We propose a new account of indicative conditionals, giving acceptability and logical closure conditions for them. We start from Adams’ Thesis: the claim that the acceptability of a simple indicative equals the corresponding conditional probability. The Thesis is widely endorsed, but arguably false and refuted by empirical research. To fix it, we submit, we need a relevance constraint: we accept a simple conditional 'If φ, then ψ' to the extent that (i) the conditional probability p(ψ|φ) (...) is high, provided that (ii) φ is relevant for ψ. How (i) should work is well-understood. It is (ii) that holds the key to improve our understanding of conditionals. Our account has (i) a probabilistic component, using Popper functions; (ii) a relevance component, given via an algebraic structure of topics or subject matters. We present a probabilistic logic for simple indicatives, and argue that its (in)validities are both theoretically desirable and in line with empirical results on how people reason with conditionals. (shrink)

This paper argues that the technical notion of conditional probability, as given by the ratio analysis, is unsuitable for dealing with our pretheoretical and intuitive understanding of both conditionality and probability. This is an ontological account of conditionals that include an irreducible dispositional connection between the antecedent and consequent conditions and where the conditional has to be treated as an indivisible whole rather than compositional. The relevant type of conditionality is found in some well-defined group of (...) conditional statements. As an alternative, therefore, we briefly offer grounds for what we would call an ontological reading: for both conditionality and conditional probability in general. It is not offered as a fully developed theory of conditionality but can be used, we claim, to explain why calculations according to the RATIO scheme does not coincide with our intuitive notion of conditional probability. What it shows us is that for an understanding of the whole range of conditionals we will need what John Heil (2003), in response to Quine (1953), calls an ontological point of view. (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)

I present new counterexamples to the asymmetry of grounding: we have prima facie reason to think that some conditional probabilities partially ground their inverse conditional probabilities, and vice versa. These new counterexamples may require that we reject the asymmetry of grounding, or alternatively may require that we reject one or more of the assumptions which enable the counterexamples. Either way, by reflecting on these purported counterexamples to grounding asymmetry we learn something important, either about the formal properties of (...) grounding, or about the nature of probability. (shrink)

Rational agents seem more confident in any possible event than in an impossible event. But if rational credences are real-valued, then there are some possible events that are assigned 0 credence nonetheless. How do we differentiate these events from impossible events then when we order events? de Finetti (1975), Hájek (2012) and Easwaran (2014) suggest that when ordering events, conditional credences and subset relations are as relevant as unconditional credences. I present a counterexample to all their proposals in this (...) paper. While their proposals order possible and impossible events correctly, they deliver the wrong verdict for disjoint possible events assigned equal positive credence. (shrink)

The purpose of this paper is to show that if one adopts conditional probabilities as the primitive concept of probability, one must deal with the fact that even in very ordinary circumstances at least some probability values may be imprecise, and that some probability questions may fail to have numerically precise answers.

Karl Popper discovered in 1938 that the unconditional probability of a conditional of the form ‘If A, then B’ normally exceeds the conditional probability of B given A, provided that ‘If A, then B’ is taken to mean the same as ‘Not (A and not B)’. So it was clear (but presumably only to him at that time) that the conditional probability of B given A cannot be reduced to the unconditional probability of (...) the material conditional ‘If A, then B’. I describe how this insight was developed in Popper’s writings and I add to this historical study a logical one, in which I compare laws of excess in Kolmogorov probability theory with laws of excess in Popper probability theory. (shrink)

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

This article is mainly composed of two discussions. First, we introduce event-expressions, individual constants of a new type referring to events, not things. We learn this from Davidson’s famous formulation of event-sentences (§§1-3). The first half or more of this article is occupied with this discussion. The second half is devoted to the creation of a new analysis of probability, especially conditional probability. As seen in the author’s other work (Kaneko 2022), probability theory can be reconstructed (...) with predicate logic in the name of inductive logic. This article includes its development and its application to the notion of conditional probability (§§4-6). (shrink)

A theory of cognitive systems individuation is presented and defended. The approach has some affinity with Leonard Talmy's Overlapping Systems Model of Cognitive Organization, and the paper's first section explores aspects of Talmy's view that are shared by the view developed herein. According to the view on offer -- the conditional probability of co-contribution account (CPC) -- a cognitive system is a collection of mechanisms that contribute, in overlapping subsets, to a wide variety of forms of intelligent behavior. (...) Central to this approach is the idea of an integrated system. A formal characterization of integration is laid out in the form of a conditional-probabilitybased measure of the clustering of causal contributors to the production of intelligent behavior. I relate the view to the debate over extended and embodied cognition and respond to objections that have been raised in print by Andy Clark, Colin Klein, and Felipe de Brigard. (shrink)

The problem of probabilities of conditionals is one of the long-standing puzzles in philosophy of language. We defend and update Adams' solution to the puzzle: the probability of an epistemic conditional is not the probability of a proposition, but a probability under a supposition. -/- Close inspection of how a triviality result unfolds in a concrete scenario does not provide counterexamples to the view that probabilities of conditionals are conditional probabilities: instead, it supports the conclusion (...) that probabilities of conditionals violate standard probability theory. -/- This does not call into question probability theory per se; rather, it calls for a more careful understanding of its role: probability theory is a theory of probabilities of propositions; but as conditionals do not express propositions, their probabilities are not subject to the standard laws. -/- We argue that both conditional probabilities and probabilities of conditionals are best understood in terms of the dynamics of supposing, modeled as a restriction operation on a probability space. This version of the suppositionalist view allows us to connect Adams' Thesis to the widely held restrictor view of the semantics of conditionals. -/- We address two common objections to Adams' view: that the relevant probabilities are ‘probabilities only in name’, and that giving up conditional propositions puts us at a disadvantage when it comes to interpreting compounds. -/- Finally, we argue that some putative counterexamples to Adams' Thesis can be diagnosed as fallacies of probabilistic reasoning: they arise from applying to conditionals laws of standard probability theory which are invalid for them. (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)

This paper explores the interaction of well-motivated (if controversial) principles governing the probability conditionals, with accounts of what it is for a sentence to be indefinite. The conclusion can be played in a variety of ways. It could be regarded as a new reason to be suspicious of the intuitive data about the probability of conditionals; or, holding fixed the data, it could be used to give traction on the philosophical analysis of a contentious notion—indefiniteness. The paper outlines (...) the various options, and shows that ‘rejectionist’ theories of indefiniteness are incompatible with the results. Rejectionist theories include popular accounts such as supervaluationism, non-classical truth-value gap theories, and accounts of indeterminacy that centre on rejecting the law of excluded middle. An appendix compares the results obtained here with the ‘impossibility’ results descending from Lewis ( 1976 ). (shrink)

This paper presents a unified theory of the truth conditions and probability of indicative conditionals and their compounds in a trivalent framework. The semantics validates a Reduction Theorem: any compound of conditionals is semantically equivalent to a simple conditional. This allows us to validate Stalnaker's Thesis in full generality and to use Adams's notion of $p$-validity as a criterion for valid inference. Finally, this gives us an elegant account of Bayesian update with indicative conditionals, establishing that despite differences (...) in meaning, it is tantamount to learning a material conditional. (shrink)

There are many scientific and everyday cases where each of Pr and Pr is high and it seems that Pr is high. But high probability is not transitive and so it might be in such cases that each of Pr and Pr is high and in fact Pr is not high. There is no issue in the special case where the following condition, which I call “C1”, holds: H 1 entails H 2. This condition is sufficient for transitivity in (...) high probability. But many of the scientific and everyday cases referred to above are cases where it is not the case that H 1 entails H 2. I consider whether there are additional conditions sufficient for transitivity in high probability. I consider three candidate conditions. I call them “C2”, “C3”, and “C2&3”. I argue that C2&3, but neither C2 nor C3, is sufficient for transitivity in high probability. I then set out some further results and relate the discussion to the Bayesian requirement of coherence. (shrink)

While we cannot ensure the occurrence of serendipity due to its nature of unexpectedness, we can try to prepare the optimal conditions to improve the possibility. This chapter first describes two types of unexpected information: within or from beyond one’s perceivable range. Next, we describe four stages of the serendipity attainment process: navigation, noticing, evaluation, and implementation. On this basis, we discuss six scenarios in the order of serendipity encounter and attainment probability, which are determined by information availability in (...) the environment and the mindset in terms of information processing. The serendipity attainment process has a higher success rate when acquiring precise navigation and employing the 3D principles of creativity (best expertise within discipline, the best expertise out of discipline, and discipline process). (shrink)

Dorothy Edgington’s work has been at the centre of a range of ongoing debates in philosophical logic, philosophy of mind and language, metaphysics, and epistemology. This work has focused, although by no means exclusively, on the overlapping areas of conditionals, probability, and paradox. In what follows, I briefly sketch some themes from these three areas relevant to Dorothy’s work, highlighting how some of Dorothy’s work and some of the contributions of this volume fit in to these debates.

*This work is no longer under development* Two major themes in the literature on indicative conditionals are that the content of indicative conditionals typically depends on what is known;1 that conditionals are intimately related to conditional probabilities.2 In possible world semantics for counterfactual conditionals, a standard assumption is that conditionals whose antecedents are metaphysically impossible are vacuously true.3 This aspect has recently been brought to the fore, and defended by Tim Williamson, who uses it in to characterize alethic necessity (...) by exploiting such equivalences as: A⇔¬A A. One might wish to postulate an analogous connection for indicative conditionals, with indicatives whose antecedents are epistemically impossible being vacuously true: and indeed, the modal account of indicative conditionals of Brian Weatherson has exactly this feature.4 This allows one to characterize an epistemic modal by the equivalence A⇔¬A→A. For simplicity, in what follows we write A as KA and think of it as expressing that subject S knows that A.5 The connection to probability has received much attention. Stalnaker suggested, as a way of articulating the ‘Ramsey Test’, the following very general schema for indicative conditionals relative to some probability function P: P = P 1For example, Nolan ; Weatherson ; Gillies. 2For example Stalnaker ; McGee ; Adams. 3Lewis. See Nolan for criticism. 4‘epistemically possible’ here means incompatible with what is known. 5This idea was suggested to me in conversation by John Hawthorne. I do not know of it being explored in print. The plausibility of this characterization will depend on the exact sense of ‘epistemically possible’ in play—if it is compatibility with what a single subject knows, then can be read ‘the relevant subject knows that p’. If it is more delicately formulated, we might be able to read as the epistemic modal ‘must’. (shrink)

The chapter is devoted to the probability and acceptability of indicative conditionals. Focusing on three influential theses, the Equation, Adams’ thesis, and the qualitative version of Adams’ thesis, Sikorski argues that none of them is well supported by the available empirical evidence. In the most controversial case of the Equation, the results of many studies which support it are, at least to some degree, undermined by some recent experimental findings. Sikorski discusses the Ramsey Test, and Lewis’s triviality proof, with (...) special attention dedicated to the popular ways of blocking it. Sikorski concludes that the role of the three theses in future studies of conditionals should be re-thought, and he presents alternative proposals. (shrink)

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

A few purported counterexamples to the Adams thesis have cropped up in the literature in the last few decades. I propose a theory that accounts for them, in a way that makes the connections between indicative conditionals and counterfactuals clearer.

Why are conditional degrees of belief in an observation E, given a statistical hypothesis H, aligned with the objective probabilities expressed by H? After showing that standard replies are not satisfactory, I develop a suppositional analysis of conditional degree of belief, transferring Ramsey’s classical proposal to statistical inference. The analysis saves the alignment, explains the role of chance-credence coordination, and rebuts the charge of arbitrary assessment of evidence in Bayesian inference. Finally, I explore the implications of this analysis (...) for Bayesian reasoning with idealized models in science. (shrink)

In my article, I present a new version of a probabilistic truth prescribing semantics for natural language indicative conditionals. The proposed truth conditions can be paraphrased as follows: an indicative conditional is true if the corresponding conditional probability is high and the antecedent is positively probabilistically relevant for the consequent or the probability of the antecedent of the conditional equals 0. In the paper, the truth conditions are defended and some of the logical properties of (...) the proposed semantics are described. (shrink)

A stereotype is a belief or claim that a group of people has a particular feature. Stereotypes are expressed by sentences that have the form of generic statements, like “Canadians are nice.” Recent work on generics lends new life to understanding generics as statements involving probabilities. I argue that generics (and thus sentences expressing stereotypes) can take one of several forms involving conditional probabilities, and these probabilities have what I call a naturalness requirement. This is the natural probability (...) theory of stereotypes. Each of the two components of the theory entails a family of fallacies that contributes to the spurious reinforcement of stereotypes: inferential slippage within and between the different generic forms, and inferential slippage from facts about frequencies of group traits to beliefs about natural propensities or dispositions of groups. Empirical research suggests that we often commit these fallacies. Moreover, this theory can referee a vitriolic debate between some psychologists, who hold that stereotypes are always false and stereotyping is always wrong, and other psychologists, who hold that stereotypes are often accurate and stereotyping is often reasonable. (shrink)

A study is reported testing two hypotheses about a close parallel relation between indicative conditionals, if A then B , and conditional bets, I bet you that if A then B . The first is that both the indicative conditional and the conditional bet are related to the conditional probability, P(B|A). The second is that de Finetti's three-valued truth table has psychological reality for both types of conditional— true , false , or void for (...) indicative conditionals and win , lose , or void for conditional bets. The participants were presented with an array of chips in two different colours and two different shapes, and an indicative conditional or a conditional bet about a random chip. They had to make judgements in two conditions: either about the chances of making the indicative conditional true or false or about the chances of winning or losing the conditional bet. The observed distributions of responses in the two conditions were generally related to the conditional probability, supporting the first hypothesis. In addition, a majority of participants in further conditions chose the third option, “void”, when the antecedent of the conditional was false, supporting the second hypothesis. (shrink)

I argue that when we use ‘probability’ language in epistemic contexts—e.g., when we ask how probable some hypothesis is, given the evidence available to us—we are talking about degrees of support, rather than degrees of belief. The epistemic probability of A given B is the mind-independent degree to which B supports A, not the degree to which someone with B as their evidence believes A, or the degree to which someone would or should believe A if they had (...) B as their evidence. My central argument is that the degree-of-support interpretation lets us better model good reasoning in certain cases involving old evidence. Degree-of-belief interpretations make the wrong predictions not only about whether old evidence confirms new hypotheses, but about the values of the probabilities that enter into Bayes’ Theorem when we calculate the probability of hypotheses conditional on old evidence and new background information. (shrink)

Stalnaker's Thesis about indicative conditionals is, roughly, that the probability one ought to assign to an indicative conditional equals the probability that one ought to assign to its consequent conditional on its antecedent. The thesis seems right. If you draw a card from a standard 52-card deck, how confident are you that the card is a diamond if it's a red card? To answer this, you calculate the proportion of red cards that are diamonds -- that (...) is, you calculate the probability of drawing a diamond conditional on drawing a red card. Skyrms' Thesis about counterfactual conditionals is, roughly, that the probability that one ought to assign to a counterfactual equals one's rational expectation of the chance, at a relevant past time, of its consequent conditional on its antecedent. This thesis also seems right. If you decide not to enter a 100-ticket lottery, how confident are you that you would have won had you bought a ticket? To answer this, you calculate the prior chance--that is, the chance just before your decision not to buy a ticket---of winning conditional on entering the lottery. The central project of this article is to develop a new uniform theory of conditionals that allows us to derive a version of Skyrms' Thesis from a version of Stalnaker's Thesis, together with a chance-deference norm relating rational credence to beliefs about objective chance. (shrink)

This paper motivates and develops a novel semantic framework for deontic modals. The framework is designed to shed light on two things: the relationship between deontic modals and substantive theories of practical rationality and the interaction of deontic modals with conditionals, epistemic modals and probability operators. I argue that, in order to model inferential connections between deontic modals and probability operators, we need more structure than is provided by classical intensional theories. In particular, we need probabilistic structure that (...) interacts directly with the compositional semantics of deontic modals. However, I reject theories that provide this probabilistic structure by claiming that the semantics of deontic modals is linked to the Bayesian notion of expectation. I offer a probabilistic premise semantics that explains all the data that create trouble for the rival theories. (shrink)

This paper discusses and relates two puzzles for indicative conditionals: a puzzle about indeterminacy and a puzzle about triviality. Both puzzles arise because of Ramsey's Observation, which states that the probability of a conditional is equal to the conditional probability of its consequent given its antecedent. The puzzle of indeterminacy is the problem of reconciling this fact about conditionals with the fact that they seem to lack truth values at worlds where their antecedents are false. The (...) puzzle of triviality is the problem of reconciling Ramsey's Observation with various triviality proofs which establish that Ramsey's Observation cannot hold in full generality. In the paper, I argue for a solution to the indeterminacy puzzle and then apply the resulting theory to the triviality puzzle. On the theory I defend, the truth conditions of indicative conditionals are highly context dependent and such that an indicative conditional may be indeterminate in truth value at each possible world throughout some region of logical space and yet still have a nonzero probability throughout that region. (shrink)

The epistemic probability of A given B is the degree to which B evidentially supports A, or makes A plausible. This paper is a first step in answering the question of what determines the values of epistemic probabilities. I break this question into two parts: the structural question and the substantive question. Just as an object’s weight is determined by its mass and gravitational acceleration, some probabilities are determined by other, more basic ones. The structural question asks what probabilities (...) are not determined in this way—these are the basic probabilities which determine values for all other probabilities. The substantive question asks how the values of these basic probabilities are determined. I defend an answer to the structural question on which basic probabilities are the probabilities of atomic propositions conditional on potential direct explanations. I defend this against the view, implicit in orthodox mathematical treatments of probability, that basic probabilities are the unconditional probabilities of complete worlds. I then apply my answer to the structural question to clear up common confusions in expositions of Bayesianism and shed light on the “problem of the priors.”. (shrink)

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 present a puzzle about knowledge, probability and conditionals. We show that in certain cases some basic and plausible principles governing our reasoning come into conflict. In particular, we show that there is a simple argument that a person may be in a position to know a conditional the consequent of which has a low probability conditional on its antecedent, contra Adams’ Thesis. We suggest that the puzzle motivates a very strong restriction on the inference of (...) a conditional from a disjunction. (shrink)

The present thesis focuses on counterfactuals. Specifically, we will address new questions and open problems that arise for the standard semantic accounts of counterfactual conditionals. The first four chapters deal with the Lewisian semantic account of counterfactuals. On a technical level, we contribute by providing an equivalent algebraic semantics for Lewis' variably strict conditional logics, which is notably absent in the literature. We introduce a new kind of algebra and differentiate between local and global versions of each of Lewis' (...) variably strict conditional logics. We study the algebraic properties of Lewis' logics and the structure theory of our newly introduced algebras. Additionally, we employ a new algebraic construction, based on the framework of Boolean algebras of conditionals, to provide an alternative semantics for Lewisian counterfactual conditionals. This semantic account allows us to establish new truth conditions for Lewisian counterfactuals, implying that Lewisian counterfactuals are definable conditionals, and each counterfactual can be characterized as a modality of a corresponding probabilistic conditional. We further extend these results by demonstrating that each Lewisian counterfactual can also be characterized as a modality of the corresponding Stalnaker conditional. The resulting formal semantic framework is much more expressive than the standard one and, in addition to providing new truth conditions for counterfactuals, it also allows us to define a new class of conditional logics falling into the broader framework of weak logics. On the philosophical side, we argue that our results shed new light on the understanding of Lewisian counterfactuals and prompt a conceptual shift in this field: Lewisian counterfactual dependence can be understood as a modality of probabilistic conditional dependence or Stalnakerian conditional dependence. In other words, whether a counterfactual connection occurs between A and B depends on whether it is "necessary" for a Stalnakerian/probabilistic dependence to occur between A and B. We also propose some ways to interpret the kind of necessity involved in this interpretation. The remaining two chapters deal with the probability of counterfactuals. We provide an answer to the question of how we can characterize the probability that a Lewisian counterfactual is true, which is an open problem in the literature. We show that the probability of a Lewisian counterfactual can be characterized in terms of belief functions from Dempster-Shafer theory of evidence, which are a super-additive generalization of standard probability. We define an updating procedure for belief functions based on the imaging procedure and show that the probability of a counterfactual A > B amounts to the belief function of B imaged on A. This characterization strongly relies on the logical results we proved in the previous chapters. Moreover, we also solve an open problem concerning the procedure to assign a probability to complex counterfactuals in the framework of causal modelling semantics. A limitation of causal modelling semantics is that it cannot account for the probability of counterfactuals with disjunctive antecedents. Drawing on the same previous works, we define a new procedure to assign a probability to counterfactuals with disjunctive antecedents in the framework of causal modelling semantics. We also argue that our procedure is satisfactory in that it yields meaningful results and adheres to some conceptually intuitive constraints one may want to impose when computing the probability of counterfactuals. (shrink)

Indicative conditionals and tendency causal claims are closely related (e.g., Frosch and Byrne, 2012), but despite these connections, they are usually studied separately. A unifying framework could consist in their dependence on probabilistic factors such as high conditional probability and statistical relevance (e.g., Adams, 1975; Eells, 1991; Douven, 2008, 2015). This paper presents a comparative empirical study on differences between judgments on tendency causal claims and indicative conditionals, how these judgments are driven by probabilistic factors, and how these (...) factors differ in their predictive power for both causal and conditional claims. (shrink)

We generalize the Kolmogorov axioms for probability calculus to obtain conditions defining, for any given logic, a class of probability functions relative to that logic, coinciding with the standard probability functions in the special case of classical logic but allowing consideration of other classes of "essentially Kolmogorovian" probability functions relative to other logics. We take a broad view of the Bayesian approach as dictating inter alia that from the perspective of a given logic, rational degrees of (...) belief are those representable by probability functions from the class appropriate to that logic. Classical Bayesianism, which fixes the logic as classical logic, is only one version of this general approach. Another, which we call Intuitionistic Bayesianism, selects intuitionistic logic as the preferred logic and the associated class of probability functions as the right class of candidate representions of epistemic states (rational allocations of degrees of belief). Various objections to classical Bayesianism are, we argue, best met by passing to intuitionistic Bayesianism—in which the probability functions are taken relative to intuitionistic logic—rather than by adopting a radically non-Kolmogorovian, for example, nonadditive, conception of (or substitute for) probability functions, in spite of the popularity of the latter response among those who have raised these objections. The interest of intuitionistic Bayesianism is further enhanced by the availability of a Dutch Book argument justifying the selection of intuitionistic probability functions as guides to rational betting behavior when due consideration is paid to the fact that bets are settled only when/if the outcome bet on becomes known. (shrink)

Dutch Book arguments have been presented for static belief systems and for belief change by conditionalization. An argument is given here that a rule for belief change which under certain conditions violates probability kinematics will leave the agent open to a Dutch Book.

Bayesianism is the position that scientific reasoning is probabilistic and that probabilities are adequately interpreted as an agent's actual subjective degrees of belief, measured by her betting behaviour. Confirmation is one important aspect of scientific reasoning. The thesis of this paper is the following: if scientific reasoning is at all probabilistic, the subjective interpretation has to be given up in order to get right confirmation—and thus scientific reasoning in general. The Bayesian approach to scientific reasoning Bayesian confirmation theory The example (...) The less reliable the source of information, the higher the degree of Bayesian confirmation Measure sensitivity A more general version of the problem of old evidence Conditioning on the entailment relation The counterfactual strategy Generalizing the counterfactual strategy The desired result, and a necessary and sufficient condition for it Actual degrees of belief The common knock-down feature, or ‘anything goes’ The problem of prior probabilities. (shrink)

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

“Negative probability” in practice. Quantum Communication: Very small phase space regions turn out to be thermodynamically analogical to those of superconductors. Macro-bodies or signals might exist in coherent or entangled state. Such physical objects having unusual properties could be the basis of quantum communication channels or even normal physical ones … Questions and a few answers about negative probability: Why does it appear in quantum mechanics? It appears in phase-space formulated quantum mechanics; next, in quantum correlations … and (...) for wave-particle dualism. Its meaning:- mathematically: a ratio of two measures (of sets), which are not collinear; physically: the ratio of the measurements of two physical quantities, which are not simultaneously measurable. The main innovation is in the mapping between phase and Hilbert space, since both are sums. Phase space is a sum of cells, and Hilbert space is a sum of qubits. The mapping is reduced to the mapping of a cell into a qubit and vice versa. Negative probability helps quantum mechanics to be represented quasi-statistically by quasi-probabilistic distributions. Pure states of negative probability cannot exist, but they, where the conditions for their expression exists, decrease the sum probability of the integrally positive regions of the distributions. They reflect the immediate interaction (interference) of probabilities common in quantum mechanics. (shrink)

Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability (...) of being true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter) examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic. (shrink)

Abstract. In this article is presented a model of the change of the probability of the global catastrophic risks in the world with exponentially evolving technologies. Increasingly cheaper technologies become accessible to a larger number of agents. Also, the technologies become more capable to cause a global catastrophe. Examples of such dangerous technologies are artificial viruses constructed by the means of synthetic biology, non-aligned AI and, to less extent, nanotech and nuclear proliferation. The model shows at least double exponential (...) growth of the probability of the global catastrophe which means that the accumulated probability of the catastrophe will grow from negligible to overwhelming at the period of a few doubling times of the technological capabilities. For biotech and AI, such doubling time roughly corresponds to the doubling period of Moore’s law and its analogues in other technologies and is around 2 years. Thus the global catastrophe in the exponential technologies world will most likely happen during a “dangerous decade”. Such a dangerous decade could start as early as in the 2020s. We also found that the double exponential growth in the model makes the model less sensitive to its initial conditions, like the number of dangerous agents or initial probabilities constants. The model also shows that smaller catastrophes could happen earlier than larger ones, and such a smaller catastrophe may be able to stop the technological growth before larger ones become possible, thus global risks will be self-limiting. However, if the growth of the number of dangerous agents will be very quick, the multiple smaller catastrophes could happen simultaneously and will be equal to a global catastrophe. (shrink)

The material interpretation of conditionals is commonly recognized as involving some paradoxical results. I here argue that the truth functional approach to natural language is the reason for the inadequacy of this material interpretation, since the truth or falsity of some pair of statements ‘p’ and ‘q’ cannot per se be decisive for the truth or falsity of a conditional relation ‘if p then q’. This inadequacy also affects the ability of the overall formal system to establish whether or (...) not arguments involving conditionals are valid. I also demonstrate that the Paradox of Indicative Conditionals does not actually involve a paradox, but instead contains some paralogistic elements that make it appear to be a paradox. The discussion of the paradox in this paper further reveals that the material interpretation of conditionals adversely affects the treatment of disjunctions. -/- Much has been said about these matters in the literature that point in the same direction. However, there seems to be some reluctance against fully complying with the arguments against the truth functional account of conditionals, since many of the alternative accounts rely on the material conditional, or at least on an understanding of the conditional as a function of antecedent and consequent in a similar sense as the material conditional. My argument against truth functionality indicates that it may in general involve similar problems to treat conditionals as such functions, whether one deals with theories of truth, assertability or probability. (shrink)

The probabilities a Bayesian agent assigns to a set of events typically change with time, for instance when the agent updates them in the light of new data. In this paper we address the question of how an agent's probabilities at different times are constrained by Dutch-book coherence. We review and attempt to clarify the argument that, although an agent is not forced by coherence to use the usual Bayesian conditioning rule to update his probabilities, coherence does require the agent's (...) probabilities to satisfy van Fraassen's [1984] reflection principle (which entails a related constraint pointed out by Goldstein [1983]). We then exhibit the specialized assumption needed to recover Bayesian conditioning from an analogous reflection-style consideration. Bringing the argument to the context of quantum measurement theory, we show that "quantum decoherence" can be understood in purely personalist terms---quantum decoherence (as supposed in a von Neumann chain) is not a physical process at all, but an application of the reflection principle. From this point of view, the decoherence theory of Zeh, Zurek, and others as a story of quantum measurement has the plot turned exactly backward. (shrink)

In this study we investigate the influence of reason-relation readings of indicative conditionals and ‘and’/‘but’/‘therefore’ sentences on various cognitive assessments. According to the Frege-Grice tradition, a dissociation is expected. Specifically, differences in the reason-relation reading of these sentences should affect participants’ evaluations of their acceptability but not of their truth value. In two experiments we tested this assumption by introducing a relevance manipulation into the truth-table task as well as in other tasks assessing the participants’ acceptability and probability evaluations. (...) Across the two experiments a strong dissociation was found. The reason-relation reading of all four sentences strongly affected their probability and acceptability evaluations, but hardly affected their respective truth evaluations. Implications of this result for recent work on indicative conditionals are discussed. (shrink)

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

Should we understand implicit attitudes on the model of belief? I argue that implicit attitudes are (probably) members of a different psychological kind altogether, because they seem to be insensitive to the logical form of an agent’s thoughts and perceptions. A state is sensitive to logical form only if it is sensitive to the logical constituents of the content of other states (e.g., operators like negation and conditional). I explain sensitivity to logical form and argue that it is a (...) necessary condition for belief. I appeal to two areas of research that seem to show that implicit attitudes fail spectacularly to satisfy this condition—although persistent gaps in the empirical literature leave matters inconclusive. I sketch an alternative account, according to which implicit attitudes are sensitive merely to spatiotemporal relations in thought and perception, i.e., the spatial and temporal orders in which people think, see, or hear things. (shrink)

In Counterfactual Conditionals, Daniel Dohrn discusses the standard account of counterfactuals, conditionals of the form ‘If A had been the case, then B would have been the case’. According to the standard account, a counterfactual is true if the then-sentence is true in all closest worlds in which the if-sentence is true. Closeness is spelled out in terms of an ordering of worlds by their similarity. Dohrn explores resources of defending the standard account against several challenges. In particular, he defends (...) the standard logics for counterfactuals. He discusses exemplary doubts as to whether conditionals have truth conditions. He inquires into the interaction between truth and probability of counterfactuals. He tackles problems with the similarity ordering. He address the interaction between counterfactuals and normalcy conditions. He closes with elaborating peculiarities of future-directed counterfactuals. (shrink)