We offer a probabilistic model of rational consequence relations (Lehmann and Magidor, 1990) by appealing to the extension of the classical Ramsey-Adams test proposed by Vann McGee in (McGee, 1994). Previous and influential models of nonmonotonic consequence relations have been produced in terms of the dynamics of expectations (Gärdenfors and Makinson, 1994; Gärdenfors, 1993).'Expectation' is a term of art in these models, which should not be confused with the notion of expected utility. The expectations of an agent are some form (...) of belief weaker than absolute certainty. Our model offers a modified and extended version of an account of qualitative belief in terms of conditional probability, first presented in (van Fraassen, 1995). We use this model to relate probabilistic and qualitative models of non-monotonic relations in terms of expectations. In doing so we propose a probabilistic model of the notion of expectation. We provide characterization results both for logically finite languages and for logically infinite, but countable, languages. The latter case shows the relevance of the axiom of countable additivity for our probability functions. We show that a rational logic defined over a logically infinite language can only be fully characterized in terms of finitely additive conditional probability. (shrink)

In the tradition of substructural logics, it has been claimed for a long time that conjunction and inclusive disjunction are ambiguous:we should, in fact, distinguish between ‘lattice’ connectives (also called additive or extensional) and ‘group’ connectives (also called multiplicative or intensional). We argue that an analogous ambiguity affects the quantifiers. Moreover, we show how such a perspective could yield solutions for two well-known logical puzzles: McGee’s counterexample to modus ponens and the lottery paradox.

Given plausible assumptions about the nature of evidence and undercutting defeat, many believe that the force of the evidential problem of evil depends on sceptical theism’s being false: if evil is...

I give a semantics for a logic with two pairs of doxastic modals and an indicative conditional connective that all nest without restriction. Sentences are evaluated as accepted, rejected, or neither. Certainty is the necessity-like modality of acceptance. Inferences may proceed from premises that are certain, or merely accepted, or a mix of both. This semantic setup yields some striking results. Notably, the existence of inferences that preserve certainty but not acceptance very directly implies both failure of modus ponens for (...) the indicative conditional in the logic of acceptance and failure of the deduction theorem for the material conditional in the logic of certainty. The latter failure dissolves, in the logic of certainty, the much - discussed tension between modus ponens and the law of import-export. (shrink)

Stephen Schiffer has argued that contextualist solutions to skepticism rest on an implausible "error theory" concerning our own semantic intentions. Similar arguments have recently been offered also by Thomas Hofweber and Patrick Rysiew. I attempt to show how contextualists can rebut these arguments. The kind of self-knowledge that contextualists are committed to denying us is not a kind of self-knowledge that we need, nor is it a kind of self-knowledge that we can plausibly be thought to possess.

Jim buys a ticket in a million-ticket lottery. He knows it is a fair lottery, but, given the odds, he believes he will lose. When the winning ticket is chosen, it is not his. Did he know his ticket would lose? It seems that he did not. After all, if he knew his ticket would lose, why would he have bought it? Further, if he knew his ticket would lose, then, given that his ticket is no different in its chances (...) of winning from any other ticket, it seems that by parity of reasoning he should also know that every other ticket would lose. But of course he doesn’t know that; in fact, he knows that not every ticket will lose. (shrink)

In making assertions one takes on commitments to the consistency of what one asserts and to the logical consequences of what one asserts. Although there is no quick link between belief and assertion, the dialectical requirements on assertion feed back into normative constraints on those beliefs that constitute one's evidence. But if we are not certain of many of our beliefs and that uncertainty is modelled in terms of probabilities, then there is at least prima facie incoherence between the normative (...) constraints on belief and the probability-like structure of degrees of belief. I suggest that the norm-governed practice relating to degrees of belief is the evaluation of betting odds. (shrink)

Starting from John MacFarlane's recent survey of answers to the question ‘What is assertion?’, I defend an account of assertion that draws on elements of MacFarlane's and Robert Brandom's commitment accounts, Timothy Williamson's knowledge norm account, and my own previous work on the normative status of logic. I defend the knowledge norm from recent attacks. Indicative conditionals, however, pose a problem when read along the lines of Ernest Adams' account, an account supported by much work in the psychology of reasoning. (...) Furthermore, there seems to be no place for degrees of belief in the accounts of belief and assertion given here. Degrees of belief do have a role in decision‐making, but, again, there is much evidence that the orthodox theory of subjective utility maximization is not a good description of what we do in decision‐making and, arguably, neither is it a good normative guide to how we ought to make decisions. (shrink)

We introduce an expected-value theory of linguistic modality that makes reference to expected utility and a likelihood-based confirmation measure for deontics and epistemics, respectively. The account is a probabilistic semantics for deontics and epistemics, yet it proposes that deontics and epistemics share a common core modal semantics, as in traditional possible-worlds analysis of modality. We argue that this account is not only theoretically advantageous, but also has far-reaching empirical consequences. In particular, we predict modal versions of reasoning fallacies from the (...) heuristics and biases literature. Additionally, we derive the modal semantics in an entirely transparent manner, as it is based on the compositional semantics of Korean modal expressions that are morphosyntactically decomposed into a conditional and an evaluative predicate. (shrink)

We reflect on lessons that the lottery and preface paradoxes provide for the logic of uncertain inference. One of these lessons is the unreliability of the rule of conjunction of conclusions in such contexts, whether the inferences are probabilistic or qualitative; this leads us to an examination of consequence relations without that rule, the study of other rules that may nevertheless be satisfied in its absence, and a partial rehabilitation of conjunction as a ‘lossy’ rule. A second lesson is the (...) possibility of rational inconsistent belief; this leads us to formulate criteria for deciding when an inconsistent set of beliefs may reasonably be retained. (shrink)

The linguistic and cognitive sciences have severely underestimated the degree of linguistic diversity in the world. Part of the reason for this is that we have projected assumptions based on English and familiar languages onto the rest. We focus on some distortions this has introduced, especially in the study of semantics.

There is the idea that rational belief for a single individual can be constructed via a process of unilateral argument. To preempt antipathy between the AI communities that can claim the idea that rational belief can be so constructed, we trace the idea to the beginning of this century, to Keynes' dispute with Russell over logic and probability. We review how Keynesian ideas were revived in AI's work on non-monotonic reasoning and parallel developments in philosophical logic.

I show that the Lottery Paradox is just a version of the Sorites, and argue that this should modify our way of looking at the Paradox itself. In particular, I focus on what I call “the Cut-off Point Problem” and contend that this problem, well known by Sorites scholars, ought to play a key role in the debate on Kyburg’s puzzle. Very briefly, I show that, in the Lottery Paradox, the premises “ticket n°1 will lose”, “ticket n°2 will lose”… “ticket (...) n°1000 will lose” are equivalent to soritical premises of the form “~(the winning ticket is in {…, (tn)}) ⊃ ~(the winning ticket is in {…, tn, (tn + 1)})” (where “⊃” is the material conditional, “~” is the negation symbol, “tn” and “tn + 1” are “ticket n°n” and “ticket n°n + 1” respectively, and “{}” identify the elements of the lottery tickets’ set. The brackets in “(tn)” and “(tn + 1)” are meant to point out that in the antecedent of the conditional we do not always have a “tn” (and, as a result, a “tn + 1” in the consequent): consider the conditional “~(the winning ticket is in {}) ⊃ ~(the winning ticket is in {t1})”). As a result, failing to believe, for some ticket, that it will lose comes down to introducing a cut-off point in a chain of soritical premises. In this paper I explore the consequences of the different ways of blocking the Lottery Paradox with respect to the Cut-off Point Problem. A heap variant of the Lottery Paradox is especially relevant for evaluating the different solutions. One important result is that the most popular way out of the puzzle, i.e., denying the Lockean Thesis, becomes less attractive. Moreover, I show that, along with the debate on whether rational belief is closed under classical logic, the debate on the validity of modus ponens should play an important role in discussions on the Lottery Paradox. -/- . (shrink)

This paper concerns the extent to which uncertain propositional reasoning can track probabilistic reasoning, and addresses kinematic problems that extend the familiar Lottery paradox. An acceptance rule assigns to each Bayesian credal state p a propositional belief revision method B p , which specifies an initial belief state B p (T) that is revised to the new propositional belief state B(E) upon receipt of information E. An acceptance rule tracks Bayesian conditioning when B p (E) = B p|E (T), for (...) every E such that p(E) > 0; namely, when acceptance by propositional belief revision equals Bayesian conditioning followed by acceptance. Standard proposals for uncertain acceptance and belief revision do not track Bayesian conditioning. The "Lockean" rule that accepts propositions above a probability threshold is subject to the familiar lottery paradox (Kyburg 1961), and we show that it is also subject to new and more stubborn paradoxes when the tracking property is taken into account. Moreover, we show that the familiar AGM approach to belief revision (Harper, Synthese 30(1-2):221-262, 1975; Alchourrón et al., J Symb Log 50:510-530, 1985) cannot be realized in a sensible way by any uncertain acceptance rule that tracks Bayesian conditioning. Finally, we present a plausible, alternative approach that tracks Bayesian conditioning and avoids all of the paradoxes. It combines an odds-based acceptance rule proposed originally by Levi (1996) with a non-AGM belief revision method proposed originally by Shoham (1987). (shrink)

“Since today is Saturday, the grocery store is open today and will be closed tomorrow; so let’s go today”. That is an example of everyday practical reasoning—reasoning directly with the propositions that one believes but may not be fully certain of. Everyday practical reasoning is one of our most familiar kinds of decisions but, unfortunately, some foundational questions about it are largely ignored in the standard decision theory: (Q1) What are the decision rules in everyday practical reasoning that connect qualitative (...) belief and desire to preference over acts? (Q2) What sort of logic should govern qualitative beliefs in everyday practical reasoning, and to what extent is that logic necessary for the purposes of qualitative decisions? (Q3) What kinds of qualitative decisions are always representable as results of everyday practical reasoning? (Q4) Under what circumstances do the results of everyday practical reasoning agree with the Bayesian ideal of expected utility maximization? This paper proposes a rigorous decision theory for answering all of those questions, which is developed in parallel to Savage’s (1954) foundation of expected utility maximization. In light of a new representation result, everyday practical reasoning provides a sound and complete method for a very wide class of qualitative decisions; and, to that end, qualitative beliefs must be allowed to be closed under classical logic plus a well-known nonmonotonic logic—the so-called system ℙ. (shrink)

This paper seeks to defend the following conclusions: The program advanced by Carnap and other necessarians for probability logic has little to recommend it except for one important point. Credal probability judgments ought to be adapted to changes in evidence or states of full belief in a principled manner in conformity with the inquirer’s confirmational commitments—except when the inquirer has good reason to modify his or her confirmational commitment. Probability logic ought to spell out the constraints on rationally coherent confirmational (...) commitments. In the case where credal judgments are numerically determinate confirmational commitments correspond to Carnap’s credibility functions mathematically represented by so—called confirmation functions. Serious investigation of the conditions under which confirmational commitments should be changed ought to be a prime target for critical reflection. The necessarians were mistaken in thinking that confirmational commitments are immune to legitimate modification altogether. But their personalist or subjectivist critics went too far in suggesting that we might dispense with confirmational commitments. There is room for serious reflection on conditions under which changes in confirmational commitments may be brought under critical control. Undertaking such reflection need not become embroiled in the anti inductivism that has characterized the work of Popper, Carnap and Jeffrey and narrowed the focus of students of logical and methodological issues pertaining to inquiry. (shrink)

This essay develops a joint theory of rational (all-or-nothing) belief and degrees of belief. The theory is based on three assumptions: the logical closure of rational belief; the axioms of probability for rational degrees of belief; and the so-called Lockean thesis, in which the concepts of rational belief and rational degree of belief figure simultaneously. In spite of what is commonly believed, this essay will show that this combination of principles is satisfiable (and indeed nontrivially so) and that the principles (...) are jointly satisfied if and only if rational belief is equivalent to the assignment of a stably high rational degree of belief. Although the logical closure of belief and the Lockean thesis are attractive postulates in themselves, initially this may seem like a formal “curiosity”; however, as will be argued in the rest of the essay, a very reasonable theory of rational belief can be built around these principles that is not ad hoc and that has various philosophical features that are plausible independently. In particular, this essay shows that the theory allows for a solution to the Lottery Paradox, and it has nice applications to formal epistemology. The price that is to be paid for this theory is a strong dependency of belief on the context, where a context involves both the agent's degree of belief function and the partitioning or individuation of the underlying possibilities. But as this essay argues, that price seems to be affordable. (shrink)

On an increasingly popular view of rationality, rationality is fundamentally about responding correctly to reasons and there is no independent rational requirement to avoid incoherence: having an incoherent combination of attitudes is irrational not because there is a fundamental requirement of rationality that prohibits it, but rather because you are guaranteed to fail to respond correctly to reasons in having it. This paper argues that any such attempt to explain the irrationality of incoherence in terms of responsiveness to reasons fails. (...) For there is something distinctively irrational about incoherence that is not explained in terms of the guaranteed failure to respond to reasons. Any adequate account of the nature of coherence requirements on belief and intention should take into account the distinctive kind of commitments involved in each type of attitude. (*published with open access). (shrink)

Many moral theories are committed to the idea that some kinds of moral considerations should be respected, whatever the cost to ‘lesser’ types of considerations. A person's life, for instance, should not be sacrificed for the trivial pleasures of others, no matter how many would benefit. However, according to the decision-theoretic critique of lexical priority theories, accepting lexical priorities inevitably leads us to make unacceptable decisions in risky situations. It seems that to operate in a risky world, we must reject (...) lexical priorities altogether. This paper argues that lexical priority theories can, in fact, offer satisfactory guidance in risky situations. It does so by equipping lexical priority theories with overlooked resources from decision theory. (shrink)

Does faith require belief? “Belief-plus” accounts of faith say yes. “Non-doxastic” accounts say no but tend to place a “no-disbelief constraint” on faith. Both sides, I argue, are mistaken for making belief explanatorily prior to faith. Indeed, both “faith” and “belief” have contextualist semantics, which leaves only a tenuous tie between the applications of the two words.

There are a number of reasons for being interested in uncertainty, and there are also a number of uncertainty formalisms. These formalisms are not unrelated. It is argued that they can all be reflected as special cases of the approach of taking probabilities to be determined by sets of probability functions defined on an algebra of statements. Thus, interval probabilities should be construed as maximum and minimum probabilities within a set of distributions, Glenn Shafer's belief functions should be construed as (...) lower probabilities, etc. Updating probabilities introduces new considerations, and it is shown that the representation of belief as a set of probabilities conflicts in this regard with the updating procedures advocated by Shafer. The attempt to make subjectivistic probability plausible as a doctrine of rational belief by making it more flowery — i.e., by adding new dimensions — does not succeed. But, if one is going to represent beliefs by sets of distributions, those sets of distributions might as well be based in statistical knowledge, as they are in epistemological or evidential probability. (shrink)

Plural definite descriptions across many languages display two well-known properties. First, they can give rise to so-called non-maximal readings, in the sense that they ‘allow for exceptions’. Second, while they tend to have a quasi-universal quantificational force in affirmative sentences, they tend to be interpreted existentially in the scope of negation. Building on previous works, we offer a theory in which sentences containing plural definite expressions trigger a family of possible interpretations, and where general principles of language use account for (...) their interpretation in various contexts and syntactic environments. Our theory solves a number of problems that these previous works encounter, and has broader empirical coverage in that it offers a precise analysis for sentences that display complex interactions between plural definites, quantifiers and bound variables, as well as for cases involving non-distributive predicates. The resulting proposal is briefly compared with an alternative proposal by Križ, which has similar coverage but is based on a very different architecture and sometimes makes subtly different predictions. (shrink)

Thomas Kroedel has recently proposed an interesting Pareto-style condition on permissible belief. Despite the condition’s initial plausibility, this paper aims at providing a counterexample to it. The example is based on the view that a proper condition on permissible belief should not give permission to believe a proposition that undermines one’s belief system or whose epistemic standing decreases in the light of one’s de facto beliefs.

Bayesian networks are computer programs which represent probabilitistic relationships graphically as directed acyclic graphs, and which can use those graphs to reason probabilistically , often at relatively low computational cost. Almost every expert system in the past tried to support probabilistic reasoning, but because of the computational difficulties they took approximating short-cuts, such as those afforded by MYCIN's certainty factors. That all changed with the publication of Judea Pearl's Probabilistic Reasoning in Intelligent Systems, in 1988, which synthesized a decade of (...) research making accurate graphical probabilistic reasoning computationally achievable.Bayesian network technology is now one of the fastest growing fields of research in artificial intelligence. That it has become a publication industry in its own right is shown by a search on Google scholar :This development, together with a parallel related growth in the use of causal discovery algorithms which automate the learning of Bayesian networks from sample data, has generated considerable interest, and controversy, within the philosophy-of-science community.Three central questions bringing together AI researchers and philosophers of science are: Are Bayesian networks Bayesian? What is the relation between probability and causality? Are the assumptions behind causal discovery of Bayesian networks realistic or fantastical?Jon Williamson, as a philosopher of science with a keen interest in the technology, asks and answers these questions in his new book. Although it is self-contained, his book is not very likely as an introduction to the technology , nor is it optimal even as an introduction to the philosophical problems in interpreting Bayesian networks . Rather Williamson's book is an attempt to move the debate forward by solving the central problems of the …. (shrink)

Im Gegensatz zur verbreiteten Auffassung über die Voraussetzungen zur Ausbildung neuen Wissens entstehen wichtige Umstellungen im Denken häufig nicht durch harte Konfrontation zwischen Hypothesen bzw. Theorien. Vielmehr sind oft allmähliche Deutungsverschiebungen bei der Auffassung einer einzigen Hypothese bzw. Theorie zu beobachten. Derartige Prozesse erstrecken sich in der Regel über vergleichsweise lange Zeiträume und sind nicht wie die erstgenannten Vorgänge von eher kurzfristiger Natur. Verglichen mit dem Ausgangspunkt der jeweiligen Entwicklung tritt uns aber eine oft an ihrem Ende tiefgreifend veränderte Wissenssituation (...) entgegen. Infolge der Langsamkeit der geschilderten Entwicklungen und der damit zugleich gegebenen Gewöhnung ihrer Vertreter an die geänderte Situation sind derartige Veränderungen des Wissens selten in einem Ausmaß vom Bewußtsein der Neuheit begleitet, wie dies bei Vorgängen der zuerst geschilderten Art der Fall ist. Trotzdem können aber gerade diese Veränderungen nachhaltiger und von grundlegenderer Art sein als jene. In der Folge sollen die beiden angedeuteten Wege der Erkenntnis hinsichtlich ihres strukturellen Unterschieds klarer erfaßt werden. Unterschiedliche Annahmen über die zugrundeliegende Ontologie sind für diese beiden Arten der Wissensbildung von besonderem Interesse. Insbesondere wird sich auch zeigen, daß die bereits in der Fristfrage berührten pragmatischen Gesichtspunkte eine wichtige Rolle spielen. Die beiden in dieser Untersuchung herauspräparierten "Wege" sind Aspekte jeweils eines Erkenntnisprozesses. So sind sie nicht gegeneinander auszuspielen. Vielmehr weist jeder derartige Vorgang Charakteristika beider modellhaft erfaßten Typen auf. (shrink)

Orthodox Probabilists hold that an inquirer ought to harbor a precise degree of confidence in each hypothesis about which she is concerned. Modest Probabilism is one of a family doctrines inspired by the thought that Orthodox Probabilists are thereby demanding that an inquirer effect a precision that is often unwarranted by her evidence. The purpose of this essay is (i) to explain the particular way in which Modest Probabilism answers to this thought, and (ii) to address an alleged counterexample to (...) Modest Probabilism meant to offer proof that Modest Probabilism is no better than its orthodox predecessor at avoiding unwarranted precision. (shrink)