There are some who defend a view of vagueness according to which there are intrinsically vague objects or attributes in reality. Here, in contrast, we defend a view of vagueness as a semantic property of names and predicates. All entities are crisp, on this view, but there are, for each vague name, multiple portions of reality that are equally good candidates for being its referent, and, for each vague predicate, multiple classes of objects that are equally good candidates for being (...) its extension. We provide a new formulation of these ideas in terms of a theory of granular partitions. We show that this theory provides a general framework within which we can understand the relation between vague terms and concepts on the one hand and correlated portions of reality on the other. We also sketch how it might be possible to formulate within this framework a theory of vagueness which dispenses with the notion of truth-value gaps and other artifacts of more familiar approaches. (shrink)

We have a variety of different ways of dividing up, classifying, mapping, sorting and listing the objects in reality. The theory of granular partitions presented here seeks to provide a general and unified basis for understanding such phenomena in formal terms that is more realistic than existing alternatives. Our theory has two orthogonal parts: the first is a theory of classification; it provides an account of partitions as cells and subcells; the second is a theory of reference (...) or intentionality; it provides an account of how cells and subcells relate to objects in reality. We define a notion of well-formedness for partitions, and we give an account of what it means for a partition to project onto objects in reality. We continue by classifying partitions along three axes: (a) in terms of the degree of correspondence between partition cells and objects in reality; (b) in terms of the degree to which a partition represents the mereological structure of the domain it is projected onto; and (c) in terms of the degree of completeness with which a partition represents this domain. (shrink)

In this paper we propose a formal theory of partitions (ways of dividing up or sorting or mapping reality) and we show how the theory can be applied in the geospatial domain. We characterize partitions at two levels: as systems of cells (theory A), and in terms of their projective relation to reality (theory B). We lay down conditions of well-formedness for partitions and we define what it means for partitions to project truly onto reality. We (...) continue by classifying well-formed partitions along three axes: (a) degree of correspondence between partition cells and objects in reality; (b) degree to which a partition represents the mereological structure of the domain it is projected onto; and (c) degree of completeness and exhaustiveness with which a partition represents reality. This classification is used to characterize three types of partitions that play an important role in spatial information science: cadastral partitions, categorical coverages, and the partitions involved in folk categorizations of the geospatial domain. (shrink)

The theory of granular partitions (TGP) is a new approach to the understanding of ontologies and other classificatory systems. The paper explores the use of this new theory in the treatment of task-based clinical guidelines as a means for better understanding the relations between different clinical tasks, both within the framework of a single guideline and between related guidelines. We used as our starting point a DAML+OIL-based ontology for the WHO guideline for hypertension management, comparing this with related (...) guidelines and attempting to show that TGP provides a flexible and highly expressive basis for the manipulation of ontologies of a sort which might be useful in providing more adequate Computer Interpretable Guideline Models (CIGMs) in the future. (shrink)

The theory of granular partitions is designed to capture in a formal framework important aspects of the selective character of common-sense views of reality. It comprehends not merely the ways in which we can view reality by conceiving its objects as gathered together not merely into sets, but also into wholes of various kinds, partitioned into parts at various levels of granularity. We here represent granular partitions as triples consisting of a rooted tree structure as first (...) component, a domain satisfying the axioms of Extensional Mereology as second component, and a mapping (called ’projection’) of the first into the second as a third component. We define ordering relations among granular partitions the resulting structures are called partition frames. We then introduce an axiomatic theory which sentences are interpreted in partition frames. (shrink)

Abstract: We propose a view of vagueness as a semantic property of names and predicates. All entities are crisp, on this semantic view, but there are, for each vague name, multiple portions of reality that are equally good candidates for being its referent, and, for each vague predicate, multiple classes of objects that are equally good candidates for being its extension. We provide a new formulation of these ideas in terms of a theory of granular partitions. We show (...) that this theory provides a general framework within which we can understand the relation between vague terms and concepts and the corresponding crisp portions of reality. We also sketch how it might be possible to formulate within this framework a theory of vagueness which dispenses with the notion of truth-value gaps and other artifacts of more familiar approaches. Central to our approach is the idea that judgments about reality involve in every case (1) a separation of reality into foreground and background of attention and (2) the feature of granularity. On this basis we attempt to show that even vague judgments made in naturally occurring contexts are not marked by truth-value indeterminacy. We distinguish, in addition to crisp granular partitions, also vague partitions, and reference partitions, and we explain the role of the latter in the context of judgments that involve vagueness. We conclude by showing how reference partitions provide an effective means by which judging subjects are able to temper the vagueness of their judgments by means of approximations. (shrink)

Published in extended form as "Endurants and Perdurants in Directly Depicting Ontologies", -/- We propose an ontological theory that is powerful enough to describe both complex spatio-temporal processes and the enduring entities that participate in such processes. For this purpose we distinguish between ontologies and metaontology. Ontologies are based on very simple directly depicting languages and fall into two major categories: ontologies of type SPAN and ontologies of type SNAP. These represent two complementary perspectives on reality and result in distinct (...) though compatible systems of categories. In a SNAP (snapshot) ontology we have the enduring entities in a given domain as they exist to be inventoried at some given moment of time. In a SPAN ontology we have perduring entities such as processes and their parts and aggregates. We argue that both kinds of ontology are required, together with the meta-ontology which joins them together. On the level of meta-ontology we are able to impose constraints on ontologies of a sort which can support efficient processing of large amounts of data. (shrink)

We provide a methodology for the creation of ontological partitions in biomedicine and we test the methodology via an application to the phenomenon of blood pressure. An ontology of blood pressure must do justice to the complex networks of intersecting pathways in the organism by which blood pressure is regulated. To this end it must deal not only with the anatomical structures and physiological processes involved in such regulation but also with the relations between these at different levels of (...) granularity. For this purpose our ontology offers a variety of distinct partitions � of substances, processes and functions � and integrates these together within a single framework via transitive networks of part-whole and dependence relations among the entities in each of these categories. The paper concludes with a comparison of this methodology with the approaches of GOTM, KEGG, DIP and BIND and provides an outline of how the methodology is currently being applied in the field of biomedical database integration. (shrink)

We propose a new account of vagueness and approximation in terms of the theory of granular partitions. We distinguish different kinds of crisp and non-crisp granular partitions and we describe the relations between them, concentrating especially on spatial examples. We describe the practice whereby subjects use regular grid-like reference partitions as a means for tempering the vagueness of their judgments, and we demonstrate how the theory of reference partitions can yield a natural account of (...) this practice, which is referred to in the literature as ‘approximation’. (shrink)

If Mont Blanc is a vague object, then its vagueness will depend on the context in which reference is made. In a geological context the mountain might include only rock, perhaps together with a certain amount of air in the crevices and tunnels which have been formed beneath its surface. In a context of soil chemistry we might include also a surrounding thin layer of organic matter. In a skiing context we might include some snow. This essay sketches in informal (...) terms the theory of granular partitions, which is designed to do justice to this context-dependence of vagueness by means of what might be described as a contextualized supervaluationism. Granularity and vagueness, it is argued, are two sides of a single coin: what is vague at one level of granularity may appear crisp at another. The resultant theory can be shown to resolve certain problems in our description of perceptual content, for example when John says that he sees the wall, but that he does not see the molecules by which the wall is constituted. (shrink)

The truthmaker theory rests on the thesis that the link between a true judgment and that in the world to which it corresponds is not a one-to-one but rather a one-to-many relation. An analogous thesis in relation to the link between a singular term and that in the world to which it refers is already widely accepted. This is the thesis to the effect that singular reference is marked by vagueness of a sort that is best understood in supervaluationist terms. (...) In what follows we show that the supervaluationist approach to singular reference, when wedded to the truthmaker idea, yields a framework of surprising power, which offers a uniform set of solutions to a range of problems regarding identity, reference and knowledge, problems which have hitherto been dealt with on an ad hoc basis. (shrink)

The truthmaker theory rests on the thesis that the link between a true judgment and that in the world to which it corresponds is not a one-to-one but rather a one-to-many relation. An analogous thesis in relation to the link between a singular term and that in the world to which it refers is already widely accepted. This is the thesis to the effect that singular reference is marked by vagueness of a sort that is best understood in supervaluationist terms. (...) In what follows we show that the supervaluationist approach to singular reference, when wedded to the truthmaker idea, yields a framework of surprising power, which offers a uniform set of solutions to a range of problems regarding identity, reference and knowledge, problems which have hitherto been dealt with on an ad hoc basis. (shrink)

Mereotopology faces problems when its methods are extended to deal with time and change. We offer a new solution to these problems, based on a theory of partitions of reality which allows us to simulate (and also to generalize) aspects of set theory within a mereotopological framework. This theory is extended to a theory of coarse- and fine-grained histories (or finite sequences of partitions evolving over time), drawing on machinery developed within the framework of the so-called ‘consistent histories’ (...) interpretation of quantum mechanics. (shrink)

The Unified Medical Language System and the Gene Ontology are among the most widely used terminology resources in the biomedical domain. However, when we evaluate them in the light of simple principles for wellconstructed ontologies we find a number of characteristic inadequacies. Employing the theory of granular partitions, a new approach to the understanding of ontologies and of the relationships ontologies bear to instances in reality, we provide an application of this theory in relation to an example drawn (...) from the context of the pathophysiology of hypertension. This exercise is designed to demonstrate how, by taking ontological principles into account we can create more realistic biomedical ontologies which will also bring advantages in terms of efficiency and robustness of associated software applications. (shrink)

Possible-worlds accounts of mental or linguistic content are often criticized for being too coarse-grained. To make room for more fine-grained distinctions among contents, several authors have recently proposed extending the space of possible worlds by "impossible worlds". We argue that this strategy comes with serious costs: we would effectively have to abandon most of the features that make the possible-worlds framework attractive. More generally, we argue that while there are intuitive and theoretical considerations against overly coarse-grained notions of content, the (...) same kinds of considerations also prohibit an overly fine-grained individuation of content. An adequate notion of content, it seems, should have intermediate granularity. However, it is hard to construe a notion of content that meets these demands. Any notion of content, we suggest, must be either implausibly coarse-grained or implausibly fine-grained (or both). (shrink)

This is a re-look at the (Indian) Partition event through the lens of Advaita Vedanta.

Implicit contextual factors mean that the boundary between causal and noncausal explanation is not as neat as one might hope: as the phenomenon to be explained is given descriptions with varying degrees of granularity, the nature of the favored explanation alternates between causal and non-causal. While it is not surprising that different descriptions of the same phenomenon should favor different explanations, it is puzzling why re-describing the phenomenon should make any difference for the causal nature of the favored explanation. I (...) argue that this is a problem for the ontic framework of causal and noncausal explanation, and instead propose a pragmatic-modal account of causal and non-causal explanation. This account has the added advantage of dissolving several important disagreements concerning the status of non-causal explanation. (shrink)

We propose an ontological theory that is powerful enough to describe both complex spatio-temporal processes (occurrents) and the enduring entities (continuants) that participate therein. The theory is divided into two major categories of sub-theories: (sub-) theories of type SPAN and (sub-)theories of type SNAP. These theories represent two complementary perspectives on reality and result in distinct though compatible systems of categories. In SNAP we have enduring entities such as substances, qualities, roles, functions; in SPAN we have perduring entities such as (...) processes and their parts and aggregates. We argue that both kinds of ontological theory are required in order to give a non-reductionism account of complex domains of reality. (shrink)

Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics (QM) is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets (or partitions) which are category-theoretically (...) dual to one another and which are developed in two mathematical logics, the usual Boolean logic of subsets and the more recent logic of partitions. Our sense-making strategy is "follow the math" by showing how the logic and mathematics of set partitions can be transported in a natural way to Hilbert spaces where it yields the mathematical machinery of QM--which shows that the mathematical framework of QM is a type of logical system over ℂ. And then we show how the machinery of QM can be transported the other way down to the set-like vector spaces over ℤ₂ showing how the classical logical finite probability calculus (in a "non-commutative" version) is a type of "quantum mechanics" over ℤ₂, i.e., over sets. In this way, we try to make sense out of objective indefiniteness and thus to interpret quantum mechanics. (shrink)

Objective: There are benefits and risks of giving patients more granular control of their personal health information in electronic health record (EHR) systems. When designing EHR systems and policies, informaticists and system developers must balance these benefits and risks. Ethical considerations should be an explicit part of this balancing. Our objective was to develop a structured ethics framework to accomplish this. -/- Methods: We reviewed existing literature on the ethical and policy issues, developed an ethics framework called a “Points (...) to Consider” (P2C) document, and convened a national expert panel to review and critique the P2C. -/- Results: We developed the P2C to aid informaticists designing an advanced query tool for an electronic health record (EHR) system in Indianapolis. The P2C consists of six questions (“Points”) that frame important ethical issues, apply accepted principles of bioethics and Fair Information Practices, comment on how questions might be answered, and address implications for patient care. -/- Discussion: The P2C is intended to clarify whatis at stake when designers try to accommodate potentially competing ethical commitments and logistical realities. The P2C was developed to guide informaticists who were designing a query tool in an existing EHR that would permit patient granular control. While consideration of ethical issues is coming to the forefront of medical informatics design and development practices, more reflection is needed to facilitate optimal collaboration between designers and ethicists. This report contributes to that discussion. (shrink)

In the present version of these lecture notes only a number of typos and a few glaring mistakes have been corrected. Thanks to Paul Dekker for his help in this respect. No attempt has been been made to update the original text or to incorporate new insights and approaches. For a more recent overview, see our ‘Questions’ in the Handbook of Logic and Language (edited by Johan van Benthem and Alice ter Meulen, Elsevier, 1997).

Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (nonempty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality--which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the idea arises of a (...) dual logic of partitions. That dual logic is described here. Partition logic is at the same mathematical level as subset logic since models for both are constructed from (partitions on or subsets of) arbitrary unstructured sets with no ordering relations, compatibility or accessibility relations, or topologies on the sets. Just as Boole developed logical finite probability theory as a quantitative treatment of subset logic, applying the analogous mathematical steps to partition logic yields a logical notion of entropy so that information theory can be refounded on partition logic. But the biggest application is that when partition logic and the accompanying logical information theory are "lifted" to complex vector spaces, then the mathematical framework of quantum mechanics is obtained. Partition logic models indefiniteness (i.e., numerical attributes on a set become more definite as the inverse-image partition becomes more refined) while subset logic models the definiteness of classical physics (an entity either definitely has a property or definitely does not). Hence partition logic provides the backstory so the old idea of "objective indefiniteness" in QM can be fleshed out to a full interpretation of quantum mechanics. (shrink)

An explicit formal-ontological representation of entities existing at multiple levels of granularity is an urgent requirement for biomedical information processing. We discuss some fundamental principles which can form a basis for such a representation. We also comment on some of the implicit treatments of granularity in currently available ontologies and terminologies (GO, FMA, SNOMED CT).

Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen as the logic of subsets (...) of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic. (shrink)

We present a minimum message length (MML) framework for trajectory partitioning by point selection, and use it to automatically select the tolerance parameter ε for Douglas-Peucker partitioning, adapting to local trajectory complexity. By examining a range of ε for synthetic and real trajectories, it is easy to see that the best ε does vary by trajectory, and that the MML encoding makes sensible choices and is robust against Gaussian noise. We use it to explore the identification of micro-activities within a (...) longer trajectory. This MML metric is comparable to the TRACLUS metric – and shares the constraint of abstracting only by omission of points – but is a true lossless encoding. Such encoding has several theoretical advantages – particularly with very small segments (high frame rates) – but actual performance interacts strongly with the search algorithm. Both differ from unconstrained piecewise linear approximations, including other MML formulations. (shrink)

In his “Bridging mainstream and formal ontology”, Augusto (2021) gives an excellent analysis of Dietrich von Freiberg’s idea of using causality as a partitioning principle for upper ontologies. For this Dietrich’s notion of extrinsic principles is crucial. The question whether causation can and indeed should be used as a partitioning principle for ontologies is discussed using mathematics and physics as examples.

In this short note many conjectures on partitions of integers as summations of prime numbers are presented, which are extension of Goldbach conjecture.

Ethics should guide the design of electronic health records (EHR), and recognized principles of bioethics can play an important role. This approach was adopted recently by a team of informaticists designing and testing a system where patients exert granular control over who views their personal health information. While this method of building ethics in from the start of the design process has significant benefits, questions remain about how useful the application of bioethics principles can be in this process, especially (...) when principles conflict. For instance, while the ethical principle of respect for autonomy supports a robust system of granular control, the principles of beneficence and non-maleficence counsel restraint due to the danger of patients being harmed by restrictions on provider access to data. Conflict between principles has long been recognized by ethicists and has even motivated attacks on approaches that state and apply principles. In this paper we show how using ethical principles can help in the design of EHRs by first, explaining how ethical principles can and should be used generally, and then by, discuss how attention to details in specific cases can show that the tension between principles is not as bad as it initially appeared. We conclude by suggesting further ways in which the application of these (and other) principles can add value to the ongoing discussion of patient involvement in their health care. This is a new approach to linking principles to informatics design that we expect will stimulate further interest. (shrink)

The purpose of this paper is to show that the mathematics of quantum mechanics is the mathematics of set partitions linearized to vector spaces, particularly in Hilbert spaces. That is, the math of QM is the Hilbert space version of the math to describe objective indefiniteness that at the set level is the math of partitions. The key analytical concepts are definiteness versus indefiniteness, distinctions versus indistinctions, and distinguishability versus indistinguishability. The key machinery to go from indefinite to (...) more definite states is the partition join operation at the set level that prefigures at the quantum level projective measurement as well as the formation of maximally-definite state descriptions by Dirac’s Complete Sets of Commuting Operators. This development is measured quantitatively by logical entropy at the set level and by quantum logical entropy at the quantum level. This follow-the-math approach supports the Literal Interpretation of QM—as advocated by Abner Shimony among others which sees a reality of objective indefiniteness that is quite different from the common sense and classical view of reality as being “definite all the way down”. (shrink)

A number of authors in favor of a unitary account of singular descriptions have alleged that the unitary account can be extrapolated to account for plural definite descriptions. In this paper I take a closer look at this suggestion. I argue that while the unitary account is clearly onto something right, it is in the end empirically inadequate. At the end of the paper I offer a new partitive account of plural definite descriptions that avoids the problems with both the (...) unitary account and standard Russellian analyses. (shrink)

Bashabi Fraser is a poet in her own right. She is also a creative translator. This is a review of her edited volume on the Partition of Bengal. The review highlights our need to read the partition event as a warning for future and ongoing genocides. The review also shows the superiority of literature over history. And finally it has something to say about translation and separately, on P Lal. For instance, this reviewer in many other reviews too insists on (...) the superiority of Fr Mignon SJ over Prof Lal's understanding of translation. (shrink)

The lattice operations of join and meet were defined for set partitions in the nineteenth century, but no new logical operations on partitions were defined and studied during the twentieth century. Yet there is a simple and natural graph-theoretic method presented here to define any n-ary Boolean operation on partitions. An equivalent closure-theoretic method is also defined. In closing, the question is addressed of why it took so long for all Boolean operations to be defined for (...) class='Hi'>partitions. (shrink)

This paper proposes a new classification algorithm for the partitioning of a conceptual space. All the algorithms which have been used until now have mostly been based on the theory of Voronoi diagrams. This paper proposes an approach based on potential theory, with the criteria for measuring similarities between objects in the conceptual space being based on the Newtonian potential function. The notion of a fuzzy prototype, which generalizes the previous definition of a prototype, is introduced. Furthermore, the necessary conditions (...) that a natural concept must meet are discussed. Instead of convexity, as proposed by Gärdenfors, the notion of geodesically convex sets is used. Thus, if a concept corresponds to a set which is geodesically convex, it is a natural concept. This definition applies, for example, if the conceptual space is an Euclidean space. As a by-product of the construction of the algorithm, an extension of the conceptual space to d-dimensional Riemannian manifolds is obtained. (shrink)

In this paper I discuss the translation of a line in Plato's description of the ‘greatest accusation’ against imitative poetry, Republic 606a3–b5. This line is pivotal in Plato's account of how poetry corrupts its audience and is one of the Republic's most complex and interesting applications of his partite psychology, but it is misconstrued in most recent translations, including the most widely used. I argue that an examination of the text and reflections on Platonic psychology settle the translation decisively.

An important part of the Unified Medical Language System (UMLS) is its Semantic Network, consisting of 134 Semantic Types connected to each other by edges formed by one or more of 54 distinct Relation Types. This Network is however for many purposes overcomplex, and various groups have thus made attempts at simplification. Here we take this work further by simplifying the relations which involve the three Semantic Types – Diagnostic Procedure, Laboratory Procedure and Therapeutic or Preventive Procedure. We define operators (...) which can be used to generate terms instantiating types from this selected set when applied to terms designating certain other Semantic Types, including almost all the terms specifying clinical tasks. Usage of such operators thus provides a useful and economical way of specifying clinical tasks. The operators allow us to define a mapping between those types within the UMLS which do not represent clinical tasks and those which do. This mapping then provides a basis for an ontology of clinical tasks that can be used in the formulation of computer-interpretable clinical guideline models. (shrink)

Abstract: This article explores the medical theories of the Dutch philosopher and physician Henricus Regius (1598-1679), who sought to provide clearer notions of medicine than the traditional theories of Jean Fernel, Daniel Sennert and Vopiscus Plempius. To achieve this, Regius overtly built upon the natural philosophy of René Descartes, in particular his theories of mechanical physiology and the corpuscular nature of matter. First, I show that Regius envisaged a novel partitioning of medicine, intended to make it independent in exposition but (...) conceptually grounded in natural philosophy. This served his overall purpose of making medicine a ‘clearer’ discipline. To this end Regius detaches the general notion of physiology as the study of the human body (which pertains to natural philosophy) from a medical physiology that concerns only health. Secondly, I show that Regius’s notions of health, disease, temperament and medicaments were the product of a Cartesian interpretation of traditional concepts, which was influenced by Santorio Santorio’s project of practicing medicine without occult elements. As a consequence, Regius’s classification of these notions is largely traditional. In conclusion, I show that Regius’s method of investigation in the area of natural philosophy consisted of a problem-solving procedure based on sensorial ideas. This approach can be traced back to Descartes, but was grounded on a purely empirical theory of knowledge, by which he rejected Descartes’s metaphysics. Regius’s order of exposition in natural philosophy and medicine, on the other hand, proceeds by definitions and divisions, and omits proofs. The reasons why Regius adopted this order were: 1) his rejection of Descartes’s metaphysical interpreta-tion of physics, and 2) his wish to emphasize the conceptual clarity and continuity of natural philosophy and medicine. (shrink)

Consider the following two (hypothetical) generic causal claims: “Living in a neighborhood with many families with children increases purchases of bicycles” and “living in an affluent neighborhood with many families with children increases purchases of bicycles.” These claims not only differ in what they suggest about how bicycle ownership is distributed across different neighborhoods (i.e., “the data”), but also have the potential to communicate something about the speakers’ values: namely, the prominence they accord to affluence in representing and making decisions (...) about the social world. Here, we examine the relationship between the level of granularity with which a cause is described in a generic causal claim (e.g., neighborhood vs. affluent neighborhood) and the value of the information contained in the causal model that generates that claim. We argue that listeners who know any two of the following can make reliable inferences about the third: 1) the level of granularity at which a speaker makes a generic causal claim, 2) the speaker’s values, and 3) the data available to the speaker. We present results of four experiments (N=1,323) in the domain of social categories that provide evidence in keeping with these predictions. (shrink)

The new logic of partitions is dual to the usual Boolean logic of subsets (usually presented only in the special case of the logic of propositions) in the sense that partitions and subsets are category-theoretic duals. The new information measure of logical entropy is the normalized quantitative version of partitions. The new approach to interpreting quantum mechanics (QM) is showing that the mathematics (not the physics) of QM is the linearized Hilbert space version of the mathematics of (...) partitions. Or, putting it the other way around, the math of partitions is a skeletal version of the math of QM. The key concepts throughout this progression from logic, to logical information, to quantum theory are distinctions versus indistinctions, definiteness versus indefiniteness, or distinguishability versus indistinguishability. The distinctions of a partition are the ordered pairs of elements from the underlying set that are in different blocks of the partition and logical entropy is defined (initially) as the normalized number of distinctions. The cognate notions of definiteness and distinguishability run throughout the math of QM, e.g., in the key non-classical notion of superposition (= ontic indefiniteness) and in the Feynman rules for adding amplitudes (indistinguishable alternatives) versus adding probabilities (distinguishable alternatives). (shrink)

In the TACITUS project for using commonsense knowledge in the understanding of texts about mechanical devices and their failures, we have been developing various commonsense theories that are needed to mediate between the way we talk about the behavior of such devices and causal models of their operation. Of central importance in this effort is the axiomatization of what might be called commonsense metaphysics. This includes a number of areas that figure in virtually every domain of discourse, such as granularity, (...) scales, time, space, material, physical objects, shape, causality, functionality, and force. Our effort has been to construct core theories of each of these areas, and then to define, or at least characterize, a large number of lexical items in terms provided by the core theories. In this paper we discuss our methodological principles and describe the key ideas in the various domains we are investigating. (shrink)

Recent developments in pure mathematics and in mathematical logic have uncovered a fundamental duality between "existence" and "information." In logic, the duality is between the Boolean logic of subsets and the logic of quotient sets, equivalence relations, or partitions. The analogue to an element of a subset is the notion of a distinction of a partition, and that leads to a whole stream of dualities or analogies--including the development of new logical foundations for information theory parallel to Boole's development (...) of logical finite probability theory. After outlining these dual concepts in mathematical terms, we turn to a more metaphysical speculation about two dual notions of reality, a fully definite notion using Boolean logic and appropriate for classical physics, and the other objectively indefinite notion using partition logic which turns out to be appropriate for quantum mechanics. The existence-information duality is used to intuitively illustrate these two dual notions of reality. The elucidation of the objectively indefinite notion of reality leads to the "killer application" of the existence-information duality, namely the interpretation of quantum mechanics. (shrink)

Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector space--which is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The notion of a partition (or quotient set or equivalence relation) is dual (in a category-theoretic sense) to (...) the notion of a subset. Hence the Boolean logic of subsets has a dual logic of partitions. Then the dual progression is from that logic of partitions to the quantum logic of direct-sum decompositions (i.e., the vector space version of a set partition) of a general vector space--which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the logic to express measurement by any self-adjoint operators rather than just the projection operators associated with subspaces. In this introductory paper, the focus is on the quantum logic of direct-sum decompositions of a finite-dimensional vector space (including such a Hilbert space). The primary special case examined is finite vector spaces over ℤ₂ where the pedagogical model of quantum mechanics over sets (QM/Sets) is formulated. In the Appendix, the combinatorics of direct-sum decompositions of finite vector spaces over GF(q) is analyzed with computations for the case of QM/Sets where q=2. (shrink)

Outcomes research in healthcare has been a topic much addressed in recent years. Efforts in this direction have been supplemented by work in the areas of guidelines for clinical practice and computer-interpretable workflow and careflow models.In what follows we present the outlines of a framework for understanding the relations between organizations, guidelines, individual patients and patient-related functions. The derived framework provides a means to extract the knowledge contained in the guideline text at different granularities, in ways that can help us (...) to assign tasks within the healthcare organization and to assess clinical performance in realizing the guideline. It does this in a way that preserves the flexibility of the organization in the adoption of the guidelines. (shrink)

Two senses of ‘ontology’ can be distinguished in the current literature. First is the sense favored by information scientists, who view ontologies as software implementations designed to capture in some formal way the consensus conceptualization shared by those working on information systems or databases in a given domain. [Gruber 1993] Second is the sense favored by philosophers, who regard ontologies as theories of different types of entities (objects, processes, relations, functions) [Smith 2003]. Where information systems ontologists seek to maximize reasoning (...) efficiency even at the price of simplifications on the side of representation, philosophical ontologists argue that representational adequacy can bring benefits for the stability and resistance to error of an ontological framework and also for its extendibility in the future. In bioinformatics, however, a third sense of ‘ontology’ has established itself, above all as a result of the successes of the Gene Ontology (hereafter: GO), which is a tool for the representation and processing of information about gene products and their biological functions [Gene Ontology Consortium 2000]. We show how Basic Formal Ontology (BFO) has established itself as an overarching ontology drawing on all three of the strands distinguished above, and describe applications of BFO especially in the treatment of biological granularity. (shrink)

The paper surveys the currently available axiomatizations of common belief (CB) and common knowledge (CK) by means of modal propositional logics. (Throughout, knowledge- whether individual or common- is defined as true belief.) Section 1 introduces the formal method of axiomatization followed by epistemic logicians, especially the syntax-semantics distinction, and the notion of a soundness and completeness theorem. Section 2 explains the syntactical concepts, while briefly discussing their motivations. Two standard semantic constructions, Kripke structures and neighbourhood structures, are introduced in Sections (...) 3 and 4, respectively. It is recalled that Aumann's partitional model of CK is a particular case of a definition in terms of Kripke structures. The paper also restates the well-known fact that Kripke structures can be regarded as particular cases of neighbourhood structures. Section 3 reviews the soundness and completeness theorems proved w.r.t. the former structures by Fagin, Halpern, Moses and Vardi, as well as related results by Lismont. Section 4 reviews the corresponding theorems derived w.r.t. the latter structures by Lismont and Mongin. A general conclusion of the paper is that the axiomatization of CB does not require as strong systems of individual belief as was originally thought- only monotonicity has thusfar proved indispensable. Section 5 explains another consequence of general relevance: despite the "infinitary" nature of CB, the axiom systems of this paper admit of effective decision procedures, i.e., they are decidable in the logician's sense. (shrink)

The logical basis for information theory is the newly developed logic of partitions that is dual to the usual Boolean logic of subsets. The key concept is a "distinction" of a partition, an ordered pair of elements in distinct blocks of the partition. The logical concept of entropy based on partition logic is the normalized counting measure of the set of distinctions of a partition on a finite set--just as the usual logical notion of probability based on the Boolean (...) logic of subsets is the normalized counting measure of the subsets (events). Thus logical entropy is a measure on the set of ordered pairs, and all the compound notions of entropy (join entropy, conditional entropy, and mutual information) arise in the usual way from the measure (e.g., the inclusion-exclusion principle)--just like the corresponding notions of probability. The usual Shannon entropy of a partition is developed by replacing the normalized count of distinctions (dits) by the average number of binary partitions (bits) necessary to make all the distinctions of the partition. (shrink)

Some authors have argued that, in order to give an account of weakness of the will, we must assume that the mind is divisible into parts. This claim is often referred to as the partitioning claim. There appear to be two main arguments for this claim. While the first is conceptual and claims that the notion of divisibility is entailed by the notion of non-rational mental causation (which is held to be a necessary condition of weakness of the will), the (...) second is explanatory and claims that the notion of divisibility is required for the causal explanation of weak-willed action. In this paper I want to argue that the partitioning claim remains unsupported, no matter how it is interpreted, and that weakness of the will can be made perfectly good sense of without the idea that the mind is divisible into parts. In fact, there are available various explanatory models each of which characterizes different psychological mechanisms that may be involved in weakness of will, none of which depends on any claims about mental division. I describe three familiar mechanisms and argue that weakness of will may occur as the result of any one of them. (shrink)

Observe that complement questions can be either directly or indirectly conjoined, but they can only be indirectly disjoined. • What theories of questions and coordination predict this difference? • Look at Partition theory (Groenendijk & Stokhof 1984) and Inquisitive Semantics (Groenendijk & Roelofsen 2009, Ciardelli et al. 2012).

I propose a revision of Cantor’s account of set size that understands comparisons of set size fundamentally in terms of surjections rather than injections. This revised account is equivalent to Cantor's account if the Axiom of Choice is true, but its consequences differ from those of Cantor’s if the Axiom of Choice is false. I argue that the revised account is an intuitive generalization of Cantor’s account, blocks paradoxes—most notably, that a set can be partitioned into a set that is (...) bigger than it—that can arise from Cantor’s account if the Axiom of Choice is false, illuminates the debate over whether the Axiom of Choice is true, is a mathematically fruitful alternative to Cantor’s account, and sheds philosophical light on one of the oldest unsolved problems in set theory. (shrink)