The volume analyses and develops David Makinson’s efforts to make classical logic useful outside its most obvious application areas. The book contains chapters that analyse, appraise, or reshape Makinson’s work and chapters that develop themes emerging from his contributions. These are grouped into major areas to which Makinsons has made highly influential contributions and the volume in its entirety is divided into four sections, each devoted to a particular area of logic: belief change, uncertain reasoning, normative systems and the resources (...) of classical logic. Among the contributions included in the volume, one chapter focuses on the “inferential preferential method”, i.e. the combined use of classical logic and mechanisms of preference and choice and provides examples from Makinson’s work in non-monotonic and defeasible reasoning and belief revision. One chapter offers a short autobiography by Makinson which details his discovery of modern logic, his travels across continents and reveals his intellectual encounters and inspirations. The chapter also contains an unusually explicit statement on his views on the role of logic in philosophy. (shrink)

The structural view of rational acceptance is a commitment to developing a logical calculus to express rationally accepted propositions sufficient to represent valid argument forms constructed from rationally accepted formulas. This essay argues for this project by observing that a satisfactory solution to the lottery paradox and the paradox of the preface calls for a theory that both (i) offers the facilities to represent accepting less than certain propositions within an interpreted artificial language and (ii) provides a logical calculus of (...) rationally accepted formulas that preserves rational acceptance under consequence. The essay explores the merit and scope of the structural view by observing that some limitations to a recent framework advanced James Hawthorne and Luc Bovens are traced to their framework satisfying the first of these two conditions but not the second. (shrink)

David Lewis has argued that impossible worlds are nonsense: if there were such worlds, one would have to distinguish between the truths about their contradictory goings-on and contradictory falsehoods about them; and this--Lewis argues--is preposterous. In this paper I examine a way of resisting this argument by giving up the assumption that ‘in so-and-so world’ is a restricting modifier which passes through the truth-functional connectives The outcome is a sort of subvaluational semantics which makes a contradiction ‘A & ~A’ false (...) even when both ‘A’ and ‘~A’ are true, just as supervaluational semantics makes a tautology ‘A v ~A’ true even when neither ‘A’ nor ‘~A’ are. Connections with discussive logics and complications of the account are discussed, and some general morals are drawn. (shrink)

The rule of adjunction is intuitively appealing and uncontroversial for deductive inference, but in situations where information can be uncertain, the rule is neither needed nor wanted for rational acceptance, as illustrated by the lottery paradox. Practical certainty is the acceptance of statements whose chances of error are smaller than a prescribed threshold parameter, when evaluated against an evidential corpus. We examine the failure of adjunction in relation to the threshold parameter for practical certainty, with an eye towards reinstating the (...) rule of adjunction in some restricted forms, by observing the conditions under which the overall chance of error of the joint statements can be variously bounded. (shrink)

In this paper I develop an account of petitio principii (the fallacy sometimes also called ‘vicious circularity’, or ‘begging the question’) which has two crucial features: it employs the notion of doxastic justification, and it takes circularity to be relative to an evidential state. According to my account, an argument will be circular relative to an evidential state if and only if having doxastic justification for the conclusion is necessary, for a subject in that evidential state, to have doxastic justification (...) for the premisses. I compare the account to some existing ones, and claim some advantages. I then rebut an objection which threatens to undermine the importance of one of those advantages. This account seems to shed new light on the old problem of characterizing petitio principii. It avoids the two obvious problems which any account of this phenomenon must face: being too narrow, for example by leaving out all arguments in which the conclusion does not appear among the premisses, and being too wide, making all valid arguments circular. (shrink)

Traditional procedures for rational updating fail when it comes to self-locating opinions, such as your credences about where you are and what time it is. This paper develops an updating procedure for rational agents with self-locating beliefs. In short, I argue that rational updating can be factored into two steps. The first step uses information you recall from your previous self to form a hypothetical credence distribution, and the second step changes this hypothetical distribution to reflect information you have genuinely (...) learned as time has passed. While the second step resembles traditional procedures of updating by conditionalization, the first is best understood by analogy to traditional models of how agents transmit self-locating opinions through ordinary interpersonal communication. (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)

This is a brief review of issues over which Henry Kyburg and I differed concerning the requirement that full beliefs should be closed under deductive consequence.

The preface paradox can be motivated by appealing to a plausible inference from an author’s reasonable assertion that her book is bound to contain errors to the author’s rational belief that her book contains errors. By evaluating and undermining the validity of this inference, I offer a resolution of the paradox. Discussions of the preface paradox have surprisingly failed to note that expressions of fallibility made in prefaces typically employ terms such as surely, undoubtedly, and bound to be. After considering (...) what these terms mean, I show that the motivating inference is invalid. Moreover, I argue that a closer consideration of our expressions of fallibility suggest that epistemically responsible authors would not be rational to believe that their books contain errors. I conclude by considering alternative expressions of fallibility that employ terms such as possible and probable, and discuss the role that expressions of fallibility play in conversation. (shrink)

– We offer a new motivation for imprecise probabilities. We argue that there are propositions to which precise probability cannot be assigned, but to which imprecise probability can be assigned. In such cases the alternative to imprecise probability is not precise probability, but no probability at all. And an imprecise probability is substantially better than no probability at all. Our argument is based on the mathematical phenomenon of non-measurable sets. Non-measurable propositions cannot receive precise probabilities, but there is a natural (...) way for them to receive imprecise probabilities. The mathematics of non-measurable sets is arcane, but its epistemological import is far-reaching; even apparently mundane propositions are liable to be affected by non-measurability. The phenomenon of non-measurability dramatically reshapes the dialectic between critics and proponents of imprecise credence. Non-measurability offers natural rejoinders to prominent critics of imprecise credence. Non-measurability even reverses some of the critics’ arguments—by the very lights that have been used to argue against imprecise credences, imprecise credences are better than precise credences. (shrink)

The Preface Paradox has led many philosophers to believe that, if it isassumed that high probability is necessary for rational acceptability, the principleaccording to which rational acceptability is closed under conjunction (CP)must be abandoned. In this paper we argue that the paradox is far less damaging to CP than is generally believed. We describe how, given certain plausibleassumptions, in a large class of cases in which CP seems to lead tocontradiction, it does not do so after all. A restricted version (...) of CP canthus be maintained. (shrink)

According to so-called epistemic theories of conditionals, the assertability/acceptability/acceptance of a conditional requires the existence of an epistemically significant relation between the conditional’s antecedent and its consequent. This paper points to some linguistic data that our current best theories of the foregoing type appear unable to explain. Further, it presents a new theory of the same type that does not have that shortcoming. The theory is then defended against some seemingly obvious objections.

The Lottery Paradox and the Preface Paradox both involve the thesis that high probability is sufficient for rational acceptability. The standard solution to these paradoxes denies that rational acceptability is deductively closed. This solution has a number of untoward consequences. The present paper suggests that a better solution to the paradoxes is to replace the thesis that high probability suffices for rational acceptability with a somewhat stricter thesis. This avoids the untoward consequences of the standard solution. The new solution will (...) be defended against a seemingly obvious objection. 1 The paradoxes of rational acceptability 2 The standard solution 3 A new solution to the paradoxes 4 Basic assumptions 5 The new solution defended 6 Conclusion 7 Appendix. (shrink)

The article focuses on representing different forms of non-adjunctive inference as sub-Kripkean systems of classical modal logic, where the inference from □A and □B to □A ∧ B fails. In particular we prove a completeness result showing that the modal system that Schotch and Jennings derive from a form of non-adjunctive inference in (Schotch and Jennings, 1980) is a classical system strictly stronger than EMN and weaker than K (following the notation for classical modalities presented in Chellas, 1980). The unified (...) semantical characterization in terms of neighborhoods permits comparisons between different forms of non-adjunctive inference. For example, we show that the non-adjunctive logic proposed in (Schotch and Jennings, 1980) is not adequate in general for representing the logic of high probability operators. An alternative interpretation of the forcing relation of Schotch and Jennings is derived from the proposed unified semantics and utilized in order to propose a more fine-grained measure of epistemic coherence than the one presented in (Schotch and Jennings, 1980). Finally we propose a syntactic translation of the purely implicative part of Jaśkowski's system D₂ into a classical system preserving all the theorems (and non-theorems) explicilty mentioned in (Jaśkowski, 1969). The translation method can be used in order to develop epistemic semantics for a larger class of non-adjunctive (discursive) logics than the ones historically investigated by Jaśkowski. (shrink)

ABSTRACT: I shall consider the phenomenon of believing ourselves to have at least one false belief: a phenomenon I call believed fallibility I shall first present a paradoxical argument which appears to show that believed fallibility is incoherent; second, note that this argument assumes that we are committed to the conjunction of all our beliefs; third, sketch a more intuitive notion of commitment in which we are not committed to the conjunction of all our beliefs and argue that the original (...) paradoxical argument is now defused; fourth, construct a new paradoxical argument showing that if we believe that we have at least one false belief we are committed to a contradiction, employing the preferable notion of commitment; and, fifth, suggest that perhaps we might avoid the new paradox by denying that closing our beliefs under conjunction is required by rationalityRÉSUMÉ: Je me pencherai ici sur le phénomène qui consiste à penser que nous avons au moins une opinion fausse: je nomme ce phénoméne «la croyance d’etre faillible». Je présente en premier lieu un argument paradoxal qui semble démontrer que la croyance d’être faillible est incohérente. Deuxièmement, je remarque que cet argument suppose que nous nous obligions à réaliser la conjonction de toutes nosopinions. Troisièmement, je propose une version plus intuitive de cette obligation selon laquelle nous ne sommes pas obligés de réaliser la conjonction de toutes nos opinions, et je soutiens que le paradoxe initial est alors désamorcé. Quatrièmement, je construis un nouvel argument paradoxal démontrant que, si nous croyons que nous avons au moins une opinion fausse, nous sommes obligés de nous contredire, selon la notion d’obligation que je préconise d’adopter. Cinquièmement, je suggère que nous pourrions peut-être éviter ce nouveau paradoxe en niant que la rationalité exige de soumettre nos opinions à la conjonction. (shrink)

There are three distinct questions associated with Simpson's paradox, (i) Why or in what sense is Simpson's paradox a paradox? (ii) What is the proper analysis of the paradox? (iii) How one should proceed when confronted with a typical case of the paradox? We propose a "formar" answer to the first two questions which, among other things, includes deductive proofs for important theorems regarding Simpson's paradox. Our account contrasts sharply with Pearl's causal (and questionable) account of the first two questions. (...) We argue that the "how to proceed question?" does not have a unique response, and that it depends on the context of the problem. We evaluate an objection to our account by comparing ours with Blyth's account of the paradox. Our research on the paradox suggests that the "how to proceed question" needs to be divorced from what makes Simpson's paradox "paradoxical.". (shrink)

There are three distinct questions associated with Simpson’s paradox. Why or in what sense is Simpson’s paradox a paradox? What is the proper analysis of the paradox? How one should proceed when confronted with a typical case of the paradox? We propose a “formal” answer to the first two questions which, among other things, includes deductive proofs for important theorems regarding Simpson’s paradox. Our account contrasts sharply with Pearl’s causal account of the first two questions. We argue that the “how (...) to proceed question?” does not have a unique response, and that it depends on the context of the problem. We evaluate an objection to our account by comparing ours with Blyth’s account of the paradox. Our research on the paradox suggests that the “how to proceed question” needs to be divorced from what makes Simpson’s paradox “paradoxical.”. (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)

Paradoxes have played an important role both in philosophy and in mathematics and paradox resolution is an important topic in both fields. Paradox resolution is deeply important because if such resolution cannot be achieved, we are threatened with the charge of debilitating irrationality. This is supposed to be the case for the following reason. Paradoxes consist of jointly contradictory sets of statements that are individually plausible or believable. These facts about paradoxes then give rise to a deeply troubling epistemic problem. (...) Specifically, if one believes all of the constitutive propositions that make up a paradox, then one is apparently committed to belief in every proposition. This is the result of the principle of classical logical known as ex contradictione (sequitur) quodlibetthat anything and everything follows from a contradiction, and the plausible idea that belief is closed under logical or material implication (i.e. the epistemic closure principle). But, it is manifestly and profoundly irrational to believe every proposition and so the presence of even one contradiction in one’s doxa appears to result in what seems to be total irrationality. This problem is the problem of paradox-induced explosion. In this paper it will be argued that in many cases this problem can plausibly be avoided in a purely epistemic manner, without having either to resort to non-classical logics for belief (e.g. paraconsistent logics) or to the denial of the standard closure principle for beliefs. The manner in which this result can be achieved depends on drawing an important distinction between the propositional attitude of belief and the weaker attitude of acceptance such that paradox constituting propositions are accepted but not believed. Paradox-induced explosion is then avoided by noting that while belief may well be closed under material implication or even under logical implication, these sorts of weaker commitments are not subject to closure principles of those sorts. So, this possibility provides us with a less radical way to deal with the existence of paradoxes and it preserves the idea that intelligent agents can actually entertain paradoxes. (shrink)

This paper presents statistical default logic, an expansion of classical (i.e., Reiter) default logic that allows us to model common inference patterns found in standard inferential statistics, including hypothesis testing and the estimation of a populations mean, variance and proportions. The logic replaces classical defaults with ordered pairs consisting of a Reiter default in the first coordinate and a real number within the unit interval in the second coordinate. This real number represents an upper-bound limit on the probability of accepting (...) the consequent of an applied default and that consequent being false. A method for constructing extensions is then defined that preserves this upper bound on the probability of error under a (skeptical) non-monotonic consequence relation. (shrink)

Modern belief revision theory is based to a large extent on partial meet contraction that was introduced in the seminal article by Carlos Alchourrón, Peter Gärdenfors, and David Makinson that appeared in 1985. In the same year, Alchourrón and Makinson published a significantly different approach to the same problem, called safe contraction. Since then, safe contraction has received much less attention than partial meet contraction. The present paper summarizes the current state of knowledge on safe contraction, provides some new results (...) and offers a list of unsolved problems that are in need of investigation. (shrink)

The paper sketches an analysis of the notion of a self-fulfilling belief in terms of doxastic modal logic. We point out a connection between self-fulfilling beliefs and Moore’s paradox. Then we look at self-fulfilling beliefs in the context of neighborhood semantics. We argue that the analysis of several interesting self-fulfilling beliefs has to make essential use of propositional quantification.

There are two ways of understanding the notion of a contradiction: as a conjunction of a statement and its negation, or as a pair of statements one of which is the negation of the other. Correspondingly, there are two ways of understanding the Law of Non-Contradiction (LNC), i.e., the law that says that no contradictions can be true. In this paper I offer some arguments to the effect that on the first (collective) reading LNC is non-negotiable, but on the second (...) (distributive) reading it is perfectly plausible to suppose that LNC may, in some rather special and perhaps undesirable circumstances, fail to hold. (shrink)