Logic for Philosophy is an introduction to logic for students of contemporary philosophy. It is suitable both for advanced undergraduates and for beginning graduate students in philosophy. It covers (i) basic approaches to logic, including proof theory and especially model theory, (ii) extensions of standard logic that are important in philosophy, and (iii) some elementary philosophy of logic. It emphasizes breadth rather than depth. For example, it discusses modal logic and counterfactuals, but does not prove the central metalogical results for (...) predicate logic (completeness, undecidability, etc.) Its goal is to introduce students to the logic they need to know in order to read contemporary philosophical work. It is very user-friendly for students without an extensive background in mathematics. In short, this book gives you the understanding of logic that you need to do philosophy. (shrink)

Many otherwise reasonable philosophers are impatient with ontology. These philosophers will probably have little time for Objects and Persons, which claims that while there do exist “atoms arranged statuewise”, there do not exist statues; while there do exist atoms arranged tablewise and atoms arranged chairwise, there exist no tables and chairs. Though I join these philosophers, at the end of the day, in rejecting Merricks’s claims, that day is long, whereas they want a quick verdict. But why? Do our impatient (...) friends think that Merricks’s claims are contradictory, analytically false, or otherwise conceptually incoherent? They may say that the conventional meaning of “there exists a statue” is “there exist atoms arranged statuewise”, but this does not stand up to scrutiny. Someone could, of course, just decide to mean such a thing by “there exists a statue”, but then her pronouncements would be irrelevant to Merricks, who intends to be using a.. (shrink)

This is an introduction to metaphysics for students and non-philosophers. (Philosophers: it's supposed to be the kind of book you can give to your friends and family, when they ask what you do for a living.) Contents: personal identity, fatalism, time, God, why not nothing?, free will, constitution, universals, necessity and possibility, what is metaphysics? (There is a second edition, which adds chapters on meta-metaphysics and the metaphysics of ethics.).

This is an introduction to metaphysics for students and non-philosophers. (Philosophers: it's supposed to be the kind of book you can give to your friends and family, when they ask what you do for a living.) Contents: personal identity, fatalism, time, God, why not nothing?, free will, constitution, universals, necessity and possibility, what is metaphysics, meta-metaphysics, the metaphysics of ethics.

The current work examines how cues traditionally used to signal environmental friendliness, specifically the color green and eco-labels, and influence product efficacy perceptions and subsequent purchase intentions. Across three experiments, we find that environmental cues used in isolation reduce perceptions of product efficacy. We argue that this efficacy discounting effect occurs because the isolated use of an environmental cue introduces category ambiguity by activating competing functionality and environmentally friendly schemas during evaluation. We discuss the implications of our findings for research (...) on environmental consumption as well as offer insight into the effective use of environmental cues on product packaging. (shrink)

Instead of an overture : no heirs -- The house in Schöne Aussicht : a Frankfurt childhood around 1910 -- From Teddie Wiesengrund to Dr. Wiesengrund-Adorno -- Adorno as "non-identical" man -- Transitions -- Bertolt Brecht : "to those who come after us" -- Theodor W. Adorno : "out of the firing-line" -- Hanns Eisler, the non-identical brother -- Fritz Lang, the American friend -- Frankfurt transfer -- Adorno as "identical" man -- The palimpsest of life.

In this foundational study, originally published in 1880, Heinrich Kihn, professor of theology at Würzburg, compared the exegetical work of two significant figures in late antiquity. Theodore, born at Antioch and a friend of John Chrysostom, was an influential bishop of Mopsuestia from 392 to 428. His work was widely regarded as heretical in the centuries following his death. A century later Junillus Africanus served as chief legal minister to the Byzantine emperor Justinian and wrote a handbook of (...) biblical exegesis that became important in the medieval Latin West. Kihn argued that Junillus' Instituta, the text of which he included in this study, showed the direct influence of Theodore's writings. All subsequent scholarly work on Junillus, and much on Theodore, depends on Kihn's edition and analysis, even when disagreeing with important aspects of his interpretation. This indispensable but hard-to-find work is now reissued in its entirety. (shrink)

On September 12, 1949, the liberal politician Theodor Heuss, party leader of the „Freie Demokratische Partei“ (FDP), was elected by the Bundesversammlung (Federal Convention) as the first Bundespräsident, i. e. head of state, of the newly founded Bundesrepublik Deutschland. As a young man Heuss had been a close friend and political ally of Friedrich Naumann, the protestant pastor and left wing liberal politician, supported by Ernst Troeltsch. Heuss then working as a political journalist for liberal newspapers and Naumann’s weekly (...) journal Die Hilfe, was an admirer of Troeltsch, and since 1910 they often met in Heidelberg, at Villa Fallenstein, Ziegelhäuser Landstraße 17, where Max and Marianne Weber as well als the Troeltschs lived; Heuss frequently visited Max Weber’s Sunday jour fixe. When in 1915 Troeltsch became a professor at the Philosophical Faculty of Berlin University Heuss regularly kept in contact with him especially through Hans Delbrück’s „Mittwochabend“, a weekly gathering of liberal intellectuals, professors, politicians and journalists discussing political reforms and the ongoing war. Both Troeltsch and Heuss, 19 years younger, demanded the democratization of the Deutsches Reich, and after the end of the war and the revolution of 1918/19 they became members of the newly founded left wing liberal party „Deutsche Demokratische Partei“ (DDP). After Troeltsch’s sudden death on February 1st, 1923, Heuss wrote an obituary, and, together with his wife Elly Heuss-Knapp, attended the funeral service conducted by Adolf von Harnack. Heuss was a member of the „Demokratischer Klub Berlin“, founded in 1919 by liberal politicians, bankers and academics to regularly discuss the political situation of the new democratic state, the Weimar Republic. Unknown until now, seven weeks after Troeltsch’s death the „Demokratische Klub“ invited its members (the club only had male mebers!) to a memorial act for Troeltsch. On this occasion Heuss delivered a speech „Zu Ernst Troeltschs Gedächtnis“ („To the Memory of Ernst Troeltsch“) which is published here for the first time. (shrink)

An unusual and thought-provoking history of humankind traces the evolution of emotions and personal relationships through the ages and among diverse cultures, discussing such varied topics as the art of conversation, inter-gender friendships, lifestyles, and cookery.

In einem Brief nennt Adorno die "Negative Dialektik" kurz nach ihrem Erscheinen unter seinen Schriften "das philosophische Hauptwerk, wenn ich so sagen darf“. Dieser herausgehobenen Bedeutung, die das Werk für Adorno hatte, entspricht nicht nur die lange Zeit, die er mit der Abfassung des Buchs beschäftigt war, sondern auch die lange Geschichte, die ihre zentralen Motive in seinem Denken haben. Philosophische Begriffsklärung, die Arbeit an "Begriff und Kategorien“ einer negativen Dialektik, versteht Adorno dabei als dialektischen Übergang in inhaltliches Denken – (...) und so betreibt er sie auch hier. Das hat Konsequenzen für die Form des kooperativen Kommentars, der in diesem Band versucht wird. Adornos "Negative Dialektik" zu kommentieren, kann nur in dem Bewußtsein der unüberbrückbaren Kluft gelingen, die den Kommentar von diesem Text trennt. (shrink)

Attempts have been made to prove God's non-existence. Often this takes the form of an appeal to the so-called Argument from Evil: if God were to exist, then he would not permit as much suffering in the world as there actually is. Hence the fact that there is so much suffering constitutes evidence for God's non-existence. In this essay I propose a variation which I shall call ‘The Argument from Non-belief’. Its basic idea is that if God were to exist, (...) then he would not permit as much non-belief in the world as there actually is. Hence the fact that there is so much non-belief constitutes evidence for God's non-existence. (shrink)

The following is the transcript of a lecture taken in shorthand by Hans-Georg Backhaus. The transcript was originally published as an appendix in Hans-Georg Backhaus, Dialektik der Wertform. Untersuchungen zur marxschen Ökonomiekritik, a complete translation of which is forthcoming in the Historical Materialism book series.

A property, F, is maximal iff, roughly, large parts of an F are not themselves Fs.' Maximality makes trouble for a recent analysis of intrinsicality by Rae Langton and David Lewis.

Traditional debate on the metaphysics of gender has been a contrast of essentialist and social-constructionist positions. The standard reaction to this opposition is that neither position alone has the theoretical resources required to satisfy an equitable politics. This has caused a number of theorists to suggest ways in which gender is unified on the basis of social rather than biological characteristics but is “real” or “objective” nonetheless – a position I term social objectivism. This essay begins by making explicit the (...) motivations for, and central assumptions of, social objectivism. I then propose that gender is better understood as a real kind with a historical essence, analogous to the biologist’s claim that species are historical entities. I argue that this proposal achieves a better solution to the problems that motivate social objectivism. Moreover, the account is consistent with a post-positivist understanding of the classificatory practices employed within the natural and social sciences. (shrink)

What accounts for the prestige of quantitative methods? The usual answer is that quantification is desirable in social investigation as a result of its successes in science. Trust in Numbers questions whether such success in the study of stars, molecules, or cells should be an attractive model for research on human societies, and examines why the natural sciences are highly quantitative in the first place. Theodore Porter argues that a better understanding of the attractions of quantification in business, government, (...) and social research brings a fresh perspective to its role in psychology, physics, and medicine. Quantitative rigor is not inherent in science but arises from political and social pressures, and objectivity derives its impetus from cultural contexts. In a new preface, the author sheds light on the current infatuation with quantitative methods, particularly at the intersection of science and bureaucracy. (shrink)

A certain conception of Hell is inconsistent with God's traditional attributes. My argument is novel in focusing on considerations involving vagueness. God is in charge of the selection procedure, so the selection procedure must be just; any just procedure will have borderline cases; but according to the traditional conception, the afterlife is binary and has no borderline cases.

According to four dimensionalism, the material world is divided into momentary stages. In a four-dimensional world, which objects are the ordinary things, the things we normally name and quantify over? Aggregates of stages, according to most four-dimensionalists, but according to stage theorists (or exdurantists), ordinary objects are instead to be identified with the stages themselves. (A temporal counterpart theoretic account of de re temporal predication is then given.) This paper argues that a stage theorist is best positioned to accept David (...) Lewis's argument from temporary intrinsics for four-dimensionalism, since stage theorists are the only four-dimensionalists who attribute monadic temporary intrinsic properties to ordinary things. (shrink)

For deliberations on the ethical and meta-ethical implications of Alfred North Whitehead's process philosophy, here are abstracts and papers from the Ethics Section of the 6th International Whitehead Conference held at the University of Salzburg in Salzburg, Austria in July 2006. In accordance with the conference schedule, there are three subsections. The subsection on "Metaphysics of Morals and Moral Theory" includes contributions from Franklin I. Gamwell (Does Morality Presuppose God?), John W. Lango (abstract only), Duane Voskuil ("Ethics' Dipolar Necessities and (...) Theistic Implications), and Theodore Walker Jr. ("Neoclassical Cosmology and Matthew 22:36-40). The subsection on "Evaluating Moral Practices" includes contributions from Frederick Ferrè (abstract only), Seung Gap Lee (Hope for the Earth: A Process Eschatological Eco-ethics for South Korea), Mary Elizabeth Mullino Moore (Compassion, Creativity, and Form: The Ethics of Institutions), and George W. Shields (Ruse, Altruism, and Process Philosophy). The subsection on "Ethics and Aesthetic Values" includes contributions from Stephen T. Franklin (abstract only), Brian G. Henning (Is There an Ethics of Creativity?), Mihàly Tòth (Art of Life and the Ethics of Life Forming), and Guorong Yang (Problems and Perspectives in the Emerging of Global Society). (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 41.1 (2003) 136-137 [Access article in PDF] John Watkins. Human Freedom after Darwin: A Critical Rationalist View. Chicago: Open Court Publishing, 1999. Pp. xi + 348. Cloth, $49.95. Paper, $24.95. John Watkins examines man's place in nature since Darwin. As a critical rationalist, using the methods of science, Watkins hopes to construct a world-view which challenges competing hypotheses and supports his own. He (...) recognizes that his view is, also, a candidate for critical rejection. All scientific views are, in principle, fallible. He divides his book into two parts: Naturalism and Freedom. [End Page 137]In evolutionary development, he accepts, as critically established, the sequence: matter before life, life before mind, and, no mind without a brain. Thus, naturalism is preferred beforesupernaturalism or divine intervention. These guides may support a continuous, steady evolutionaryprocess or discontinuity, mutations, and various stages of change. Darwin acceptedthe former and held physical changes were unaffected by consciousness. All change is, in principle,present from the beginning and unfolds over time. Darwin's view, then, is deterministic.Watkins challenges this position and argues for consciousness affecting physical change in new and unpredictable ways. It is herein that the possibility of animal and human freedom lies. Watkins gives examples of the actions of the hunter and hunted among animals, and of strategies developed by humans during crises, to support the view that there are novel and unpredictable actions by conscious beings. These actions can hardly be called predetermined in the nature of things. There is a huge difference between a general genetic structure for kinds of behavior and specific actions during crises (not that these are the only instances of freedom). At any rate, there is a reciprocal action between an individual and its environment and not merely a unilinear one.In Part II, Freedom, Watkins examines some classical, philosophical views of human freedom. He gives brief, but penetrating, analyses of Hobbes, Descartes, Hume, Kant, Schopenhauer, and Spinoza. Either they account for human action in strictly physical terms that are causally deterministic or they turn to non-empirical sources that are not verifiable. Since the latter make no difference in our account of behavior, they are entities multiplied beyond necessity. Those who give a completely deterministic view—Hobbes, Hume, and Schopenhauer—offer no continuum between cause and chance. Chance turns out to depend upon our ignorance of causes; further examination would show it causally determined.Watkins seems most sympathetic to Spinoza's treatment. In his hard-headed naturalism, Spinoza rejects any non-natural or supernatural accounts of human behavior. He is a strict determinist where man's confrontation with nature is concerned. Human emotions are passive and react to our environment; they do not affect it. Human reason makes no difference, but in its capacity to contemplate eternal truths it offers a way out. Instead of reacting to nature, man may transcend it. This transcendence is the key to Spinoza's view of the intellectual love of God, and human freedom. Watkins criticizes this flight from man's place in nature and the limited use of reason. In this vein, Spinoza seems to abandon the confrontation of nature for a flight to eternal truths. For Watkins, however, reasons must be used in confronting nature. As we saw, he claims that reason may have an effect upon the environment that is novel and unpredictable. Human agency is a part of the evolutionary story. Human agency is not an either/or proposition. There are degrees of effectiveness in human response to nature.In his analysis of human freedom, Watkins turns to the works of Newton, Faraday, Einstein, and others. In studying the problems posed by the consideration of natural phenomena during their time, they showed intellectual perspicacity of the highest order. Faraday's unified field theory brought electrical and magnetic phenomena, previously thought to be disparate, together. Newton's laws of motion brought celestial and terrestrial mechanics under one theory. Einstein changed the way in which space, time, mass, energy, and light were conceived. In a real sense, they were... (shrink)

Contributes a black Atlantic perspective to postmodernism, theology, and metaphysics.

In "Sider on Existence" (Noužs, 2007), David Liebesman and Matti Eklund argue that my "indeterminacy argument", according to which quantifiers are never vague, clashes with my "naturalness argument", according to which quantifiers "carve at the joints". There is, I argue, no outright inconsistency. But Liebesman and Eklund have shown that my arguments are not as independent as it may have appeared. The best defense of the indeterminacy argument is via the naturalness argument.

The kinds of real or natural kinds that support explanation and prediction in the social sciences are difficult to identify and track because they change through time, intersect with one another, and they do not always exhibit their properties when one encounters them. As a result, conceptual practices directed at these kinds will often refer in ways that are partial, equivocal, or redundant. To improve this epistemic situation, it is important to employ open-ended classificatory concepts, to understand when different research (...) programs are tracking the same real kind, and to maintain an ongoing commitment to interact causally with real kinds to focus reference on those kinds. A tempting view of these non-idealized epistemic conditions should be avoided: that they signal an ontological structure of the social world so plentiful that it would permit ameliorated classificatory schemes to achieve their normative aims regardless of whether they defer to real-kind classificatory schemes. To ground these discussions, the essay appeals to an overlooked convergence in the systematic naturalistic frameworks of Richard Boyd and Ruth Millikan. (shrink)

Brackets and tables, circles and maps, 1554-1872 -- Early botanical networks and trees, 1766-1815 -- The first evolutionary tree, 1786-1820 -- Diverse and unusual trees of the early nineteenth century, 1817-1834 -- The rule of five, 1819-1854 -- Pre-Darwinian branching diagrams, 1828-1858 -- Evolution and the trees of Charles Darwin, 1837-1868 -- The trees of Ernst Haeckel, 1866-1905 -- Post-Darwinian nonconformists, 1868-1896 -- More late-nineteenth-century trees, 1874-1897 -- Trees of the early twentieth century, 1901-1930 -- The trees of Alfred Sherwood (...) Romer, 1933-1966 -- Additional trees of the mid-twentieth century, 1931-1943 -- The trees of William King Gregory, 1938-1951 -- Hints of new approaches, 1954-1969 -- Phenograms and cladograms, 1958-1966 -- Early molecular trees, 1962-1987 -- Notable trees of the past four decades, 1970-2010 -- Primeval branches and universal trees of life, 1997-2010. (shrink)

Contains a collection of the essays of the great Viennese Hellenist Theodor Gomperz concerning the Herculanean Papyri of Epicurus and Philodemus. The Introduction presents a concise biography of Gomperz and a careful discussion of his work on Hellenistic philosophy. Das Buch enthalt eine Auswahl der _Herculanensia minora_ von Theodor Gomperz. Die kurze Einleitung erhebt sich nicht den Anspruch, ein systematisches und vollständiges Bild der herkulanensischen Studien von Gomperz zu zeichnen; sie soll lediglich zur Orientierung und Einführung in die Lektüre dienen.

There is growing support for the view that social categories like men and women refer to “objective types” (Haslanger 2000, 2006, 2012; Alcoff 2005). An objective type is a similarity class for which the axis of similarity is an objective rather than nominal or fictional property. Such types are independently real and causally relevant, yet their unity does not derive from an essential property. Given this tandem of features, it is not surprising why empirically-minded researchers interested in fighting oppression and (...) marginalization have found this ontological category so attractive: objective types have the ontological credentials to secure the reality (and thus political representation) of social categories, and yet they do not impose exclusionary essences that also naturalize and legitimize social inequalities. This essay argues that, from the perspective of these political goals of fighting oppression and marginalization, the category of objective types is in fact a Trojan horse; it looks like a gift, but it ends up creating trouble. I argue that objective type classifications often lack empirical adequacy, and as a result they lack political adequacy. I also provide, and in reference to the normative goals described above, several arguments for preferring a social ontology of natural kinds with historical essences. (shrink)

We show that every nontrivial interval in the recursively enumerable degrees contains an incomparable pair which have an infimum in the recursively enumerable degrees.

An historian and cultural critic explores the relationships between psychology, ecology, and new scientific insights into systems in nature. Drawing on our understanding of the evolutionary, self-organizing universe, Roszak discusses our rootedness in the greater web of life and explores the relationship between our own sanity and the larger-than-human world.

The paper discusses the active part in the process of perceiving, usually expressed by the Greek word krinein. It is argued that krinein in one of its uses means "to judge" in the sense of judging a case, i. e. deciding it. It is not used for making statements. A second meaning of the Greek word is that of discerning or discriminating, and it is this meaning that plays a central part in Aristotle's theory of perception.

There is broad agreement among social researchers and social ontologists that the project of dividing humans into social kinds should be guided by at least two methodological commitments. First, a commitment to what best serves moral and political interests, and second, a commitment to describing accurately the causal structures of social reality. However, researchers have not sufficiently analyzed how these two commitments interact and constrain one another. In the absence of that analysis, several confusions have set in, threatening to undermine (...) shared goals for the responsible modeling of social kinds of humans. This essay first explains the source and substance of these confusions. Then, by distinguishing different value-laden investigative questions into the classification of social kinds of humans, it sets out specific relations of dependence and constraint between empirically-driven investigations and value-driven investigations into social kinds of humans. The result is a more detailed and fruitful framework for thinking about the classification of social kinds that respects both normative interests and mind-independent causal regularities. (shrink)

In domains like stock brokerage, clinical psychiatry, and long‐term political forecasting, experts generally fail to outperform novices. Empirical researchers agree on why this is: experts must receive direct or environmental learning feedback during training to develop reliable expertise, and these domains are deficient in this type of feedback. A growing number of philosophers resource this consensus view to argue that, given the absence of direct or environmental philosophical feedback, we should not give the philosophical intuitions or theories of expert philosophers (...) greater credence than those of novice philosophers. This article has three objectives. The first is to explore several overlooked issues concerning the strategy of generalizing from empirical studies of non‐philosophical expertise to the epistemic status of philosophical expertise. The second is to explain why empirical research into a causal relationship between direct learning feedback and enhanced expert performance does not provide good grounds for abandoning a default optimism about the epistemic superiority of expert philosophical theories. The third is to sketch a positive characterization of learning feedback that addresses developmental concerns made salient by the empirical literature on expert performance for specifically theory‐driven or “armchair” domains like philosophy. (shrink)

The question as to what makes a perfect Aristotelian syllogism a perfect one has long been discussed by Aristotelian scholars. G. Patzig was the first to point the way to a correct answer: it is the evidence of the logical necessity that is the special feature of perfect syllogisms. Patzig moreover claimed that the evidence of a perfect syllogism can be seen for Barbara in the transitivity of the a-relation. However, this explanation would give Barbara a different status over the (...) other three first figure syllogisms. I argue that, taking into account the role of the being-contained-as-in-a-whole formulation, transitivity can be seen to be present in all four first figure syllogisms. Using this wording will put the negation sign with the predicate, similar to the notation in modern predicate calculus. (shrink)

We answer three questions: 1. Can we give a wholly mathematical explanation of a physical phenomenon? 2. Can we give a wholly mathematical explanation for a whole physical theory? 3. What is gained or lost in giving a wholly, or partially, mathematical explanation of a phenomenon or a scientific theory? To answer these questions we look at a project developed by Hajnal Andréka, Judit Madarász, István Németi and Gergely Székely. They, together with collaborators, present special relativity theory in a three-sorted (...) first-order formal language. (shrink)

Although popular music is increasingly recognized as an important area of inquiry in philosophy of art, many organizing principles have been taken over from other fields without scrutiny. This article selects heavy metal as an example of the value of applying philosophy of criticism to discourse about popular music. Metal is now in its fifth decade, and its combination of longevity and diversity have made it an attractive topic in popular music studies. In accounts of metal by musicologists and social (...) science researches, metal is almost always referred to as a genre of music, and the genre is said to unify diverse metal subcultures, which are associated with diverse styles of metal. An examination of the concepts of genre and subculture as they are typically understood in popular music studies generally, and in metal studies in particular, suggests that they do not function to delineate metal as a unitary category. However, contemporary philosophy of art provides several theoretical principles that provide criteria for identifying a unified genre. The relevant theories are historical definitions of art and Kendall Walton's account of style. Viewed in terms of these philosophical developments, metal should be seen as an evolving assemblage of related styles, only some of which are associated with fans who constitute a subculture. However, it is not a "category" in Walton's sense of that term. Video abstract. (shrink)

We prove that every countable relation on the enumeration degrees, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document}, is uniformly definable from parameters in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document}. Consequently, the first order theory of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document} is recursively isomorphic to the second order theory of arithmetic. By an effective version of coding lemma, we show that the first order (...) theory of the enumeration degrees of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Sigma^0_2$\end{document} sets is not decidable. (shrink)