If we try to evaluate the sentence on line 1 we ¯nd ourselves going in an unending cycle. For this reason alone we may conclude that the sentence is not true. Moreover we are driven to this conclusion by an elementary argument: If the sentence is true then what it asserts is true, but what it asserts is that the sentence on line 1 is not true. Consequently the sentence on line 1 is not true. But when we write this (...) true conclusion on line 2 we ¯nd ourselves repeating the very same sentence. It seems that we are unable to deny the truth of the sentence on line 1 without asserting it at the same time. (shrink)

This article seeks to establish that the social sciences have an important contribution to make to the study of ethics. The discussion is framed around three questions: (i) what theoretical work can the social sciences contribute to the understanding of ethics? (ii) what empirical work can the social sciences contribute to the understanding of ethics? And (iii) how does this theoretical and empirical work combine, to enhance the understanding of how ethics, as a field of analysis and debate, is socially (...) constituted and situated? Through these questions the argument goes beyond the now commonly cited objection to the over‐simplistic division between normative and descriptive ethics (that assigns the social sciences the ‘handmaiden’ role of simply providing the ‘facts’). In extending this argument, this article seeks to establish, more firmly and in more detail, that: (a) the social sciences have a longstanding theoretical interest analysing the role that a concern with ethics plays in explanations of social change, social organisation and social action; (b) the explanations that are based on the empirical investigations conducted by social scientists exemplify the interplay of epistemological and methodological analyses so that our understanding of particular substantive issues is extended beyond the conventional questions raised by ethicists, and (c) through this combination of theoretical and empirical work, social scientists go beyond the specific ethical questions of particular practices to enquire further into the social processes that lie behind the very designation of certain matters as being ‘ethical issues’. (shrink)

Many philosophers in the field of meta-ethics believe that rational degrees of confidence in moral judgments should have a probabilistic structure, in the same way as do rational degrees of belief. The current paper examines this position, termed “moral Bayesianism,” from an empirical point of view. To this end, we assessed the extent to which degrees of moral judgments obey the third axiom of the probability calculus, ifP(A∩B)=0thenP(A∪B)=P(A)+P(B), known as finite additivity, as compared to degrees of beliefs on the one (...) hand and degrees of desires on the other. Results generally converged to show that degrees of moral judgment are more similar to degrees of belief than to degrees of desire in this respect. This supports the adoption of a Bayesian approach to the study of moral judgments. To further support moral Bayesianism, we also demonstrated its predictive power. Finally, we discuss the relevancy of our results to the meta-ethical debate between moral cognitivists and moral non-cognitivists. (shrink)

In his classic book “the Foundations of Statistics” Savage developed a formal system of rational decision making. The system is based on (i) a set of possible states of the world, (ii) a set of consequences, (iii) a set of acts, which are functions from states to consequences, and (iv) a preference relation over the acts, which represents the preferences of an idealized rational agent. The goal and the culmination of the enterprise is a representation theorem: Any preference relation that (...) satisfies certain arguably acceptable postulates determines a (finitely additive) probability distribution over the states and a utility assignment to the consequences, such that the preferences among acts are determined by their expected utilities. Additional problematic assumptions are however required in Savage's proofs. First, there is a Boolean algebra of events (sets of states) which determines the richness of the set of acts. The probabilities are assigned to members of this algebra. Savage's proof requires that this be a σ-algebra (i.e., closed under infinite countable unions and intersections), which makes for an extremely rich preference relation. On Savage's view we should not require subjective probabilities to be σ-additive. He therefore finds the insistence on a σ-algebra peculiar and is unhappy with it. But he sees no way of avoiding it. Second, the assignment of utilities requires the constant act assumption: for every consequence there is a constant act, which produces that consequence in every state. This assumption is known to be highly counterintuitive. The present work contains two mathematical results. The first, and the more difficult one, shows that the σ-algebra assumption can be dropped. The second states that, as long as utilities are assigned to finite gambles only, the constant act assumption can be replaced by the more plausible and much weaker assumption that there are at least two non-equivalent constant acts. The second result also employs a novel way of deriving utilities in Savage-style systems -- without appealing to von Neumann-Morgenstern lotteries. The paper discusses the notion of “idealized agent" that underlies Savage's approach, and argues that the simplified system, which is adequate for all the actual purposes for which the system is designed, involves a more realistic notion of an idealized agent. (shrink)

The goal of this paper is a comprehensive analysis of basic reasoning patterns that are characteristic of vague predicates. The analysis leads to rigorous reconstructions of the phenomena within formal systems. Two basic features are dealt with. One is tolerance: the insensitivity of predicates to small changes in the objects of predication (a one-increment of a walking distance is a walking distance). The other is the existence of borderline cases. The paper shows why these should be treated as different, though (...) related phenomena. Tolerance is formally reconstructed within a proposed framework of contextual logic, leading to a solution of the Sorites paradox. Borderline-vagueness is reconstructed using certain modality operators; the set-up provides an analysis of higher order vagueness and a derivation of scales of degrees for the property in question. (shrink)

Haim Gordon; Nietzsche's Zarathustra as Educator, Journal of Philosophy of Education, Volume 14, Issue 2, 30 May 2006, Pages 181–192, https://doi.org/10.1111/j.

The semantic paradoxes, whose paradigm is the Liar, played a crucial role at a crucial juncture in the development of modern logic. In his 1908 seminal paper, Russell outlined a system, soon to become that of the Principia Mathematicae, whose main goal was the solution of the logical paradoxes, both semantic and settheoretic. Russell did not distinguish between the two and his theory of types was designed to solve both kinds in the same uniform way. Set theoreticians, however, were content (...) to treat only the set-theoretic paradoxes, putting aside the semantic ones as a non-mathematical concern. This separation was explicitly proposed, eighteen years after Russell’s paper, by Ramsey, though he, like Russell, advocated a system that addresses both kinds. Since then, the semantic paradoxes have been viewed within the perspective of the theory of truth, where they have occupied a respectable niche, but one of rather specialized interest. (shrink)

There are three sections in this paper. The first is a philosophical discussion of the general problem of reasoning under limited deductive capacity. The second sketches a rigorous way of assigning probabilities to statements in pure arithmetic; motivated by the preceding discussion, it can nonetheless be read separately. The third is a philosophical discussion that highlights the shifting contextual character of subjective probabilities and beliefs.

In this work we give a complete answer as to the possible implications between some natural properties of Lebesgue measure and the Baire property. For this we prove general preservation theorems for forcing notions. Thus we answer a decade-old problem of J. Baumgartner and answer the last three open questions of the Kunen-Miller chart about measure and category. Explicitly, in \S1: (i) We prove that if we add a Laver real, then the old reals have outer measure one. (ii) We (...) prove a preservation theorem for countable-support forcing notions, and using this theorem we prove (iii) If we add ω 2 Laver reals, then the old reals have outer measure one. From this we obtain (iv) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + \neg B(m) + \neg U(m) + U(c))$ . In \S2: (i) We prove a preservation theorem, for the finite support forcing notion, of the property " $F \subseteq ^\omega\omega$ is an unbounded family." (ii) We introduce a new forcing notion making the old reals a meager set but the old members of ω ω remain an unbounded family. Using this we prove (iii) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + U(m) + \neg B(c) + \neg U(c) + C(c))$ . In \S3: (i) We prove a preservation theorem, for the finite support forcing notion, of a property which implies "the union of the old measure zero sets is not a measure zero set," and using this theorem we prove (ii) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + \neg U(m) + C(m) + \neg C(c))$. (shrink)

Non-standard models were introduced by Skolem, first for set theory, then for Peano arithmetic. In the former, Skolem found support for an anti-realist view of absolutely uncountable sets. But in the latter he saw evidence for the impossibility of capturing the intended interpretation by purely deductive methods. In the history of mathematics the concept of a nonstandard model is new. An analysis of some major innovations–the discovery of irrationals, the use of negative and complex numbers, the modern concept of function, (...) and non-Euclidean geometry–reveals them as essentially different from the introduction of non-standard models. Yet, non-Euclidean geometry, which is discussed at some length, is relevant to the present concern; for it raises the issue of intended interpretation. The standard model of natural numbers is the best candidate for an intended interpretation that cannot be captured by a deductive system. Next, I suggest, is the concept of a wellordered set, and then, perhaps, the concept of a constructible set. One may have doubts about a realistic conception of the standard natural numbers, but such doubts cannot gain support from non-standard models. Attempts to utilize non-standard models for an anti-realist position in mathematics, which appeal to meaning-as-use, or to arguments of the kind proposed by Putnam, fail through irrelevance, or lead to incoherence. Robinson’s skepticism, on the other hand, is a coherent position, though one that gives up on providing a detailed philosophical account. The last section enumerates various uses of non-standard models. (shrink)

In this paper we study the question assuming MA+⌝CH does Sacks forcing or Laver forcing collapse cardinals? We show that this question is equivalent to the question of what is the additivity of Marczewski's ideals 0. We give a proof that it is consistent that Sacks forcing collapses cardinals. On the other hand we show that Laver forcing does not collapse cardinals.

Naïve realism is the view that veridical experiences are fundamentally relations of acquaintance to external objects and their features, and multidisjunctivism is the conjunction of naïve realism and the view that hallucinatory experiences don’t share a common fundamental kind. Multidisjunctivism allegedly removes the screening-off worry over naïve realism, and the relevant literature suggests that multidisjunctivism is one of the naïve realist responses to the worry. The present paper argues that the multidisjunctive solution is implicitly changing the subject, so the impression (...) that the multidisjunctivist is addressing the screening-off problem is illusory. (shrink)

We present experimental evidence that supports the thesis :602–625, 2015, Br J Philos Sci 70:77–102, 2019; Bradley in Decisions theory with a human face, Cambridge University Press, Cambridge, 2017; Goldschmidt and Nissan-Rozen in Synthese 198:7553–7575, 2021) that people might positively or negatively desire risky prospects conditional on only some of the prospects’ outcomes obtaining. We argue that this evidence has important normative implications for the central debate in normative decision theory between two general approaches on how to rationalize several common (...) patterns of preference, which are ruled out as irrational by orthodox decision theory, namely the re-individuation approach and the non-expected utility approach. (shrink)

The paper is concerned with the way in which “ontology” and “realism” are to be interpreted and applied so as to give us a deeper philosophical understanding of mathematical theories and practice. Rather than argue for or against some particular realistic position, I shall be concerned with possible coherent positions, their strengths and weaknesses. I shall also discuss related but different aspects of these problems. The terms in the title are the common thread that connects the various sections.

In a recent paper S. McCall adds another link to a chain of attempts to enlist Gödel’s incompleteness result as an argument for the thesis that human reasoning cannot be construed as being carried out by a computer.1 McCall’s paper is undermined by a technical oversight. My concern however is not with the technical point. The argument from Gödel’s result to the no-computer thesis can be made without following McCall’s route; it is then straighter and more forceful. Yet the argument (...) fails in an interesting and revealing way. And it leaves a remainder: if some computer does in fact simulate all our mathematical reasoning, then, in principle, we cannot fully grasp how it works. Gödel’s result also points out a certain essential limitation of self-reflection. The resulting picture parallels, not accidentally, Davidson’s view of psychology, as a science that in principle must remain “imprecise”, not fully spelt out. What is intended here by “fully grasp”, and how all this is related to self-reflection, will become clear at the end of this comment. (shrink)

We trace self-reference phenomena to the possibility of naming functions by names that belong to the domain over which the functions are defined. A naming system is a structure of the form ,{ }), where D is a non-empty set; for every a∈ D, which is a name of a k-ary function, {a}: Dk → D is the function named by a, and type is the type of a, which tells us if a is a name and, if it is, (...) the arity of the named function. Under quite general conditions we get a fixed point theorem, whose special cases include the fixed point theorem underlying Gödel's proof, Kleene's recursion theorem and many other theorems of this nature, including the solution to simultaneous fixed point equations. Partial functions are accommodated by including “undefined” values; we investigate different systems arising out of different ways of dealing with them. Many-sorted naming systems are suggested as a natural approach to general computatability with many data types over arbitrary structures. The first part of the paper is a historical reconstruction of the way Gödel probably derived his proof from Cantor's diagonalization, through the semantic version of Richard. The incompleteness proof–including the fixed point construction–result from a natural line of thought, thereby dispelling the appearance of a “magic trick”. The analysis goes on to show how Kleene's recursion theorem is obtained along the same lines. (shrink)

The technique of minimizing information (infomin) has been commonly employed as a general method for both choosing and updating a subjective probability function. We argue that, in a wide class of cases, the use of infomin methods fails to cohere with our standard conception of rational degrees of belief. We introduce the notion of a deceptive updating method and argue that non-deceptiveness is a necessary condition for rational coherence. Infomin has been criticized on the grounds that there are no higher (...) order probabilities that ‘support’ it, but the appeal to higher order probabilities is a substantial assumption that some might reject. Our elementary arguments from deceptiveness do not rely on this assumption. While deceptiveness implies lack of higher order support, the converse does not, in general, hold, which indicates that deceptiveness is a more objectionable property. We offer a new proof of the claim that infomin updating of any strictly-positive prior with respect to conditional-probability constraints is deceptive. In the case of expected-value constraints, infomin updating of the uniform prior is deceptive for some random variables but not for others. We establish both a necessary condition and a sufficient condition (which extends the scope of the phenomenon beyond cases previously considered) for deceptiveness in this setting. Along the way, we clarify the relation which obtains between the strong notion of higher order support, in which the higher order probability is defined over the full space of first order probabilities, and the apparently weaker notion, in which it is defined over some smaller parameter space. We show that under certain natural assumptions, the two are equivalent. Finally, we offer an interpretation of Jaynes, according to which his own appeal to infomin methods avoids the incoherencies discussed in this paper. (shrink)

We construct a Borel maximal cofinitary group.

John Forrester’s book Thinking in Cases does not provide one ultimate definition of what it means to ‘think in cases’, but rather several alternatives: a ‘style of reasoning’ (Hacking), ‘paradigms’ or ‘exemplars’ (Kuhn), and ‘language games’ (Wittgenstein), to mention only a few. But for Forrester, the stories behind each of the figures who suggested these different models for thinking (in cases) are as important as the models themselves. In other words, the question for Forrester is not only what ‘thinking in (...) cases’ is, but also who might be considered a ‘thinker in cases’. Who could serve as a case study for such a thinker? The major candidates that Forrester considers in his book to be ‘thinkers in cases’ are Kuhn, Foucault, Freud, and Winnicott. In what follows, I will argue that one name is missing from this list, as well as from Forrester’s book more generally: Michael Balint. This name is missing not only because Balint was a great ‘thinker in cases’, but also because we have some reasons to believe that Forrester himself thought so and wished to add him to the list. Forrester, I will argue, found in Balint an exemplar for a thinker in cases that combined elements from Winnicott’s psychoanalytic theory and Foucault’s philosophy of the case-based sciences. (shrink)

Since its origins in the mid-1980s, DNA profiling has become the most powerful tool for identification in contemporary society. Practitioners have deployed it to determine parentage, verify claims to identity in various civil contexts, identify bodies in wars and mass disasters, and infer the identity of individuals who have left biological traces at crime scenes. Thus DNA profiling can be used to implicate or exonerate individuals from participation in particular social relations and activities; this affords it a growing importance in (...) major social institutions such as the family, the criminal justice system, immigration services, and health services. There are key state, security, civil liberty, personal, and commercial considerations surrounding the reliability and social implications of DNA profiling in establishing the identities of “family members,” “claimants,” “customers,” “suspects,” and “citizens.”Given that DNA profiling is increasingly influential in forensic inquiries, the recently-developed practice in the UK of “familial searching” of DNA databases has the potential to become a significant aspect of investigations. (shrink)

Savage's framework of subjective preference among acts provides a paradigmatic derivation of rational subjective probabilities within a more general theory of rational decisions. The system is based on a set of possible states of the world, and on acts, which are functions that assign to each state a consequence€. The representation theorem states that the given preference between acts is determined by their expected utilities, based on uniquely determined probabilities (assigned to sets of states), and numeric utilities assigned to consequences. (...) Savage's derivation, however, is based on a highly problematic well-known assumption not included among his postulates: for any consequence of an act in some state, there is a "constant act" which has that consequence in all states. This ability to transfer consequences from state to state is, in many cases, miraculous -- including simple scenarios suggested by Savage as natural cases for applying his theory. We propose a simplification of the system, which yields the representation theorem without the constant act assumption. We need only postulates P1-P6. This is done at the cost of reducing the set of acts included in the setup. The reduction excludes certain theoretical infinitary scenarios, but includes the scenarios that should be handled by a system that models human decisions. (shrink)

The impetus driving this article is the uncritical uses of ‘culture’ as an explanatory variable in public health research of ‘suicide’, regarding its conceptualization and operationalization as a mentally riddled phenomenon clamped in nomothetic and epidemiological nomenclature. This reduction of suicide to its presumed ‘evidence based’ figures and graphs under the guise of the lingo of culture requires and yields not only ‘thin’ understandings but also non-committal conclusions. Thus, ‘culture’ merely appears as a ‘thing’ made of shared norms and values (...) that arguably can be measured, computed, and even predicted in relation to tables of suicide rates. With the aim of critically un-disciplining and unleashing these two intersecting fuzzy concepts from the straits of their epistemological habitus, we set an alternative scene of relations between them that contribute to the re-conceptualization of both. ‘Suicide cultures’, a term we coined for highlighting temporal, agentive, performative, and social justice questions of suicide and culture, place the preoccupation with mortality and terminality at its core. This turn in the analysis of public health sheds a non-perfunctory light on the cultural parameters of counting of and accounting for the incidence of suicide in different social settings. (shrink)

This engaging work of comparative philosophy puts the Chinese and American philosophical traditions into a mutually informative and transformative philosophical dialogue on the way to developing a new form of Confucian pragmatism.

Self-reference in semantics, which leads to well-known paradoxes, is a thoroughly researched subject. The phenomenon can appear also in decision theoretic situations. There is a structural analogy between the two and, more interestingly, an analogy between principles concerning truth and those concerning rationality. The former can serve as a guide for clarifying the latter. Both the analogies and the disanalogies are illuminating.

The moral status of the human embryo has gained much attention in debates over the acceptability, or otherwise, of human embryonic stem cell research. Far less attention has been paid to the suppliers of those embryos: people who have undergone IVF treatment to produce embryos to assist them to have a baby. It is sociologically and ethically important to understand their views and experiences of being asked to donate embryos for research if we are to fully understand the wider social (...) and regulatory aspects of hESC science. This paper reports on parallel studies investigating these issues in the UK and in Switzerland. The studies reveal the inextricable entangling of the social and moral status of embryos. Since donors participate in different discursive domains and contexts (public, clinic, family) that shape their perception of “what” an embryo is, their views of embryos embody conflicting ideas and ambivalences. (shrink)

Dummett’s The Logical Foundations of Metaphysics (LFM) outlines an ambitious project that has been at the core of his work during the last forty years. The project is built around a particular conception of the theory of meaning (or philosophy of language), according to which such a theory should constitute the corner stone of philosophy and, in particular, provide answers to various metaphysical questions. The present paper is intended as a critical evaluation of some of the main features of that (...) approach. My negative answer to the title’s question notwithstanding, I find Dummett’s analyses, which both inform and are guided by his project, of very high value. Among the subjects to be discussed here, which relate to but are not fully reflected in the title, are the concept of a full-blooded theory (in 4.) Davidson’s program (in 5.) and holism, to which the last third of this paper (section 6.) is devoted. That section can be read independently; to some extent, this is also true of 4. and 5. taken together. (shrink)

The Cambridge Malting House, an experimental school, serves here as a case study for investigating the tensions within 1920s liberal elites between their desire to abandon some Victorian and Edwardian sets of values in favour of more democratic ones, and at the same time their insistence on preserving themselves as an integral part of the English upper class. Susan Isaacs, the manager of the Malting House, provided the parents – some of whom were the most famous scientists and intellectuals of (...) their age – with an opportunity to fulfil their ‘fantasy’ of bringing up children in total freedom. In retrospect, however, she deeply criticized those from their milieu for not fully understanding the real socio-cultural implications of their ideological decision to make independence and freedom the core values in their children’s education. Thus, 1920s progressive education is a paradigmatic case study of the cultural and ideological inner contradictions within liberal thought in the interwar era. The article also shows how psychoanalysis – which attracted many progressive educators – played a crucial role in providing liberals of all sorts with a new language to articulate their political visions, but, at the same time, explored the limits of the liberal discourse as a whole. (shrink)

The presence of nonobservational vocabulary is shown to be necessary for wide application of a conservative principle of theory revision.

We show the Borel Conjecture is consistent with the continuum large.

This article deploys a Bourdieusian framework to analyse the process of how children are located in, and attached to, families. The focus is on children whose placement is problematic for some reason (such as adoption, egg and semen donation, surrogacy and so on). Through a detailed examination of four case studies in which the placement of children is disputed, I show how notions of embodied spaces (such as the womb) are part of the repertoire of arguments used for establishing claims (...) as to the appropriate placing of such children. However, I also show how the significance of such spaces is relative to the importance attached to other spaces (e.g. geographical, domestic, metaphorical). The analysis demonstrates the usefulness of Bourdieu’s central concepts but also shows how those concepts might be both supplemented and challenged. (shrink)

In this book, best-selling author and mathematician Haim Shapira presents an introduction to mathematical theories which deal with the most beautiful concept ever invented by humankind: infinity. Written in clear, simple language and aimed at a lay audience, this book also offers some strategies that will allow readers to try their ability at solving truly fascinating mathematical problems. Infinity is a deeply counter-intuitive concept that has inspired many great thinkers. In this book we will meet many sages, both familiar (...) and unfamiliar: Zeno and Pythagoras, Georg Cantor and Bertrand Russell, Sofia Kovalevskaya and Emmy Noether, al-Khwarizmi and Euclid, Sophie Germain and Srinivasa Ramanujan. The world of infinity is inhabited by many paradoxes, and so is this book: Zeno paradoxes, Hilbert's "Infinity Hotel", Achilles and the gods paradox, the paradox of heaven and hell, the Ross-Littlewood paradox involving tennis balls, the Galileo paradox and many more. Aimed at the curious but non-technical reader, this book refrains from using any fearsome mathematical symbols. It uses only the most basic operations of mathematics: adding, subtracting, multiplication, division, powers and roots - that is all. But that doesn't mean that a bit of deep thinking won't be necessary and rewarding. Writing with humour and lightness of touch, Haim Shapira banishes the chalky pallor of the schoolroom and offers instead a truly thrilling intellectual journey. Fasten your seatbelt - we are going to Infinity, and beyond! (shrink)

We improve a theorem of Raisonnier by showing that Cons(ZFC+every Σ 2 1 -set of reals in Lebesgue measurable+every Π 2 1 -set of reals isK σ-regular) implies Cons(ZFC+there exists an inaccessible cardinal). We construct, fromL, a model where every Δ 3 1 -sets of reals is Lebesgue measurable, has the property of Baire, and every Σ 2 1 -set of reals isK σ-regular. We prove that if there exists a Σ n+1 1 unbounded filter on ω, then there exists (...) a nonK σ-regular Π 2 1 -subset. (shrink)

This article is a response to McLeod and Baylis (2007) who speculate on the dangers of requesting fresh ‘spare’ embryos from IVF patients for human embryonic stem cell (hESC) research, particularly when those embryos are good enough to be transferred back to the woman. They argue that these embryos should be frozen instead. We explore what is meant by ‘spare’ embryos. We then provide empirical evidence, from a study of embryo donation and of embryo donors' views, to substantiate some of (...) their speculations about the problems associated with requesting fresh embryos. However, we also question whether such problems are resolved by embryo freezing, since further empirical evidence suggests that this raises other social and ethical problems for patients. There is little evidence that the request for embryos for research, in itself, causes patients distress. We suggest, however, that no requests for fresh embryos should be made in the first cycle of IVF treatment. Deferring the request to a later cycle ensures that potential donors are better informed (by experience and reflection) about the possible destinations of their embryos and about the definition of ‘spare embryos’. Both this article, and that by McLeod and Baylis, emphasize the need to consider the views and experiences of embryo donors when evaluating the ethics of embryo donation for hESC research. (shrink)

Chinese Philosophy provides a clear, accessible conception of the Chinese philosophical sensibility and its evolution throughout history.

We define the ideal with the property that a real omits all Borel sets in the ideal which are coded in a transitive model if and only if it is an amoeba real over this model. We investigate some other properties of this ideal. Strolling through the "amoeba forest" we gain as an application a modification of the proof of the inequality between the additivities of Lebesgue measure and Baire category.