In the literature, different frameworks of probabilistic epistemic logic have been proposed. Most of these frameworks define knowledge or belief by relational structure. In this paper, we explore the relationship between probability and belief, based on the Lockean thesis, and adopt neighborhood semantics that defines belief directly using probability. We provide a sound and weakly complete axiomatization for our framework. We also try to explain the lottery paradox by modelling it within our framework. Moreover, the paper presents (...) findings concerning the relationship between knowledge, belief and probability within various frameworks. (shrink)

This paper aims to extend in two directions the probabilistic dynamic epistemic logic provided in Kooi’s paper (J Logic Lang Inform 12(4):381–408, 2003) and to relate these extensions to ones made in van Benthem et al. (Proceedings of LOFT’06. Liverpool, 2006). Kooi’s probabilistic dynamic epistemic logic adds to probabilistic epistemic logic sentences that express consequences of public announcements. The paper (van Benthem et al., Proceedings of LOFT’06. Liverpool, 2006) extends (Kooi, (...) J Logic Lang Inform 12(4):381–408, 2003) to using action models, but in both papers, the probabilities are discrete, and are defined on trivial σ -algebras over finite sample spaces. The first extension offered in this paper is to add a previous-time operator to a probabilistic dynamic epistemic logic similar to Kooi’s in (J Logic Lang Inform 12(4):381–408, 2003). The other is to involve non-trivial σ -algebras and continuous probabilities in probabilistic dynamic epistemic logic. (shrink)

The paper offers a formalization of reasoning about distributed multi-agent systems. The presented propositional probabilistic temporal epistemic logic |$\textbf {PTEL}$| is developed in full detail: syntax, semantics, soundness and strong completeness theorems. As an example, we prove consistency of the blockchain protocol with respect to the given set of axioms expressed in the formal language of the logic. We explain how to extend |$\textbf {PTEL}$| to axiomatize the corresponding first-order logic.

In this paper I combine the dynamic epistemic logic ofGerbrandy (1999) with the probabilistic logic of Fagin and Halpern (1994). The resultis a new probabilistic dynamic epistemic logic, a logic for reasoning aboutprobability, information, and information change that takes higher orderinformation into account. Probabilistic epistemic models are defined, and away to build them for applications is given. Semantics and a proof systemis presented and a number of examples are discussed, including (...) the MontyHall Dilemma. (shrink)

We study a propositional probabilistic temporal epistemic logic |$\textbf {PTEL}$| with both future and past temporal operators, with non-rigid set of agents and the operators for agents’ knowledge and for common knowledge and with probabilities defined on the sets of runs and on the sets of possible worlds. A semantics is given by a class |${\scriptsize{\rm Mod}}$| of Kripke-like models with possible worlds. We prove decidability of |$\textbf {PTEL}$| by showing that checking satisfiability of a formula in (...) |${\scriptsize{\rm Mod}}$| is equivalent to checking its satisfiability in a finite set of finitely representable structures. The same procedure can be applied to the class of all synchronous |${\scriptsize{\rm Mod}}$|-models. We give an upper complexity bound for the satisfiability problem for |${\scriptsize{\rm Mod}}$|⁠. (shrink)

This paper studies Aumann’s agreeing to disagree theorem from the perspective of dynamic epistemic logic. This was first done by Dégremont and Roy (J Phil Log 41:735–764, 2012) in the qualitative framework of plausibility models. The current paper uses a probabilistic framework, and thus stays closer to Aumann’s original formulation. The paper first introduces enriched probabilistic Kripke frames and models, and various ways of updating them. This framework is then used to prove several agreement theorems, which (...) are natural formalizations of Aumann’s original result. Furthermore, a sound and complete axiomatization of a dynamic agreement logic is provided, in which one of these agreement theorems can be derived syntactically. These technical results are used to show the importance of explicitly representing the dynamics behind the agreement theorem, and lead to a clarification of some conceptual issues surrounding the agreement theorem, in particular concerning the role of common knowledge. The formalization of the agreement theorem thus constitutes a concrete example of the so-called dynamic turn in logic. (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)

In the literature there are at least two main formal structures to deal with situations of interactive epistemology: Kripke models and type spaces. As shown in many papers :149–225, 1999; Battigalli and Siniscalchi in J Econ Theory 106:356–391, 2002; Klein and Pacuit in Stud Log 102:297–319, 2014; Lorini in J Philos Log 42:863–904, 2013), both these frameworks can be used to express epistemic conditions for solution concepts in game theory. The main result of this paper is a formal comparison (...) between the two and a statement of semantic equivalence with respect to two different logical systems: a doxastic logic for belief and an epistemic–doxastic logic for belief and knowledge. Moreover, a sound and complete axiomatization of these logics with respect to the two equivalent Kripke semantics and type spaces semantics is provided. Finally, a probabilistic extension of the result is also presented. A further result of the paper is a study of the relationship between the epistemic–doxastic logic for belief and knowledge and the logic STIT by Belnap and colleagues. (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)

Is logical consistency required for a set of beliefs or propositions to be categorically coherent? An affirmative answer is often assumed by mainstream epistemologists, and yet it is unclear why. Cases like the lottery and the preface call into question the assumption that beliefs must be consistent in order to be epistemically rational. And thus it is natural to wonder why all inconsistent sets of propositions are incoherent. On the other hand, Easwaran and Fitelson have shown that particular kinds of (...) inconsistency entail the epistemically ‘irrationality’ of holding certain sets of beliefs. In cases of the latter kind of inconsistency, it seems more reasonable to insist that such sets of beliefs or propositions are categorically incoherent. What the precise relationship is between coherence and consistency depends on the nature of the coherence relation. We shall examine recent attempts to explicate the coherence relation in terms of probabilistic measures of confirmation or agreement to see what they can teach us about the relationship between coherence and consistency. We shall show that some probabilistic measures of coherence allow for inconsistent sets to be categorically coherent, while satisfying plausible epistemic rationality constraints. Other probabilistic measures of coherence impose very strong logical consistency requirements, and some measures are tolerant of most forms of inconsistency. As we try to understand what distinguishes coherence measures in this respect, we will also draw some important lessons about Bayesian confirmation measures and differences in the way that they treat contradictory propositions. (shrink)

In this article, we present a new logical framework to think about surprise. This research does not just aim to better understand, model and predict human behaviour, but also attempts to provide tools for implementing artificial agents. Moreover, these artificial agents should then also be able to reap the same epistemic benefits from surprise as humans do. We start by discussing the dominant literature regarding propositional surprise and explore its shortcomings. These shortcomings are of both an empirical and a (...) conceptual nature. Next, we propose a philosophical solution to the problems that ail these systems, based on the notion of issue of epistemic interest. Finally, we give a formal framework to think about surprise. More specifically, we develop a probabilistic dynamic epistemic logic that succeeds at formalizing the relevant philosophical concepts. This will be done through an issue management system grounded in topology. As an added bonus, the additional expressive power allows us to capture a richer variety of scenarios, and it also enables a more careful analysis of said scenarios. (shrink)

I present two logical systems to show the “analogy of proportionality„ common to several interpretations: modality (necessity and possibility), quantification, truth-functional relations, moral attitudes (deontic logic), states of knowledge (epistemic logic), and states of belief (doxastic logic). To display the two underlying analogical relations, I call upon the originally Scholastic convention, recently put to use again, of using squares, hexagons, and octagons “of opposition„. A combined epistemic–deontic logic happens to be found in the traditional (...) “probabilist„ theory of the “good conscience„, and I shall then briefly explain how this is so. (shrink)

Traditional philosophical discussions of knowledge have focused on the epistemic status of full beliefs. In this book, Moss argues that in addition to full beliefs, credences can constitute knowledge. For instance, your .4 credence that it is raining outside can constitute knowledge, in just the same way that your full beliefs can. In addition, you can know that it might be raining, and that if it is raining then it is probably cloudy, where this knowledge is not knowledge of (...) propositions, but of probabilistic contents. -/- The notion of probabilistic content introduced in this book plays a central role not only in epistemology, but in the philosophy of mind and language as well. Just as tradition holds that you believe and assert propositions, you can believe and assert probabilistic contents. Accepting that we can believe, assert, and know probabilistic contents has significant consequences for many philosophical debates, including debates about the relationship between full belief and credence, the semantics of epistemic modals and conditionals, the contents of perceptual experience, peer disagreement, pragmatic encroachment, perceptual dogmatism, and transformative experience. In addition, accepting probabilistic knowledge can help us discredit negative evaluations of female speech, explain why merely statistical evidence is insufficient for legal proof, and identify epistemic norms violated by acts of racial profiling. Hence the central theses of this book not only help us better understand the nature of our own mental states, but also help us better understand the nature of our responsibilities to each other. (shrink)

There is a well-documented paradigm-shift in eighteenth century Jesuit philosophy and science, at the very least in Central Europe: traditional scholastic version(s) of Aristotelianism were replaced by early modern rationalism (Wolff's systematisation of Leibnizian philosophy) and early modern science and mathematics. In the field of probability, this meant that the traditional Jesuit engagement with probability, uncertainty, and truthlikeness (in particular, as applied to moral theology) could translate into mathematical language, and can be analysed against the background of the accounts of (...) probability, pre-mathematical Jesuit logic, Wolff's conceptual analysis, and Bernoullian mathematisation. The works of two Jesuit philosophers, Berthold Hauser and Sigismund Storchenau, can be related to this context. The core of their logic of (epistemic) probability is the account of negation (or ‘contradiction’) and implication (or ‘argument’), in particular, the algorithms for computing the reliability of one piece of evidence when compared to the respective counter-evidence and for computing the probability of a conclusion given the probability of its premises. (shrink)

A common view among nontheists combines the de jure objection that theism is epistemically unacceptable with agnosticism about the de facto objection that theism is false. Following Plantinga, we can call this a “proper” de jure objection—a de jure objection that does not depend on any de facto objection. In his Warranted Christian Belief, Plantinga has produced a general argument against all proper de jure objections. Here I first show that this argument is logically fallacious (it makes subtle probabilistic (...) fallacies disguised by scope ambiguities), and proceed to lay the groundwork for the construction of actual proper de jure objections. (shrink)

The paper starts with the distinction between conjunction-of-parts accounts and disjunction-of-possibilities accounts to truthlikeness. In Sect. 3, three distinctions between kinds of truthlikeness measures are introduced: comparative versus numeric t-measures, t-measures for qualitative versus quantitative theories, and t-measures for deterministic versus probabilistic truth. These three kinds of truthlikeness are explicated and developed within a version of conjunctive part accounts based on content elements. The focus lies on measures of probabilistic truthlikeness, that are divided into t-measures for statistical probabilities (...) and single case probabilities. The logical notion of probabilistic truthlikeness can be treated as a subcase of deterministic truthlikeness for quantitative theories. In contrast, the epistemic notion of probabilistic truthlikeness creates genuinely new problems, especially for hypotheses about single case probabilities that are evaluated not by comparison to observed frequencies, but by comparison to the truth values of single event statements. By the method of meta-induction, competing theories about single case probabilities can be aggregated into a combined theory with optimal predictive success and epistemic truthlikeness. (shrink)

We investigate the discrete (finite) case of the Popper–Renyi theory of conditional probability, introducing discrete conditional probabilistic models for knowledge and conditional belief, and comparing them with the more standard plausibility models. We also consider a related notion, that of safe belief, which is a weak (non-negatively introspective) type of “knowledge”. We develop a probabilistic version of this concept (“degree of safety”) and we analyze its role in games. We completely axiomatize the logic of conditional belief, knowledge (...) and safe belief over conditional probabilistic models. We develop a theory of probabilistic dynamic belief revision, introducing probabilistic “action models” and proposing a notion of probabilistic update product, that comes together with appropriate reduction laws. (shrink)

When one thinks—knows, believes, imagines—that something is the case, one’s thought has a topic: it is about something, towards which one’s mind is directed. What is the logic of thought, so understood? This book begins to explore the idea that, to answer the question, we should take topics seriously. It proposes a hyperintensional account of the propositional contents of thought, arguing that these are individuated not only by the set of possible worlds at which they are true, but also (...) by their topic: what they are about. The book then builds epistemic, doxastic, probabilistic, and conditional logics based on this view. It applies them to issues ranging from dogmatism, scepticism, and epistemic fallibilism, to imagination and suppositional reasoning, belief revision, framing effects, and the acceptability of indicative conditionals. (shrink)

In this paper, we investigate the social herding phenomenon known as informational cascades, in which sequential inter-agent communication might lead to epistemic failures at group level, despite availability of information that should be sufficient to track the truth. We model an example of a cascade, and check the correctness of the individual reasoning of each agent involved, using two alternative logical settings: an existing probabilistic dynamic epistemic logic, and our own novel logic for counting evidence. (...) Based on this analysis, we conclude that cascades are not only likely to occur but are sometimes unavoidable by "rational" means: in some situations, the group’s inability to track the truth is the direct consequence of each agent’s rational attempt at individual truth-tracking. Moreover, our analysis shows that this is even so when rationality includes unbounded higher-order reasoning powers, as well as when it includes simpler, non-Bayesian forms of heuristic reasoning. (shrink)

Probabilism is a Scholastic discussion of conscience beginning in the sixteenth century and lasting over four hundred years. To tackle historical issues in normative ethics, the participants had to work out a general "metaethical" theory relating epistemic and deontic logics, the logics of knowledge and virtue. May I act if I am unsure that I may? How, when I am in doubt, can I acquire the critical mass of rationality that virtue demands? The normative aspect of the controversy has (...) been studied in its historical setting, but its interesting and original metaethics has been largely ignored outside the Scholastic circle itself. The discussion is obviously relevant, even practical, today; in fact, any treatment of conscience and relativism already includes the material of probabilist analyses. (shrink)

Some propositions are structurally unknowable for certain agents. Let me call them ‘Moorean propositions’. The structural unknowability of Moorean propositions is normally taken to pave the way towards proving a familiar paradox from epistemic logic—the so-called ‘Knowability Paradox’, or ‘Fitch’s Paradox’—which purports to show that if all truths are knowable, then all truths are in fact known. The present paper explores how to translate Moorean statements into a probabilistic language. A successful translation should enable us to derive (...) a version of Fitch’s Paradox in a probabilistic setting. I offer a suitable schematic form for probabilistic Moorean propositions, as well as a concomitant proof of a probabilistic Knowability Paradox. Moreover, I argue that traditional candidates to play the role of probabilistic Moorean propositions will not do. In particular, we can show that violations of the so-called ‘Reflection Principle’ in probability need not yield structurally unknowable propositions. Among other things, this should lead us to question whether violating the Reflection Principle actually amounts to a clear case of epistemic irrationality, as it is often assumed. This result challenges the importance of the principle as a tool to assess both synchronic and diachronic rationality—a topic which is largely independent of Fitch’s Paradox—from a somewhat unexpected source. (shrink)

The author uses a series of examples to illustrate two versions of a new, nonprobabilist principle of epistemic rationality, the special and general versions of the metacognitive, expected relative frequency principle. These are used to explain the rationality of revisions to an agent’s degrees of confidence in propositions based on evidence of the reliability or unreliability of the cognitive processes responsible for them—especially reductions in confidence assignments to propositions antecedently regarded as certain—including certainty-reductions to instances of the law of (...) excluded middle or the law of noncontradiction in logic or certainty-reductions to the certainties of probabilist epistemology. The author proposes special and general versions of the MERF principle and uses them to explain the examples, including the reasoning that would lead to thoroughgoing fallibilism—that is, to a state of being certain of nothing. The author responds to the main defenses of probabilism: Dutch Book arguments, Joyce’s potential accuracy defense, and the potential calibration defenses of Shimony and van Fraassen by showing that, even though they do not satisfy the probability axioms, degrees of belief that satisfy the MERF principle minimize expected inaccuracy in Joyce’s sense; they can be externally calibrated in Shimony and van Fraassen’s sense; and they can serve as a basis for rational betting, unlike probabilist degrees of belief, which, in many cases, human beings have no rational way of ascertaining. The author also uses the MERF principle to subsume the various epistemic akrasia principles in the literature. Finally, the author responds to Titelbaum’s argument that epistemic akrasia principles require that we be certain of some epistemological beliefs, if we are rational. (shrink)

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

Probability theory, epistemically interpreted, provides an excellent, if not the best available account of inductive reasoning. This is so because there are general and definite rules for the change of subjective probabilities through information or experience; induction and belief change are one and same topic, after all. The most basic of these rules is simply to conditionalize with respect to the information received; and there are similar and more general rules. 1 Hence, a fundamental reason for the epistemological success of (...) probability theory is that there at all exists a well-behaved concept of conditional probability. Still, people have, and have reasons for, various concerns over probability theory. One of these is my starting point: Intuitively, we have the notion of plain belief; we believe propositions2 to be true (or to be false or neither). Probability theory, however, offers no formal counterpart to this notion. Believing A is not the same as having probability 1 for A, because probability 1 is incorrigible3; but plain belief is clearly corrigible. And believing A is not the same as giving A a probability larger than some 1 - c, because believing A and believing B is usually taken to be equivalent to believing A & B.4 Thus, it seems that the formal representation of plain belief has to take a non-probabilistic route. Indeed, representing plain belief seems easy enough: simply represent an epistemic state by the set of all propositions believed true in it or, since I make the common assumption that plain belief is deductively closed, by the conjunction of all propositions believed true in it. But this does not yet provide a theory of induction, i.e. an answer to the question how epistemic states so represented are changed tbrough information or experience. There is a convincing partial answer: if the new information is compatible with the old epistemic state, then the new epistemic state is simply represented by the conjunction of the new information and the old beliefs. This answer is partial because it does not cover the quite common case where the new information is incompatible with the old beliefs. It is, however, important to complete the answer and to cover this case, too; otherwise, we would not represent plain belief as conigible. The crucial problem is that there is no good completion. When epistemic states are represented simply by the conjunction of all propositions believed true in it, the answer cannot be completed; and though there is a lot of fruitful work, no other representation of epistemic states has been proposed, as far as I know, which provides a complete solution to this problem. In this paper, I want to suggest such a solution. In [4], I have more fully argued that this is the only solution, if certain plausible desiderata are to be satisfied. Here, in section 2, I will be content with formally defining and intuitively explaining my proposal. I will compare my proposal with probability theory in section 3. It will turn out that the theory I am proposing is structurally homomorphic to probability theory in important respects and that it is thus equally easily implementable, but moreover computationally simpler. Section 4 contains a very brief comparison with various kinds of logics, in particular conditional logic, with Shackle's functions of potential surprise and related theories, and with the Dempster - Shafer theory of belief functions. (shrink)

The goal of this paper is to review and critically discuss the philosophical aspects of probabilistic seismic hazard analysis (PSHA). Given that estimates of seismic hazard are typically riddled with uncertainty, diferent epistemic values (related to the pursuit of scientifc knowledge) compete in the selection of seismic hazard models, in a context infuenced by non-epistemic values (related to practical goals and aims) as well. We frst distinguish between the diferent types of uncertainty in PSHA. We claim that (...) epistemic and nonepistemic considerations are closely related in the selection of the appropriate estimate of seismic hazard by the experts. Finally, we argue that the division of scientifc responsibility among the experts can lead to responsibility gaps. This raises a problem for the ownership of the results (“no one’s model” problem) similar to the “problem of many hands” in the ethics of technology. We conclude with a plea for a close collaboration between philosophy and engineering. (shrink)

Using epistemic logic, we provide a non-probabilistic way to formalise payoff uncertainty, that is, statements such as ‘player i has approximate knowledge about the utility functions of player j.’ We show that on the basis of this formalisation common knowledge of payoff uncertainty and rationality (in the sense of excluding weakly dominated strategies, due to Dekel and Fudenberg (1990)) characterises a new solution concept we have called ‘mixed iterated strict weak dominance.’.

I argue for an epistemic conception of voting, a conception on which the purpose of the ballot is at least in some cases to identify which of several policy proposals will best promote the public good. To support this view I first briefly investigate several notions of the kind of public good that public policy should promote. Then I examine the probability logic of voting as embodied in two very robust versions of the Condorcet Jury Theorem and some (...) related results. These theorems show that if the number of voters or legislators is sufficiently large and the average of their individual propensities to select the better of two policy proposals is a little above random chance, and if each person votes his or her own best judgment (rather than in alliance with a block or faction), then the majority is extremely likely to select the better alternative. Here ‘better alternative’ means that policy or law that will best promote the public good. I also explicate a Convincing Majorities Theorem, which shows the extent to which the majority vote should provide evidence that the better policy has been selected. Finally, I show how to extend all of these results to judgments among multiple alternatives through the kind of sequential balloting typical of the legislative amendment process. (shrink)

To understand something involves some sort of commitment to a set of propositions comprising an account of the understood phenomenon. Some take this commitment to be a species of belief; others, such as Elgin and I, take it to be a kind of cognitive policy. This paper takes a step back from debates about the nature of understanding and asks when this commitment involved in understanding is epistemically appropriate, or ‘acceptable’ in Elgin’s terminology. In particular, appealing to lessons from the (...) lottery and preface paradoxes, it is argued that this type of commitment is sometimes acceptable even when it would be rational to assign arbitrarily low probabilities to the relevant propositions. This strongly suggests that the relevant type of commitment is sometimes acceptable in the absence of epistemic justification for belief, which in turn implies that understanding does not require justification in the traditional sense. The paper goes on to develop a new probabilistic model of acceptability, based on the idea that the maximally informative accounts of the understood phenomenon should be optimally probable. Interestingly, this probabilistic model ends up being similar in important ways to Elgin’s proposal to analyze the acceptability of such commitments in terms of ‘reflective equilibrium’. (shrink)

General epistemic polarization arises when the beliefs of a population grow further apart, in particular when all agents update on the same evidence. Epistemic factionalization arises when the beliefs grow further apart, but different beliefs also become correlated across the population. I present a model of how factionalization can emerge in a population of ideally rational agents. This kind of factionalization is driven by probabilistic relations between beliefs, with background beliefs shaping how the agents’ beliefs evolve in (...) the light of new evidence. Moreover, I show that in such a model, the only possible outcomes from updating on identical evidence are general convergence or factionalization. Beliefs cannot spread out in all directions: if the beliefs overall polarize, then it must result in factionalization. (shrink)

The paper addresses charges of risk and loss aversion as well as of irrationality directed against the precautionary principle, by providing an epistemic analysis of its specific role in the safety law system. In particular, I contend that: 1) risk aversion is not a form of irrational or biased behaviour; 2) both risk and loss aversion regard the form of the utility function, whereas PP rather regards the information on which to base the decision; 3) thus PP has formally (...) nothing to do with risk or loss aversion but rather with risk awareness; 4) PP removes a fictional construct in the legal system, according to which any hazard should be ignored and denied until it is scientifically proven; 5) the quandary originates in the tension between current methods of evidence evaluation, and the logic underlying PP which demands for a probabilistic epistemology. (shrink)

This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability andcomparative probability. The goal is to identify complete logics for reasoning about uncertainty in a comparative probabilistic language whose semantics is given in terms of imprecise probability. Comparative probability operators are interpreted as quantifying over a set of probability measures. Modal and dynamic operators are added for reasoning about epistemic possibility and updating sets of probability measures.

This paper calls for a re-appraisal of McGee's analysis of the semantics, logic and probabilities of indicative conditionals presented in his 1989 paper Conditional probabilities and compounds of conditionals. The probabilistic measures introduced by McGee are given a new axiomatisation built on the principle that the antecedent of a conditional is probabilistically independent of the conditional and a more transparent method of constructing such measures is provided. McGee's Dutch book argument is restructured to more clearly reveal that it (...) introduces a novel contribution to the epistemology of semantic indeterminacy, and shows that its more controversial implications are unavoidable if we want to maintain the Ramsey Test along with the standard laws of probability. Importantly, it is shown that the counterexamples that have been levelled at McGee's analysis|generating a rather wide consensus that it yields `unintuitive' or `wrong' probabilities for compounds |fail to strike at their intended target; for to honour the intuitions of the counterexamples one must either give up the Ramsey Test or the standard laws of probability. It will be argued that we need to give up neither if we take the counterexamples as further evidence that the indicative conditional sometimes allows for a non-epistemic `causal' interpretation alongside its usual epistemic interpretation. (shrink)

We reconsider the pragmatic interpretation of intuitionistic logic [21] regarded as a logic of assertions and their justi cations and its relations with classical logic. We recall an extension of this approach to a logic dealing with assertions and obligations, related by a notion of causal implication [14, 45]. We focus on the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on polarized bi-intuitionistic logic as a (...) class='Hi'>logic of assertions and conjectures: looking at the S4 modal translation, we give a de nition of a system AHL of bi-intuitionistic logic that correctly represents the duality between intuitionistic and co-intuitionistic logic, correcting a mistake in previous work [7, 10]. A computational interpretation of cointuitionism as a distributed calculus of coroutines is then used to give an operational interpretation of subtraction.Work on linear co-intuitionism is then recalled, a linear calculus of co-intuitionistic coroutines is de ned and a probabilistic interpretation of linear co-intuitionism is given as in [9]. Also we remark that by extending the language of intuitionistic logic we can express the notion of expectation, an assertion that in all situations the truth of p is possible and that in a logic of expectations the law of double negation holds. Similarly, extending co-intuitionistic logic, we can express the notion of conjecture that p, de ned as a hypothesis that in some situation the truth of p is epistemically necessary. (shrink)

This paper develops a probabilistic model of belief change under interpretation shifts, in the context of a problem case from dynamic epistemic logic. Van Benthem [4] has shown that a particular kind of belief change, typical for dynamic epistemic logic, cannot be modelled by standard Bayesian conditioning. I argue that the problems described by van Benthem come about because the belief change alters the semantics in which the change is supposed to be modelled: the new (...) information induces a shift in the interpretation of the sentences. In this paper I show that interpretation shifts can be modeled in terms of updating by conditioning. The model derives from the knowledge structures developed by Fagin et al [8], and hinges on a distinction between the propositional and informational content of sentences. Finally, I show that Dempster-Shafer theory provides the appropriate probability kinematics for the model. (shrink)

According to Kratzer’s influential account of epistemic must and might, these operators involve quantification over domains of possibilities determined by a modal base and an ordering source. Recently, this account has been challenged by invoking contexts of ‘epistemic tension’: i.e., cases in which an assertion that must\ is conjoined with the possibility that \, and cases in which speakers try to downplay a previous assertion that must\, after finding out that \. Epistemic tensions have been invoked from (...) two directions. Von Fintel and Gillies :351–383, 2010) propose a return to a simpler modal logic-inspired account: must and might still involve universal and existential quantification, but the domains of possibilities are determined solely by realistic modal bases. In contrast, Lassiter :117–163, 2016), following Swanson, proposes a more revisionary account which treats must and might as probabilistic operators. In this paper, we present a series of experiments to obtain reliable data on the degree of acceptability of various contexts of epistemic tension. Our experiments include novel variations that, we argue, are required to make progress in this debate. We show that restricted quantificational accounts à la Kratzer fit the overall pattern of results better than either of their recent competitors. In addition, our results help us identify the key components of restricted quantificational accounts, and on that basis propose some refinements and general constraints that should be satisfied by any account of the modal auxiliaries. (shrink)

In this article, the epistemological interpretation of the relationship between concepts of relativism, beliefs, and probability ensures a defense of two theses, namely, (i) epistemic relativism refers to attitudes that depend on the repetition and anchoring of probabilistic beliefs, and (ii) Popper’s propensity interpretation of probability discloses the connections between relativity, probability, and collective subjectivity. The propensity interpretation brings a framework for describing the role of collective subjectivity in epistemic systems. This approach, as an acceptable epistemological stance, (...) is related to the variability of anchored collective beliefs. The repetition and anchoring of subjective probabilistic beliefs becomes a relative reference frame for evaluating the elements of knowledge. With this in mind, each particular way of considering probability indicates a manner of distinguishing epistemic systems. In epistemic systems, the statements are collective points of view expressed in the form ‘p is justified when belief in p is probably reliable.’ The statements can be accurate in a particular epistemic system but inaccurate in other systems. As a result, by avoiding the ‘all statements must be true or false’ requirement, each proponent of a certain epistemic system can build a probabilistic picture of the world. In this picture, not every statement needs to be ‘true’ when and if the statement expresses a collective point of view that has a propensity to be reliable and justified. (shrink)

Using epistemic logic, we provide a non-probabilistic way to formalise payoff uncertainty, that is, statements such as ‘player i has approximate knowledge about the utility functions of player j.’ We show that on the basis of this formalisation common knowledge of payoff uncertainty and rationality (in the sense of excluding weakly dominated strategies, due to Dekel and Fudenberg (1990)) characterises a new solution concept we have called ‘mixed iterated strict weak dominance.’.

You have higher-order uncertainty iff you are uncertain of what opinions you should have. I defend three claims about it. First, the higher-order evidence debate can be helpfully reframed in terms of higher-order uncertainty. The central question becomes how your first- and higher-order opinions should relate—a precise question that can be embedded within a general, tractable framework. Second, this question is nontrivial. Rational higher-order uncertainty is pervasive, and lies at the foundations of the epistemology of disagreement. Third, the answer is (...) not obvious. The Enkratic Intuition---that your first-order opinions must “line up” with your higher-order opinions---is incorrect; epistemic akrasia can be rational. If all this is right, then it leaves us without answers---but with a clear picture of the question, and a fruitful strategy for pursuing it. (shrink)

The aim of this paper is to provide a philosophical interpretation of bilateralism in terms of probabilism. In particular, to interpret the main concepts of bilateralism –acceptance, rejection and incoherence– in terms of the probabilistic notions of degree of belief and coherence. According to bilateralism, the meaning of logical connectives is determined by the acceptance and rejection conditions of the sentences in which they are involved, where acceptance and rejection cannot be reduced to one another. I will focus on (...) a variant of bilateralism that understands logical consequence as the statement that it is incoherent to accept all the premises of a valid argument while rejecting all its conclusions. On the other hand, probabilism states that it is possible to interpret our degrees of belief in terms of probabilities. The aim of this work is then to interpret the concept of incoherence in terms of probability functions and determine when it is coherent to accept or to reject a proposition according to some threshold defined in terms of degrees of belief. To achieve this goal, we need both an interpretation of the concept of incoherence coined by the bilateralists as well as an interpretation of acceptance and rejection. I will show that a good interpretation of coherence in probabilistic terms can already be found in the literature. Then, I will give an interpretation of acceptance and rejection in terms of degrees of belief. In particular, I will show that it is possible to interpret these concepts in accordance with Locke’s thesis, the thesis that states that there is some threshold r such that if you believe some sentence in degree equal or higher than r you should accept it, without falling into epistemic paradoxes. (shrink)

It seems intuitive that our credal states are improved if we obtain evidence favoring truth over any falsehood. In this regard, Fallis and Lewis have recently provided and discussed some formal versions of such an intuition, which they name ‘the Monotonicity Principle’ and ‘Elimination’. They argue, with those principles in hand, that the Brier rule, one of the most popular rules of accuracy, is not a good measure, and that accuracy-firsters cannot underwrite both probabilism and conditionalization. In this paper, I (...) will argue that their conclusions are somewhat hasty. Specifically, I will demonstrate that there is another version of the Monotonicity Principle that can be satisfied by some additive rules of accuracy, such as the Brier rule. Moreover, it will also be argued that their version of the principle has some undesirable features regarding the epistemic betterness. Therefore, their criticisms can hardly jeopardize accuracy-firsters until any further justification of their versions of the Monotonicity Principle and Elimination is provided. (shrink)

It has been the dominant view that probabilistic explanations of particular facts must be inductive in character. I argue here that this view is mistaken, and that the aim of probabilistic explanation is not to demonstrate that the explanandum fact was nomically expectable, but to give an account of the chance mechanism(s) responsible for it. To this end, a deductive-nomological model of probabilistic explanation is developed and defended. Such a model has application only when the probabilities occurring (...) in covering laws can be interpreted as measures of objective chance, expressing the strength of physical propensities. Unlike inductive models of probabilistic explanation, this deductive model stands in no need of troublesome requirements of maximal specificity or epistemic relativization. (shrink)

It's widely supposed that unification is an epistemic virtue: the degree to which a theory is unified contributes to its overall confirmation. However, this supposition has consequences which haven't been noted, and which undermine the leading accounts of unification. For, given Hempel's equivalence condition, any epistemic virtue must be such that logically equivalent theories must equally well unify any body of evidence, and logically equivalent bodies of evidence must be equally well unified by any theory. Yet the leading (...) accounts of unification in Bayesian terms, or those in terms of argument patterns, cannot satisfy these constraints conditions. The reason for this runs deep: these accounts of unification make unity depend on factors that vary between equivalent theories: the probabilistic relations of their components, or their relations to argument patterns. The solution is to abandon such accounts and instead adopt an account of unity based on worldly relations such as causation, rather than inferential relations. Such an account effortlessly satisfies the equivalence conditions, and so may describe the epistemic virtue of unity. (shrink)

Current dynamic-epistemic logics model different types of information change in multi-agent scenarios. We generalize these logics to a probabilistic setting, obtaining a calculus for multi-agent update with three natural slots: prior probability on states, occurrence probabilities in the relevant process taking place, and observation probabilities of events. To match this update mechanism, we present a complete dynamic logic of information change with a probabilistic character. The completeness proof follows a compositional methodology that applies to a much (...) larger class of dynamic-probabilistic logics as well. Finally, we discuss how our basic update rule can be parameterized for different update policies, or learning methods. (shrink)

Current dynamic-epistemic logics model different types of information change in multi-agent scenarios. We generalize these logics to a probabilistic setting, obtaining a calculus for multi-agent update with three natural slots: prior probability on states, occurrence probabilities in the relevant process taking place, and observation probabilities of events. To match this update mechanism, we present a complete dynamic logic of information change with a probabilistic character. The completeness proof follows a compositional methodology that applies to a much (...) larger class of dynamic-probabilistic logics as well. Finally, we discuss how our basic update rule can be parameterized for different update policies, or learning methods. (shrink)

We introduce an epistemic theory of truth according to which the same rational degree of belief is assigned to Tr(. It is shown that if epistemic probability measures are only demanded to be finitely additive (but not necessarily σ-additive), then such a theory is consistent even for object languages that contain their own truth predicate. As the proof of this result indicates, the theory can also be interpreted as deriving from a quantitative version of the Revision Theory of (...) Truth. (shrink)

We show in this paper that the AGM postulates are too weak to ensure the rational preservation of conditional beliefs during belief revision, thus permitting improper responses to sequences of observations. We remedy this weakness by proposing four additional postulates, which are sound relative to a qualitative version of probabilistic conditioning. Contrary to the AGM framework, the proposed postulates characterize belief revision as a process which may depend on elements of an epistemic state that are not necessarily captured (...) by a belief set. We also show that a simple modification to the AGM framework can allow belief revision to be a function of epistemic states. We establish a model-based representation theorem which characterizes the proposed postulates and constrains, in turn, the way in which entrenchment orderings may be transformed under iterated belief revision. (shrink)

Human reasoning often involves explanation. In everyday affairs, people reason to hypotheses based on the explanatory power these hypotheses afford; I might, for example, surmise that my toddler has been playing in my office because I judge that this hypothesis delivers a good explanation of the disarranged state of the books on my shelves. But such explanatory reasoning also has relevance far beyond the commonplace. Indeed, explanatory reasoning plays an important role in such varied fields as the sciences, philosophy, theology, (...) medicine, forensics, and law. -/- This dissertation provides an extended study into the logic of explanatory reasoning via two general questions. First, I approach the question of what exactly we have in mind when we make judgments pertaining to the explanatory power that a hypothesis has over some evidence. This question is important to this study because these are the sorts of judgments that we constantly rely on when we use explanations to reason about the world. Ultimately, I introduce and defend an explication of the concept of explanatory power in the form of a probabilistic measure. This formal explication allows us to articulate precisely some of the various ways in which we might reason explanatorily. -/- The second question this dissertation examines is whether explanatory reasoning constitutes an epistemically respectable means of gaining knowledge. I defend the following ideas: The probability theory can be used to describe the logic of explanatory reasoning, the normative standard to which such reasoning attains. Explanatory judgments, on the other hand, constitute heuristics that allow us to approximate reasoning in accordance with this logical standard while staying within our human bounds. The most well known model of explanatory reasoning, Inference to the Best Explanation, describes a cogent, nondeductive inference form. And reasoning by Inference to the Best Explanation approximates reasoning directly via the probability theory in the real world. Finally, I respond to some possible objections to my work, and then to some more general, classic criticisms of Inference to the Best Explanation. In the end, this dissertation puts forward a clearer articulation and novel defense of explanatory reasoning. (shrink)

Many have claimed that whenever an investigation might provide evidence for a claim, it might also provide evidence against it. Similarly, many have claimed that your credence should never be on the edge of the range of credences that you think might be rational. Surprisingly, both of these principles imply that you cannot rationally be modest: you cannot be uncertain what the rational opinions are.

How does a Muslim jurist think the law and how, accordingly, he judges a fact? Using Alice in Wonderland as hermeneutical device to explore the logic of fiqh, this article identifies a divergence between Western and Islamic legal thinking in the application of abduction as key form of inference in the law of Islam. In particular, looking at the fact/law relation in symbolic terms, the article highlights how, while a dichotomy between fact and law characterizes Western legal thinking, fiqh (...) upholds a connection between the “real” and the “right”, where the effort in understanding sharī‘ah postulates the actualization of the “rule” in God’s creation. Thus, if sharī‘ahpre-scribes the Law, not only is the rule discovered through the sources, but the right has to be justified through a verdict de-scribing the fact, for the law to be validly stated for the given situation. In this sense, abduction as explanatory “hypothesis” and “inference to the best explanation” of sharī‘ah provides an account for the probabilistic nature of fiqh, its ramification through verdicts, as well as for the epistemic and narrative function of the tradition as core aspects of the logic of Islamic law. At the same time, doubts can be raised about the compatibility between this logic and the deductive logic of modern state law, as a sub-product of Western legal thinking. (shrink)

What is the epistemological structure of situations where many small risks amount to a large one? Lottery and preface paradoxes and puzzles about quantum-mechanical blips threaten the idea that competent deduction is a way of extending our knowledge. Seemingly, everyday knowledge involves small risks, and competently deducing the conjunction of many such truths from them yields a conclusion too risky to constitute knowledge. But the dilemma between scepticism and abandoning MPC is false. In extreme cases, objectively improbable truths are known. (...) Safety is modal, not probabilistic, in structure, with closure and factiveness conditions. It is modelled using closeness of worlds. Safety is analogous to knowledge. It suggests an interpretation of possible worlds semantics for epistemic logic. To avoid logical omniscience, a relation of epistemic counterparthood between formulas is introduced. This supports a safety conception of knowledge and formalizes how extending knowledge by deduction depends on logical competence. (shrink)