Schopenhauer is acknowledged as “the philosopher of pessimism” and author of a system that teaches how art and morality can help humans navigate life in “the worst of all possible worlds.” This dominant image has cut off an important branch of Schopenhauer’s tree of philosophy—metaphysics of nature and its constant dialogue with the sciences of the time. Beginning with a reappraisal of Schopenhauer’s system as a whole—which he defined as a “single thought”—this book interprets his metaphysics as a knowledge that (...) brings value and meaning to our lives. More specifically, Schopenhauer’s writings show that he saw science as adding value to metaphysical knowledge. Understanding Schopenhauer’s metaphysics therefore requires both an understanding of the dialogue he maintained with the natural sciences and an appreciation of the role of the natural sciences in his philosophical project. And exploring the relationship between science and metaphysics provides a new perspective on his metaphysics and its development between the first edition of The World as Will and Representation (1819) and 1844, when Schopenhauer added a second volume of Supplements to his work. In 1819, he valued Schelling’s philosophy of nature as a model for establishing a link between metaphysical speculation and science. In the years that followed, he assessed the implications for metaphysics of the transformation of scientific research into a practice increasingly independent of philosophy. Such an assessment also affected his metaphysics, and even the will and Ideas underwent a process of revision. (shrink)

Let T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that the residue field of such a convex hull has a natural expansion to a model of T. We give a quantifier elimination relative to T for the theory of pairs (R, V) where $\mathscr{R} \models T$ and V ≠ R is the convex hull of an elementary substructure of R. We deduce (...) that the theory of such pairs is complete and weakly o-minimal. We also give a quantifier elimination relative to T for the theory of pairs (R, N) with R a model of T and N a proper elementary substructure that is Dedekind complete in R. We deduce that the theory of such "tame" pairs is complete. (shrink)

This paper introduces a logical analysis of convex combinations within the framework of Łukasiewicz real-valued logic. This provides a natural link between the fields of many-valued logics and decision theory under uncertainty, where the notion of convexity plays a central role. We set out to explore such a link by defining convex operators on MV-algebras, which are the equivalent algebraic semantics of Łukasiewicz logic. This gives us a formal language to reason about the expected value of bounded random variables. (...) As an illustration of the applicability of our framework we present a logical version of the Anscombe–Aumann representation result. (shrink)

Convexity predicates and the convex hull operator continue to play an important role in theories of spatial representation and reasoning, yet their first-order axiomatization is still a matter of controversy. In this paper, we present a new approach to adding convexity to mereotopological theory with boundary elements by specifying first-order axioms for a binary segment operator s. We show that our axioms yields a convex hull operator h that supports, not only the basic properties of convex regions, but (...) also complex properties concerning region alignment. We also argue that h is stronger than convex hull operators from existing axiomatizations and show how to derive the latter from our axioms for s. (shrink)

Natural languages vary in their quantity expressions, but the variation seems to be constrained by general properties, so-calleduniversals. Their explanations have been sought among constraints of human cognition, communication, complexity, and pragmatics. In this article, we apply a state-of-the-art language coordination model to the semantic domain of quantities to examine whether two quantity universals—monotonicity and convexity—arise as a result of coordination. Assuming precise number perception by the agents, we evolve communicatively usable quantity terminologies in two separate conditions: a numeric-based (...) condition in which agents communicate about a number of objects and a quotient-based condition in which agents communicate about the proportions. We find out that both universals take off in all conditions but only convexity almost entirely dominates the emergent languages. Additionally, we examine whether the perceptual constraints of the agents can contribute to the further development of universals. We compare the degrees of convexity and monotonicity of languages evolving in populations of agents with precise and approximate number sense. The results suggest that approximate number sense significantly reinforces monotonicity and leads to further enhancement of convexity. Last but not least, we show that the properties of the evolved quantifiers match certain invariance properties from generalized quantifier theory. (shrink)

We propose a natural definition of what it means in a constructive context for a Banach space to be reflexive, and then prove a constructive counterpart of the Milman-Pettis theorem that uniformly convex Banach spaces are reflexive.

The notion of conceptual space, proposed by Gärdenfors as a framework for the representation of concepts and knowledge, has been highly influential over the last decade or so. One of the main theses involved in this approach is that the conceptual regions associated with properties, concepts, verbs, etc. are convex. The aim of this paper is to show that such a constraint—that of the convexity of the geometry of conceptual regions—is problematic; both from a theoretical perspective and with regard (...) to the inner workings of the theory itself. On the one hand, all the arguments provided in favor of the convexity of conceptual regions rest on controversial assumptions. Additionally, his argument in support of a Euclidean metric, based on the integral character of conceptual dimensions, is weak, and under non-Euclidean metrics the structure of regions may be non-convex. Furthermore, even if the metric were Euclidean, the convexity constraint could be not satisfied if concepts were differentially weighted. On the other hand, Gärdenfors’ convexity constraint is brought into question by the own inner workings of conceptual spaces because: some of the allegedly convex properties of concepts are not convex; the conceptual regions resulting from the combination of convex properties can be non-convex; convex regions may co-vary in non-convex ways; and his definition of verbs is incompatible with a definition of properties in terms of convex regions. Therefore, the mandatory character of the convexity requirement for regions in a conceptual space theory should be reconsidered. (shrink)

I solve here some problems left open in “T-convexity and Tame Extensions” [9]. Familiarity with [9] is assumed, and I will freely use its notations. In particular,Twill denote a completeo-minimal theory extending RCF, the theory of real closed fields. Let (,V) ⊨Tconvex, let=V/m(V)be the residue field, with residue class mapx↦:V↦, and let υ:→ Γ be the associated valuation. “Definable” will mean “definable with parameters”.The main goal of this article is to determine the structure induced by(,V)on its residue fieldand on (...) its value group Γ. In [9] we expanded the ordered fieldto a model ofTas follows. Take a tame elementary substructure′ ofsuch thatR′ ⊆VandR′maps bijectively ontounder the residue class map, and make this bijection into an isomorphism′ ≌ofT-models. (We showed such′ exists, and that this gives an expansion ofto aT-model that is independent of the choice of′.). (shrink)

I solve here some problems left open in “T-convexity and Tame Extensions” [9]. Familiarity with [9] is assumed, and I will freely use its notations. In particular,Twill denote a completeo-minimal theory extending RCF, the theory of real closed fields. Let (,V) ⊨Tconvex, let=V/m(V)be the residue field, with residue class mapx↦:V↦, and let υ:→ Γ be the associated valuation. “Definable” will mean “definable with parameters”.The main goal of this article is to determine the structure induced by(,V)on its residue fieldand on (...) its value group Γ. In [9] we expanded the ordered fieldto a model ofTas follows. Take a tame elementary substructure′ ofsuch thatR′ ⊆VandR′maps bijectively ontounder the residue class map, and make this bijection into an isomorphism′ ≌ofT-models. (We showed such′ exists, and that this gives an expansion ofto aT-model that is independent of the choice of′.). (shrink)

This paper deals with decision makers who choose among information systems. It shows that the properties of Schur convexity and of quasi-convexity are equivalent, even when general preferences are considered. Since Schur convexity is closely related to having a willingness to accept information and since quasi-convexity is closely related to having a preference for early resolution of the uncertainty about which information system prevails, then it follows that the equivalence implies that decision makers prefer more information (...) to less if, and only if, they prefer early resolution of uncertainty to later resolution. (shrink)

The model of incomplete cooperative games incorporates uncertainty into the classical model of cooperative games by considering a partial characteristic function. Thus the values for some of the coalitions are not known. The main focus of this paper is 1-convexity under this framework. We are interested in two heavily intertwined questions. First, given an incomplete game, how can we fill in the missing values to obtain a complete 1-convex game? Second, how to determine in a rational, fair, and efficient (...) way the payoffs of players based only on the known values of coalitions? We illustrate the analysis with two classes of incomplete games—minimal incomplete games and incomplete games with defined upper vector. To answer the first question, for both classes, we provide a description of the set of 1-convex extensions in terms of its extreme points and extreme rays. Based on the description of the set of 1-convex extensions, we introduce generalisations of three solution concepts for complete games, namely the τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}-value, the Shapley value and the nucleolus. For minimal incomplete games, we show that all of the generalised values coincide. We call it the average value and provide different axiomatisations. For incomplete games with defined upper vector, we show that the generalised values do not coincide in general. This highlights the importance and also the difficulty of considering more general classes of incomplete games. (shrink)

Economic theory has generally acknowledged the role that institutions have in shaping economic space. The distinction, however, between physical and institutional descriptions of economic activity has not received adequate attention within the mainstream paradigm. In this paper I show how a proper distinction between the physical and institutional space in economic models will help clarify the concept of externality and provide a better interpretation of the relationship between externality and nonconvexity. I argue that within the Arrow-Debreu framework externality should be (...) viewed as incongruence between the physical and institutional descriptions of the economic space. I also argue that, contrary to conventional wisdom, detrimental externality has no special association with nonconvexity. Starrett's (1972) fundamental nonconvexity has to do with the specific institutional structure of Arrow markets rather than the detrimental nature of externality. Indeed, Arrow markets may not eliminate externalities. In a similar vein, it is not detrimental externality, however intense, that causes the production possibility set to become nonconvex, as argued by Baumol and Bradford (1972), but the particular interpretation of intensity that would make even conventional production possibility sets nonconvex. These points become apparent when one distinguishes between the convexity of the physical and institutional production sets. (shrink)

We investigate risk associated with the violation of a constraint, which is desirable but hardly satisfiable in all possible states of nature.

We show constructively that every quasi-convex, uniformly continuous function \ with at most one minimum point has a minimum point, where C is a convex compact subset of a finite dimensional normed space. Applications include a result on strictly quasi-convex functions, a supporting hyperplane theorem, and a short proof of the constructive fundamental theorem of approximation theory.

Let T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that the residue field of such a convex hull has a natural expansion to a model of T. We give a quantifier elimination relative to T for the theory of pairs (R, V) where $\mathscr{R} \models T$ and V ≠ R is the convex hull of an elementary substructure of R. We deduce (...) that the theory of such pairs is complete and weakly o-minimal. We also give a quantifier elimination relative to T for the theory of pairs (R, N) with R a model of T and N a proper elementary substructure that is Dedekind complete in R. We deduce that the theory of such "tame" pairs is complete. (shrink)

Gärdenfors (Conceptual spaces, 2000) argues that the semantic domains that natural language deals with have a geometrical structure. He gives evidence that simple natural language adjectives usually denote natural properties, where a natural property is a convex region of such a “conceptual space.” In this paper I will show that this feature of natural categories need not be stipulated as basic. In fact, it can be shown to be the result of evolutionary dynamics of communicative strategies under very general assumptions.

Inspired by locale theory, we propose “pointfree convex geometry”. We introduce the notion of convexity algebra as a pointfree convexity space. There are two notions of a point for convexity algebra: one is a chain-prime meet-complete filter and the other is a maximal meet-complete filter. In this paper we show the following: the former notion of a point induces a dual equivalence between the category of “spatial” convexity algebras and the category of “sober” convexity spaces (...) as well as a dual adjunction between the category of convexity algebras and the category of convexity spaces; the latter notion of point induces a dual equivalence between the category of “m-spatial” convexity algebras and the category of “m-sober” convexity spaces. We finally argue that the former notion of a point is more useful than the latter one from a category theoretic point of view and that the former notion of a point actually represents a polytope and the latter notion of a point properly represents a point. We also remark on the close relationships between pointfree convex geometry and domain theory. (shrink)

Image denoising is one of the important problems in the research field of computer vision, artificial intelligence, 3D vision, and image processing, where the fundamental aim is to recover the original image features from a noisy contaminated image. The camera sensor additive noise present in the holographic recording process reduces the quality of the retrieved image. Even though various techniques have been developed to minimize the noise in digital holography, the noise reduction still remains a challenging task. This article presents (...) a compressive sensing technique to minimize the additive noise in the digital holographic reconstruction process. We demonstrate the reduction of additive noise using complex wave retrieval method as a sensing matrix in the CS model. The proposed CS method to suppress the noise during the reconstruction process is illustrated using numerical simulations. Only 50% of the pixel measurements are considered in the noisy hologram, which is far less than the original complex object pixels. The impact of additive gaussian noise in the recording plane on the reconstruction accuracy of both intensity and phase distribution is analysed. The CS method denoises and estimates the complex object information accurately. The numerical simulation results have shown that the proposed CS method has effectively minimized the noise in the reconstructed image and has greatly improved the quality of both intensity and phase information. (shrink)

This note responds to some criticisms of my recent book In Defence of Objective Bayesianism that were provided by Gregory Wheeler in his ‘Objective Bayesian Calibration and the Problem of Non-convex Evidence’.

It is well known that the convex hull of the classical truth value functions contains all and only the probability functions. Work by Paris and Williams has shown that this also holds for various kinds of nonclassical logics too. This note summarises the formal details of this topic and extends the results slightly.

Two classically equivalent, but constructively inequivalent, strict convexity properties of a preference relation are discussed, and conditions given under which the stronger notion is a consequence of the weaker. The last part of the paper introduces uniformly rotund preferences, and shows that uniform rotundity implies strict convexity. The paper is written from a strictly constructive point of view, in which all proofs embody algorithms. MSC: 03F60, 90A06.

We report a solution to an open problem regarding the axiomatization of the convex hull of a type of nonclassical evaluations. We then investigate the meaning of this result for the larger context of the relation between rational credence functions and nonclassical probability. We claim that the notions of bets and Dutch Books typically employed in formal epistemology are of doubtful use outside the realm of classical logic, eventually proposing two novel ways of understanding Dutch Books in nonclassical settings.

In this paper I argue that the analysis of natural properties as convex subsets of a metric space in which the distances are degrees of dissimilarity is incompatible with both the definition of degree of dissimilarity as number of natural properties not in common and the definition of degree of dissimilarity as proportion of natural properties not in common, since in combination with either of these definitions it entails that every property is a natural property, which is absurd. I suggest (...) it follows that we should think of the convex class analysis of natural properties as a variety of resemblance nominalism. (shrink)

The present note first discusses the concept of s-convex pain functions in decision theory. Then, the economic behavior of an agent with such a pain function is represented through the comparison of some recursive lotteries.

This paper analyzes concepts of independence and assumptions of convexity in the theory of sets of probability distributions. The starting point is Kyburg and Pittarelli’s discussion of “convex Bayesianism” (in particular their proposals concerning E-admissibility, independence, and convexity). The paper offers an organized review of the literature on independence for sets of probability distributions; new results on graphoid properties and on the justification of “strong independence” (using exchangeability) are presented. Finally, the connection between Kyburg and Pittarelli’s results and (...) recent developments on the axiomatization of non-binary preferences, and its impact on “complete” independence, are described. (shrink)

The work on prototypes in ontologies pioneered by Rosch [10] and elaborated by Lakoff [8] and Freund [3] is related to vagueness in the sense that the more remote an instance is from a prototype the fewer people agree that it is an example of that prototype. An intuitive example is the prototypical “mother”, and it is observed that more specific instances like ”single mother”, “adoptive mother”, “surrogate mother”, etc., are less and less likely to be classified as “mothers” by (...) experimental subjects. From a different direction Gärdenfors [4] provided a persuasive account of natural predicates to resolve paradoxes of induction like Goodman’s “Grue” predicate [5]. Gärdenfors proposed that “quality dimensions” arising from human cognition and perception impose topologies on concepts such that the ones that appear “natural” to us are convex in these topologies. We show that these two cognitive principles — prototypes and predicate convexity — are equivalent to unimodal (convex) fuzzy characteristic functions for sets. Then we examine the case when the fuzzy set characteristic function is not convex, in particular when it is multi-modal. We argue that this is an indication that the fuzzy concept should really be regarded as a super concept in which the decomposed components are subconcepts in an ontological taxonomy. (shrink)

A connected graph is called Hamilton-connected if there exists a Hamiltonian path between any pair of its vertices. Determining whether a graph is Hamilton-connected is an NP-complete problem. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering. The detour index of a graph is defined to be the sum of lengths of detours between all the unordered pairs of vertices. The detour index has diverse applications in chemistry. Computing the detour index for a graph is also (...) an NP-complete problem. In this paper, we study the Hamilton-connectivity of convex polytopes. We construct three infinite families of convex polytopes and show that they are Hamilton-connected. An infinite family of non-Hamilton-connected convex polytopes is also constructed, which, in turn, shows that not all convex polytopes are Hamilton-connected. By using Hamilton connectivity of these families of graphs, we compute exact analytical formulas of their detour index. (shrink)

Jon Williamson's Objective Bayesian Epistemology relies upon a calibration norm to constrain credal probability by both quantitative and qualitative evidence. One role of the calibration norm is to ensure that evidence works to constrain a convex set of probability functions. This essay brings into focus a problem for Williamson's theory when qualitative evidence specifies non-convex constraints.

Modern engineering has included the basic sciences and their accompanying mathematical theories among its primary tools. The theory of probability is one of the more recent entries into standard engineering practice in various technological disciplines. Probability and statistics serve useful functions in the solution of many engineering problems. However, not all technological manifestations of uncertainty are amenable to probabilistic representation. In this paper we identify the conceptual limitations of probabilistic and related theories as they occur in a wide range of (...) engineering tasks. We discuss the structure and properties of an alternative, non-probabilistic, method — convex modelling — for quantitatively representing uncertain phenomena. (shrink)

We approach the semantics of prepositions from the perspective of conceptual spaces. Focusing on purely spatial locative and directional prepositions, we analyze both types of prepositions in terms of polar coordinates instead of Cartesian coordinates. This makes it possible to demonstrate that the property of convexity holds quite generally in the domain of prepositions of location and direction, supporting the important role that this property plays in conceptual spaces.

Let W be a recursively enumerable vector space over a recursive ordered field. We show the Turing equivalence of the following sets: the set of all tuples of vectors in W which are linearly dependent; the set of all tuples of vectors in W whose convex closures contain the zero vector; and the set of all pairs of tuples in W such that the convex closure of X intersects the convex closure of Y. We also form the analogous sets consisting (...) of tuples with given numbers of elements, and prove similar results on the Turing equivalence of these. (shrink)

The principle of differential marginality (Casajus in Theory and Decis 71(2):163-–174) for cooperative games is a very appealing property that requires equal productivity differentials to translate into equal payoff differentials. In this paper we apply this property to axiomatic characterizations of values. We show that differential marginality implies additivity and symmetry under certain conditions. Based on this result, we propose new characterizations of the equal division and the equal surplus division values. Finally, we characterize two classes of convex combinations of (...) values, i.e., α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}-egalitarian Shapley values and α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}-equal surplus division values, by employing differential marginality and establishing the uniqueness of these values on inessential games. (shrink)

Related Works: Original Paper: Lou Van Den Dries. $T$-Convexity and Tame Extensions II. J. Symbolic Logic, Volume 62, Issue 1 , 14--34. Project Euclid: euclid.jsl/1183745182.

We study properties of definable sets and functions in power-bounded T -convex fields, proving that the latter have the multidimensional Jacobian property and that the theory of T -convex fields is b -minimal with centers. Through these results and work of I. Halupczok we ensure that a certain kind of geometrical stratifications exist for definable objects in said fields. We then discuss a number of applications of those stratifications, including applications to Archimedean o-minimal geometry.

Discounted utility theory and its generalizations use discount functions for weighting utilities of outcomes received in different time periods. We propose a new simple test of convexity–concavity of discount function. This test can be used with any utility function and any preferences over risky lotteries. The data from a controlled laboratory experiment show that about one third of experimental subjects reveal a concave discount function and another one third of subjects reveal a convex discount function.

Based on a pre-order on a set, we axiomatize the Egli-Milner order on the power set and show how it is related to the Lewis order. Moreover, we consider a strict version of the Egli-Milner order and show how it can be related to the semantics of conditionals based on a priority structure.

We prove completeness of preferential conditional logic with respect to convexity over finite sets of points in the Euclidean plane. A conditional is defined to be true in a finite set of points if all extreme points of the set interpreting the antecedent satisfy the consequent. Equivalently, a conditional is true if the antecedent is contained in the convex hull of the points that satisfy both the antecedent and consequent. Our result is then that every consistent formula without nested (...) conditionals is satisfiable in a model based on a finite set of points in the plane. The proof relies on a result by Richter and Rogers showing that every finite abstract convex geometry can be represented by convex polygons in the plane. (shrink)