In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the (...) part” applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses a computational device called the Infinity Computer (patented in USA and EU) working numerically (recall that traditional theories work with infinities and infinitesimals only symbolically) with infinite and infinitesimal numbers that can be written in a positional numeral system with an infinite radix. It is argued that numeral systems involved in computations limit our capabilities to compute and lead to ambiguities in theoretical assertions, as well. The introduced methodology gives the possibility to use the same numeral system for measuring infinite sets, working with divergent series, probability, fractals, optimization problems, numerical differentiation, ODEs, etc. (recall that traditionally different numerals lemniscate; Aleph zero, etc. are used in different situations related to infinity). Numerous numerical examples and theoretical illustrations are given. The accuracy of the achieved results is continuously compared with those obtained by traditional tools used to work with infinities and infinitesimals. In particular, it is shown that the new approach allows one to observe mathematical objects involved in the Hypotheses of Continuum and the Riemann zeta function with a higher accuracy than it is done by traditional tools. It is stressed that the hardness of both problems is not related to their nature but is a consequence of the weakness of traditional numeral systems used to study them. It is shown that the introduced methodology and numeral system change our perception of the mathematical objects studied in the two problems. (shrink)

ABSTRACTRecent rejections of epistemic consequentialism, like those from Firth, Jenkins, Berker, and Greaves, have argued that consequentialism is committed to objectionable trade-offs and suggest...

The principle of the common cause claims that if an improbable coincidence has occurred, there must exist a common cause. This is generally taken to mean that positive correlations between non-causally related events should disappear when conditioning on the action of some underlying common cause. The extended interpretation of the principle, by contrast, urges that common causes should be called for in order to explain positive deviations between the estimated correlation of two events and the expected value of their correlation. (...) The aim of this paper is to provide the extended reading of the principle with a general probabilistic model, capturing the simultaneous action of a system of multiple common causes. To this end, two distinct models are elaborated, and the necessary and sufficient conditions for their existence are determined. (shrink)

A cause may influence its effect via multiple paths. Paradigmatically (Hesslow [1974]), taking birth control pills both decreases one’s risk of thrombosis by preventing pregnancy and increases it by producing a blood chemical. Building on Pearl ([2001]), I explicate the notion of a path-specific effect. Roughly, a path-specific effect of C on E via path P is the degree to which a change in C would change E were they to be transmitted only via P. Facts about such effects may (...) be gleaned from the structural equations commonly used to represent the causal relationships among variables. I contrast my analysis of the Hesslow case with those given by theorists of probabilistic causality, who mistakenly link it to issues of causal heterogeneity, token-causation and indeterminism. The reason probabilistic theories misdiagnose this case is that they pay inadequate attention to the structural relationships among variables. (shrink)

State of art paper on the topic causation, around the problem of the nature of causation. Central theories of contemporary philosophical literature are discussed and analysed, namely, regularity theories of Hume and Mackie, counterfactual theories of Lewis, probabilistic theories of Reichenbach, Lewis and Menzies and causal processes theories of Salmon and Dowe.

The likelihood principle (LP) is a core issue in disagreements between Bayesian and frequentist statistical theories. Yet statements of the LP are often ambiguous, while arguments for why a Bayesian must accept it rely upon unexamined implicit premises. I distinguish two propositions associated with the LP, which I label LP1 and LP2. I maintain that there is a compelling Bayesian argument for LP1, based upon strict conditionalization, standard Bayesian decision theory, and a proposition I call the practical relevance principle. In (...) contrast, I argue that there is no similarly compelling argument for or against LP2. I suggest that these conclusions lead to a restrictedly pluralistic view of Bayesian confirmation measures. (shrink)

There are three questions associated with Simpson’s Paradox (SP): (i) Why is SP paradoxical? (ii) What conditions generate SP?, and (iii) What should be done about SP? By developing a logic-based account of SP, it is argued that (i) and (ii) must be divorced from (iii). This account shows that (i) and (ii) have nothing to do with causality, which plays a role only in addressing (iii). A counterexample is also presented against the causal account. Finally, the causal and logic-based (...) approaches are compared by means of an experiment to show that SP is not basically causal. (shrink)

Disagreement is a hot topic in epistemology. A fast-growing literature centers around a dispute between the ‘steadfast’ view, on which one may maintain one’s beliefs even in the light of disagreement with epistemic peers who have all the same evidence, and the ‘conciliationist’ view, on which such disagreement requires a revision of attitudes. In this paper, however, I argue that there is less separating the main rivals in the debate about peer disagreement than is commonly thought. The extreme versions of (...) both views are clearly indefensible, while more moderate versions of the views converge on the idea that how much revision of belief is called for by an instance of peer disagreement varies from case to case. Those tempted by this diagnosis are sometimes pessimistic about the prospects for giving a unified account which clearly predicts when more or less extensive revisions will be called for. By contrast, in this paper I give an account that aspires to such unity and predictive power, centering on the notion of the net resilience of your estimate of your own reliability against your estimate of your interlocutor’s reliability. The view I present thus amounts to a new, moderate theory of how one should respond to disagreement. I argue that ultimately, when we weaken conciliationism and the steadfast view to account for exception cases and to make them adequately plausible, they end up converging on the moderate view I present. Much of the seeming disagreement about disagreement is, then, illusory. (shrink)

This chapter contains sections titled: Highlights of Past Literature Current Work Future Work.

Normative theories can be useful in developing descriptive theories, as when normative subjective expected utility theory is used to develop descriptive rational choice theory and behavioral game theory. “Ought” questions are also the essence of theories of moral reasoning, a domain of higher mental processing that could not survive without normative considerations.

As van Fraassen pointed out in his opening remarks, Henry Kyburg's lottery paradox has long been known to raise difficulties in attempts to represent full belief as a probability greater than or equal to p, where p is some number less than 1. Recently, Patrick Maher has pointed out that to identify full belief with probability equal to 1 presents similar difficulties. In his paper, van Fraassen investigates ways of representing full belief by personal probability which avoid the difficulties raised (...) by Maher's measure-theoretic version of the lottery paradox. Van Fraassen's more subtle representation dissolves the simple identification of full belief with maximal personal probability. His investigation exploits the richer resources for representing opinion provided by taking conditional, rather than unconditional, personal probability as fundamental. It has interesting implications for equivalent alternative approaches based on non-Archimedean probability, as well as for equivalent approaches in which assumption contexts representing full belief relative to suppositions are taken as fundamental. (shrink)

A paper on how to adapt your probabilisitc beliefs when learning a conditional.

Randomized controlled trials (RCTs) are widely taken as the gold standard for establishing causal conclusions. Ideally conducted they ensure that the treatment ‘causes’ the outcome—in the experiment. But where else? This is the venerable question of external validity. I point out that the question comes in two importantly different forms: Is the specific causal conclusion warranted by the experiment true in a target situation? What will be the result of implementing the treatment there? This paper explains how the probabilistic theory (...) of causality implies that RCTs can establish causal conclusions and thereby provides an account of what exactly that causal conclusion is. Clarifying the exact form of the conclusion shows just what is necessary for it to hold in a new setting and also how much more is needed to see what the actual outcome would be there were the treatment implemented. (shrink)

We consider the dispute between causal decision theorists and evidential decision theorists over Newcomb-like problems. We introduce a framework relating causation and directed graphs developed by Spirtes et al. (1993) and evaluate several arguments in this context. We argue that much of the debate between the two camps is misplaced; the disputes turn on the distinction between conditioning on an event E as against conditioning on an event I which is an action to bring about E. We give the essential (...) machinery for calculating the effect of an intervention and consider recent work which extends the basic account given here to the case where causal Knowledge is incomplete. (shrink)

Efforts to formalize qualitative accounts of inference to the best explanation (IBE) confront two obstacles: the imprecise nature of such accounts and the unusual logical properties that explanations exhibit, such as contradiction-intolerance and irreflexivity. This paper aims to surmount these challenges by utilising a new, more precise theory that treats explanations as expressions that codify defeasible inferences. To formalise this account, we provide a sequent calculus in which IBE serves as an elimination rule for a connective that exhibits many of (...) the properties associated with the behaviour of the English expression ‘That... best explains why ... ’. We first construct a calculus that encodes these properties at the level of the turnstile, i.e. as a metalinguistic expression for classes of defeasible consequence relations. We then show how this calculus can be conservatively extended over a language that contains a best-explains-why operator. (shrink)

A probability distribution is regular if no possible event is assigned probability zero. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson (2017) and Benci et al. (2016) have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson’s (2007) “isomorphic” events are not in fact isomorphic, but Howson is speaking (...) of set-theoretic representations of events in a probability model. While those sets are not isomorphic, Williamson’s physical events are, in the relevant sense. Benci et al. claim that all three arguments rest on a conflation of different models, but they do not. They are founded on the premise that similar events should have the same probability in the same model, or in one case, on the assumption that a single rotation-invariant distribution is possible. Having failed to refute the symmetry arguments on such technical grounds, one could deny their implicit premises, which is a heavy cost, or adopt varying degrees of instrumentalism or pluralism about regularity, but that would not serve the project of accurately modelling chances. (shrink)

This paper shows how a particular resiliency-centered approach to chance lends support for two conditions characterizing chance. The first condition says that the present chance of some proposition A conditional on the proposition about some later chance of A should be set equal to that later chance of A. The second condition requires the present chance of some proposition A to be equal to the weighted average of possible later chances of A. I first introduce, motivate, and make precise a (...) resiliency-centered approach to chance whose basic idea is that any chance distribution should be maximally invariant under variation of experimental factors. Second, I show that any present chance distribution that violates the two conditions can be replaced by another present chance distribution that satisfies them and is more resilient under variation of experimental factors. This shows that the two conditions are an essential feature of chances that maximize resiliency. Finally, I explore the relationship between the idea of resilient chances so understood and so-called Humean accounts of chance—one of the most promising recent philosophical accounts of chance. (shrink)

This paper discusses how to update one’s credences based on evidence that has initial probability 0. I advance a diachronic norm, Kolmogorov Conditionalization, that governs credal reallocation in many such learning scenarios. The norm is based upon Kolmogorov’s theory of conditional probability. I prove a Dutch book theorem and converse Dutch book theorem for Kolmogorov Conditionalization. The two theorems establish Kolmogorov Conditionalization as the unique credal reallocation rule that avoids a sure loss in the relevant learning scenarios.

It seems natural to think of an unwilling addict as having a pattern of preferences that she does not endorse—preferences that, in some sense, she does not ‘identify’ with. Following Frankfurt (1971), Jeffrey (1974) proposed a way of modeling those features of an agent’s preferences by appealing to preferences among preferences.Th„e addict’s preferences are preferences she does not prefer to have. I argue that this modeling suggestion will not do, for it follows from plausible assumptions that a minimally rational agent (...) must prefer those first-order preferences she actually has. I close by considering two different but related ways to think about the initial phenomenon. (shrink)

In “A Subjectivist’s Guide to Objective Chance,” David Lewis says that he is “led to wonder whether anyone but a subjectivist is in a position to understand objective chance.” The present essay aims to motivate this same Lewisean attitude, and a similar degree of modest subjectivism, with respect to objective causation. The essay begins with Newcomb problems, which turn on an apparent tension between two principles of choice: roughly, a principle sensitive to the causal features of the relevant situation, and (...) a principle sensitive only to evidential factors. Two-boxers give priority to causal beliefs, and one-boxers to evidential beliefs. The essay notes that a similar issue can arise when the modality in question is chance, rather than causation. In this case, the conflict is between decision rules based on credences guided solely by chances, and rules based on credences guided by other sorts of probabilistic evidence. Far from excluding cases of the latter kind, Lewis’s Principal Principle explicitly allows for them, in the form of the caveat that credences should follow beliefs about chances only in the absence of “inadmissible evidence.” The essay then exhibits a tension in Lewis’s views on these two matters, by presenting a class of decision problems—some of them themselves Newcomb problems—in which Lewis’s view of the relevance of inadmissible evidence seems in tension with his causal decision theory. It offers a diagnosis for this dilemma and proposes a remedy, based on an extension of a proposal due to Ned Hall and others from the case of chance to that of causation. The remedy suggests a new view of the relation between causal decision theory and evidential decision theory, namely, that they stand to each other much as chance stands to credence, being objective and subjective faces of the same practical coin. This has much the same metaphysical benefits as Lewis’s own view of chance and also throws interesting new light on Newcomb problems, providing an irenic resolution of the apparent disagreement between causal and evidential decision rules. (shrink)

Fallibilism is an attractive epistemological position, avoiding the Scylla of rationalism, and the Charybdis of scepticism. Acknowledging, on the one hand, human imperfection, yet claiming that science and rational inquiry are possible. Fallibilism is a thesis, but equally importantly – an epistemological recommendation. that we should never be absolutely sure of anything. My aim in this paper is to drive a wedge between the thesis and the recommendation. The (eminently plausible) doctrine, I shall argue, cannot be used to ground the (...) epistemological prescription for caution. (shrink)

It is well known that aggregating the degree-of-belief functions of different subjects by linear pooling or averaging is subject to a commutativity dilemma: other than in trivial cases, conditionalizing the individual degree-of-belief functions on a piece of evidence E followed by linearly aggregating them does not yield the same result as rst aggregating them linearly and then conditionalizing the resulting social degree- of-belief function on E. In the present paper we suggest a novel way out of this dilemma: adapting the (...) method of update or learning such that linear pooling com- mutes with it. As it turns out, the resulting update scheme – imaging on the evidence – is well-known from areas such as the study of conditionals and cau- sal decision theory, and a formal result from which the required commutativity property is derivable was supplied already by Gärdenfors in a different con- text. We end up determining under which conditions imaging would seem to be right method of update, and under which conditions, therefore, group update would not be affected by the commutativity dilemma. (shrink)

You should rather play hide-and-seek against someone who cannot predict where you hide than against someone who can, as the article illustrates in connection with a high-stakes example. Causal Decision Theory denies this. So Causal Decision Theory is false.

Reliabilist theories propose to analyse epistemic justification in terms of reliability. This paper argues that if we pay attention to the details of probability theory we find that there is no concept of reliability that can possibly play the role required by reliabilist theories. A distinction is drawn between the general reliability of a process and the single case reliability of an individual belief, And it is argued that neither notion can serve the reliabilist adequately.

In this paper I offer an 'integrating account' of singular causation, where the term 'integrating' refers to the following program for analysing causation. There are two intuitions about causation, both of which face serious counterexamples when used as the basis for an analysis of causation. The 'process' intuition, which says that causes and effects are linked by concrete processes, runs into trouble with cases of 'misconnections', where an event which serves to prevent another fails to do so on a particular (...) occasion and yet the two events are linked by causal processes. The chance raising intuition, according to which causes raise the chance of their effects, easily accounts for misconnections but faces the problem of chance lowering causes, a problem easily accounted for by the process approach. The integrating program attempts to provide an analysis of singular causation by synthesising the two insights, so as to solve both problems. In this paper I show that extant versions of the integrating program due to Eells, Lewis, and Menzies fail to account for the chance-lowering counterexample. I offer a new diagnosis of the chance lowering case, and use that as a basis for an integrating account of causation which does solve both cases. In doing so, I accept various assumptions of the integrating program, in particular that there are no other problems with these two approaches. As an example of the process account, I focus on the recent CQ theory of Wesley Salmon (1997). (shrink)

Recent advances in causal reasoning have given rise to a computational model that emulates the process by which humans generate, evaluate, and distinguish counterfactual sentences. Contrasted with the “possible worlds” account of counterfactuals, this “structural” model enjoys the advantages of representational economy, algorithmic simplicity, and conceptual clarity. This introduction traces the emergence of the structural model and gives a panoramic view of several applications where counterfactual reasoning has benefited problem areas in the empirical sciences.