outline two further aspects of evolutionary epistemology which may benefit historical theorists. (shrink)

The sum, difference, product and quotient of two functions with different domains are usually defined only on their common domain. This paper extends these definitions so that the sum and other operations are essentially defined anywhere that at least one of the components is defined. This idea is applied to propositions and events, expressed as indicator functions, to define conditional propositions and conditional events as three-valued indicator functions that are undefined when their condition is false. Extended operations of "and", (...) "or", "not" and "conditioning" are then defined on these conditional events with variable conditions. The probabilities of the disjunction and of the conjunction of two conditionals are expressed in terms of the conditional probabilities of the component conditionals. In a special case, these are shown to be weighted averages of the component conditional probabilities where the weights are the relative probabilities of the various conditions. Next, conditional random variables are defined to be random variables X whose domain has been restricted by a condition on a second random variable Y. The extended sum, difference, product and conditioning operations on functions are then applied to these conditional random variables. The expectation of a random variable and the conditional expectation of a conditional random variable are recounted. Theorem 1 generalizes the standard result that the conditional expectation of the sum of two conditional random variables with disjoint and exhaustive conditions is a weighted sum of the conditional expectations of the component conditional random variables. Because of the extended operations, the theorem is true for arbitrary conditions. Theorem 2 gives a formula for the expectation of the product of two conditional random variables. After the definition of independence of two random variables is extended to accommodate the extended operations, it is applied to the formula of Theorem 2 to simplify the expectation of a product of conditional random variables. Two examples end the paper. The first concerns a work force of n workers of different output levels and work shifts. The second example involves two radars with overlapping surveillance regions and different detection error rates. One radar's error rate is assumed to be sensitive to fog and the other radar's error rate is assumed to be sensitive to air traffic density. The combined error rate over the combined surveillance region given heavy fog and moderate air traffic is computed. (shrink)

The paper is devoted to the discussion of some philosophical and historical problems connected with the theorem on the undefinability of the notion of truth. In particular the problem of the priority of proving this theorem will be considered. It is claimed that Tarski obtained this theorem independently though he made clear his indebtedness to Gödel’s methods. On the other hand, Gödel was aware of the formal undefinability of truth in 1931, but he did not publish this result. Reasons for (...) that are also considered. (shrink)

Among the main theorems obtained in mathematical logic in this century are the so called limitation theorems, i.e., the Löwenheim-Skolem theorem on the cardinality of models of first-order theories, Gödel’s incompleteness theorems and Tarski’s theorem on the undefinability of truth. Problems connected with the latter are the subject of this paper. In Section 1 we shall consider Tarski’s theorem. In particular the original formulation of it as well as some specifications will be provided. Next various meanings of the notion of (...) a satisfaction predicate will be studied. In Section 2 the problem of definability of the notion of truth, in particular of the notion of truth for the language of Peano arithmetic PA, will be discussed. It will be explicitly shown that the notion of satisfaction for the language of PA can be defined in a certain weak fragment of the second order arithmetic. Finally the axiomatic characterization of satisfaction and truth as well and its mathematical and philosophical meaning will be discussed. (shrink)

Many attempts have been made to define analytic philosophy in a nonhistorical or otherwise deictic way, and to provide a set of necessary and sufficient conditions for a piece of philosophical work to be part of analytic philosophy. This is more difficult than might appear, for the conditions appealed to are normative and must be claimed by non-analytic philosophers to apply to their production as well. In fact, no such set of conditions has been forthcoming, and it is unlikely that (...) it ever will. Instead, I offer a holistic characterization of analytic philosophy as a mode of organization of philosophical work modeled on science. This accounts for analytic philosophy’s success in the academic world, its pedagogic virtues, and its ability to expand beyond its initial boundaries. Two related questions remain: one about the indirect justification analytic philosophy might be granted from outside of philosophy, the other about its ability to make contact with cultural realms other than those with which it has until now established a fruitful exchange. The answers are crucial for the potential renewal of analytic philosophy. (shrink)

This paper develops an information-sensitive theory of the semantics and probability of conditionals and statements involving epistemic modals. The theory validates a number of principles linking probability and modality, including the principle that the probability of a conditional If A, then C equals the probability of C, updated with A. The theory avoids so-called triviality results, which are standardly taken to show that principles of this sort cannot be validated. To achieve this, we deny that rational (...) agents update their credences via conditionalization. We offer a new rule of update, Hyperconditionalization, which agrees with Conditionalization whenever nonmodal statements are at stake but differs for modal and conditional sentences. (shrink)

The theory of lower previsions is designed around the principles of coherence and sure-loss avoidance, thus steers clear of all the updating anomalies highlighted in Gong and Meng's "Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss, and Simpson's Paradox" except dilation. In fact, the traditional problem with the theory of imprecise probability is that coherent inference is too complicated rather than unsettling. Progress has been made simplifying coherent inference by demoting sets of probabilities from fundamental building blocks to secondary (...) representations that are derived or discarded as needed. (shrink)

We show that (contrary to the parallel case of intuitionistic logic, see [7], [4]) there does not exist a translation fromS42 (the propositional modal systemS4 enriched with propositional quantifiers) intoS4 that preserves provability and reduces to identity for Boolean connectives and.

I describe a realist, ontologically objective interpretation of probability, "far-flung frequency (FFF) mechanistic probability". FFF mechanistic probability is defined in terms of facts about the causal structure of devices and certain sets of frequencies in the actual world. Though defined partly in terms of frequencies, FFF mechanistic probability avoids many drawbacks of well-known frequency theories and helps causally explain stable frequencies, which will usually be close to the values of mechanistic probabilities. I also argue that it's (...) a virtue rather than a failing of FFF mechanistic probability that it does not define single-case chances, and compare some aspects of my interpretation to a recent interpretation proposed by Strevens. (shrink)

We discuss Robinson's model theoretic proof of Tarski's theorem on undefinability of truth. We present two other “diagonal-free” proofs of Tarski's theorem, and we compare undefinability of truth to other forms of undefinability in nonstandard models of arithmetic.