Probabilistic Reasoning

Many tasks in statistical and causal inference can be construed as problems of entailment in a suitable formal language. We ask whether those problems are more difficult, from a computational perspective, for causal probabilistic languages than for pure probabilistic (or “associational”) languages. Despite several senses in which causal reasoning is indeed more complex—both expressively and inferentially—we show that causal entailment (or satisfiability) problems can be systematically and robustly reduced to purely probabilistic problems. Thus there is no jump in computational complexity. (...) Along the way we answer several open problems concerning the complexity of well-known probability logics, in particular demonstrating the ${\exists \mathbb {R}}$ -completeness of a polynomial probability calculus, as well as a seemingly much simpler system, the logic of comparative conditional probability. (shrink)

This essay gives an account of epistemic deference for agents with imprecise credences. I look at the two main imprecise deference principles in the literature, known as Identity Reflection and Pointwise Reflection (Moss, 2021). I show that Pointwise Reflection is strictly weaker than Identity Reflection, and argue that, if you are certain you will update by conditionalisation, you should defer to your future self according to Identity Reflection. Then I give a more general justification for Pointwise and Identity Reflection from (...) the assumption that you defer to someone whenever you consider their doxastic state to be better than yours, in the sense of leading to better decision-making. (shrink)

Credences are beliefs about evidential probabilities. We give the view an assessment-sensitive formulation, show how it evades the standard objections, and give several arguments in support.

Traditionally epistemologists have taken doxastic states to come in three varieties—belief, disbelief, and suspension. Recently many epistemologists have taken our doxastic condition to be usefully represented by credences—quantified degrees of belief. Moreover, some have thought that this new credal picture is sufficient to account for everything we want to explain with the old traditional picture. Therefore, belief, disbelief, and suspension must map onto the new picture somehow. In this paper I challenge that possibility. Approaching the question from the angle of (...) suspension, I argue that all possible credal accounts face serious challenges. They either (i) falsify central claims that uphold the credal picture itself or (ii) do not permit suspension in cases where it is permissible or (iii) rule out the possibility of plainly possible confidence comparisons. (shrink)

Trước những khó khăn và thách thức của các nhà khoa học trẻ và các nhà khoa học ở những nước đang phát triển do thiếu tài nguyên và cơ hội được tiếp cận với các kiến thức và phương pháp nghiên cứu bài bản…, các nhà khoa học Việt Nam đã phát triển nền tảng Cổng thông tin SM3D hỗ trợ đào tạo phương pháp nghiên cứu khoa học xã hội cho các nhà khoa học. Sau gần 2 năm (...) được triển khai, Cổng thông tin SM3D được đánh giá là một trong những sáng kiến đóng góp tiêu biểu của khoa học Việt Nam cho cộng đồng khoa học quốc tế, đặc biệt là đối với các nước đang phát triển. (shrink)

Many philosophers accept that, as well as having a right that others not harm us, we also have a right that others not subject us to a risk of harm. And yet, when we attempt to spell out precisely what this ‘right against risk imposition’ involves, we encounter a series of notorious puzzles. Existing attempts to deal with these puzzles have tended to focus on the nature of rights – but I propose an approach that focusses instead on the nature (...) of risk. The key move is to distinguish two different ways in which to conceptualise the risk that a given action presents – one of which is linked to the notion of probability and the other to the notion of normalcy. (shrink)

Do people tend to be overconfident? Many think so. They’ve run studies on whether people are calibrated: whether their average confidence in their opinions matches the proportion of those opinions that are true. Under certain conditions, people are systematically ‘over-calibrated’—for example, of the opinions they’re 80% confident in, only 60% are true. From this empirical over-calibration, it’s inferred that people are irrationally overconfident. My question: When and why is this inference warranted? Answering it requires articulating a general connection between being (...) rational and being right—something extant studies have not done. I show how to do so using the notion of deference. This provides a theoretical foundation for calibration research, but also reveals a flaw: the connection between being rational and being right is much weaker than is standardly assumed, since rational people can often be expected to be miscalibrated. Thus we can’t test whether people are overconfident by simply testing whether they are over-calibrated; instead, we must try to predict the rational deviations from calibration, and then compare those predictions to people’s performance. I show how this can be done—and that doing so complicates the interpretation of robust empirical effects. (shrink)

The nature of evidence is a problem for epistemology, but I argue that this problem intersects with normative decision theory in a way that I think is underappreciated. Among some decision theorists, there is a presumption that one can always ignore the nature of evidence while theorizing about principles of rational choice. In slogan form: decision theory only cares about the credences agents actually have, not the credences they should have. I argue against this presumption. In particular, I argue that (...) if evidence can be unspecific, then an alleged counterexample to causal decision theory fails. This implies that what theory of decision we think is true may depend on our opinions regarding the nature of evidence. Even when we are theorizing about subjective theories of rationality, we cannot set aside questions about the objective nature of evidence. (shrink)

The best accuracy arguments for probabilism apply only to credence functions with finite domains, that is, credence functions that assign credence to at most finitely many propositions. This is a significant limitation. It reveals that the support for the accuracy-first program in epistemology is a lot weaker than it seems at first glance, and it means that accuracy arguments cannot yet accomplish everything that their competitors, the pragmatic (Dutch book) arguments, can. In this paper, I investigate the extent to which (...) this limitation can be overcome. Building on the best arguments in finite domains, I present two accuracy arguments for probabilism that are perfectly general—they apply to credence functions with arbitrary domains. I then discuss how the arguments’ premises can be challenged. We will see that it is particularly difficult to characterize admissible accuracy measures in infinite domains. (shrink)

Reasoning from inconclusive evidence, or ‘induction’, is central to science and any applications we make of it. For that reason alone it demands the attention of philosophers of science. This Element explores the prospects of using probability theory to provide an inductive logic, a framework for representing evidential support. Constraints on the ideal evaluation of hypotheses suggest that overall support for a hypothesis is represented by its probability in light of the total evidence, and incremental support, or confirmation, indicated by (...) an increased probability given the evidence than otherwise. This proposal is shown to have the capacity to reconstruct many canons of the scientific method and inductive inference. Along the way, significant objections are discussed, such as the challenge from inductive scepticism and the objection that the probabilistic approach makes evidential support arbitrary. (shrink)

In natural language, conditionals are frequently used for giving explanations. Thus the antecedent of a conditional is typically understood as being connected to, being relevant for, or providing evidential support for the conditional's consequent. This aspect has not been adequately mirrored by the logics that are usually offered for the reasoning with conditionals: neither in the logic of the material conditional or the strict conditional, nor in the plethora of logics for suppositional conditionals that have been produced over the past (...) 50 years. In this paper I survey some recent attempts to come to terms with the problem of encoding evidential support or relevance in the logic of conditionals. I present models in a qualitative-modal and in a quantitative-probabilistic setting. Focussing on some particular examples, I show that no perfect match between the two kinds of settings has been achieved yet. (shrink)

Safety purports to explain why cases of accidentally true belief are not knowledge, addressing Gettier cases and cases of belief based on statistical evidence. However, problems arise for using safety as a condition on knowledge: safety is not necessary for knowledge and cannot always explain the Gettier cases and cases of statistical evidence it is meant to address. In this paper, I argue for a new modal condition designed to capture the non-accidental relationship between facts and evidence required for knowledge: (...) causal safety. I argue that possible errors in belief can be captured by accounting for deviations in causal relationships and that there is a natural way to characterize which causal errors are relevant in an epistemic situation. Using this, I develop a causal analogue to safety, where one’s belief in p is causally safe if it is true in all causally relevant worlds where one believes p. Causal safety, I argue, can better explain the cases safety is meant to address and can avoid the arguments raised against the necessity of safety. (shrink)

Habilitation thesis in psychology. The book consists of a collection of reasoning studies. The experimental investigations will take us from people’s reasoning about probabilities, entailments, pragmatic factors, argumentation, and causality to morality. An overarching theme of the book is norm pluralism and individual differences in rationality research.

La epistemología bayesiana tiene como concepto capital la condicionalización simple. Para comprender de buena forma cómo opera esta regla, se debe dar cuenta de la concepción subjetiva de la probabilidad. Sobre la base de lo anterior es posible esclarecer alcances y límites de la condicionalización simple. En general, cuando esta regla enfrenta una dificultad se hacen esfuerzos por resolver dicha particular cuestión, pero no es usual encontrar propuestas unificadas con la intención de resolver varias de las complicaciones subyacentes al bayesianismo (...) ortodoxo. Por lo mismo, el propósito de esta investigación es justamente proponer aquello: un marco bayesiano ampliado que tendrá por objeto solucionar más de una cuestión particular. Más específicamente, el problema de la incorregibilidad, el problema del surgimiento del cero, el problema de la evidencia antigua y el problema de las teorías nuevas. Lo anterior se logrará adicionando al bayesianismo ortodoxo dos cuestiones, a saber, la condicionalización épsilon y una reconstrucción racional. Palabras clave: Probabilidad Subjetiva, Grados de Creencia, Dogmatismo Bayesiano, Evidencia Antigua, Nuevas Teorías. (shrink)

Academia is a competitive environment. Early Career Researchers (ECRs) are limited in experience and resources and especially need achievements to secure and expand their careers. To help with these issues, this book offers a new approach for conducting research using the combination of mindsponge innovative thinking and Bayesian analytics. This is not just another analytics book. 1. A new perspective on psychological processes: Mindsponge is a novel approach for examining the human mind’s information processing mechanism. This conceptual framework is used (...) to construct models in studies. The framework is highly flexible and widely applicable for many different types of information processes. The mindsponge approach can help researchers discover interesting ideas or even formulate their very own theories when investigating psychosocial phenomena. This approach brings a fresh wind to the current landscape of social sciences and humanities (SSH). 2. Easy-to-follow analysis protocol: The Bayesian Mindsponge Framework (BMF analytics) is useful in terms of computing and visualizing power but also easy to learn and apply. Contrary to being intimidating, the Bayesian analytics section of this book is presented in a reader-friendly manner with a detailed yet clear step-by-step procedure. Examples are from published BMF articles, allowing readers to immediately practice the method and quickly create their own applications. With educational purposes in mind, the book is very suitable for ECRs who are looking to innovate their research methods. 3. Advocating for low-cost, high-quality research: Doing science can be very costly. Mindsponge innovative thinking and BMF analytics help produce impactful studies using openly available data on online repositories. This is based on the authors’ previous works and experiences. The book presents examples of employing the open R package bayesvl on secondary data from different sources. With less financial constraints, researchers can have more freedom of thought to pursue their curiosity and creativity. ECRs in low- and middle-income countries may find this aspect crucial in their careers. 4. Support and collaboration: The authors share their insights from experiences in the academic publishing system to help readers get through the processes of manuscript writing and peer-reviewing more easily. The authors are also ready to support other researchers with further inquiries and collaboration opportunities at the following website, mindsponge(dot)info. This book is for: a) ECRs whose only abundant resources are their innovation capacity and strength of will; b) Researchers in SSH who want to explore a novel approach to thinking and study conducting; c) Low- and middle-income countries’ researchers looking for a cost-effective research protocol; and, d) Innovative thinkers who want to turn their interesting thoughts into good publications. (shrink)

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)

This dissertation is concerned with two of the largest questions that we can ask about the nature of physical reality: first, whether physical reality begin to exist and, second, what criteria would physical reality have to fulfill in order to have had a beginning? Philosophers of religion and theologians have previously addressed whether physical reality began to exist in the context of defending the Kal{\'a}m Cosmological Argument (KCA) for theism, that is, (P1) everything that begins to exist has a cause (...) for its beginning to exist, (P2) physical reality began to exist, and, therefore, (C) physical reality has a cause for its beginning to exist. While the KCA has traditionally been used to argue for God's existence, the KCA does not mention God, has been rejected by historically significant Christian theologians such as Thomas Aquinas, and raises perennial philosophical questions -- about the nature and history of physical reality, the nature of time, the nature of causation, and so on -- that should be of interest to all philosophers and, perhaps, all humans. While I am not a religious person, I am interested in the questions raised by the KCA. In this dissertation, I articulate three necessary conditions that physical reality would need to fulfill in order to have had a beginning and argue that, given the current state of philosophical and scientific inquiry, we cannot determine whether physical reality began to exist. -/- Friends of the KCA have sought to defend their view that physical reality began to exist in two distinct ways. As I discuss in chapter 2, the first way in which friends of the KCA have sought to defend their view that physical reality began to exist involves a family of a priori arguments meant to show that, as a matter of metaphysical necessity, the past must be finite. If the past is necessarily finite, then the past history of physical reality is necessarily finite. And if having a finite past suffices for having a beginning, then, since the past history of physical reality is necessarily finite, physical reality necessarily began to exist. I show that the arguments which have been offered thus far for the view that the past is necessarily finite do not succeed. Moreover, as I elaborate on in chapter 5, having a finite past does not suffice for having a beginning. -/- As I discuss in chapter 3, the second way in which friends of the KCA have sought to defend their view that physical reality began to exist involves a family of a posteriori arguments meant to show that we have empirical evidence that physical reality has a finite past history. For example, the big bang is sometimes claimed to have been the beginning of physical reality and, since we have excellent empirical evidence for the big bang, we have excellent empirical evidence for the beginning of physical reality. The big bang can be understood in two ways. On the one hand, the big bang can be understood as a theory about the history and development of the observable universe. Understood in that sense, then I agree that the big bang is supported by excellent empirical evidence and by a scientific consensus. On the other hand, some authors (particularly science popularizers, science journalists, and religious apologists) have wrongly interpreted big bang theory as a theory about the beginning of the whole of physical reality. As I argue, while a beginning of physical reality may be consistent with classical big bang theory, classical big bang theory does not provide good reason for thinking that physical reality began to exist. -/- In part II, I turn to discussing three necessary, but not necessarily sufficient, conditions for physical reality to have a beginning. Before discussing the three conditions, in chapter 4, I introduce three metaphysical accounts of the nature of time (A-theory, B-theory, and C-theory) as well as some formal machinery that will subsequently become useful in the dissertation. I introduce the first of the three conditions in chapter 5. According to the Modal Condition, physical reality began to exist only if, at the closest possible worlds without time, physical reality does not exist. I show that this condition helps us to make sense of various views in both theology and philosophy of physics. In chapter 6}, I introduce the second of my three conditions, the Direction Condition, according to which, roughly, physical reality began to exist only if all space-time points agree about the direction of time, so that all space-time points can agree that physical reality's putative beginning took place in their objective past. In chapter 7, I discuss the third condition, the Boundary Condition, according to which physical reality began to exist only if there is a past temporal boundary such that physical reality did not exist before the boundary. I show that there are two senses in which physical reality could be said to have had a past temporal boundary. Lastly, in chapter 8, I show that there is a relationship between my three conditions and classical big bang theory, even though the relationship is not the one usually identified in the literature. -/- In part III, I present four arguments for the view that, at the present stage of philosophical and scientific inquiry, we cannot know whether physical reality satisfies the three necessary conditions to have had a beginning and, consequently, we cannot know whether physical reality had a beginning. As I will prove in chapter 9, no set of observations that we currently have, when conjoined with General Relativity, entails that physical reality satisfies the Direction or Boundary Conditions. As I show in chapter 10, considerations in the philosophical foundations of statistical mechanics entail either that the Cosmos violates the Modal Condition or else that there is a transcendental condition on the possibility of our knowledge of the past that prevents our access to data we would need to gather to determine whether physical reality satisfies the Boundary Condition. In chapter 11, I show that there are a number of live cosmological models according to which physical reality does not satisfy the Boundary Condition. As long as we don't know whether any of those cosmological models are correct, we do not know whether physical reality satisfies the Boundary Condition. Lastly, I turn to confirmation theory and show that, at our present stage of inquiry, ampliative inferences for the conclusion that physical reality satisfies the Modal, Direction, and Boundary Conditions are not successful. (shrink)

I argue that we can get evidence for the temporal ontology of the universe simply by looking at the time. The argument is an extension of the ‘epistemic objection’ towards Growing Block theories.

A standard way to challenge convergence-based accounts of inductive success is to claim that they are too weak to constrain inductive inferences in the short run. We respond to such a challenge by answering some questions raised by Juhl (1994). When it comes to predicting limiting relative frequencies in the framework of Reichenbach, we show that speed-optimal convergence—a long-run success condition—induces dynamic coherence in the short run.

This paper models knowledge in cases where an agent has multiple experiences over time. Using this model, we introduce a series of observations that undermine the pretheoretic idea that the evidential significance of experience depends on the extent to which that experience matches the world. On the basis of these observations, we model knowledge in terms of what is likely given the agent’s experience. An agent knows p when p is implied by her epistemic possibilities. A world is epistemically possible (...) when its probability given the agent’s experiences is not significantly lower than the probability of the actual world given that experience. (shrink)

The present paper focuses on the work of Adolphe Quetelet (1796-1874), the Belgian author of the Social Physics who worked in the tradition of the French mathématique sociale, and of Otto Neurath (1882-1945), the Vienna Circle’s member who supported a “sociology within physicalism”. They shared some important philosophical and methodological positions: an empiricist approach to the social sciences, a unitary conception of the natural and the social sciences, and the appreciation of statistics as a tool for investigating and also reforming (...) society. My paper analyses these elements of continuity between the two authors, in order to highlight how an empirical-quantitative approach to society was (at least partially) inherited by Austrian sociology from an earlier French tradition. On the contrary, most German statisticians refused such an approach at the time in which social sciences were developing in Germany, and they strongly believed in an in-principle difference between natural and social sciences. From a systematic point of view, my paper devotes a special attention to the relationship between the application of statistics (praxis) and a unitary conception of the sciences (theory). Both in Quetelet’s and in Neurath’s work these two aspects support each other. Nevertheless Neurath can be said to have turned Quetelet’s reasoning upside-down: While Quetelet appealed to statistics to claim that the social sciences can reach the same objectivity and determinacy as the natural ones, Neurath referred to statistics to argue that the natural sciences – exactly like the social ones – are characterized by a certain degree of indeterminacy and underdetermination. (shrink)

My first goal is to motivate a distinctively metaphysical approach to the problem of induction. I argue that there is a precise sense in which the only way that orthodox Humean and non-Humean views can justify induction is by appealing to extremely strong and unmotivated probabilistic biases. My second goal is to sketch what such a metaphysical approach could possibly look like. After sketching such an approach, I consider a toy case that illustrates the way in which such a metaphysics (...) can help us make progress on the problem of induction. (shrink)

This paper briefly outlines some advancements in paraconsistent logics for modelling knowledge representation and reasoning. Emphasis is given on the so-called Logics of Formal Inconsistency (LFIs), a class of paraconsistent logics that formally internalize the very concept(s) of consistency and inconsistency. A couple of specialized systems based on the LFIs will be reviewed, including belief revision and probabilistic reasoning. Potential applications of those systems in the AI area of KRR are tackled by illustrating some examples that emphasizes the importance of (...) a fine-tuned treatment of consistency in modelling reputation systems, preferences, argumentation, and evidence. (shrink)

We present an interdisciplinary approach to study systematic relations between logical form and attacks between claims in an argumentative framework. We propose to generalize qualitative attack principles by quantitative ones. Specifically, we use coherent conditional probabilities to evaluate the rationality of principles which govern the strength of argumentative attacks. Finally, we present an experiment which explores the psychological plausibility of selected attack principles.

Video gaming has been rising rapidly to become one of the primary entertainment media, especially during the COVID-19 pandemic. Playing video games has been reported to associate with many psychological and behavioral traits. However, little is known about the connections between game players' behaviors in the virtual environment and environmental perceptions. Thus, the current data set offers valuable resources regarding environmental worldviews and behaviors in the virtual world of 640 Animal Crossing: New Horizons (ACNH) game players from 29 countries around (...) the globe. The data set consists of six major categories: 1) socio-demographic profile, 2) COVID-19 concern, 3) environmental perception, 4) game-playing habit, 5) in-game behavior, and 6) game-playing feeling. By making this data set open, we aim to provide policymakers, game producers, and researchers with valuable resources for understanding the interactions between behaviors in the virtual world and environmental perceptions, which could help produce video games in compliance with the United Nations (UN) Sustainable Development Goals. (shrink)

We provide counterexamples to some purported characterizations of dilation due to Pedersen and Wheeler :1305–1342, 2014, ISIPTA ’15: Proceedings of the 9th international symposium on imprecise probability: theories and applications, 2015).

We investigate epistemic independence for choice functions in a multivariate setting. This work is a continuation of earlier work of one of the authors [23], and our results build on the characterization of choice functions in terms of sets of binary preferences recently established by De Bock and De Cooman [7]. We obtain the independent natural extension in this framework. Given the generality of choice functions, our expression for the independent natural extension is the most general one we are aware (...) of, and we show how it implies the independent natural extension for sets of desirable gambles, and therefore also for less informative imprecise-probabilistic models. Once this is in place, we compare this concept of epistemic independence to another independence concept for choice functions proposed by Seidenfeld [22], which De Bock and De Cooman [1] have called S-independence. We show that neither is more general than the other. (shrink)

This paper is about two requirements on wish reports whose interaction motivates a novel semantics for these ascriptions. The first requirement concerns the ambiguities that arise when determiner phrases, such as definite descriptions, interact with ‘wish’. More specifically, several theorists have recently argued that attitude ascriptions featuring counterfactual attitude verbs license interpretations on which the determiner phrase is interpreted relative to the subject’s beliefs. The second requirement involves the fact that desire reports in general require decision-theoretic notions for their analysis. (...) The current study is motivated by the fact that no existing account captures both of these aspects of wishing. I develop a semantics for wish reports that makes available belief-relative readings but also allows decision-theoretic notions to play a role in shaping the truth conditions of these ascriptions. The general idea is that we can analyze wishing in terms of a two-dimensional notion of expected utility. (shrink)

Once upon a time, some thought that indicative conditionals could be effectively analyzed as material conditionals. Later on, an alternative theoretical construct has prevailed and received wide acceptance, namely, the conditional probability of the consequent given the antecedent. Partly following critical remarks recently ap- peared in the literature, we suggest that evidential support—rather than conditional probability alone—is key to understand indicative conditionals. There have been motivated concerns that a theory of evidential conditionals (unlike their more tra- ditional counterparts) cannot generate (...) a sufficiently interesting logical system. Here, we will describe results dispelling these worries. Happily, and perhaps surprisingly, appropriate technical variations of Ernst Adams’s classical approach allow for the construction of a logic of evidential conditionals with distinctive fea- tures, which is also well-behaved and reasonably strong. (shrink)

El estudio actual de la argumentación se encuentra distanciado de la lógica. En este artículo sostengo que restaurar el vínculo del estudio de la argumentación con esta disciplina podría resultar benéfico para la metas descriptivas y normativas de este campo de investigación. Tras destacar algunos aspectos del surgimiento la teoría de la argumentación contemporánea, enfatizando la idea de "perspectivas", explico cómo el reconocimiento de sus objetivos y tareas volvió problemática la coexistencia de varios enfoques o aproximaciones para el estudio de (...) la realidad argumentativa. Sugiero algunas hipótesis explicativas sobre el abandono del enfoque lógico en favor de otras alternativas y, a continuación, ofrezco razones para encauzar el estudio sistemático de la argumentación desde este enfoque. Además de las consideraciones normativas y las herramientas descriptivas que pueden aducirse en su favor, sostengo que el enfoque lógico puede proporcionar una clase de integración a este campo investigación de maneras que no están disponibles para otras aproximaciones. (shrink)

Necessity and sufficiency are the building blocks of all successful explanations. Yet despite their importance, these notions have been conceptually underdeveloped and inconsistently applied in explainable artificial intelligence (XAI), a fast-growing research area that is so far lacking in firm theoretical foundations. Building on work in logic, probability, and causality, we establish the central role of necessity and sufficiency in XAI, unifying seemingly disparate methods in a single formal framework. We provide a sound and complete algorithm for computing explanatory factors (...) with respect to a given context, and demonstrate its flexibility and competitive performance against state of the art alternatives on various tasks. (shrink)

We deepen the study of conjoined and disjoined conditional events in the setting of coherence. These objects, differently from other approaches, are defined in the framework of conditional random quantities. We show that some well known properties, valid in the case of unconditional events, still hold in our approach to logical operations among conditional events. In particular we prove a decomposition formula and a related additive property. Then, we introduce the set of conditional constituents generated by $n$ conditional events and (...) we show that they satisfy the basic properties valid in the case of unconditional events. We obtain a generalized inclusion-exclusion formula, which can be interpreted by introducing a suitable distributive property. Moreover, under logical independence of basic unconditional events, we give two necessary and sufficient coherence conditions. The first condition gives a geometrical characterization for the coherence of prevision assessments on a family F constituted by n conditional events and all possible conjunctions among them. The second condition characterizes the coherence of prevision assessments defined on $F\cup K$, where $K$ is the set of conditional constituents associated with the conditional events in $F$. Then, we give some further theoretical results and we examine some examples and counterexamples. Finally, we make a comparison with other approaches and we illustrate some theoretical aspects and applications. (shrink)

Can perceptual experiences be states of uncertainty? We might expect them to be, if the perceptual processes from which they're generated, as well as the behaviors they help produce, take account of probabilistic information. Yet it has long been presumed that perceptual experiences purport to tell us about our environment, without hedging or qualifying. Against this long-standing view, I argue that perceptual experiences may well occasionally be states of uncertainty, but that they are never probabilistically structured. I criticize a powerful (...) line of reasoning that we should expect perceptual experience to be probabilistic, given their interfaces with unconscious probabilistic information, with behavior responsive to it, and with credences. (shrink)

Permissivism is the view that, sometimes, there is more than one doxastic attitude that is perfectly rationalised by the evidence. Impermissivism is the denial of Permissivism. Several philosophers, with the aim to defend either Impermissivism or Permissivism, have recently discussed the value of (im)permissive rationality. This paper focuses on one kind of value-conferring considerations, stemming from the so-called “truth-connection” enjoyed by rational doxastic attitudes. The paper vindicates the truth-connected value of permissive rationality by pursuing a novel strategy which rests on (...) two main planks: first, there is a distinction between a fine-grained and a coarse-grained type-individuation of belief-forming methods. Secondly, different kinds of decision-theoretic reasoning, i.e. expected-accuracy reasoning and accuracy-domination reasoning, must be paired with a fine-grained and a coarse-grained type-individuation of methods, respectively. I argue that while the first pair is wholly irrelevant to the question of the truth-connection, the second affords the means to a permissivist explanation of the truth-connected value of rationality. (shrink)

This paper is about the alethic aspect of epistemic rationality. The most common approaches to this aspect are either normative (what a reasoner ought to/may believe?) or evaluative (how rational is a reasoner?), where the evaluative approaches are usually comparative (one reasoner is assessed compared to another). These approaches often present problems with blindspots. For example, ought a reasoner to believe a currently true blindspot? Is she permitted to? Consequently, these approaches often fail in describing a situation of alethic maximality, (...) where a reasoner fulfills all the alethic norms and could be used as a standard of rationality (as they are, in fact, used in some of these approaches). I propose a function α, which accepts a set of beliefs as inputand returns a numeric alethic value. Then I use this function to define a notion of alethic maximality that is satisfiable by finite reasoners (reasoners with cognitive limitations) and does not present problems with blindspots. Function α may also be used in alethic norms and evaluation methods (comparative and non-comparative) that may be applied to finite reasoners and do not present problems with blindspots. A result of this investigation isthat the project of providing purely alethic norms is defective. The use of function α also sheds light on important epistemological issues, such as the lottery and the preface paradoxes, and the principles of clutter avoidance and reflection. (shrink)

The principle of epistemic closure is the claim that what is known to follow from knowledge is known to be true. This intuitively plausible idea is endorsed by a vast majority of knowledge theorists. There are significant problems, however, that have to be addressed if epistemic closure – closed knowledge – is endorsed. The present essay locates the problem for closed knowledge in the separation it imposes between knowledge and evidence. Although it might appear that all that stands between knowing (...) the truth of the premises of a valid inference and knowledge of its conclusion is inferring it from the premises, the evidence for each of the premises may jointly count against the conclusion. The intuitive view regarding inferred knowledge says one thing, the evidence says another. One epistemological framework that seems to have the resources to resolve this tension endorses the view that knowledge always requires conclusive evidence. A second framework resolves the tension by limiting the scope of the closure principle. Only inferences drawn directly from propositions contained in the scope of a single knowledge operator are considered closed. The aim of the present essay is to revive the unpopular third option, the idea that knowledge is open. The essay proceeds by arguing that in different ways the two former frameworks only succeed in relocating the problem, not in resolving it. The first framework, the infallibilist view, relocates the problem to a sharp separation between knowledge of the occurrences of events from knowledge of their chance of occurring, a separation leading to several significant additional problems. The fallibilist view, the second framework, in endorsing closure neglects to take into full account the ways in which evidence fails to be transitive. For instance, evidence can count in favor of a conjunction while counting against each of its conjuncts. This fact, which is argued for in the essay on probabilistic as well as non-probabilistic grounds, is used as the foundation of an argument against closed knowledge that can be used as a way to understand several of the most fundamental challenges of epistemology. Not only can an open knowledge view that is based on open evidence resolve all these problems in a simple and natural way, it can also respond to formidable challenges that significantly hinder other open knowledge views. There are good reasons, then, to view both knowledge and evidence as open. (shrink)

The principle of epistemic closure is the claim that what is known to follow from knowledge is known to be true. This intuitively plausible idea is endorsed by a vast majority of knowledge theorists. There are significant problems, however, that have to be addressed if epistemic closure – closed knowledge – is endorsed. The present essay locates the problem for closed knowledge in the separation it imposes between knowledge and evidence. Although it might appear that all that stands between knowing (...) the truth of the premises of a valid inference and knowledge of its conclusion is inferring it from the premises, the evidence for each of the premises may jointly count against the conclusion. The intuitive view regarding inferred knowledge says one thing, the evidence says another. One epistemological framework that seems to have the resources to resolve this tension endorses the view that knowledge always requires conclusive evidence. A second framework resolves the tension by limiting the scope of the closure principle. Only inferences drawn directly from propositions contained in the scope of a single knowledge operator are considered closed. The aim of the present essay is to revive the unpopular third option, the idea that knowledge is open. The essay proceeds by arguing that in different ways the two former frameworks only succeed in relocating the problem, not in resolving it. The first framework, the infallibilist view, relocates the problem to a sharp separation between knowledge of the occurrences of events from knowledge of their chance of occurring, a separation leading to several significant additional problems. The fallibilist view, the second framework, in endorsing closure neglects to take into full account the ways in which evidence fails to be transitive. For instance, evidence can count in favor of a conjunction while counting against each of its conjuncts. This fact, which is argued for in the essay on probabilistic as well as non-probabilistic grounds, is used as the foundation of an argument against closed knowledge that can be used as a way to understand several of the most fundamental challenges of epistemology. Not only can an open knowledge view that is based on open evidence resolve all these problems in a simple and natural way, it can also respond to formidable challenges that significantly hinder other open knowledge views. There are good reasons, then, to view both knowledge and evidence as open. (shrink)

Conditional probability is often used to represent the probability of the conditional. However, triviality results suggest that the thesis that the probability of the conditional always equals conditional probability leads to untenable conclusions. In this paper, I offer an interpretation of this thesis in a possible worlds framework, arguing that the triviality results make assumptions at odds with the use of conditional probability. I argue that these assumptions come from a theory called the operator theory and that the rival restrictor (...) theory can avoid these problematic assumptions. In doing so, I argue that recent extensions of the triviality arguments to restrictor conditionals fail, making assumptions which are only justified on the operator theory. (shrink)

The logic of indicative conditionals remains the topic of deep and intractable philosophical disagreement. I show that two influential epistemic norms—the Lockean theory of belief and the Ramsey test for conditional belief—are jointly sufficient to ground a powerful new argument for a particular conception of the logic of indicative conditionals. Specifically, the argument demonstrates, contrary to the received historical narrative, that there is a real sense in which Stalnaker’s semantics for the indicative did succeed in capturing the logic of the (...) Ramseyan indicative conditional. (shrink)

Suppositions can be introduced in either the indicative or subjunctive mood. The introduction of either type of supposition initiates judgments that may be either qualitative, binary judgments about whether a given proposition is acceptable or quantitative, numerical ones about how acceptable it is. As such, accounts of qualitative/quantitative judgment under indicative/subjunctive supposition have been developed in the literature. We explore these four different types of theories by systematically explicating the relationships canonical representatives of each. Our representative qualitative accounts of indicative (...) and subjunctive supposition are based on the belief change operations provided by AGM revision and KM update respectively; our representative quantitative ones are offered by conditionalization and imaging. This choice is motivated by the familiar approach of understanding supposition as `provisional belief revision' wherein one temporarily treats the supposition as true and forms judgments by making appropriate changes to their other opinions. To compare the numerical judgments recommended by the quantitative theories with the binary ones recommended by the qualitative accounts, we rely on a suitably adapted version of the Lockean thesis. Ultimately, we establish a number of new results that we interpret as vindicating the often-repeated claim that conditionalization is a probabilistic version of revision, while imaging is a probabilistic version of update. (shrink)

This paper argues that if knowledge is defined in terms of probabilistic tracking then the benefits of epistemic closure follow without the addition of a closure clause. (This updates my definition of knowledge in Tracking Truth 2005.) An important condition on this result is found in "Closure Failure and Scientific Inquiry" (2017).

In actuarial parlance, the price of an insurance policy is considered fair if customers bearing the same risk are charged the same price. The estimate of this fair amount hinges on the expected value obtained by weighting the different claims by their probability. We argue that, historically, this concept of actuarial fairness originates in an Aristotelian principle of justice in exchange (equality in risk). We will examine how this principle was formalized in the 16th century and shaped in life insurance (...) during the following two hundred years, in two different interpretations. The Domatian account of actuarial fairness relied on subjective uncertainty: An agreement on risk was fair if both parties were equally ignorant about the chances of an uncertain event. The objectivist version grounded any agreement on an objective risk estimate drawn from a mortality table. We will show how the objectivist approach collapsed in the market for life annuities during the 18th century, leaving open the question of why we still speak of actuarial fairness as if it were an objective expected value. (shrink)

A probabilistic Chomsky–Schützenberger hierarchy of grammars is introduced and studied, with the aim of understanding the expressive power of generative models. We offer characterizations of the distributions definable at each level of the hierarchy, including probabilistic regular, context-free, (linear) indexed, context-sensitive, and unrestricted grammars, each corresponding to familiar probabilistic machine classes. Special attention is given to distributions on (unary notations for) positive integers. Unlike in the classical case where the "semi-linear" languages all collapse into the regular languages, using analytic tools (...) adapted from the classical setting we show there is no collapse in the probabilistic hierarchy: more distributions become definable at each level. We also address related issues such as closure under probabilistic conditioning. (shrink)

Enjoying great popularity in decision theory, epistemology, and philosophy of science, Bayesianism as understood here is fundamentally concerned with epistemically ideal rationality. It assumes a tight connection between evidential probability and ideally rational credence, and usually interprets evidential probability in terms of such credence. Timothy Williamson challenges Bayesianism by arguing that evidential probabilities cannot be adequately interpreted as the credences of an ideal agent. From this and his assumption that evidential probabilities cannot be interpreted as the actual credences of human (...) agents either, he concludes that no interpretation of evidential probabilities in terms of credence is adequate. I argue to the contrary. My overarching aim is to show on behalf of Bayesians how one can still interpret evidential probabilities in terms of ideally rational credence and how one can maintain a tight connection between evidential probabilities and ideally rational credence even if the former cannot be interpreted in terms of the latter. By achieving this aim I illuminate the limits and prospects of Bayesianism. (shrink)

Games of chance are developed in their physical consumer-ready form on the basis of mathematical models, which stand as the premises of their existence and represent their physical processes. There is a prevalence of statistical and probabilistic models in the interest of all parties involved in the study of gambling – researchers, game producers and operators, and players – while functional models are of interest more to math-inclined players than problem-gambling researchers. In this paper I present a structural analysis of (...) the knowledge attached to mathematical models of games of chance and the act of modeling, arguing that such knowledge holds potential in the prevention and cognitive treatment of excessive gambling, and I propose further research in this direction. (shrink)

Justification logics are constructive analogues of modal logics. They are often used as epistemic logics, particularly as models of evidentialist justification. However, in this role, justification (and modal) logics are defective insofar as they represent justification with a necessity-like operator, whereas actual evidentialist justification is usually probabilistic. This paper first examines and rejects extant candidates for solving this problem: Milnikel’s Logic of Uncertain Justifications, Ghari’s Hájek–Pavelka-Style Justification Logics and a version of probabilistic justification logic developed by Kokkinis et al. It (...) then proposes a new solution to the problem in the form of a justification logic that incorporates the essential features of both a fuzzy logic and a probabilistic logic. (shrink)

According to the Fine-Tuning Argument, the existence of life in our universe confirms the Multiverse Hypothesis. A standard objection to FTA is that it violates the Requirement of Total Evidence. I argue that RTE should be rejected in favor of the Predesignation Requirement, according to which, in assessing the outcome of a probabilistic process, we should only use evidence characterizable in a manner available before observing the outcome. This produces the right verdicts in some simple cases in which RTE leads (...) us astray, and, when applied to FTA, it shows that our evidence does confirm HM. (shrink)

Deduction is important to scientific inquiry because it can extend knowledge efficiently, bypassing the need to investigate everything directly. The existence of closure failure—where one knows the premises and that the premises imply the conclusion but nevertheless does not know the conclusion—is a problem because it threatens this usage. It means that we cannot trust deduction for gaining new knowledge unless we can identify such cases ahead of time so as to avoid them. For philosophically engineered examples we have “inner (...) alarm bells” to detect closure failure, but in scientific investigation we would want to use deduction for extension of our knowledge to matters we don’t already know that we couldn’t know. Through a quantitative treatment of how fast probabilistic sensitivity is lost over steps of deduction, I identify a condition that guarantees that the growth of potential error will be gradual; thus, dramatic closure failure is avoided. Whether the condition is fulfilled is often obvious, but sometimes it requires substantive investigation. I illustrate that not only safe deduction but the discovery of dramatic closure failures can lead to scientific advances. (shrink)

Integrated Information Theory (IIT) is one of the most influential theories of consciousness, mainly due to its claim of mathematically formalizing consciousness in a measurable way. However, the theory, as it is formulated, does not account for contextual observations that are crucial for understanding consciousness. Here we put forth three possible difficulties for its current version, which could be interpreted as a trilemma. Either consciousness is contextual or not. If contextual, either IIT needs revisions to its axioms to include contextuality, (...) or it is inconsistent. If consciousness is not contextual, then IIT faces an empirical challenge. Therefore, we argue that IIT in its current version is inadequate. (shrink)

In this paper we consider conditional random quantities (c.r.q.’s) in the setting of coherence. Based on betting scheme, a c.r.q. X|H is not looked at as a restriction but, in a more extended way, as \({XH + \mathbb{P}(X|H)H^c}\) ; in particular (the indicator of) a conditional event E|H is looked at as EH + P(E|H)H c . This extended notion of c.r.q. allows algebraic developments among c.r.q.’s even if the conditioning events are different; then, for instance, we can give a (...) meaning to the sum X|H + Y|K and we can define the iterated c.r.q. (X|H)|K. We analyze the conjunction of two conditional events, introduced by the authors in a recent work, in the setting of coherence. We show that the conjoined conditional is a conditional random quantity, which may be a conditional event when there are logical dependencies. Moreover, we introduce the negation of the conjunction and by applying De Morgan’s Law we obtain the disjoined conditional. Finally, we give the lower and upper bounds for the conjunction and disjunction of two conditional events, by showing that the usual probabilistic properties continue to hold. (shrink)