The ‘Reflection Principle’ is what rules out predictable persuasion and biases in updating, requiring your current probabilities to match your expectation for your future ones. It’s often claimed to be a theorem when Bayesians update by conditioning on the true cell of a finite partition. It’s not. I show that it robustly fails when a Bayesian’s priors are ambiguous, i.e. are probabilistically uncertainty about what their own priors are. Contrary to recent discussions, Bayesians can be predictably persuaded and biased (...) by information. (shrink)

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)

Paper 1 establishes that persistence-based frameworks — the Free Energy Principle, Global Workspace Theory, and adjacent approaches — rely on batteries of dynamical observables insufficient, on at least one substrate where the frameworks would naturally apply them, to individuate the systems they characterize. The supplementary structural condition Paper 1 §7 calls a partition-warrant must be sensitive to substrate properties not invariant under the admissible transformation group of the dynamics. The present paper develops closure activation as a candidate filling for (...) that slot. The formal predicate W_CA is partition-evaluative: for each subsystem under a partition P, it returns a verdict structure (τ₀, μ*, κ) recording whether the activation event T₀ has been crossed, the maintenance margin μ* under graded perturbation, and the resulting classification (closed, closure-competent, subthreshold). The activation conditions — state-referential constraint, internally governed corrective response, internally viable maintenance budget — are evaluated relative to the partition's imposed boundary. The predicate's two-axis structure (binary T₀, scalar T*) is phase-structured in a way that distinguishes closure activation from classical autopoietic operational closure. W_CA is non-G-invariant at the predicate level: the causal support graph relative to the imposed boundary is partition-relative and not preserved under admissible transformations. The predicate operates in three structural modes — rejection, selection, and stratification — depending on the substrate's closure-bearing properties under G-related partitions. Evaluation on the Paper 1 witness yields a rejection-mode verdict: neither partition is closure-bearing, and the uniform negative output is the predicate's correctness behavior on a substrate where the activation conditions cannot be satisfied under any admissible labeling. Distinctions from optimization-based (Integrated Information Theory), structural-definitional (Global Workspace Theory), statistical-structural (Free Energy Principle in nonlinear regimes), and process-based (autopoietic operational closure) alternatives are developed structurally and by operational-mode coverage. The route × mode cross-tabulation supplies a meta-taxonomy for partition-warrants more generally. Falsification conditions are stated at predicate, verdict-structure, and discriminative-scope levels. In conjunction with Paper 1, the present paper separates three questions persistence-based frameworks have often conflated: what persists dynamically, what individuates structurally, and what sustains recursive closure. The class of closure-bearing systems is a proper subset of the class of persistence-bearing systems under the battery characterization, and the structural rarity of closure-bearing organization is a derivable consequence of the activation conditions' joint stringency. (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 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 which are category-theoretically dual to one another (...) and which are developed in two dual 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 mathematics of set partitions can be transported in a natural way to complex vector spaces where it yields the mathematical machinery of QM. And then we show how the machinery of QM can be transported the other way down to set-like vector spaces over ℤ₂ yielding a rather fulsome "toy" or pedagogical model of "quantum mechanics over sets." In this way, we try to make sense out of objective indefiniteness and thus to interpret quantum mechanics. (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)

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

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)

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)

Imagine that you are standing on a bridge above a highway checking off the makes and models of the cars that are passing underneath; or a laboratory technician sorting samples of bacteria into species and subspecies; or you are making a list of the fossils in your museum. In each of these cases, you are employing a certain grid of labeled units, and you are recognizing certain objects as being located in those units. Such a grid of labeled units is (...) an example of what we shall call a granular partition. We shall argue that granular partitions are involved in all naming, listing, sorting, counting, cataloguing, and mapping activities, activities that are performed by human beings in their traffic with the world. Partitions are the cognitive devices designed and built by human beings to fulfill these various listing, mapping, and classifying purposes. (shrink)

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)

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.

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)

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 purpose of this paper is to show that the mathematics of quantum mechanics (QM) is the mathematics of set partitions (which specify indefiniteness and definiteness) 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)

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).

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)

Objective: This study aims to criticize the historical narrative of the 1947 Partition of India that focuses too much on political and state aspects and ignores women's personal experiences. In addition, the concept of intimate archive was introduced to understand how women build meaning in life after violence. Theoretical framework: Approaches used include Feminist Historiography, Trauma Theory, and Critical Archive Studies to show that archives and testimonies are not objective, but rather the result of constructions influenced by power relations (...) and individual experiences. Literature review: Previous literature has tended to focus on macro aspects such as politics and conflict. Instead, this study highlights the work of Saadat Hasan Manto and Amrita Pritam as well as feminist oral histories that describe women's experiences in a more personal and profound way. Methods: The method used is qualitative through close reading of literary works and oral history analysis to explore hidden meanings in women's experiences. Results: Three forms of intimate archive were found, namely meaning in objects, memories in the body, and the use of silence as a strategy. These findings show that women have an active role in shaping identity and meaning in life. Implications: The findings of the study expand the perspective of archives, not only as official data but also as representations of personal experiences. The concept of an intimate archive can also be applied to other studies that address marginalized experiences. Novelty: Novelty lies in the concept of an intimate archive that views testimony as an active process in building meaning, not just a record of experience. (shrink)

Formal theories of belief-credence interaction that satisfy the standard logical requirements on belief, such as conjunctive closure, face the problem of partition-sensitivity. According to these theories, a rational agent can believe X relative to one partitioning of possibilities, but the same belief may not be rational relative to some other partitioning, even when the agent’s evidence remains the same. Focusing on Leitgeb’s stability theory (Leitgeb, The stability of belief: How rational belief coheres with probability, Oxford University Press, 2017), which (...) exemplifies this problem, this paper aims to go beyond the simple partition-sensitivity of such theories. To address this and related problems, I connect the issues surrounding belief-credence interaction with Bayesian network theory and the notion of epistemic basing. The view that I defend and explicate is that rational belief corresponds to stably high probability within a network of interconnected propositions, where the network is structured by the relation of direct explanatory priority. As I argue, this view provides solutions to important problems regarding partition-sensitivity, specifically the problems involving irrelevant propositions and large partitions. In the final section, I extend the proposed theory and articulate the network-based notion of robustness of belief, which differentiates lottery-type beliefs from more ordinary beliefs. (shrink)

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

This paper presents the mechanics of the Substrate-Neutral Structural Foundation Theory (SNSFT), executed via a high-performance, deterministic JavaScript runtime engine known as the Octobeam core. Operating completely independently of heuristic, probabilistic, or stochastic frameworks, the engine functions as a real-time formal proof synthesizer for high-density spatial interactions. By tracking and resolving multi-beam data intersections—specifically the Two-Beam (GAMCollider), Four-Beam (Quadbeam) and Eight-Beam (Octobeam) architectures—the script executes exact structural computations and translates state reductions directly into machine-verifiable Lean 4 proof structures. To unify (...) these state transitions across variable physical and computational domains, we utilize the foundational PNBA Framework (Pattern, Narrative, Behavior, Action) to map interaction sequences into immutable structural primitives. -/- The system introduces a lightweight, purely deterministic software architecture capable of processing dense multi-beam intersection matrices using structural foundation laws. The core engine, known as the Collider (scaling from the four-dimensional Quadbeam implementation to the eight-dimensional Octobeam architecture), relies entirely on pure execution loops and bitwise spatial partitioning. Rather than simply rendering or storing spatial state coordinates, the JavaScript engine operates as an immediate runtime compiler, systematically transforming systemic state transitions and interaction boundaries directly into formal, rigorous proof components written in Lean 4 (such as the four-beam fusion theorem and the eight-beam fusion theorem files). By relying on explicit, exact code blocks, the framework maintains absolute mathematical validity within the execution loop, optimizing computation cycles and ensuring perfect predictability. -/- The computational environment structures space-time variables as clean coordinate fields, treating incoming data vectors as continuous directional streams (Beams) across a strictly sequential four-stage deterministic data pipeline: (1) Data Boundary Ingestion, (2) Deterministic Partitioning via exact matrix equations, (3) State Reduction to Primitives, and (4) Proof Synthesis. A core pillar of this architecture is its absolute substrate neutrality; mathematical definitions behave identically across kinetic, network, or identity domains. This is sustained by mapping all data transitions into four structural primitives: Pattern (P), Narrative (N), Behavior (B), and Action/Primitive (A). Expressing a multi-beam fusion event mathematically, the localized collision behaves as a strict reduction mapping where intersecting patterns collapse into a singular validated historical state vector where a function maps multiple incoming patterns through a behavioral operator directly into a fused narrative and resolved action state. -/- Finally, the paper maps the geometric scaling mechanics from a baseline four-stream Quadbeam system to an eight-axis Octobeam expansion. To bypass the exponential O(N^2) computational overhead standard in multi-stream tracking, the underlying engine implements explicit bitwise mask filters and localized coordinate bounding boxes. This ensures that the engine only computes multi-beam alignment properties when the localized vectors are within critical intersection range, instantly triggering the generation of formal verification terms upon perfect convergence. (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)

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 partitions.

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)

Cloud storage gives consumers easy, on-demand access to top-notch cloud apps while allowing them to save and retrieve data remotely. This ensures effective data storage on cloud servers while doing away with the need to manage local hardware and software. Users can gain access to nearly limitless computation and storage capabilities by utilizing cloud computing to get around resource limitations like memory and storage restrictions. The necessity of scalable data- partitioning strategies in remote cloud systems is underscored by the increasing (...) dependence on cloud platforms for data- intensive operations, including real-time processing and data analytics. This study examines a number of data-partitioning strategies meant to improve scalability, optimize load distribution, and maximize system efficiency. The study intends to enhance cloud-based data processing procedures by investigating these approaches, allowing businesses to optimize cloud computing's potential for data-driven projects. (shrink)

This work provides the first systematic quantification and geometric evidence of the astronomical precision limitations that linear methods of celestial partition present compared to a non-linear method in oblique latitudes. As opposed to previous critiques, which have focused upon textual or philosophical exegesis, we set forth the necessary mathematical evidence through applied spherical geometry at 57º N. Computational analysis revealed that a linear assumption upon a fundamentally non-linear phenomenon produces a cumulative temporal error of up to 30 minutes (Alcabitius, (...) Regiomontanus) and of up to 72 minutes (Campanus, Koch) between two cusps. These results demonstrate that the underlying assumption—inherent in all uniform partitions—is geometrically untenable in oblique horizons. This error, quantifiable and quantified, clarifies the mathematical justification pursuant to which methods that assume uniformity (Regiomontanus, in particular), were historically superseded, requiring a historical reassessment of their presumed physical accuracy. The analysis confirms that the method of diurnal motion, by recognising all possible circles of declination at any latitude (including polar circles), establishes itself as the historical coordinate system naturally compatible with the true and non-uniform passage of time across the celestial sphere. The Alcabitius and Koch partition methods constitute simplified uniform variations of the complex oblique calculation—originally set forth by Ptolemy in the second century—whereas that of Regiomontanus an also linear variation mathematically equivalent to that used by Campanus in the thirteenth century. Because the time of arrival of the sun or its ecliptic degrees is a daily changing, non-uniform physical reality, this work demonstrates that only a method that is not uniform-dependent can accurately reflect arrival times. (shrink)

Philosophical arguments for the rationality of Bayesian conditioning often use assumptions and concepts from epistemic logic without fully explicating them. In this work, we present a general dynamic epistemic logic framework to analyze logical properties of (Bayesian) learning scenarios. We argue that Perfect-Recall is an important epistemic principle at the heart of the project of justifying Bayesian conditioning. In such epistemic logic formulation, Perfect-Recall is not really about memory, but about the agent’s general ability to know how they came to (...) know what they know. Following the existing literature, we use Monty Hall style cases to demonstrate the connection between PerfectRecall and the commitments of Bayesian conditioning. (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)

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.

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)

The free energy principle holds that any system persisting across a statistical boundary can be described as though it minimizes a single quantity, the variational free energy, and it has been offered as a unifying account of mind and life. Its status is contested: critics have questioned whether the Markov blanket construction is well defined and whether the principle explains anything or merely redescribes it. This paper brings a different kind of test to that debate. Because the variational free energy (...) is built from probabilities it is dimensionless, and dimensional analysis supplies a unit-independent, theory-neutral criterion for physical content: a quantity that genuinely couples to the world must appear as a function of the physical dimensionless groups of the system, whereas a quantity whose value can be changed without moving any such group is not tracking the physics. Applied in a concrete, physically bounded system, bacterial concentration sensing, whose Markov blanket is an actual membrane, the criterion shows that the variational free energy partitions into three parts: a leading term equal to a known physical quantity, the Berg-Purcell measurement number; a dominant correction that is independent of every physical group and can be moved at will by the agent's priors; and a subleading term that does couple to the physics but vanishes as sensing improves. In this system the principle is a bookkeeping device laid over a coupling law that dimensional analysis exhibits directly. The test generalizes: any postulated coupling quantity can be asked whether its dimensional content collapses onto a physical group or floats free of them all. (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)

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)

Starting at the logical level of the logic of partitions, dual to the usual Boolean logic of subsets, the notion of logical entropy, i.e., information as distinctions, is developed as the quantification of the distinctions of partitions—just as probability theory starts with the quantification of elements of subsets. Logical entropy is compared and contrasted with the usual notion of Shannon entropy. Then a semi-algorithmic procedure (from the mathematical folklore) is used to translate the notion of logical entropy at the set (...) level to the corresponding notion of quantum logical entropy at the (Hilbert) vector space level. Examples and some important propositions relate quantum logical entropy to projective measurement and to the Hilbert-Schmidt distance. Overall, the approach demonstrates the logical basis and naturality of this notion of entropy for quantum mechanics. (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)

There is a fundamental subsets–partitions duality that runs through the exact sciences. In more concrete terms, it is the duality between elements of a subset and the distinctions of a partition. In more abstract terms, it is the reverse-the-arrows of category theory that provides a major architectonic of mathematics. The paper first develops the duality between the Boolean logic of subsets and the logic of partitions. Then, probability theory and information theory (as based on logical entropy) are shown to (...) start with the quantitative versions of subsets and partitions. Some basic universal mapping properties in the category of Sets are developed that precede the abstract duality of category theory. But by far the main application is to the clarification and interpretation of quantum mechanics. Since classical mechanics illustrates the Boolean worldview of full distinctness, it is natural that quantum mechanics would be based on the indefiniteness of its characteristic superposition states, which is modeled at the set level by partitions (or equivalence relations). This approach to interpreting quantum mechanics is not a jury-rigged or ad hoc attempt at the interpretation of quantum mechanics but is a natural application of the fundamental duality running throughout the exact sciences. (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)

A general mathematical framework, based on countable partitions of Natural Numbers [1], is presented, that allows to provide a Semantics to propositional languages. It has the particularity of allowing both the valuations and the interpretation Sets for the connectives to discriminate complexity of the formulas. This allows different adequacy criteria to be used to assess formulas associated with the same connective, but that differ in their complexity. The presented method can be adapted potentially infinite number of connectives and truth values, (...) therefore, it can be considered a general framework to provide semantics to several of the known logic systems (eg, LC, L3 LP, FDE). The presented semantics allow to converge to different standard semantics if the separation complexity procedure is annulled. Therefore, it can be understood as a framework that allows greater precision (in complexity terms) with respect to formula satisfaction. Naturally, because of how it is built, it can be incorporated into non-deterministic semantics. The presented procedure also allows generating valuations that grant a different truth value to each formula of propositional language. As a positive side effect, our method allows a constructive proof of the equipotence between N and N^n for all Natural n. (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)

Saadat Hasan Manto is one of the most celebrated and controversial figures in Urdu literature. His writing, fueled by the trauma, and disillusionment of the Partition of the Indian subcontinent in 1947, cut through polite society’s decorum to reveal brutal, unsettling truths. Among his many stories that deal with the horrors of Partition, “Thanda Gosht” (Cold Meat) stands apart for its unflinching depiction of sexual violence, the disintegration of the moral self, and the psychological aftermath upon both perpetrator (...) and victim. This paper attempts to analyse representations of trauma and violence found in "Thanda Gosht,". It employs literary criticism, trauma theory, and Partition studies. It will address the gendered, bodily, and psychological facets of violence. It will also carefully look at Manto's somewhat unorthodox narrative style that minimizes the divide between victim and perpetrator, thus implicating all parties in their shared journey into suffering and unfeeling. (shrink)

Persistence-based accounts of identity-bearing systems — HAOS-type frameworks, the Free Energy Principle, Global Workspace Theory, and related approaches — rely on batteries of dynamical observables to characterize which substrates constitute persisting systems. We show that such batteries are insufficient, on at least one substrate where frameworks would naturally apply them, to individuate the systems they are used to identify. We construct an existence witness: two coupled harmonic oscillators embedded in a weak thermal bath, admitting two substrate partitions related by an (...) orthogonal transformation in the admissible transformation group of the dynamics. The partitions are distinct — one groups the oscillators by physical location, the other by normal-mode index — and satisfy each framework's systemhood criterion under its own definition. An eight-item persistence/coherence battery (autocorrelation, return statistics, spectral stability, transport, phase-locking, Kuramoto order parameter, correlation-matrix coherence, broadcast-access) takes invariant values on both partitions. The identity-sets they pick out differ. The battery therefore does not individuate within the transformation group, and any individuation drawn from the battery alone is incomplete without a supplementary structural condition we call a partition-warrant. We survey four families of candidate partition-warrants present (sometimes implicitly) in the existing literature — optimization-based, structural-definitional, statistical-structural, and process-based — and state a minimum requirement any warrant must satisfy: sensitivity to substrate properties that are not invariant under the admissible transformation group. Integrated Information Theory's exclusion postulate is the clearest example of an explicit partition-warrant; the paper's argument locates other frameworks relative to this structural slot. The result is methodological: it reclassifies some consciousness-adjacent disputes as partition-choice disputes, clarifies the division of labor between battery computation and individuation in empirical research, and makes tractable a comparative assessment of candidate warrants across frameworks. A companion paper develops one specific warrant — closure activation — as a positive proposal. -/- . (shrink)

A basic duality arises throughout the mathematical and natural sciences. Traditionally, logic is thought to be based on the Boolean logic of subsets, but the development of category theory in the mid-twentieth century shows the duality between subsets and partitions (or equivalence relations). Hence, there is an equally fundamental dual logic of partitions. At a more basic or granular level, the elements of a subset are dual to the distinctions (pairs of elements in different blocks) of a partition. The (...) quantitative version of subset logic is probability theory (as developed by Boole), and the quantitative version of the logic of partitions is information theory re-founded on the notion of logical entropy. The subset side of the duality uses a one-sample (or one-element) approach, e.g., the mean of a random variable; the partition side uses a two-sample (or pair-of-elements) approach. This paper gives a new derivation of the variance (and covariance) based on the two-sample approach, which positions the variance on the partition and information theory side of the duality and thus dual to the mean. (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 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 hard problem of consciousness rests on a step whose architectural commitments have not been drawn out. The standard argument treats phenomenal qualities — the redness of red, the painfulness of pain — as ontological surplus that any physical description fails to absorb. This paper offers a thought experiment that dissolves the framing rather than answering it. Consider a species constitutively without visual apparatus: not blinded, but never possessed of the biological architecture in which colour vision could be instantiated. For (...) such beings, the redness of red does not present itself as an unexplained property. There is nothing for it to be. The hard problem, in their world, is not unsolved; it is unformed. Generalising the diagnosis: every instance of the hard problem has the same structure — phenomenally constituted bearers notice that their experience exceeds their physical descriptions of it, and infer that consciousness itself resists physical explanation. The constitutively blind case reveals the unexamined step. The problem-space is not given by the universe; it is generated by the bearer's constitutive organisation. Phenomenality is what occurs when one finite partition registers another through the gradient of its own boundedness. Qualia are not surplus; they are bearer-relative physical properties of a constituted bearer in registration. The redness of red is real because we are real, but no view-from-nowhere recovers it. What was never there cannot await solution. Keywords: qualia; hard problem of consciousness; phenomenality; bearer-relativity; constitutively blind; partition; ontology-first; explanatory gap; zombie argument . (shrink)

“Almost everyone,” Ronald Dworkin wrote in Sovereign Virtue, “assumes that democracy means equal voting power.” What, then, is voting power? The standard view defines it as the probability that a vote changes the outcome assuming that each possible combination of votes is equiprobable. This has significant implications: institutions like the US Electoral College give disproportionate voting power to voters in large states. But the standard view cannot be true. As an a priori account, it rests on the principle of indifference, (...) and thus faces the problem of multiple partitions: we could assign an equal probability to each possible combination of votes or to each possible vote share. The choice to partition in terms of combinations of votes is arbitrary, and makes the standard view objectionably overconfident. These problems of arbitrariness and overconfidence undermine the standard view’s main implications. And they call into question whether equal voting power should play a central role in democratic theory. (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)