It is well known that Gödel’s incompleteness theorems hold for ∑1-definable theories containing Peano arithmetic. We generalize Gödel’s incompleteness theorems for arithmetically definable theories. First, we prove that every ∑n+1-definable ∑n-sound theory is incomplete. Secondly, we generalize and improve Jeroslow and Hájek’s results. That is, we prove that every consistent theory having ∏n+1set of theorems has a true but unprovable ∏nsentence. Lastly, we prove that no ∑n+1-definable ∑n-sound theory can prove its own ∑n-soundness. These three results are (...) generalizations of Rosser’s improvement of the first incompleteness theorem, Gödel’s first incompleteness theorem, and the second incompleteness theorem, respectively. (shrink)

The book uses both environmental movements and political theory to help define what is meant by environmental and ecological justice. It will be attractive to anyone interested in environmental politics, environmental movements, and justice theory.

In Outline of a Theory of Truth, Kripke introduces many of the central concepts of the logical study of truth and paradox. He informally defines some of these—such as groundedness and paradoxicality—using modal locutions. We introduce a modal language for regimenting these informal definitions. Though groundedness and paradoxicality are expressible in the modal language, we prove that intrinsicality—which Kripke emphasizes but does not define modally—is not. This follows from a characterization of the modally definable sets and relations and an (...) attendant axiomatization of the modal semantics. (shrink)

A pointwise definable model is one in which every object is \loos definable without parameters. In a model of set theory, this property strengthens $V=\HOD$, but is not first-order expressible. Nevertheless, if \ZFC\ is consistent, then there are continuum many pointwise definable models of \ZFC. If there is a transitive model of \ZFC, then there are continuum many pointwise definable transitive models of \ZFC. What is more, every countable model of \ZFC\ has a class forcing extension that is pointwise (...) definable. Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters. (shrink)

In times of crisis, when current theories are revealed as inadequate to task, and new physics is thought to be required—physics turns to re-evaluate its principles, and to seek new ones. This paper explores the various types, and roles of principles that feature in the problem of quantum gravity as a current crisis in physics. I illustrate the diversity of the principles being appealed to, and show that principles serve in a variety of roles in all stages of the crisis, (...) including in motivating the need for a new theory, and defining what this theory should be like. In particular, I consider: the generalised correspondence principle, UV-completion, background independence, and the holographic principle. I also explore how the current crisis fits with Friedman’s view on the roles of principles in revolutionary theory-change, finding that while many key aspects of this view are not represented in quantum gravity (at the current stage), the view could potentially offer a useful diagnostic, and prescriptive strategy. This paper is intended to be relatively non-technical, and to bring some of the philosophical issues from the search for quantum gravity to a more general philosophical audience interested in the roles of principles in scientific theory-change. (shrink)

We study definable types in the theory of closed ordered differential fields. We show a condition for a type to be definable, then we prove that definable types are dense in the Stone space of CODF.

Numerous theories have attempted to overcome the anti-essentialist scepticism about the possibility of defining art. While significant advances have been made in this field, it seems that most modern definitions fail to successfully address the issue of the ever-changing nature of art raised by Morris Weitz, and rarely even attempt to provide an account which would be valid in more than just the modern Western context. This thesis looks at the most successful definitions currently defended, determines their strengths and weaknesses, (...) and offers a new, cultural definition which can preserve the good elements of other theories, solve or avoid their problems, and have a scope wide enough to account for art of different times and cultures. The resulting theory is a synthetic one in that it preserves the essential institutionalism of Dickie's institutional views, is inspired by the historical and functional determination of artistic phenomena present in Levinson's historicism and Beardsley's functionalism, and presents the reasons for something becoming art in a disjunctive form of Gaut's cluster account. Its strengths lie in the ability to account for the changing art-status of objects in various cultures and at various times, providing an explanation of not only what is or was art, but also how and why the concept 'art' changes historically and differs between cultures, and successfully balancing between the over-generalisations of ahistorical and universalist views, and the uninformativeness of relativism. More broadly, the cultural theory stresses the importance of treating art as a historical phenomenon embedded in particular social and cultural settings, and encourages cooperation with other disciplines such as anthropology and history of art. (shrink)

We continue the study of simple theories begun in [3] and [5]. We first find the right analogue of definability of types. We then develop the theory of generic types and stabilizers for groups definable in simple theories. The general ideology is that the role of formulas (or definability) in stable theories is replaced by partial types (or ∞-definability) in simple theories.

Energy is a crucial concept within classical and quantum physics. An essential tool to quantify energy is the Hamiltonian. Here, we consider how to define a Hamiltonian in general probabilistic theories—a framework in which quantum theory is a special case. We list desiderata which the definition should meet. For 3-dimensional systems, we provide a fully-defined recipe which satisfies these desiderata. We discuss the higher dimensional case where some freedom of choice is left remaining. We apply the definition to example (...) toy theories, and discuss how the quantum notion of time evolution as a phase between energy eigenstates generalises to other theories. (shrink)

The Joint Non-Trivialization Theorem, two Definability Theorems and the generalized Quantifier Elimination Theorem are proved for J 3-theories. These theories are three-valued with more than one distinguished truth-value, reflect certain aspects of model type logics and can. be paraconsistent. J 3-theories were introduced in the author's doctoral dissertation.

The paper provides a selective analysis of the main theories of trust and e-trust (that is, trust in digital environments) provided in the last twenty years, with the goal of preparing the ground for a new philosophical approach to solve the problems facing them. It is divided into two parts. The first part is functional toward the analysis of e-trust: it focuses on trust and its definition and foundation and describes the general background on which the analysis of e-trust rests. (...) The second part focuses on e-trust, its foundation and ethical implications. The paper ends by synthesising the analysis of the two parts. (shrink)

In his thesis 'Para uma Teoria Geral dos Homomorfismos' (1944) the Portuguese mathematician José Sebastião e Silva constructed an abstract or generalized Galois theory, that is intimately linked to F. Klein’s Erlangen Program and that foreshadows some notions and results of today’s model theory; an analogous theory was independently worked out by M. Krasner in 1938. In this paper, we present a version of the theory making use of tools which were not at Silva’s disposal. At (...) the same time, we tried to keep in mind, so much as possible, the gist of his standpoint. (shrink)

This book is a specialized monograph on interpolation and definability, a notion central in pure logic and with significant meaning and applicability in all areas where logic is applied, especially computer science, artificial intelligence, logic programming, philosophy of science and natural language. Suitable for researchers and graduate students in mathematics, computer science and philosophy, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism (second edition), J. M. Dunn and G. Hardegree's (...) Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning, P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2, and David J. Pym and Eike Ritter's Reductive Logic and Proof Search: Proof theory, semantics and control. (shrink)

We analyze the effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity. We find, for example, that the class of hereditarily ordinal algebraic sets is the same as the class of hereditarily ordinal definable sets; that is, $\mathrm{HOA}=\mathrm{HOD}$. Moreover, we show that every algebraic model of $\mathrm{ZF}$ is actually pointwise definable. Finally, we consider the implicitly constructible universe Imp—an algebraic analogue of the constructible universe—which is obtained by iteratively adding (...) not only the sets that are definable over what has been built so far, but also those that are algebraic over the existing structure. While we know that $\mathrm{Imp}$ can differ from $L$, the subtler properties of this new inner model are just now coming to light. Many questions remain open. (shrink)

Recent advances in the brain sciences have dramatically improved our understanding of brain function. As we find out more and more about what makes us tick, we must stop and consider the ethical implications of this new found knowledge. This ground-breaking book on the emerging field of neuroethics answers many pertinent questions, such as: What makes monitoring and manipulating the human brain so ethically challenging? Will having a new biology of the brain through imaging make us less responsible for our (...) behavior and lose our free will? Should certain brain scan studies be disallowed on the basis of moral grounds? A distinguished group of contributors from a wide range of fields make this an invaluable introduction to neuroethics. (shrink)

In How States Think, John Mearsheimer and Sebastian Rosato argue that expected-utility maximization is too subjective to serve as the basis for making rational decisions in the realm of national security. They claim that rationality in security studies should instead be defined by whether leaders conduct deliberative, theory-driven reasoning. This essay explains why Mearsheimer and Rosato’s critique of expected-utility theory is unpersuasive, and how their conception of theory-driven reasoning ignores key aspects of decision-making that national security officials (...) can feasibly address. Lyndon Johnson’s decision to take the United States to war in Vietnam provides a vivid example of how leaders can meet Mearsheimer and Rosato’s criteria for rationality despite clear shortfalls of analytic reasoning. (shrink)

In the philosophical debate over the epistemic status of conspiracy theories, the view that each theory ought to be judged on its own merits, ‘particularism’, has the upper hand. But challenges to this view continue to be put forth; this paper summarizes that debate and reaffirms the particularist perspective. In this paper, we address how different conceptions of what counts as a ‘conspiracy theory’ impact how one might evaluate particularism, with specific emphasis on (1) a ‘simple definition’ of (...) a conspiracy theory as a theory that posits a conspiracy, and (2) ‘contrarian conspiracy theories’ (those that contrast with an official account). (shrink)

This paper studies definability within the theory of institutions, a version of abstract model theory that emerged in computing science studies of software specification and semantics. We generalise the concept of definability to arbitrary logics, formalised as institutions, and we develop three general definability results. One generalises the classical Beth theorem by relying on the interpolation properties of the institution. Another relies on a meta Birkhoff axiomatizability property of the institution and constitutes a source for (...) many new actual definability results, including definability in (fragments of) classical model theory. The third one gives a set of sufficient conditions for 'borrowing' definability properties from another institution via an 'adequate' encoding between institutions. The power of our general definability results is illustrated with several applications to (many-sorted) classical model theory and partial algebra, leading for example to definability results for (quasi-)varieties of models or partial algebras. Many other applications are expected for the multitude of logical systems formalised as institutions from computing science and logic. (shrink)

Given an o-minimal structure M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal M}$$\end{document} with a group operation, we show that for a properly convex subset U, the theory of the expanded structure M′=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal M}'=$$\end{document} has definable Skolem functions precisely when M′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal M}'$$\end{document} is valuational. As a corollary, we get an elementary proof that the theory of any such M′\documentclass[12pt]{minimal} (...) \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal M}'$$\end{document} does not satisfy definable choice. (shrink)

We point out how the "Fundamental Theorem of Stability Theory", namely the equivalence between the "non order property" and definability of types, proved by Shelah in the 1970s, is in fact an immediate consequence of Grothendieck's "Criteres de compacite" from 1952. The familiar forms for the defining formulae then follow using Mazur's Lemma regarding weak convergence in Banach spaces.

The idea of a property's being supervenient on a class of properties is familiar from much philosophical literature. We give this idea a linguistic turn by converting it into the idea of a predicate symbol's being supervenient on a set of predicate symbols relative to a (first order) theory. What this means is that according to the theory, any individuals differing in respect to whether the given predicate applies to them also differ in respect to the application of (...) at least one of the predicates in the set. The latter relationship we show turns out to coincide with something antecedently familiar from work on definability: with what is called the piecewise (or modelwise) definability, in the theory in question, of the given predicate in terms of those in the set. (shrink)

We investigate expressiveness and definability issues with respect to minimal models, particularly in the scope of Circumscription. First, we give a proof of the failure of the Löwenheim-Skolem Theorem for Circumscription. Then we show that, if the class of P; Z-minimal models of a first-order sentence is Δ-elementary, then it is elementary. That is, whenever the circumscription of a first-order sentence is equivalent to a first-order theory, then it is equivalent to a finitely axiomatizable one. This means that (...) classes of models of circumscribed theories are either elementary or not Δ-elementary. Finally, using the previous result, we prove that, whenever a relation Pi is defined in the class of P; Z-minimal models of a first-order sentence Φ and whenever such class of P; Z-minimal models is Δ-elementary, then there is an explicit definition ψ for Pi such that the class of P; Z-minimal models of Φ is the class of models of Φ ∧ ψ. In order words, the circumscription of P in Φ with Z varied can be replaced by Φ plus this explicit definition ψ for Pi. (shrink)

The series is devoted to the publication of high-level monographs on all areas of mathematical logic and its applications. It is addressed to advanced students and research mathematicians, and may also serve as a guide for lectures and for seminars at the graduate level.

The present paper suggests relative truth definability as a tool for comparing conceptual aspects of axiomatic theories of truth and gives an overview of recent developments of axiomatic theories of truth in the light of it. We also show several new proof-theoretic results via relative truth definability including a complete answer to the conjecture raised by Feferman in [13].

We investigate the definability in monadic ∑11 and monadic Π11 of the problems REGk, of whether there is a regular subgraph of degree k in some given graph, and XREGk, of whether, for a given rooted graph, there is a regular subgraph of degree k in which the root has degree k, and their restrictions to graphs in which every vertex has degree at most k, namely REGkk and XREGkk, respectively, for k ≥ 2 . Our motivation partly stems (...) from the fact that REGkk and XREGkk are logspace equivalent to CONN and REACH, respectively, for k ≥ 3, where CONN is the problem of whether a given graph is connected and REACH is the problem of whether a given graph has a path joining two given vertices. We use monadic first - order reductions, monadic ∑11 games and a recent technique due to Fagin, Stockmeyer and Vardi to almost completely classify whether these problems are definable in monadic ∑11 and monadic Π11, and we compare the definability of these problems. (shrink)

The bounded arithmetic theories R2i, S2i, and T2i are closely connected with complexity theory. This paper is motivated by the questions: what are the Σi+1b-definable multifunctions of R2i? and when is one theory conservative over another? To answer these questions we consider theories , and where induction is restricted to prenex formulas. We also define which has induction up to the 0 or 1-ary L2-terms in the set τ. We show and and for . We show that the (...) -multifunctions of are FPΣip and that those of are FPΣip. For -definability we get FPΣi+k+1p for all these theories. Write for the set of terms 2min,t) where ℓ is a finite product of terms in τ and tL2. We prove and we show - INDτ provided τO2. This gives a proof theoretic proof that S2iΔi+1b-LIND and -LLIND solving an open problem. For τO2, we define using weak replacement axioms and show . We show if or if or if where τ′ has an unbounded term then PH=B. We separate PΣip from PΣip for behaved ℓ and deduce theory separations. We lastly introduce a notion of a model separating two theories and derive some consequences. (shrink)

A new games model of the language FPC, a type theory with products, sums, function spaces and recursive types, is described. A definability result is proved, showing that every finite element of the model is the interpretation of some term of the language.

Ever since the early decades of this century, there have emerged a number of competing schools of ecology that have attempted to weave the concepts underlying natural resource management and natural-historical traditions into a formal theoretical framework. It was widely believed that the discovery of the fundamental mechanisms underlying ecological phenomena would allow ecologists to articulate mathematically rigorous statements whose validity was not predicated on contingent factors. The formulation of such statements would elevate ecology to the standing of a rigorous (...) scientific discipline on a par with physics. However, there was no agreement as to the fundamental units of ecology. Systems ecologists sought to identify the fundamental organization that tied the physical and biological components of ecosystems into an irreducible unit: the ecosystem was their fundamental unit. Population ecologists sought, instead, to identify the biological mechanisms regulating the abundance and distribution of plant and animal species: to these ecologists, the individual organism was the fundamental unit of ecology, and the physical environment was nothing more than a stage upon which the play of individuals in perennial competition took place. As Joel Hagen has pointed out, the two schools were thus dividied by fundamentally different and irreconcilable assumptions about the nature of ecosystems.Notwithstanding these divisive efforts to elevate the image of ecology, the discipline remained in the shadows of American academia until the mid-1960s, when systems ecologists succeeded in projecting ecology onto the national scene. They did so by seeking closer involvement with practical problems: they argued before Congress that their approach to the theoretical problems of ecology was uniquely suited to the solution of the impending “environmental crisis.” With the establishment of the International Biological Program, they succeeded in attracting unprecedented levels of funding for systems ecology research. Theoretical population ecologists, on the other hand, found themselves consigned to the outer regions of this new institutional landscape. The systems ecologists' successful capture of the limelight and the purse brought the divisions between them and population ecologists into sharper relief — hence the hardening of the division of ecology observed by Hagen.45I have argued that the population biologist Richard Levins, prompted by these institutional developments, sought to challenge the social position of systems ecology, and to assert the intellectual priority of theoretical population ecology. He attempted to do so by articulating a nontrivial and rather carefully thought out classification of ecological models that led to the disqualification of systems analysis as a legitimate approach to the study of ecological phenomena. I have suggested that — ultimately —Levins's case against systems analysis in ecology rested on the view that an aspiration to realism and prediction was incompatible with an interest in theoretical issues, a concern that he equated with the search for generality. He sought to reinforce this argument by exploiting the fact that systems ecologists had staked their future on the provision of technical solutions to the problems of the “environmental crisis”: he associated systems ecologists' aspiration to realism and precision with a concern for practical issues, trading on the widely accepted view that practical imperatives are incompatible with the aims of scientific inquiry.46 These are plausible, but nonetheless questionable, claims which have now become an integral part of ecological knowledge. And finally, I hope to have shown how even the most abstract levels of scientific argument are shaped by political considerations, and how discussions of the conceptual development of modern ecology might benefit from a greater consideration of its historical and social dimensions.47. (shrink)

Combining two results from machine learning theory we prove that a formula is NIP if and only if it satisfies uniform definability of types over finite sets. This settles a conjecture of La...

After an introduction which demonstrates the failure of the equational analogue of Beth’s definability theorem, the first two sections of this paper are devoted to an elementary exposition of a proof that a functional constant is equationally definable in an equational theory iff every model of the set of those consequences of the theory that do not contain the functional constant is uniquely extendible to a model of the theory itself.Sections three, four and five are devoted (...) to applications and extensions of this result.Topics considered here include equational definability in first order logic, an extended notion of definability in equational logic and the synonymy of equational theories.The final two sections briefly review some of the history of equational logic. (shrink)

In this paper I am going to inquire to what extent the main requirements of a minimalist theory of truth and falsity (as formulated, for example, by Horwich and Field) can be consistently implemented in a formal theory. I will discuss several of the existing logical theories of truth, including Tarski-type (un)definability results, Kripke’s partial interpretation of truth and falsity, Barwise and Moss’ theory based upon non-well-founded sets, McGee’s treatment of truth as a vague predicate, and (...) Hintikka’s languages of imperfect information, to see which axioms of the minimalist theory they satisfy or fail to satisfy. Finally, I will discuss the relation between the minimalist program and compositionality. (shrink)

Let be a relational system whereby D is a nonempty set and P1 is an m1-ary relation on D. With we associate the (weak) monadic second-order theory consisting of the first-order predicate calculus with individual variables ranging over D; monadic predicate variables ranging over (finite) subsets of D; monadic predicate quantifiers; and constants corresponding to P1, P2, …. We will often use ambiguously to mean also the set of true sentences of.

We give two theories, Th1 and Th2, which are explicitly definable over each other (i.e. the relation symbols of one theory are explicitly definable in the other, and vice versa), but are not definitionally equivalent. The languages of the two theories are disjoint. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim).

Although the theory of definability had many important antecedents—such as the descriptive set theory initiated by the French semi-intuitionists in the early 1900s—the main ideas were first laid out in precise mathematical terms by Alfred Tarski beginning in 1929. We review here the basic notions of languages, explicit definability, and grammatical complexity, and emphasize common themes in the theories of definability for four important languages underlying, respectively, descriptive set theory, recursive function theory, classical (...) pure logic, and finite-universe logic. We review the history of previous studies of the similarities and differences in the theories of definability of the first three of these four languages. A seminal role leading toward unification of the theories has been played by the separation principles introduced by Nikolai Luzin in 1927. Emphasizing analogies and driving toward further unification embracing finite-universe logic we concentrate on a simple example—the first and second separation principles for existential-universal first-order sentences. Using this as a test case for the fundamental problem of how to “finitize” arguments in classical pure logic to the finite-universe case, we are led to the analogous negative solution by using the theory of certain special graphs: a graph is - special for any positive integers m, n, p, q iff it is bipartite with m red points and n blue points and for every p -tuple of red points there is a blue point to which they are all connected. As an aside we introduce for further study a natural “Ramseyesque” increasing sequence A of positive integers, where A is the least positive integer n for which an -special graph exists. (shrink)

The Defining Issues Test (DIT) has been the dominant measure of moral development. The DIT has its roots in Kohlberg’s original stage theory of moral judgment development and asks respondents to rank a set of stage typed statements in order of importance on six stories. However, the question to what extent the DIT-data match the underlying stage model was never addressed with a statistical model. Therefore, we applied item response theory (IRT) to a large data set (55,319 cases). (...) We found that the ordering of the stages as extracted from the raw data fitted the ordering in the underlying stage model good. Furthermore, difficulty differences of stages across the stories were found and their magnitude and location were visualized. These findings are compatible with the notion of one latent moral developmental dimension and lend support to the hundreds of studies that have used the DIT-1 and by implication support the renewed DIT-2. (shrink)

We prove that the two-variable fragment of first-order logic has the weak Beth definability property. This makes the two-variable fragment a natural logic separating the weak and the strong Beth properties since it does not have the strong Beth definability property.

We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst nonforking types.