The first-order theory of the lattice of recursively enumerable closed subsets of an effective topological space is proved undecidable using the undecidability of the first-order theory of the lattice of recursively enumerable sets. In particular, the first-order theory of the lattice of recursively enumerable closed subsets of Euclidean n -space, for all n , is undecidable. A more direct proof of the undecidability of the lattice of recursively enumerable closed subsets of Euclidean n -space, n ⩾ 2, is provided using (...) the method of reduction and the recursive inseparability of the set of all formulae satisfiable in every model of the theory of SIBs and the set of all formulae refutable in some finite model of the theory of SIBs. (shrink)

This paper is a continuation of the authors' work , where the main problem considered was whether a given recursive structure is recursively isomorphic to a polynomial-time structure. In that paper, a recursive Abelian group was constructed which is not recursively isomorphic to any polynomial-time Abelian group. We now show that if every element of a recursive Abelian group has finite order, then the group is recursively isomorphic to a polynomial-time group. Furthermore, if the orders are bounded, then the group (...) is recursively isomorphic to a polynomial-time group with universe A being the set of tally representations of natural numbers Tal = s{;1s};* or the set of binary representations of the natural numbers Bin. We also construct a recursive Abelian group with all elements of finite order but which has elements of arbitrary large finite order which is not isomorphic to any polynomial-time group with universe Tal or Bin. Similar results are obtained for structures , where f is a permutation on the set A. (shrink)

The central problem considered in this paper is whether a given recursive structure is recursively isomorphic to a polynomial-time structure. Positive results are obtained for all relational structures, for all Boolean algebras and for the natural numbers with addition, multiplication and the unary function 2x. Counterexamples are constructed for recursive structures with one unary function and for Abelian groups and also for relational structures when the universe of the structure is fixed. Results are also given which distinguish primitive recursive structures, (...) exponential-time structures and structures with honest witnesses. (shrink)

A Π01 class is an effectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of the members of a class P. Given an effective enumeration {Pe:e < ω} of the Π01 classes, the index set I for a certain property is the set of indices e such that Pe has the property. For example, the index set of binary Π01 classes of positive measure is Σ02 complete. (...) Various notions of boundedness are discussed and classified. For example, the index set of the recursively bounded classes is Σ03 complete and the index set of the recursively bounded classes which have infinitely many recursive members is Π04 complete. Consideration of the Cantor-Bendixson derivative leads to index sets in the transfinite levels of the hyperarithmetic hierarchy. (shrink)

An ω-language is a set of infinite sequences on a countable language, and corresponds to a set of real numbers in a natural way. Languages may be described by logical formulas in the arithmetical hierarchy and also may be described as the set of words accepted by some type of automata or Turing machine. Certain families of languages, such as the equation image languages, may enumerated as P0, P1, … and then an index set associated to a given property R (...) of languages is just the set of e such that Pe has the property. The complexity of index sets for 7 types of languages is determined for various properties related to the size of the language. (shrink)

In this paper, we study the lattice of r.e. subspaces of a recursively presented vector space V ∞ with regard to the various complexity-theoretic speed-up properties such as speedable, effectively speedable, levelable, and effectively levelable introduced by Blum and Marques. In particular, we study the interplay between an r.e. basis A for a subspace V of V ∞ and V with regard to these properties. We show for example that if A or V is speedable , then V is levelable (...) . We also show that if A is an r.e. subset of a recursive basis for V ∞ , then A is levelable iff V is speedable while it is not the case that A is levelable iff V is speedable. We also provide several contrasts between the lattice of r.e. subspaces and the lattice of r.e. sets with respect to these speed-up properties. For example, every maximal set is levelable but we show that there exist supermaximal spaces which are nonspeedable in all possible r.e. degrees as well as supermaximal spaces which are levelable in all r.e. degrees which contain levelable sets. (shrink)

We investigate the complexity of finding solutions to infinite recursive constraint satisfaction problems. We show that, in general, the problem of finding a solution to an infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through a recursive tree. We also identify natural classes of infinite recursive constraint satisfaction problems where the problem of finding a solution to the infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through (...) finitely branching recursive trees or recursive binary trees. There are a large number of results in the literature on the complexity of the problem of finding an infinite path through a recursive tree. Our main result allows us to automatically transfer such results to give equivalent results about the complexity of the problem of finding a solution to a recursive constraint satisfaction problem. (shrink)

The problem of when a recursive graph has a recursive k-coloring has been extensively studied by Bean, Schmerl, Kierstead, Remmel, and others. In this paper, we study the polynomial time analogue of that problem. We develop a number of negative and positive results about colorings of polynomial time graphs. For example, we show that for any recursive graph G and for any k, there is a polynomial time graph G′ whose vertex set is {0,1}* such that there is an (...) effective degree preserving correspondence between the set of k-colorings of G and the set of k-colorings of G′ and hence there are many examples of k-colorable polynomial time graphs with no recursive k-colorings. Moreover, even though every connected 2-colorable recursive graph is recursively 2-colorable, there are connected 2-colorable polynomial time graphs which have no primitive recursive 2-coloring. We also give some sufficient conditions which will guarantee that a polynomial time graph has a polynomial time or exponential time coloring. (shrink)

We give resource bounded versions of the Completeness Theorem for propositional and predicate logic. For example, it is well known that every computable consistent propositional theory has a computable complete consistent extension. We show that, when length is measured relative to the binary representation of natural numbers and formulas, every polynomial time decidable propositional theory has an exponential time (EXPTIME) complete consistent extension whereas there is a nondeterministic polynomial time (NP) decidable theory which has no polynomial time complete consistent extension (...) when length is measured relative to the binary representation of natural numbers and formulas. It is well known that a propositional theory is axiomatizable (respectively decidable) if and only if it may be represented as the set of infinite paths through a computable tree (respectively a computable tree with no dead ends). We show that any polynomial time decidable theory may be represented as the set of paths through a polynomial time decidable tree. On the other hand, the statement that every polynomial time decidable, relative to the tally representation of natural numbers and formulas, is equivalent to P = NP. For predicate logic, we develop a complexity theoretic version of the Henkin construction to prove a complexity theoretic version of the Completeness Theorem. Our results imply that that any polynomial space decidable theory △ possesses a polynomial space computable model which is exponential space decidable and thus △ has an exponential space complete consistent extension. Similar results are obtained for other notions of complexity. (shrink)

This paper discusses an extension of Answer Set Programming called Hybrid Answer Set Programming which allows the user to reason about dynamical systems that exhibit both discrete and continuous aspects. The unique feature of Hybrid ASP is that it allows the use of ASP type rules as controls for when to apply algorithms to advance the system to the next position. That is, if the prerequisites of a rule are satisfied and the constraints of the rule are not violated, then (...) the algorithm associated with the rule is invoked. (shrink)

We examine the effective categoricity of equivalence structures via Ershov's difference hierarchy. We explore various kinds of categoricity available by distinguishing three different notions of isomorphism available in this hierarchy. We prove several results relating our notions of categoricity to computable equivalence relations: for example, we show that, for such relations, computable categoricity is equivalent to our notion of weak ω-c.e. categoricity, and that $\Delta _2^0 $ -categoricity is equivalent to our notion of graph-ω-c.e. categoricity.

We study computability theoretic properties of and equivalence structures and how they differ from computable equivalence structures or equivalence structures that belong to the Ershov difference hierarchy. Our investigation includes the complexity of isomorphisms between equivalence structures and between equivalence structures.

A computable graph is constructed which is not computably isomorphic to any polynomial-time graph with a standard universe . Conditions are given under which a computable graph is computably isomorphic to a polynomial-time graph with a standard universe — for example, if every vertex has finite degree. Two special types of graphs are studied. It is shown that any computable tree is recursively isomorphic to a p-time tree with standard universe and that any computable equivalence relation is computably isomorphic to (...) a p-time equivalence relation with standard universe. (shrink)

Index sets are used to measure the complexity of properties associated with the differentiability of real functions and the existence of solutions to certain classic differential equations. The new notion of a locally computable real function is introduced and provides several examples of Σ04 complete sets.

We investigate notions of randomness in the space ${{\mathcal C}(2^{\mathbb N})}$ of continuous functions on ${2^{\mathbb N}}$ . A probability measure is given and a version of the Martin-Löf test for randomness is defined. Random ${\Delta^0_2}$ continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any ${y \in 2^{\mathbb N}}$ , (...) there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set. (shrink)

We develop a theory of LOGSPACE structures and apply it to construct a number of examples of Abelian Groups which have LOGSPACE presentations. We show that all computable torsion Abelian groups have LOGSPACE presentations and we show that the groups ${\mathbb {Z}, Z(p^{\infty})}$ , and the additive group of the rationals have LOGSPACE presentations over a standard universe such as the tally representation and the binary representation of the natural numbers. We also study the effective categoricity of such groups. For (...) example, we give conditions are given under which two isomorphic LOGSPACE structures will have a linear space isomorphism. (shrink)

Jeffrey E. Brower presents and explains the hylomorphic conception of the material world developed by Thomas Aquinas, according to which material objects are composed of both matter and form. In addition to presenting and explaining Aquinas's views, Brower seeks wherever possible to bring them into dialogue with the best recent literature on related topics. Along the way, he highlights the contribution that Aquinas's views make to a host of contemporary metaphysical debates, including the nature of change, composition, material constitution, (...) the ontology of stuff vs. things, the proper analysis of ordinary objects, the truthmakers for essential vs. accidental predication, and the metaphysics of property possession. (shrink)

Closest-possible-world analyses of counterfactuals suffer from what has been called the ‘problem of counterpossibles’: some counterfactuals with metaphysically impossible antecedents seem plainly false, but the proposed analyses imply that they are all (vacuously) true. One alleged solution to this problem is the addition of impossible worlds. In this paper, I argue that the closest possible or impossible world analyses that have recently been suggested suffer from the ‘new problem of counterpossibles’: the proposed analyses imply that some plainly true counterpossibles (viz., (...) ‘counterlogicals’) are false. After motivating and presenting the ‘new problem’, I give reasons to think that the most plausible objection to my argument is not compelling. (shrink)

This is annotated bibliography of the field of business ethics. It identifies and summarizes useful journals, textbooks, anthologies, and articles.

What can we know about God by reason alone? Philosophical theology is the attempt to obtain such knowledge. An ancient tradition, which is perhaps more influential now than ever, tries to derive the attributes of God from the principle that God is the greatest possible being. Jeff Speaks argues that that constructive project is a failure. He also argues that the related view that the concept of God is the concept of a greatest possible being is a mistake. In the (...) last chapter, he sketches an alternative path forward. (shrink)

This Element has two aims. The first is to discuss arguments philosophers have made about the difference God's existence might make to questions of general interest in metaethics. The second is to argue that it is a mistake to think we can get very far in answering these questions by assuming a thin conception of God, and to suggest that exploring the implications of thick theisms for metaethics would be more fruitful.

This article suggests an iconic turn in cultural sociology. Icons can be seen, it is argued, as symbolic condensations that root social meanings in material form, allowing the abstractions of cognition and morality to be subsumed, to be made invisible, by aesthetic shape. Meaning is made iconically visible, in other words, by the beautiful, sublime, ugly, or simply by the mundane materiality of everyday life. But it is via the senses that iconic power is made. This new approach to meaning (...) is compared with others — with materialism, semiotics, aestheticism, moralism, realism, and spiritualism. (shrink)

Preface : plant theory? -- The first birth of biopower : from plant to animal life in Foucault -- Thinking plants, with Aristotle and Heidegger -- Animal and plant, life and world in Derrida, or, The plant and the sovereign -- From the world to the territory : vegetable life in Deleuze and Guattari, or, What is a rhizome? -- Coda : what difference does it make?

Jeffrey Bloechl traces the evolution of Levinas's thought to argue that his conception of God is dependent on his existential phenomenology.

Justice requires giving people what they deserve. Or so many philosophers – and according to many of those philosophers, everyone else – thought for centuries. In the 1970’s and 1980’s, however, perhaps under the influence of Rawls’s (1971) desert-less theory, desert was largely cast out of discussions of distributive justice. Now it is making a comeback. In this chapter I consider recent research on the concept of desert, arguments for its requital, and connections between desert and other distributive ideals. I (...) suggest that desert-sensitive theories of distributive justice, though they face obstacles, have a promising future. (shrink)

Recently in this journal, Jessica Morley and colleagues (AI & SOC 2023 38:411–423) review AI ethics and education, suggesting that a cultural shift is necessary in order to prepare students for their responsibilities in developing technology infrastructure that should shape ways of life for many generations. Current AI ethics guidelines are abstract and difficult to implement as practical moral concerns proliferate. They call for improvements in ethics course design, focusing on real-world cases and perspective-taking tools to immerse students in challenging (...) situations with relatable personal, social and environmental implications. The present paper considers the ACTWith model of moral cognition for use in such training contexts. Morley and colleagues' paper and contemporary literature are reviewed. Whistleblowing as a source for relatable case studies is discussed, and the purpose of ethics education considered. The ACTWith model is introduced, and correlations with contemporaries noted. The ACTWith model has been successfully employed in a broad range of technology ethics courses in the Netherlands and South Korea. Illustrations are offered. Correlations with contemporary efforts are drawn in the context of software engineering. Summary discussion considers strengths and limitations of the ACTWith program. Overall, the paper emphasizes training high-level moral reasoning through guided inquiry education informing lifetime professional practice, and indicates an outstanding need for teachers able to do the job. (shrink)

Soren Kierkegaard's Fear and Trembling is one of the most widely read works of Continental philosophy and the philosophy of religion. While several commentaries and critical editions exist, Jeffrey Hanson offers a distinctive approach to this crucial text. Hanson gives equal weight and attention to all three of Kierkegaard’s "problems," dealing with Fear and Trembling as part of the entire corpus of Kierkegaard's production and putting all parts into relation with each other. Additionally, he offers a distinctive analysis of (...) the Abraham story and other biblical texts, giving particular attention to questions of poetics, language, and philosophy, especially as each relates to the aesthetic, the ethical, and the religious. Presented in a thoughtful, well-informed, and fresh manner, Hanson’s claims are original and edifying. This new reading of Kierkegaard will stimulate fruitful dialogue on well-traveled philosophical ground. (shrink)

We prove several results about groups of finite Morley rank without unipotent p-torsion: p-torsion always occurs inside tori, Sylow p-subgroups are conjugate, and p is not the minimal prime divisor of our approximation to the "Weyl group". These results are quickly finding extensive applications within the classification project.

Introduction -- 1. Problem of the New -- 2. Problem of Relations -- 3. Problem of Emergence -- 4. Problem of One and Many -- 5. Plato and the Third Man Argument -- 6. Bradley and the Problem of Relations -- 7. Moore, Russell and the Birth of Analytic Philosophy -- 8. Russell and Deleuze on Leibniz -- 9. On Problematic Fields -- 10. Kant and Problematic Ideas -- 11. Armstrong and Lewis on the Problem of One and Many -- (...) 12. Determinables and Determinates -- 13. The Limits of Representational Thought -- 14. Learning from a Cup of Coffee -- 15. Carnap and the Fate of Metaphysics -- 16. Truth and Relevance -- Conclusion. (shrink)

The works of Hannah Arendt and Albert Camus--two of the most compelling political thinkers of the "resistance generation" that lived through World War II--can still provide penetrating insights for contemporary political reflection. Jeffrey C. Isaac offers new interpretations of these writers, viewing both as engaged intellectuals who grappled with the possibilities of political radicalism in a world in which liberalism and Marxism had revealed their inadequacy by being complicit in the rise of totalitarianism. According to Isaac, self-styled postmodern writers (...) who proclaim the death of grandiose ideologies often fail to recognize that such thinkers as Camus and Arendt had already noted this. But unlike many postmodernists, these two sought to preserve what was worthy in modern humanism--the idea of a common human condition and a commitment to human rights and the dignity of individuals. Isaac shows that both writers advanced the idea of a democratic civil society made up of self-limiting groups. Although they criticized the typical institutions of mass democratic politics, they endorsed alternative forms of local and international organization that defy the principle of state sovereignty. Isaac also shows how Arendt's writings on the Middle East, and Camus's on Algeria, urged the creation of such institutions. The vision of a "rebellious politics" that Arendt and Camus shared is of great relevance to current debates in democratic theory and to the transformations taking place in Europe and the states of the former Soviet Union. (shrink)

Packed with examples, this book offers a clear and engaging overview of ethical issues in business. -/- It begins with a discussion of foundational issues, including the objectivity of ethics, the content of ethical theories, and the debate between capitalism and socialism, making it suitable for the beginning student. It then examines ethical issues in business in three broad areas. The first is the market. Issues explored are what can be sold (the limits of markets) and how it can be (...) sold (ethics in marketing). The second is work. Topics in this area are health and safety, meaningful work, compensation, hiring and firing, privacy, and whistleblowing. The third area is the firm in society. Here readers explore corporate social responsibility, corporate political activity, and the set of ethical challenges that attend international business. -/- Issues are introduced through real-world examples that underscore their importance and make them come alive. Arguments for opposing positions are given fair hearings and students are encouraged to develop and defend their own views. (shrink)

I elucidate a frame of mind that David Wiggins callsrespect for nature, which he understands as a special attitude toward asui generisobject, Natureas such. A person with this frame of mind takes nature to impose defeasible limits on her action, so that there are some courses of action that she will refuse even to entertain, except in circumstances of dire exigency. I defend the reasonableness of respect for nature, drawing upon considerations in Wiggins's work. But I argue that the natural (...) systems that comprise the proper object of respect for nature are notsui generis; they are kindred, for practical reason, to complex social, political, and economic systems that we inhabit. I argue that it is reasonable to treat all such valuable systems with a similar respect, and that this respect is continuous with the respect we owe to persons and to valuable objects more generally. In all of these cases, respect consists, in part, in a disposition to defeasible constraints on practical deliberation. (shrink)

The present paper shows how one might model Everettian quantum mechanics using hyperfinitely many worlds. A hyperfinite model allows one to consider idealized measurements of observables with continuous-valued spectra where different outcomes are associated with possibly infinitesimal probabilities. One can also prove hyperfinite formulations of Everett’s limiting relative-frequency and randomness properties, theorems he considered central to his formulation of quantum mechanics. Finally, this model provides an intuitive framework in which to consider no-collapse formulations of quantum mechanics more generally.

The present paper shows how one might model Everettian quantum mechanics using hyperfinitely many worlds. A hyperfinite model allows one to consider idealized measurements of observables with continuous-valued spectra where different outcomes are associated with possibly infinitesimal probabilities. One can also prove hyperfinite formulations of Everett’s limiting relative-frequency and randomness properties, theorems he considered central to his formulation of quantum mechanics. Finally, this model provides an intuitive framework in which to consider no-collapse formulations of quantum mechanics more generally.

Until the 1970s the history of sexuality was a marginalized practice. Today it is a flourishing field, increasingly integrated into the mainstream and producing innovative insights into the ways in which societies shape and are shaped by sexual values, norms, identities and desires. In this book, Jeffrey Weeks, one of the leading international scholars in the subject, sets out clearly and concisely how sexual history has developed, and its implications for our understanding of the ways we live today. The (...) emergence of a new wave of feminism and lesbian and gay activism in the 1970s transformed the subject, heavily influenced by new trends in social and cultural history, radical sociological insights and the impact of Michel Foucault’s work. The result was an increasing emphasis on the historical shaping of sexuality, and on the existence of many different sexual meanings and cultures on a global scale. With chapters on, amongst others, lesbian, gay and queer history, feminist sexual history, the mainstreaming of sexual history, and the globalization of sexual history, What is Sexual History? is an indispensable guide to these developments. (shrink)